CERTIFICATE OF APPROVAL. PH.D. THESIS. This is to certify that the Ph.D. thesis of ... quantum mechanics. pneumonic for â¡ to 17 digits. -Sir James Jeans.
University of Iowa
Iowa Research Online Theses and Dissertations
2014
Applications of the holographic principle in string theory Bradly Kevin Button University of Iowa
Copyright 2014 Bradly Kevin Button This dissertation is available at Iowa Research Online: http://ir.uiowa.edu/etd/1299 Recommended Citation Button, Bradly Kevin. "Applications of the holographic principle in string theory." PhD (Doctor of Philosophy) thesis, University of Iowa, 2014. http://ir.uiowa.edu/etd/1299.
Follow this and additional works at: http://ir.uiowa.edu/etd Part of the Physics Commons
APPLICATIONS OF THE HOLOGRAPHIC PRINCIPLE IN STRING THEORY
by Bradly Kevin Button
A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa
August 2014
Thesis Supervisor: Professor Vincent G. J. Rodgers
Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL
PH.D. THESIS
This is to certify that the Ph.D. thesis of Bradly Kevin Button has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the August 2014 graduation. Thesis Committee: Vincent Rodgers, Thesis Supervisor
Yannick L. Meurice
Leopoldo A. Pando Zayas
Wayne N. Polyzou
Craig E. Pryor
To my wife Anna, daughter Rebecca, and son Miles, I love you with all my heart.
ii
⇡ ⇡ How I need a drink, alcoholic of course, after all those lectures involving quantum mechanics. pneumonic for ⇡ to 17 digits -Sir James Jeans
Nelson Munce: Lisa Simpson: Nelson Munce:
She can do the kind of math that has letters. Watch! What’s x, Lisa? Well, that depends. Sorry... She did it yesterday. The Simpsons, "MoneyBART" (2010) -Nelson Munce referring to Lisa Simpson
iii
ACKNOWLEDGEMENTS
As I have progressed through my graduate studies at the University of Iowa, I have realized that my efforts are only partial contributions to the completion of my Ph.D. Many people from all aspects of my life have contributed to my success at Iowa and beyond. I would like to take the time to thank all of them, however this would require more space than is allotted. First, I would like to thank the members of my family. My daughter Rebecca, I thank you for being a source of joy and inspiration on this long journey, and for the many sacrifices that you have made in order for me to complete my degree. I love you very much. My son Miles, I love you and cannot imagine a world without you. My dearest wife Anna, you have sacrificed and given the most without any hesitation. Words cannot fully express my gratitude. You have been my rock, my motivation, and my truest companion. I love you more now than ever. Thank you for everything. I would like to thank Professor Vincent Rodgers for taking me under his guidance. I could not have accomplished any of this work without his support. I would also like to thank all the members of the Diffeomorphisms and Geometry Group. Specifically Tuna Yildirim for his friendship and encouragement at Iowa and in life. It has helped tremendously, and in more ways than you know. Chris Doran, you were always welcoming and friendly to me when I first entered the D&G Group. It has been a pleasure to know you. Catherine Whiting for her constant cheerful demeanor
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and willingness to help me every time I have asked for it. Dr. Leo Rodriguez for his friendship and tutelage. You have helped me in so many ways in which I know I will never have the opportunity to repay. Delalcan Kilic, your friendship has meant the world to me. I will miss our vibrant discussions and debates on the finer details of all things physics, string theory, and mathematics. Dr. Kory Stiffler for his time and effort in my initial training in the k-string project. Dr. Xiaolong Liu for his friendship and encouragement. I would also like to take the time to thank Marc Herrmann, you have been one of my best friends at Iowa and in my life. I am truly grateful to have you as a friend. Professor Mary-Hall Reno, in no small way was she integral for my success at Iowa. Professor Wayne Polyzou, for devoting his time for an entire year to a readings course in QFT. Professor Aldo Migone of SIUC, only recently have I began to understand the significance of your support, and for that I am in your debt. Dr. Andrei Kolmakov of NIST, your friendship, guidance, tutelage, and support was one of the most significant contributions to my current place in life, and therefore any place that follows. I will always remember your encouragement and kindness. I would like to take the time to acknowledge Yannick Meurice, Craig Pryor, Debbie Foreman, Heather Mineart, Jeane Mullen, Chris Stevens, Arron Votroubek, Sujeev Wickramasekara, Usha Malik, and the many others whom have made my time at Iowa one of the best parts of my life.
v
ABSTRACT
The holographic principle has become an extraordinary tool in theoretical physics, most notably in the form of the Anti-deSitter Conformal Field Theory (AdS/CF T ) correspondence, in which classical gravitational degrees of freedom in N -dimensions are related quantum field theory degrees of freedom in N
1 dimen-
sions in the limit of a large number of fields. Here we present an account of the AdS/CF T correspondence, also known as the gauge/gravity duality, from its origins in the large N ’tHooft expansion, up to Maldacena’s proposal that type IIB string theory in the presence of D-branes at low energy is dual to an N = 4, d = 4, U (N ) super Yang-Mills on AdS 5 ⇥ S5 . We begin with an extensive review of (super)string theory including D-branes. We then present the general formulation of the AdS/CF T in the supergravity background of AdS 5 ⇥ S5 , along with several examples of how it is used in terms of the identification of bulk fields with operators on the boundary of a CFT. We move on to discuss two applications of the gauge/gravity duality. The first is the application of the holographic gauge/gravity correspondence to the QCD k-string. The second applies the AdS/CF T formalism to a Kerr black hole solution embedded in 10-dimensional heterotic sting theory. These two applications of the holographic gauge/gravity duality comprise the original work presented here. We follow with summaries and discussions of the background material, the original work, and future investigations.
vi
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1
. . . . . . . . . . . .
1 2 6 9 13 15 21 21 26 27 30 40
2 STRING THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
1.2 1.3
1.4 1.5
2.1 2.2
2.3
2.4
The Holographic Principle . . . . . . . . . . . . . 1.1.1 Suggestive Evidence of Holography . . . . 1.1.2 Large N and String Theory . . . . . . . . The Conformal Group, Algebra, and AdS Space 1.2.1 The Virasoro Algebra . . . . . . . . . . . 1.2.2 Anti-de Sitter Space . . . . . . . . . . . . Asymptotic Symmetries in AdS . . . . . . . . . 1.3.1 Canonical Asymptotic Symmetries . . . . 1.3.2 AdS and Asymptotically AdS Space . . . 1.3.2.1 Ex : AdS3 Space . . . . . . . . Black Hole Entropy and Temperature via Cardy QCD k-strings . . . . . . . . . . . . . . . . . . .
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1
Outline of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . The Classical Bosonic String . . . . . . . . . . . . . . . . . . . . 2.2.1 Solutions to the classical wave equation . . . . . . . . . . 2.2.1.1 Intermission, Comments on gauge symmetries . 2.2.1.2 Back to general solutions of the classical wave equation . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Witt Algebra and Global Symmetries . . . . . . . . . 2.2.3 Free Bosonic String Quantization . . . . . . . . . . . . . . The RNS Superstring . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 RNS Constraint Equations . . . . . . . . . . . . . . . . . 2.3.2 RNS Boundary Conditions and Mode Expansions . . . . . 2.3.3 RNS Superstring Quantization . . . . . . . . . . . . . . . 2.3.3.1 The Quantum Super-Virasoro Algebra . . . . . . 2.3.3.2 The Faddeev-Popov Prescription . . . . . . . . . 2.3.4 BRST Quantization . . . . . . . . . . . . . . . . . . . . . 5 Consistent Superstring Theories . . . . . . . . . . . . . . . . . 2.4.1 Type IA . . . . . . . . . . . . . . . . . . . . . . . . . . .
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44 45 50 51 54 56 60 63 66 69 72 75 76 81 85 85
. . . . . .
86 86 87 89 91 94
3 THE HOLOGRAPHIC CORRESPONDENCE . . . . . . . . . . . . .
97
2.5
3.1 3.2 3.3
3.4 3.5
2.4.2 Type IIA and IIB . . . . . . . . . . . . . . . . 2.4.3 Spin32/Z2 and E8 ⇥ E8 Heterotic Superstrings D-Branes . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Kaluza-Klein Compactification on S 1 . . . . . 2.5.2 T-duality and Closed Bosonic Strings . . . . . 2.5.3 T-duality and Open Bosonic Strings (D-branes)
. . . . . .
AdS/CFT and Gauge/Gravity Duality . . . . . . . . . . AdS/CF T and the SYM Spectrum . . . . . . . . . . . . 3.2.1 Symmetry Matching N = 4 SYM and AdS 5 ⇥ S5 3.2.2 Massless SYM Spectrum . . . . . . . . . . . . . CFT Boundary Operators From Bulk Fields . . . . . . . 3.3.1 Ex : The Gravity Action in AdS 5 ⇥ S5 . . . . . . 3.3.2 Gauge Partition Functions . . . . . . . . . . . . 3.3.3 Ex : Massless Scalar Fields . . . . . . . . . . . . 3.3.4 Ex : Gauge Field Correlator . . . . . . . . . . . 3.3.5 Ex : Massive Scalars . . . . . . . . . . . . . . . . 3.3.6 Comments on Asymptotic AdS Space . . . . . . Holographic Renormalization . . . . . . . . . . . . . . . Holographic Wilson Loops . . . . . . . . . . . . . . . . 3.5.1 Super Yang-Mills Wilson loops . . . . . . . . . .
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97 103 104 106 109 110 113 115 117 119 122 126 130 130
4 HOLOGRAPHIC K-STRINGS . . . . . . . . . . . . . . . . . . . . . . 134 4.1 4.2 4.3 4.4
4.5
The Dp-brane action . . . . . . . . . . . . . . . . . 4.1.1 The MN and MNa backgrounds . . . . . . The k-string Tension . . . . . . . . . . . . . . . . . 4.2.1 The Tension Law . . . . . . . . . . . . . . Tensions from Various Methods . . . . . . . . . . . 4.3.1 Analysis of tension results . . . . . . . . . . The Quantum D5-brane in MN/MNa Backgrounds 4.4.1 The Geometry of the Minimized Solution . 4.4.2 Quadratic fluctuations . . . . . . . . . . . . 4.4.3 Effective Lagrangian . . . . . . . . . . . . Massless Modes and Lüscher Term . . . . . . . . . 4.5.1 Lüscher term universality . . . . . . . . . .
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134 136 138 146 149 150 151 152 153 154 157 159
5 THE ADS/CFT OF THE KERR-SEN BLACK HOLE . . . . . . . . . 160 5.1
The Kerr/CF T Corrspondence . . . . . . . . . . . . . . . . . . 160 5.1.1 The Kerr Space-time . . . . . . . . . . . . . . . . . . . . 161 5.1.2 Near Horizon Extremal Kerr (N HEK ) . . . . . . . . . 163
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5.2 5.3 5.4 5.5 5.6
5.1.3 The Asymptotic Symmetries of the N HEK Geometry 5.1.4 N HEK Temperature and Entropy . . . . . . . . . . . Kerr-Sen Space Time . . . . . . . . . . . . . . . . . . . . . . Near Horizon Near-Extremal Kerr-Sen Geometry . . . . . . Quantum Fields in N HN EKS . . . . . . . . . . . . . . . . . Asymptotic Symmetries . . . . . . . . . . . . . . . . . . . . . AdS2 /CF T1 and Near-Exremal Kerr-Sen Thermodynamics .
. . . . . . .
. . . . . . .
165 168 169 172 175 179 183
6 SUMMARY, CONCLUSIONS, AND FUTURE WORK . . . . . . . . 186 6.1 6.2 6.3 6.4
Summary of Chapters 1,2,3 . . . . . . . . . . . . . . . . . . . . . k-String Summary and Conclusions . . . . . . . . . . . . . . . . 6.2.1 Holographic k-String Conclusions . . . . . . . . . . . . . . 6.2.2 Open Questions and Future Work in k-strings . . . . . . . Kerr-Sen Summary and Conclusions . . . . . . . . . . . . . . . 6.3.1 Kerr-Sen Future Considerations . . . . . . . . . . . . . . Recent Developments and Limitations of the AdS/CF T Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Black Hole Complimentarity . . . . . . . . . . . . . . . . 6.4.2 Firewalls, Not Black Hole Complimentarity . . . . . . . . 6.4.3 Firewalls and Fuzzballs . . . . . . . . . . . . . . . . . . .
186 188 188 189 191 194 198 198 199 204
APPENDIX A THE ADM FORMALISM . . . . . . . . . . . . . . . . . . . . . . . . 207 B GAUGE FIXED PATH INTEGRAL . . . . . . . . . . . . . . . . . . . 210 C VIRASORO ANOMALIES . . . . . . . . . . . . C.1 Anomaly From the Old Covariant Method . C.2 The Faddeev-Popov Prescription . . . . . . C.3 Virasoro Anomaly of the Bosonic Matter . C.4 Virasoro Anomaly from (Anti)Ghost Matter
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212 212 215 219 221
D BRST QUANTIZATION . . . . . . . . . . . . . . . . . . . . . . . . . 226 E THE CHAN-PATON METHOD . . . . . . . . . . . . . . . . . . . . . 229 F YOUNG DIAGRAMS . . . . . . . . . . . . . . F.1 Ex : SU (2) ⇥ SU (2) . . . . . . . . . . . . F.2 Dimensionality of Higher Representations F.3 Ex : SU (3) . . . . . . . . . . . . . . . . . F.4 SO(N ) and Sp(2N ) Groups . . . . . . . .
ix
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233 236 238 239 240
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
x
LIST OF TABLES
1.1
Conformal transformations and infinitesimal conformal generators. . . . .
11
1.2
Conformal algebra commutation relations. . . . . . . . . . . . . . . . . .
11
1.3
Superconformal algebra (anti)commutation relations. . . . . . . . . . . .
12
3.1
Massless SYM states and Helicity . . . . . . . . . . . . . . . . . . . . . . 108
4.1
Comparison of 2 + 1 k-string tensions from various methods. . . . . . . . 149
xi
LIST OF FIGURES
1.1
c-number source for internal quark lines. ni , pi , i form 3 different representations of SU (N ), i = 1, . . . , N . The index lines continue on through an external source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2
A conformal mapping of AdS2 ! R ⇥ [ ⇡/2, ⇡, 2]. . . . . . . . . . . . . .
1.3
On the left, a hadronized grouping of a more general k-string configuration, and the right is a single q q¯ forms the simplest k-string. . . . . . . . . .
42
2.1
The world-sheet of an open string embedded in D = 10 dimensions . . .
49
2.2
Open strings with endpoints on D-branes which allow Yang-Mills degrees of freedom, where the open string endpoints act as Chan-Paton color sources. 88
3.1
Commuting diagram for the gauge/gravity correspondence between type IIB SUGRA and low energy type IIB string theory in AdS 5 ⇥ S5 . . . . . 102
4.1
A graphic representation of embedding and pullback . . . . . . . . . . . 136
4.2
The N = 6, d = 2 + 1 k-string tension (MNa D5-brane) for various gauge/gravity models compared to the Casimir and sine laws. . . . . . . 147
4.3
The N = 6, d = 3 + 1 k-string tension (MN D5-brane) for various gauge/gravity models compared to the Casimir and sine laws. . . . . . . 148
6.1
The semi-classical description of Hawking radiation occurring at the black hole horizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.2
A Penrose (conformal) Diagram of bulk Hawking modes outside of the horizon b, beyond the horizon ˜b, and the corresponding CFT boundary operators ˆb and ˆ˜b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
E.1 Oriented qN
q¯N¯ pair has N gauge degrees of freedom at each endpoint.
17
229
E.2 Primitive M -point scattering amplitudes. . . . . . . . . . . . . . . . . . . 231 F.1 Example of 2 hook-lengths for an arbitrary tableau shape. . . . . . . . . 239
xii
1 CHAPTER 1 INTRODUCTION
I can’t believe it. You’ve actually found a practical use for geometry!
The Simpsons, "Dead Putting Society" (1990) -Bart Simpson to Lisa Simpson 1.1
The Holographic Principle
In a very real sense (and in an abuse of language), the (quantum) information contained in the universe is holographic. It is quite possible that the most intriguing and also surprising realizations of modern theoretical physics is that the number of degrees of freedom of a quantum system with gravity should not scale proportionally with the space-time volume, but rather it should scale proportionally with the area which encloses the quantum system [2, 3]. This is (crudely) analogous to Guass’s law in electrostatics, where the electric charge in a finite volume can be determined from the flux lines which emanate normal to a gaussian surface enclosing the charge. This is a radical re-interpretation of how information in systems with gravity can be formulated. To be more specific, the holographic principle claims that a gravitational theory in d-dimensions is contains the information of a quantum theory in d 1-dimenstions. However in hindsight, it is curious that a holographic formulation of quantum information and gravity was a nearly 25 year endeavor starting with the early works of ’tHooft, Hawking and Bekenstein [4–7] through the proposition of
2 a mismatch between gravitational and quantum degrees of freedom by ’tHooft in 1993 [2] and the proposition of a "holographic" principle in string theory by Suskind in 1994 [3], and finally realized concretely by Maldecena in 1997 [8] with the AntideSitter-conformal field theory (AdS/CFT) correspondence. 1.1.1 Suggestive Evidence of Holography Early evidence suggestive of a holographic description came circa the mid 1970’s when Bekenstein proposed that the entropy of a static black hole should be proportional to the event horizon area over the Planck area and not the volume [7], Sbh /
Abh ln 2 · . 2 4⇡
(1.1)
Soon after, Hawking determined the exact expected equality [6] of, SBH = where the Planck length, lp =
q
~GN , c3
kB Abh , 4lp2
(1.2)
or in natural unints (c = ~ = 1,) =
A , 4GN
(1.3)
where GN is Newton’s gravitational constant, and the entropy is known as the Bekenstein-Hawking entropy. Given that boolean binary information (bits) per plank area can be written in terms of mass-squared energy (this is the origin of the ln 2 in (1.1)) and thus entropy (as the Bekenstein Bound Sbh 2⇡kb Rbh /~c and I 2⇡cRbh /~ ln 2 [7, 9]), there was an established natural relation between quantum gravity and quantum information reminiscent of the holographic principle. Let us look somewhat heuristically at the arguments of [2,3,9] for a holographic description between gravity and quantum information. The simplest of arguments put
3 forth in one form or another by ’tHooft, Susskind, and Bekenstein demonstrates a implied violation of the second law of thermodynamics. We will briefly recount the explicit argument of [3], though the origin of the argument is from Bekenstein. Suppose that a 3-dimensional lattice of spacing lp carries binary spin degrees of freedom. The number of distinct orthogonal quantum states in a finite region of space with volume V is N (V ) = 2n ,
(1.4)
and n is the number of spin sites in V . The log of (1.4) gives the maximum possible entropy contained in V , Smax = log N (V ) = n log 2 =V
(1.5)
log 2 . lp3
From this, one may (incorrectly) conclude that for a bounded energy density, Smax is proportional to the volume of the region, V . It was shown by Bekenstein and Hawking in [6, 7] and again re-derived by Bekenstein in [9] that entropy of a black hole is correctly given by (1.2), and we will assume it’s validity here. We note that Susskind claims that most of the states of (1.5) of sufficient energy that they would form a black hole B, of a size VB > V . We suppose that there exists is a space contained in V with an entropy SB such that SB > SBH but with smaller energy EBH > EB , so that it was maximally contained in V and any additional matter added would form a black hole. Since the
4 entropy of the black hole B, created by the addition of any new matter would be less than that of the original black hole, then by (1.2), the second law of thermodynamics would be violated as SB would decrease. Admittedly, the connection between this argument and the implication of a holographic description of matter is not made rigorous, however it does give credence to the claim that quantum information is encoded by the area of a black hole and not its volume. Susskind goes on further in [3] to build his case. He demonstrates that information contained along the circumferential length of a closed string with a projected image onto a screen which lies coplanar to the string’s radius, naturally encodes the information of the string on the 2-dimensional screen surface. He carefully argues that if the string is relativistically boosted in a direction normal but away from the screen position, that the string’s "size" grows transversely as the momentum is increased without bound. The details of the argument are lengthy and will not be given here, but the summary of it is the following. Analogous to the infinitely boosted string, the same process occurs if the string falls into a black hole. As the string’s transverse size grows with it’s longitudinal momentum, it’s size must also be bounded by the energy of the black hole which accelerates the string. The quantum information contained by the string must by extension be bounded by the maximum entropy of the black hole, and thus its area. The final claim is that the information contained by the incident string must spread over the surface of the event horizon with a maximum causally allowed
5 spread time. The quantum information spread time is obtained from arguments which are only consistent with Bekenstein entropy. Therefore Susskind demonstrates a "proof of concept" of the holographic description of matter. In [8] Maldacena uses the arguments of ’tHooft, Susskind and others to establish the first rigorous realization of the holographic principle, in which he makes the now famous conjecture, that N = 4, d=4, supersymmetric Yang-Mills (SYM) is dual to a low energy type IIB superstring theory with D-branes in AdS 5 ⇥ S5 . Maldacena’s conjecture establishes a mapping between a strongly coupled SYM gauge theory in a supergravity (SUGRA) background and a weakly coupled string theory with U (N ) gauge degrees of freedom. The main idea is built upon the intuition that for a black hole with U (N ) gauge group, a distant observer would see the black hole horizon (a strongly coupled gravitational theory) and observe a redshifted (low energy) SYM gauge theory near the horizon. Thus a strongly coupled gravity theory in AdS 5 ⇥ S5 corresponds to weakly coupled CFT that lives on the boundary of the AdS 5 space. Conversely a strongly couple SYM gauge theory should correspond to a weakly coupled gravitational theory. The correspondence becomes an equivalence if the states of one side of the theory (weak gravity/strong gauge theory) can be successfully mapped to states on other side (weak gauge theory/strong gravity) without encountering divergent excitations (i.e. phase transitions) [8, 10–13].
6 1.1.2 Large N and String Theory One necessary component of the AdS/CFT correspondence is ’tHooft’s largeN expansion [4, 8] 25 years prior to Maldacena’s conjecture, and is again suggestive of a quantum gauge theory description of in lower dimensions. For instance in YangMills, a power of the gauge coupling gY M is usually absorbed into the field definitions so that the Lagrangian density may be written as L ⇠
T r 4gF2
2
YM
[14].
In [4], ’tHooft considers internal gauge diagrams for SU (N ) theories with F faces, P -propagators (internal lines), and V -vertices. The faces are initially written as F = L + I, where L is the number of quark loops, I-the number of index loops, and V =
P
n
Vn is the number of n-point vertices.
quarks: ni , pi ,
i
n J (x) (c-number source J)
p Figure 1.1: c-number source for internal quark lines. ni , pi ,
i
form 3 different
representations of SU (N ), i = 1, . . . , N . The index lines continue on through an external source.
For a simple example see Figure 1.1. Each diagram the has an associated
7 factor r, (1.6)
r = g V3 +2V4 N I ,
where g is the gauge coupling. A dot is associated with the end of each internal line, then the number of propagators P is related to the number of n point vertices by 2P =
P
n
=F
nVn . When this is combined with (1.6) and the Euler character formula [14],
E+V = 2
2H (where H is the number of holes in a surface), then (1.6)
becomes, r = g2N
1 V +V4 2 3
N2
2H L
(1.7)
.
More generally, we note that each vertex contributes a factor of g 2 , each propagator a factor of g 2 , and each loop L contributes a factor of N . Thus a general Feynman diagram takes the form [15] 2
r= g N
E V
N
F
! g
2 E V
F
N =
✓
N
=N
◆E E V
V
NF (1.8) .
If we take the limit for fixed , N ! 1,
g ! 0,
g2N =
(fixed),
with the sources coupled to quarks (this implies that there is at least 1 loop: L
(1.9) 1),
then the leading diagram is planar defined by H = 0, and L = 1. In Yang-Mills, Feynman diagrams have an overall factor of N/ , with factor of N and propagators have a factor of 1/N [4, 14].
= gY2 M N , vertices have a
8 From (1.8) we see that the expansion terms arrange themselves into orders of 1/N . The 1/N expansion has an important realization in string perturbation theory where the scattering amplitude is given by Zstring ⇡ C with
Z
(1.10)
D D⌧ exp Sstring ,
and ⌧ the string world-sheet parameters and interactions are characterized or-
der by order according to the world-sheet genus (i.e. Euler character) [14–17]. Therefore in the limit of a large numbers of colors(Nc ), the large-Nc expansion corresponds to a planar diagram expansion of color interactions, and expansion of Nc in super Yang-Mills in string theory [8, 13, 15]. In particular gY2 M N ⌘ ,
and gstring = gY2 M =
N
.
(1.11)
It is this identification of couplings that allows string theory to be identified with large-N SYM gauge theories [8, 12, 14, 18]. The weak/strong duality discussed at the end of Section 1.1.1 is then broken down into two incompatible limits of validity. In the original construction of the AdS/CFT [8], the AdS 5 ⇥ S5 space has radius of curvature R for both AdS 5 and S5 . From the identifications of (1.11), field theory (SYM) perturbative calculations are valid when, SYM limit:
gY2 M N ⇠ gs N ⇠
R4 ⌧ 1, ls4
(1.12)
with ls the fundamental string length. Conversely when R becomes much larger then
9 ls , a classical gravity description is valid [12], Gravity limit:
gY2 M N ⇠ gs N ⇠
R4 ls4
1.
(1.13)
We will apply the tools provided by the holographic principle, the AdS/CFT correspondence, and more generally of the supposed gauge/gravity duality to study the QCD k-string, and find AdS dual CFT’s, Bekenstein-Hawking entropy and Hawking temperature of the Kerr-Sen black hole metric. These investigations and analysis will comprise of the results based applications of the gauge/gravity correspondence and built on the background formalisms prior to those discussions. 1.2
The Conformal Group, Algebra, and AdS Space
As we will see encounter many instances in the discussions about AdS/CFT where we will match symmetries of a particular group with those of the conformal group (and their respective algebras of generators) on some conformal boundary, it is worth taking the time to introduce them early on. The conformal group is the group of transformations which reparametrize the metric up to an overall multiplicative scale factor [12]: gµ⌫ (x) ! ⌦2 (x)gµ⌫ (x), µ, ⌫ = 0, . . . , d
1.
(1.14)
The conformal group is the minimal inclusion of the Poincaré group plus inversion symmetry (xµ ! xµ /x2 ). String theory is a 2-dimensional CFT on the world-sheet i.e. the string is parametrized by the set
µ
, such that
µ
= {⌧, |µ = 0, 1} parametrizes the local
10 string coordinates, and the space-time coordinates are functions of the world-sheet coordinates such that X ↵ (
µ
), ↵ = 0, 1, . . . , d
1, where d is the total space-time
dimension. From these definitions we identify ⌧ as a timelike coordinate and
as a
spacelike coordinate. Thus the space-time coordinates serve as embedding functions on the target manifold defined by the space-time metric gµ⌫ (x) as in (1.14). If we let x 2
µ
:
µ
2 R2 we may complexify the world-sheet parameters in
terms of holomorphic and anit-holomorphic parameters by the mapping of x ! (z, z¯), where z =
+ i⌧ , and z¯ =
i⌧ . Then for a field (z, z¯) (or coordinate X ↵ (z, z¯)),
¯ under the scaling transformations of is said to have conformal dimension of (h, h) z, z¯ ! z, ¯ z¯ defined by, (z, z¯) !
0
(z, z¯) =
¯ h ¯h
( z, ¯ z¯).
(1.15)
Similarly, if a field transforms under conformal transformations z, z¯ ! f (z), f¯(¯ z) according to (z, z¯) ! the
0
(z, z¯) =
✓
@f @z
◆h ✓ ¯◆h¯ @f (f (z), f¯(¯ z )), @ z¯
¯ is said to have conformal dimension (h, h). Furthermore, if (1.16) is true
only for global conformal transformations (i.e. f, f¯ 2 SL(2, C)/Z2 ), then quasi
(1.16)
is called
primary. Finally if
is purely holomorphic (anti-holomorphic) such that
= (z) (or
= (¯ z )), then (z)( (¯ z )) is called chiral (anti-chiral) [19]. The conformal group admits representations defined on a compactification of Minkowski space R1,p , which are the group actions of the Poincaré, scale, and spe-
11 cial conformal transformations and have the corresponding generators given in Table 1.1 [19]. The conformal algebra generators annihilate the vacuum of a conformal
Transformations translation xµ ! xµ + aµ dilatation xµ ! axµ rotation xµ ! M µ⌫ x⌫ x µ x ⌫ x ⌫ bµ SCT xµ ! 1 2b⌫ x⌫ + (b⌫ b⌫ )(x⌘ x⌘ )
Generators Pµ = i@µ D = ixµ @µ Lµ⌫ = i(xµ @⌫ x⌫ @µ ) Kµ =
i {2xµ x⌫ @⌫
(x⌘ x⌘ )@µ }
Table 1.1: Conformal transformations and infinitesimal conformal generators.
theory, and is given by the non-vanishing commutators in Table 1.2 [12]. The confor-
[Mµ⌫ , P⇢ ] = i (⌘µ⇢ P⌫ [Mµ⌫ , M⇢ ] = i⌘µ⇢ M⌫ [D, Pµ ] = iPµ ,
Conformal Algebra ⌘⌫⇢ Pµ ) , [Mµ⌫ , K⇢ ] = i (⌘µ⇢ K⌫ ⌘⌫⇢ Kµ ) , ± permutations, [D, Kµ ] = iKµ , [Pµ , K⌫ ] = 2iMµ⌫ 2i⌘µ⌫ D,
Table 1.2: Conformal algebra commutation relations.
mal algebra is isomorphic to the algebra of SO(2, d) with signature ( , +, . . . , +, ). This can be shown by defining the conformal algebra in terms of the SO(2, d) generators, Jab with a, b = 0, 1, . . . , d + 1 Jµ⌫ = Mµ⌫ ,
Jµd =
1 (Kµ 2
Pµ ) ,
Jµ(d+1) =
1 (Kµ + Pµ ) , 2
J(d+1)d = D,
(1.17)
12 where we again have µ, ⌫ = 0, 1, . . . , d
1. We note that in Euclidean space the
conformal group is SO(1, d + 1), which is the connected component of the conformal group minus the inversion symmetry. Compactified Rd is conformally equivalent to S d , thus a field theory with a point added at infinity in Rd is isomorphic to a theory on S d [12]. This will be useful as we will discuss the tree level Euclidean path integral in string theory by mapping it to the Riemann sphere in Chapter 2 and the AdS/CFT correspondence in Chapter 3. For completeness we give the supersymmetric extension of the conformal algebra above. The full classification of the superconformal algebras was given by [20], where it was shown that they can exist in dimensions d 6. There are two species of generators of the superconformal groups and superconformal algebras in addition to those of Table 1.1. They are the fermionic generators S (one fore each supersymmetry generator), and sometimes the R-symmetry generators R [12]. The S generators are manifest in the commutator of the supersymmetry generator Q and Kµ , where as the R-symmetry generators arise in the anticommutator of Q and S. The generators Q, S are in the fundamental representation of the superconformal algebra. An outline of the superconformal (anti)commutator relations are
Superconformal Algegra [D, Q] = 2i Q, [D, S] = 2i S, [K, Q] = i ' S, [P, S] ' Q, {Q, Q} ' P, {S, S} ' K, Table 1.3: Superconformal algebra (anti)commutation relations.
13 given in Table 1.3. The index structure is suppressed as exact form is dependent on the dimensionality, and for different R-symmetry groups [12]. For free theories without gravity (fields do not have spin greater than 1) and interacting theories, it is thought that the maximum number of supercharges is 16. Therefore the maximum possible number of fermionic generators is 32. The field theories with 32 fermionic generators exist in dimensions d = 3, 4, 6 [12]. 1. d=3: The R-symmetry group is Spin(8), and the fermionic generators are in the (4, 8) representation of SO(3, 2) ⇥ Spin(8). 2. d=4: The R-symmetry group is SU (4), and the fermionic generators are in the (4, 4)
¯ 4) ¯ representation of SO(4, 2) ⇥ SU (4). (4,
3. d=6: The R-symmetry group is Sp(2) ' SO(5), and the fermionic generators are in the (8, 4) representation of SO(6, 2) ⇥ Sp(2). 1.2.1 The Virasoro Algebra ˜ m ) , m 2 Z is The generators of the conformal group in two dimensions (Lm , L ˜ m) infinite dimensional and known as the Virasoro algebra [15, 16, 21–23]. Here Lm (L is the left (right) moving mode. We will discuss the Virasoro and super-Virasora algebras in great detail in Chapter 2 and Appendix C. For now we give a short introduction to it. Let X ↵ (
µ
) be classical solution of the world-sheet string equation (we assume
the boundary conditions of a closed string here) such that ✓
@2 @ 2
@2 @⌧ 2
◆
X ↵ = 0.
(1.18)
14 We make a change of variables on the world-sheet coordinates to move into light-cone coordinates:
±
= ⌧ ± , which implies that (1.18) becomes @± = 1/2(@⌧ ± @ ),
and the solution becomes X ↵ (
µ
) = XR↵ (
) + XL↵ (
+
). The Virasoro algebra is
given as the Fourier transform of the energy-momentum tensor (EMT), T↵ which is symmetric and traceless. Under light-cone coordinates, the EMT becomes T++ =@+ X ↵ @+ X↵ = 0
(1.19)
=@ X ↵ @ X = 0,
(1.20)
T
where (1.19) and (1.20) must be zero due (1.18) which acts as a constraint , and the traceless condition (g ↵ T↵ ) implies that T+ = T
+
(1.21)
= 0.
As stated above, the Virasoro algebra is defined as the Fourier transform of the EMT T Lm = 2 ˜m = T L 2
Z Z
⇡
d e
2im
(1.22)
T
0 ⇡
(1.23)
d e2im T++ , 0
where T in the front of the integrals is the string tension. The generators satisfy the commutation relations [Lm , Ln ] =(m
n)Lm+n + c(m)
m+n,0 ,
˜ m, L ˜ n ] =(m [L
˜ m+n + c(m) n)L
m+n,0 ,
˜ n ] =0, [Lm , L
(1.24)
15 which define the Virasoro algebra. c(m) in (1.24) is the central extension of the algebra and is crucial to the defining characteristics of a CFT, as we will soon see. We note that for the classical theory c(m) = 0 and is called the Witt algebra. Upon quantization c(m) 6= 0, and is then referred to as the Virasoro algebra [19]. As (1.22) are generators, by Noether’s theorem they must correspond to conserved charges. In particular we note that L
1
˜ and L
1
are the generators of translations,
˜ 0 are the generators of rotations and dilations [16, 24, 25]. and L0 and L We also note that we have actually two copies of the Virasoro algebra (left and right moving modes). The genesis of the two copies lies in the solution of (1.18), where for closed strings the left and right moving modes are independent wave functions. Conversely, the open string only has one copy of the algebra. Dirichlet boundary conditions must be imposed in order to maintain energy-momentum conservation. Therefore the left and right modes must combine to form a standing wave [15, 16, 21– 23]. 1.2.2 Anti-de Sitter Space If we consider Einstein-Hilbert action in D-dimensions with a cosmological constant ⇤, the Lagrangian density is L=
p
g(R
2⇤).
(1.25)
Note that with the presence of ⇤, the flat Minkowski space metric, ⌘µ⌫ is no longer solution to Einstein’s equations. Instead we may consider "maximally symmetric"
16 solutions to the field equations of (1.25) which satisfy the conditions [14] R
µ⌫
=⌥
2 (g gµ⌫ R2
or
g ⌫ gµ ),
Rµ⌫ = ±
2⇤ gµ⌫ (D 2)
(1.26)
where (D
R2 =
1)(D 2⇤
2)
(1.27)
.
AdS space corresponds to a space-time with a negative radius of curvature which is proportional to the cosmological constant ⇤ < 0 R=
r
1 , ⇤
(1.28)
and has the general spherical form for the metric in D-dimensions ds2 =
f (r)dt2 +
and f (r) is usually a function of
r R
1 dr2 + r2 d⌦2D 2 , f (r)
(1.29)
[12, 14, 26].
At the heart of the AdS/CFT correspondence, is the identification of the isometry group of AdSp+2 with the conformal group R1,p [12]. AdS space in p + 2dimensions, can be globally written as a hyperboloid R
2
=X02
+
2 Xp+2
p+1 X
(1.30)
Xi2 ,
i=1
embedded in flat D = p + 3-space 2
ds =
dX02
2 dXD 1
+
D X2
dXi2 .
(1.31)
i=1
Upon further inspection of the line element we see that the space has the isometry group SO(2, D
2), which is homogeneous and isotropic [12].
17 r = constant
r=1
r=0
=
2
=
2
Figure 1.2: A conformal mapping of AdS2 ! R ⇥ [ ⇡/2, ⇡, 2].
We may apply a coordinate transformation to (1.30) in order to put the metric in Poincaré coordinates, which we will encounter with our review of the AdS/CFT correspondence in Chapter 3. The transformation to Poincaré coordinates is X0 =
1 1 + r2 (R2 + ~x2 2r
X i = Rrxi XD
2
XD
1
1 = 1 2r
(i = 1, . . . , D r2 (R2
t2 ) , 3),
(1.32)
~x2 + t2 ) ,
= Rrt,
where the line element becomes, 2
ds = R
2
✓
◆ dr2 2 2 2 + r ( dt + d~x ) . r2
(1.33)
We note that this representation covers only one half of the hyperboloid in (1.30), as can be seen for the case of p = 0 in Figure 1.2 [12]. The Poincaré coordinates (1.32) have the manifest SO(2, D group with subgroups ISO(1, D
2) isometry
3) (Euclidean group: xµ ! Oµ⌫ xµ + b⌫ ) which
18 is the Poincaré transformation on (t , ~x), and SO(1, 1) (dilatation D: (t, ~x, r) ! (at, a~x, r/a) for a > 0). The organization of this paper will have the following content outline. The remainder of Chapter 1 will review some of the formalisms typically used in quantum gravity, but have become relevant in string theory via the AdS/CF T correspondence. These formalisms include the the asymptotic symmetries of AdS space and their relations to conformal field theories, and the Cardy formula relating the central charges of CFT’s to the thermodynamic microstates of statistical systems. Finally we will give a cursory background of the QCD k-string in SU (N ) gauge theories. The background discussion in these sections will be relevant to the results of the original work presented in chapters 4 [27] and 5 [28] respectively, where the gauge/gravity duality is applied in both cases. Chapter 2 will present and thorough and pedagogical review of string theory. We begin with the classical bosonic string and cover the major achievements of modern superstring theory through the introduction of Polchinski’s D-branes. We highlight the conformal and reparmetrization (i.e. diffeomorphism) symmetries which are fundamental to logical underpinnings of superstring theory as a fundamental theory of nature. In Chapter 2 we also discuss in detail, quantization of the bosonic and superstring, the conformal anomaly arising from normal ordering ambiguities in the Virasoro algebra and the super-Virasoro algebra, as well as the anomaly cancellation. These discussions are based on the quantization of the path integral and employ the
19 BRST formalism which restricts the Fock space of states to a smaller Fock space of physical states where they tachyonic states are removed from the spectrum. Chapter 3 begins the discussion of the fundamental aspects of holography. Here we will discuss the AdS/CF T correspondence in anti-deSitter space (AdS 5 ⇥S5 ) as the first rigorous gauge/gravity duality. We will then cover the mapping of quantum fields in the bulk of a supergravity theory to the CFT operators to which they correspond on the conformal boundary. We will then discuss how stacks of parallel D-branes with open strings in type IIB string theory on AdS 5 ⇥ S5 and charged under super Yang-Mills gauge degrees of freedom, admits representations in terms of Young diagrams which correspond to physical configurations of D-branes. After this we will then discuss holographic renormalization in order to give a more complete operational picture of the correspondence.There we will also use some of the quantum gravity technology which is relevant in chapter 5. In Chapter 4 we apply the gauge/gravity duality to k-strings embedded in the Maldacena-Nastase (MNa) and Maldecena-Nuñez type IIB supergravity (SUGRA) backgrounds [27]. The MNa and MN SUGRA backgrounds have respectively, d=2+1 and d=3+1 extended space-time sectors. We will calculate the k-string tension law which is valid in both cases for SU (N ) gauge theories and compare the results with those from theoretical and numerical communities outside of string theory, and discuss the relevant irreducible representations of our solutions. In Chapter 5 we discuss the Kerr/CF T correspondence which adapts and ex-
20 tends the AdS/CF T correspondence in string theory to applications in 4-dimensional quantum gravity. We then apply the Kerr/CF T tools to Kerr-Sen black hole solution of low energy heterotic string theory compactified to 4 dimensions [28]. We obtain an effective 2-dimensional scalar action from the dimensionally reduced d=4 metric, where gµ⌫ (t, r, ✓, ) ! gµ⌫ (t, r) + U (1) gauge terms. We will then use the gauge/gravity correspondence to compute the d=1 dual CFT from the reduced AdS 2 geometry. We will also compute the Bekenstein-Hawking entropy and Hawking temperature via the Cardy formula, which relates these statistical thermodynamical quantities to the central charge of the dual CFT. Chapter 6 will summarize the results of Chapter 4 and Chapter 5, discuss recent developments in the gauge/gravity duality correspondence, and outline future efforts. Many of the topics covered in this work require more formalism than any reasonable attempt at a compact and concise presentation could allow. Often shorter discussions, which allow rapid progression and developments of the basic framework were abandoned in favor of completeness and self-contained discussions. That is my own personal preference. My perspective throughout the writing process has been the following; If at a later time, some poor misguided graduate student were to pick up on any of the projects discussed in this work (as was the case for me), then this thesis could serve as a self-contained "How to..." manual. This would hopefully save that lost soul some time in toiling through the vast ocean of literature and texts, before getting down to business. That being said, there are infinitesimally modest attempts
21 at brevity. For instance, some background formalisms and calculations of the string theory discussions contained in Chapter 2 have been included as appendices. T 1.3
Asymptotic Symmetries in AdS
In this section we apply the methods of canonical General Relativity or the Hamiltonian formulation of GR [29], which is typically referred to as the ADM (Arnowitt, Deser, Misner) formalism. A short review of the formalism is included in Appendix A. We will repeatedly encounter the ADM technology in various points throughout the later chapters. The aim of the ADM formalism is to formulate GR in terms of a Hamiltonian where manifest Lorentz invariance is lost due to the time component of the metric being singled out from the spatial components. 1.3.1 Canonical Asymptotic Symmetries In this subsection we will build on the ADM formalism reviewed in Appendix A. Specifically we will review the work of [26] in the case of AdS3 space and discuss how diffeomorphisms preserve the conformal structure of spaces that are asymptotically AdS. These diffeomorphisms can be identified as the generators of the conformal group in (1 + 1) AdS space, which is just the Virasoro algebra that we will encounter in Chapter 2. It is this work that seems to have laid the foundation for Maldacena’s conjecture and is seen as the precursor of the AdS/CFT correspondence. The aim of [26] is to show that even at a classical level, symmetries of an action which are only asymptotically maintained in a region far away from the action source, can induce a projective representation of the asymptotic symmetries (locally)
22 which manifest as central charges in the symmetry algebra. "Asymptotic symmetries are by definition those gauge transformations which leave the field configurations under consideration invariant, and... that they are essential for a definition of total ("global") charges of a theory" [26]. Global charges are the generators of asymptotic symmetries of some theory. If ⇠ is such a generator, then there is a phase space function H(⇠) (the Hamiltonian) which generates the corresponding transformation of the canonical variable. In their two papers, Brown and Henneaux prove that the Poisson brackets of the global charges in general produce a projective representation of the asymptotic symmetries which are isomorphic to some Lie algebra with a central extension [26, 30]. Their arguments are based on a theorem proved in [30] which states that if H(⇠) and H(⌘) are two well defined generators of the transformations stated above, then their Poisson brackets is also a well defined generator up to surface term which can at most by a constant K [⇠, ⌘]. Let ⇠, ⌘ be two global charges of some asymptotic symmetry group, then {H(⇠), H(⌘)}P B = H([⇠, ⌘]Lie ) + K [⇠, ⌘] ,
(1.34)
where [⇠, ⌘]Lie is the Lie bracket which defines the algebra, and K [⇠, ⌘] is the central charge or central extension of the Lie algebra. The central charges represent a nonuniqueness of the canonical generators in corresponding the phase vector field [26]. The significance of the last statement will be demonstrated in the following sections. For gauge theories with an asymptotic symmetry ⇠ A , the Hamiltonian generator H(⇠ A ) is defined as the functional integral over ⇠ A and a linear combination of
23 canonical constraints
A
plus a surface term, H(⇠) =
Z
dn x⇠ A (x)
A (x)
(1.35)
+ J(⇠),
where J(⇠) is the surface term of the functional and is needed in order to make the Hamiltonian well defined over the full phase space [31]. The same form of (1.35) also holds for gravitational theories as well with
A (x)
! Hµ (x), where Hµ (x) is the
linear combination of gravitational constraints in the canonical formalism [26, 29]. One way of computing the central charges (the standard way prior to [26, 30]) is carried out by taking the variation of the Hamiltonian (1.35) with
A (x)
! Hµ (x),
⇠ A ! ⇠ µ , and setting J(⇠) = 0, while keeping the surface terms from the integration by parts. Thus we have, H(⇠) = = where
Z
Z
dn x⇠ µ (x)Hµ (1.36) n
d x⇠
0µ
(x)Hµ0 ⇠,µ
+ Hµsurf ace ,
of Hµsurf ace is implicit, and the first term on the RHS of (1.36) vanishes at
the asymptotic boundary. For Hµsurf ace we have Hµsurf ace =
lim
r!1
I
⇥ dn 1 Sl Gijkl ⇠ ? (rk gij )
+ 2⇠i ⇡ il + 2⇠ i ⇡ kl
⇠ l ⇡ ik
gik ,
@k ⇠ ? gij
⇤
(1.37)
and, Gijkl =
1p g g ik g jl + g il g jk 2
2g ij g kl ,
(1.38)
and ⇠ ? is the time component of the isometry preserving vector field ⇠ µ . In other words, ⇠ ? is flow of the spatial slice along the time direction.
24 We then take the (assumed) asymptotic limits of the fields gij , ⇡ ij and vectors ⇠ µ and insert their values into (1.37). We may rewrite (1.37) in terms of an overall variation of the surface integral (which we now denote the linear combination of surface term constraints by Hµsurf ace ! Hµasymptotic ). We may then identify the negative of the variation of the asymptotic form of the surface terms of (1.37) as the charges: J(⇠) =
I
µ dn 1 ⇠asymptotic Hµasymptotic .
(1.39)
We note that the vector ⇠ µ correspond to the allowable surface deformations which are determined by asymptotic symmetries of the space-time. The geometric interpretation of the quantity Hµasymptotic is as the extrinsic curvature of the boundary surface. From (1.34) the central charge J(⇠) ⇠ K [⇠, ⌘], that differ by at most a constant. The second method is shown by Brown and Henneaux to be equivalent to the above prescription. Brown and Henneaux’s theorem states that if H(⇠) and H(⌘) are of the form in (1.35), then their Possion brackets are those of (1.34) and also define a new generator ⇣: {H(⇠), H(⌘)}P B = H([⇠, ⌘]Lie ) + K [⇠, ⌘] = H(⇣) + K [⇠, ⌘]
(1.40) (1.41)
where ⇠ and ⌘ are surface deformation vectors with their corresponding deformation algebra given by the Lie bracket [⇠, ⌘]Lie . [26, 30] show that ⇣ is determined to be a well defined generator in the same group of conformal algebra of surface deformations by the leading order contributions in 1/r (up to gauge transformations which fall off faster). Moreover, they also show that any transformation which is pure gauge will
25 not contribute to a central charge (under allowable field redefinitions). The Dirac brackets are a modified form of the Poisson brackets, such that they accommodate local symmetries and their corresponding constraints [32]. They are defined for a discrete set of constraints Ci for two fields A, B as {A, B}DB = {A, B}P B
(1.42)
{A, Ci }P B {Ci , Cj } 1P B {Cj , B}P B .
Brown and Henneaux noticed that their are two ways in which one may interpret the variation of the central charge J(⇠). 1. The Dirac bracket may be interpreted as the change in the charge J(⇠) under the unit magnitude surface deformation generated by J(⌘): ⌘ J(⇠)
(1.43)
= {J(⇠), J(⌘)}DB .
2. The charges J(⇠) also form a central extension of the conformal group algebra: ⌘ J(⇠)
(1.44)
= J ([⇠, ⌘]) + K [⇠, ⌘] .
The charges may most easily be found by evaluating (1.37) for a global AdS spacetime restricted to hypersurface at t = 0. For this configuration, one has the freedom to define the charges such that the integral will vanish. This implies that J([⇠, ⌘]) = 0. The charge J(⇠), before the surface is deformed, is also then zero. Then the central charge K [⇠, ⌘] reduces to the value of the charge J(⇠)SD =
⌘ J(⇠),
which
is the charge on the boundary surface deformed by ⌘. Then prescription outlined above reduces to a simple statement about the
26 asymptotic symmetry group (ASG) [32]: ASG =
allowed deformations . trivial deformations
(1.45)
1.3.2 AdS and Asymptotically AdS Space We can think of asymptotically AdS space as the deviation from global AdS space from the Lie flow along a canonical group vector which is an infinitesimal AdS symmetry. Let ⇠ be the generator as in Section 1.3.2, then we may write gµ⌫asymptotic (t = t) = gµ⌫exact (t = 0) + L⇠ gµ⌫exact (t = 0).
(1.46)
From this definition, we get an idea of what asymptotic symmetries are geometrically. They are the local boundary symmetries of the space defined by the "vacuum" solution, or in other words the local symmetries asymptotically far removed from any source contributions. Thus for a central extension K [⇠, ⌘], with surface deformation vectors ⇠, ⌘, then we identify ⇣ in the asymptotic limit as ⇣ µ (x) ! [⇠ µ (x), ⌘ µ (x)]SD ,
(1.47)
K [⇠, ⌘] ! J(⇠).
(1.48)
and
27 1.3.2.1
Ex : AdS3 Space
To be explicit let us take the example given in [26], the asymptotic symmetries of AdS3 . We begin by defining a local and global metric for AdS3 by ✓ 2 ◆ r 2 2 i) dslocal = + ↵ dt2 + 2A↵dtd 2 R ✓ 2 ◆ 1 r A2 2 + +↵ dr2 + r2 A2 d 2 2 R ✓ 2◆ r R2 2 2 2 ii) dsglobal = dt + 2 dr + r2 d 2 , 2 R r
(1.49)
where ↵, A correspond to energy and angular momentum/unit mass parameters. We can get a working understanding of asymptotically AdS space here if we note that in the limit as A ! 0 in (1.49), then i) ! ii). We then transform either i) or ii) by all possible AdS group transformations. Finding these transformations amounts to calculating the Lie derivative of (1.49) along the flow of a deformation or equivalently, solving Killing’s equations for the Killing vectors. We then on the metric with these vectors, which correspond with conformal group symmetries. This action produces the necessary boundary conditions to be respected. They are gtt = grr =
r2 R2
R2 r2
+ O(1), gtr = O + O r14 , gr = O
1 r3 1 r3
, gt = O(1), , g = r2 + O(1).
(1.50)
These transformations are essentially gauge transformations for a gravitational theory, thus they preserve the symmetries of the AdS group and account for all gauge degrees of freedom in the theory. We now consider our new metric to be the gauge transformed metric of (1.50), with which we calculate the Lie derivative. We take a surface deformation vector ⇠ µ which is a symmetry of the conformal
28 geometry, and has components
⇠ , ↵ = t, r, . We calculate the Lie derivative
(3) ↵
of (1.50) with respect to ⇠ µ and solve for the components
⇠ . This gives us a series
(3) ↵
of coupled PDE’s. The Lie vector components are (3) t
⇠ =RT (t, ) +
R3 T (t, ) + O(1/r4 ) r2 (1.51)
(3) r
⇠ =rR(t, ) + O(1/r)
(3)
⇠ = (t, ) +
R2 (t, ) + O(1/r4 ), r2
and the corresponding set of differential equations is R@t T (t, ) =@
(t, ) =
R(t, )
R@t (t, ) =@ T (t, ) T (t, ) =
R @t R(t, ) 2
(1.52)
1 (t, ) = @ R(t, ). 2 We note that the O(1/r4 ) terms for the t,
components and the O(1/r) term
in the r component of (1.51) terms are pure gauge contributions from the gauge transformations of (1.50), and are therefore unphysical. We also assume that the differential equations in (1.52) have been solved and do not care about their explicit solutions here. At this point we may proceed in one of two different directions in order obtain the symmetries of the ASG, both of which are equivalent. 1. We can take conformal Killings equations (1.52) and Fourier analyze them due to the periodicity in . This gives us an integer based spectrum for the asymptotic symmetries (1.51).
29 We then write the asymptotic symmetries in terms of the ADM lapse and shift functions (canonical variables) as in Section A and restrict their evolution to spacelike hypersurfaces. Upon imposing proper restrictions consistent with Einsteins equations (here Gµ⌫ = 0) to ensure the asymptotic conformal structure is preserved, we then calculate the Hamiltonian from (1.35) with J(⇠) = 0 initially. The variation of the Hamiltonian produce surface terms constrained under the asymptotic symmetries. Evaluation of the surface integral (1.37) produce the central charges as in (1.39). 2. We can take our asymptotic symmetries for (1.51) and (1.52) and as above, analyze them in a Fourier basis in order to solve for the central charges directly. We may calculate the Lie brackets of the Fourier modes which gives the classical asymptotic Lie algebra (i.e without central extension). We also calculate the Dirac brackets of two charges in order to produce the local central extension equivalent to
⌘ J(⇠),
where we evolve off of a t = 0 spatial
slice under the Lie flow in terms of canonical variables. In [26], Brown and Henneaux demonstrate the equivalence of the two methods, and that both reproduce the conformal group algebra (Virasoro algebra) on the boundary. We will not derive the remaining aspects of their work here, but rather state the results. We will discuss the details of the second process summarized above with respect to the Kerr black hole [33–35] in Chapter 5. The central charges for asymptotically AdS3 and the associated surface diffeo-
30 morphism vectors are ✓
JA =2⇡R 1 J↵ =
↵
4⇡↵A ⇠
2
A2 R2
◆
⇠ R
d , dt
d . d
(1.53) (1.54)
These are precisely the central charges for energy JA /R angular momentum/unit mass J↵ , and energy ↵, on the conformal boundary (d = 1 + 1) in AdS3 . We make note JA is the charge obtained as a result of the metric given in (1.49) i) which have been transformed from a properly normalized metric at the asymptotic boundary. Essential (1.49) i) and therefore JA is not normalized with respect to the proper time. Thus the proper asymptotic energy is JA /R [26]. 1.4
Black Hole Entropy and Temperature via Cardy
The seminal works of [5–7,9,29,31,36] along with those of [26,30] in Section 1.3 are among the major contributions to the foundations of the field of quantum gravity. In particular the results of [26] and the later refinements in [37, 38] showed that any consistent formulation of a quantum theory of gravity must have the Virasoro algebra as it’s asymptotic symmetry group [39]. The afore mentioned works have launched an entire program of the study and classification of the black holes [37, 39–43] in terms of quantum entropy (i.e. microstates) on or near the horizon. It has been shown by Hawking that entropy, angular momentum, mass, and electrostatic charge are the only canonically independent quantities with which black holes may be characterized. This can be seen from perturbations of the energy of a
31 black hole [5] dE =
dA + ⌦dJ + Qd , 8⇡
which is analogous to the 1st law of thermodynamics, dE = T dS
(1.55) pdV + µdN . Here
is the surface gravity, ⌦, J are the angular velocity and momentum, and Q,
are
the electrostatic charge and potential. From the work of Strominger and Vafa in [42], exact agreement of the entropy and temperature to those predicted by [5–7, 9], was calculated from supersymmetric black holes in type IIB string theory. This gave the first theoretical support for the analogy between (1.55) and the 1st law of thermodynamics to become an identification of the parameters of (1.55) with thermodynamic quantities: T ⇠ ,
A ⇠ S,
⌦ ⇠ p,
J ⇠ V,
Q ⇠ µ,
(1.56)
⇠ N.
An extremely powerful and universal tool developed by Cardy [40,44] gave the theoretical framework for the developments of the study of black hole microstates for spacetimes which have the conformal group as their asymptotic symmetry group and thus generate a boundary CF T in (1+1) dimensions. The universality of Card’ys formula was shown by Strominger in [39] to be independent of quantum gravity or string theoretical foundations, and generalized by Carlip in [43] for arbitrary dimensions using the formalism of [26] as discussed in Section 1.3. The Cardy formula was later generalized in terms of Maldacena’s AdS/CFT correspondence in [45].
32 These developments launched a fever of papers related to the study of black hole microstates and their corresponding dual CF T ’s in string theory and quantum gravity [28, 33, 34, 46–57]. We now wish to discuss the black hole entropy and temperature in terms more suited to the holographic principle, where the central extension the CFT will play an important role. The central charge of a CFT is associated with the number of massless particle species in a theory [45]. In [40, 44], Cardy derived a general relation for quantum CFT’s in (1+1) dimensions where the ratios of the scaling amplitudes in the transfer matrix set on a torus had a general form which are expressible in terms of their operator product expansions. The toriodal compactification yields non-trivial constraints on the CFT. The form of Cardy’s partition function was Z(l, l0 ) = ef A+c⇡
/6
X
e(En
E0 )l0
,
(1.57)
n
where l, l0 are the lengths of the sides of a parallelogram with the sides identified and is homeomorphic to T 1 and in the limit l, l0 ! 1 the corresponding ratio l0 /l ⌘
is
kept fixed. c represents the central charge of the CFT. The result of Cardy’s analysis show that their must be a finite set of primary CFT operators which contribute to the operator product expansion. c is parametrized ¯ with by a rational number m and the set of operators with scaling dimensions (h, h) conformal character
¯ and scaling dimensions given by the Kac formula =h+h hp,q
(p(m + 1) qm)2 = , 4m(m + 1)
(1.58)
¯ p¯,¯q (where the bar denotes anti-holomorphic). Given the above defiwhere hp,q , andh
33 nitions c is parametrized as c=1
6 , m(m + 1)
(1.59)
and if the theory is unitary then 0 < c 1, and m 2 Z > 2. The final form of Cardy’s formula for the ratio of the partition functions was then found to be, Z(l, l0 ) = ef A+c⇡Re
/6
X
p,q,¯ p,¯ q
N (p, q, p¯, q¯)
p,q (
)
p¯,¯ q(
⇤)
(1.60)
where N (p, q, p¯, q¯) is just a number function of the conformal dimension of a particular term in the OPE. In [40, 44] Cardy also showed how (1.60) is bounded from above, which constrains the OPE to consist of a finite number of operator forms in a given OPE. From Cardy’s result of the partition function, one may define quantum statistical quantities such as entropy and temperature. We may take a simplified version of (1.60), and consider a closed string on a Euclidean world-sheet compactified on a torus, so the parameters ⌧ 2 [0, ), [0, 2⇡) are periodic, and
2
= 1/T where T is the temperature. The Free energy
partition function of this type of configuration is [25] Z( ) = T r(e
H
)=e
F
,
where H is the Hamiltonian. If we consider the limit as
(1.61) ! 1, this corresponds
to low temperatures where the lowest state of the spectrum will dominate. This will correspond to the ground state of the string. Let come back to the statement above after a bit of discussion about the Hamiltonian in (1.61). The ground state can be found from a few clever applications of conformal mappings.
34 We first consider the string on the cylinder, parametriced by
! w =
2 [0, 2⇡). We can conformally map the cylinder to the complex plane by
+ i⌧, z = e
+
iw
. In effect we are considering a circle of radius z on the complex to be
identified as a discrete energy level. In this description ⌧ !
1 ⇠ low energy. It is
shown in [25] that the cylindrical EMT (T (w)) is conformally related to the z plane EMT (T (z)) by Tcyl (w) =
z 2 Tplane (z) +
c . 24
(1.62)
The Hamiltonian can be computed in this configuration from a coordinate transformation of (1.19) and (1.20) in terms of complex coordinates to solve for T⌧ ⌧ : H⌘
Z
(1.63)
d T⌧ ⌧ .
Evaluation of (1.63) with respect to the w parametrization where T⌧ ⌧ ! (Tww + T w¯w¯ ) gives the low energy configuration with H = E: E=
2⇡(c + c˜) E =) = 24 2⇡
c + c˜ , 24
(1.64)
and c = c˜. (1.64) has the interpretation of the Casimir self energy, as can be seen from (1.62) in the limit as z ! 0. This is also the ground state of the string. Now we return to the partition function (1.61) with a ground state energy calculated. In the limit as
! 1 then lim Z ! e !1
We perform a coordinate redefinition by
Hgs
= ec
/12
.
(1.65)
$ ⌧ , followed by a conformal scaling (since
35 the domain of the spatial direction is ⌧!
2⇡
2 [0, 2⇡)), the rescaling is accomplished by ⌧,
!
2⇡
.
(1.66)
The combination of the reparametrization and scaling is just a modular transformation of the torus. At this point
2 [0, 4⇡ 2 / ⌘
0
) and is now a timelike direction. Conformal
invariance requires that (1.67)
Z( 0 ) = Z( ), and therefore, lim Z( 0 ) ! ec⇡ 0 !0
2 /3 0
.
(1.68)
We see that low temperatures of (1.65) correspond to high temperatures of Z( 0 ) in the limit as
0
! 0, where the high energy states dominate.
From (1.68) and (1.60) we can directly identify the thermal form of the Cardy formula, which is in terms of a general CFT in (1 + 1)-dimensions [35] is SCardy =
⇡2 c ⇡2 = cT, 3 0 3
(1.69)
where we have expressed the entropy in terms of the temperature T = 1/ 0 . We can also find the statistical Cardy formula in terms of energy and thus for strings [25, 45, 58] using the above results and rewriting the entropy in terms of the free energy. The density of states of a statistical system has the form ⇢(E) = eS(E) ,
36 with S(E) the entropy. The free energy can then be written as Z F e = dE⇢(E)e E =
Z
dEe
S(E)
E
(1.70)
.
p In (1+1) dimensional systems S(E) ! N E at large energies, where N is the number of degrees freedom. We can now approximate the integral of (1.70) using the saddle point method and the 1st law of thermodynamics, since @E/@S = T =) @S/@E = . Evaluating the integral at the saddle point for large energies gives (1.71)
F ⇠ N 2T 2. p Thus S ⇠ N E is equivalent to F ⇠ N 2 T 2 in (1+1) dimensions.
Using the scaled value from the modular transformation in (1.68), implies the statistical Cardy formula at high energies can be written in terms of the ground state for CFT’s as S0 = 2⇡
r
c E0 = 2⇡ 6
r
c Q0 , 6
(1.72)
where Q0 is the operator eigen-mode which corresponds to the eigen-value E0 . Thus ˜ 0 }, we have of for open in terms of free strings with Virasoro operators Q0 = {L0 , L and closed strings [25, 39, 45, 49] r ⇣ r c c c⌘ ˜0 Sclosed ⇡ 2⇡ Eclosed = 2⇡ L0 + L , 3 3 12 r ⇣ r c c c⌘ Sopen ⇡ 2⇡ Eopen = 2⇡ L0 , 6 6 24 where the ground state Hamiltonian is given by Hopen = L0 ˜0 and Hclosed = L0 + L
(1.73) (1.74)
c/24 for the open string
c/12 for the closed string, and the ground state for the closed
string is required to satisfy L0
˜0 c/24 | i = L
c/24 | i = 0.
37 The holographic generalization of Cardy’s formula [45] to CFT’s of dimension n + 1 was derived from Freidman-Robertson-Walker (FRW) cosmological models. FRW models generally describe a radiation dominated, expanding universe with the cosmological radius R parametrized as a function of time t. They may be described by free or weakly interacting massless particles. Thus it is possible to use an interacting CFT to describe radiation [45]. In [45], Verlinde states that from energy/momentum conservation, and the FRW equations in (n + 1) dimensions, that their form can be written as H2 =
16⇡GN E n(n 1) V
d H˙ = H = dt
1 R2 ✓
8⇡GN n 1
◆ E 1 + p + 2, V R
(1.75)
where E and later S are the total energy and entropy, H the Hubble parameter, R the cosmological radius at time t, and the equation of state p = E/(nV ), that E/V and p decrease as 1/R(n+1) . Verlinde conjectures that there is a deep connection between the Cardy formula of (1.74),(1.73) and the FRW equations. He notes in [45] that with the identification of the components of the ground state Virasoro operator (L0 ) and entropy to those of (1.75) 2⇡L0 ⇠
2⇡ ER n
2⇡c ⇠(n S ⇠(n
V 4GN R HV 1) , 4GN 1)
followed by the substitution of (1.76) into (1.74) reproduces (1.75) exactly.
(1.76)
38 In [9], Bekenstein shows that a macroscopic system with limited self gravity (HR 1 for the closed FRW universe of Verlinde’s argument), that the total entropy is bounded by constant multiple times the total energy and linear size, which in the context of [45] is, Stot SB ⌘
2⇡ ER, n
(1.77)
where SB is the macroscopic Bekenstein entropy bound. However Verlinde notes that SB does not bound the total entropy in his argument but rather the sub
extensive
part of the entropy which is associated with the Casimir self energy of the CFT. Therefor a system may be determined to of limited self gravity if SB is compared to the holographic Bekenstein-Hawking entropy SBH ⌘ (n
1)V /4GN R.
From the 1st law of thermodynamics (T S = E + pV ), Verlinde defines the deviation from the 1st due to the Casimir self energy of the CFT as EC ⌘ n(E + pV
(1.78)
T S),
and defines the total energy to be the sum of the extensive and Casimir energies: Etot (S, V ) = EE (S, V ) + (1/2)EC (S, V ), and notes that EC under conformal scalings goes as EC ( S, V ) =
1 2/n
(1.79)
EC (S, V ).
Given that one knows the scaling of EE and EC , conformal invariance constrains ER to be a function of S only such that, EE =
a 1+1/n S , 4⇡R
EC =
b 1 S 2⇡R
1/n
.
(1.80)
39 Upon substitution of (1.80) into (1.74), Verlinde establishes the form of the Cardy formula for the entropy to be, 2⇡R p S= p EC (2Etot ab
EC ),
(1.81)
where a, b are constants to be fixed. In [59], Witten argues at high temperatures the entropy, energy and temperature of a CFT can be identified with those of entropy, mass, and Hawking temperature of an AdS black hole previously considered by Hawking and Page. Using the AdS/CFT, Verlinde argues that the energy and entropy for a CFT on the manifold R ⇥ Sn is given as, c V 12 Ln ✓ ◆ c n L2 V E= 1+ 2 , 12 4⇡L R Ln S=
(1.82)
where L is the length scale of the AdS space curvature. Upon solving for the temperature T from the 1st law of thermodynamics, and solving for the Casimir energy, Verlinde finds, EC =
c n V . (n 12 2⇡R L 1) R
(1.83)
Verlinde eliminates c and L, and finds with the expressions of (1.80) and (1.74), the AdS result of the Cardy formula for the holographic entropy, S=
2⇡R p EC (2Etot n
EC ),
(1.84)
where the constants a, b of (1.80) have now been fixed(constrained) such that ab = n2 .
40 The various forms of the Cardy formulas (1.69),(1.74),(1.73) and (1.84) will be referenced in the analysis of Chapter 5, where the entropy and temperature of the Kerr black hole [33, 34, 34], Kerr-Sen black hole of low-energy heterotic string theory [28,52,60], is derived and found to agree with the Bekenstein-Hawking entropy and temperature. 1.5
QCD k-strings
In the study of quark interactions one of the quantities of interest is the QCD string tension between a quark/anti-quark pairs. k-strings are heavy quarks and antiquarks pairs which act as color sources separated by a color flux tube of distance L in the fundamental representation with respect to some gauge group [61]. The gauge group relevant here is SU (N ) with N interpreted as the number of colors which is a free parameter however the quantity g 2 N is kept fixed, where g is the gauge coupling. N -ality is a concept in k-strings where k is referred to as the N -ality of the color representation such that k = |l
m| ,
(1.85)
where l is the number of quarks in the fundamental representation and m is the number of quarks in the anti-fundamental representation and k N since color singlet objects are possible. Thus k-ality is difined mod N . There are two classes of proposed k-string tension laws, the Casimir formula and the Sine formula. Also, the tension of the fundamental string depends only on the N -ality of the representation.
41 The Casimir formula is given by CR ⌧f Cf und
⌧k =
(1.86)
where CR is the quadratic Casimir operator for the representation R and is defined by, T a T b = CR I R where IR is the identity and the T a ’s are the generators of SU (N ) in the representation R. For antisymmetric representation of index k, the Casimir formula becomes, ✓
⌧k = k 1
k N
1 1
◆
(1.87)
⌧f
Since we are interested in the gauge group SU (N ) and anticipating that we will want to consider the tension in the large N limit, it will be useful to expand (1.87) for large N which yields, ⌧k = k N
1
k+O N
(1.88)
⌧f .
The k-string Sine formula is given as sin ⌧k = ⌧ f sin
⇡k N ⇡ N
(1.89)
,
where the large N expansion takes (1.89) to the form ✓
⌧k = k 1
⇡2 2 (k 6N 2
1) + O(N
4
◆
) ⌧f .
(1.90)
The central charge can also be related to the energy of the QCD string. Let the flux tube between a quark/anti-quark pair is of length L and has a string tension
42
⁞
L
Figure 1.3: On the left, a hadronized grouping of a more general k-string configuration, and the right is a single q
q¯ forms the simplest k-string.
of T . The string will confine quark/anti-quark pair if T L . m, where m is the mass of the lightest quark. The Casimir energy of the string can be written from the result of (1.64), and the string energy has an expansion as a function of L given by [25, 61–63] E(L) = T L + a
⇡c + ..., 24L
(1.91)
where a is an undetermined constant. We pause to comment briefly on the above. We note that (1.88) is an expansion in powers of 1/N where as in (1.90) contains only even powers. From analysis of [61] and [64] the pure Yang-Mills power expansion is expected to be even powers of 1/N which suggests that sine formula is more accurate. Similarly [61] and [65] claim that an exact sine formula was given due to an analysis from supergravity, and [66] obtain a result that is within a few percent of a sine formula but not exact. [64] and [61]
43 exclude the exact Casimir formula as a possibility however they concede that there does not exist as of present, a proof of the sine formula as the correct tension. Deriving the relevant tension formulas for k-string configurations from dual gravity theories will be a large portion of the results presented [27] in Chapter 4. Understanding quark interactions is a problem which remains relatively intractable in many situations. Perturbation theory is only valid at high energy scales where the coupling parameter is weak. Even at these scales lattice QCD calculations are cumbersome and require large computational resources. Other toy models such as Yang-Mills in 2 + 1 dimensions have been used [67], [68], [69], [70], however are far from a realistic 3 + 1 calculation. The limitations of these methods is one motive for the investigations of Chapter 4. We will use technology from AdS/CFT in non-conformal SUGRA backgrounds in to probe these same questions raised above.
44 CHAPTER 2 STRING THEORY
Your theory of a doughnut-shaped universe is intriguing... I may have to steal it.
The Simpsons, "They Saved Lisa’s Brain" (1999) -Professor Stephen Hawking to Homer Simpson 2.1
Outline of Chapter 2
In this chapter we review bosonic string theory which is defined on a 1 + 1 dimensional world-sheet. We begin by illustrating a pedagogical presentation (primarily based on the rigorous discussions in [16]) of the classical bosonic string on a world-sheet of flat Minkowski space. We will progress through open and closed string, equations of motion and boundary conditions, while emphasizing the classical and local symmetries of the theory before moving on to the quantization of the free string. We will discuss the classical and quantum Virasoro algebras that arise from the Poisson brackets of the classical field and the equal time canonical commutation relations respectively. The classical algebra acquires a normal ordering ambiguity which leads to an anomaly upon quantization. We will employ the manifestly covariant Faddeev-Popov prescription for path integral in order to cancel the anomaly. This will maintain Weyl and Diffeomorphism invariance of the classical theory at the quantum level, however the cost of this method of quantization is the introduction of unphysical Faddeev-Popov ghost states. We define a Hilbert space of physical states
45 by restricting the full Hilbert space of the theory with the BRST formalism, which identifies the physical states of mass m
0.
After completing the free bosonic theory, we will briefly discuss the superstring generalization. Here we will touch on the topics of supersymmetry and superspace formalism on the world sheet in order to motivate the quantum superstring. The classical and quantized bosonic string and superstring theories will be be presented from the perspective of maintaining space-time Poincaré and diffeomorphism (reparametrization) invariances, and world-sheet conformal (Weyl) and diffeomorphism invariances. Once this is completed, we will move on to briefly summarize the 5 consistent string theories, type I, type IIA, IIB, and the E8 ⇥ E8 and SO(32) heterotic string theories. We will then discuss more modern results from string theory such as Dbranes and the symmetries which they respect such as T and S-duality, and gauge symmetries. These will become important in the formulation the holographic AdSCFT correspondence conjectured by Maldecena [8]. 2.2
The Classical Bosonic String
We consider here, the classical bosonic string which is a 1-dimensional spatially extended object. The bosonic string sweeps out a 2d world area (sheet) parameterized by 2 world-sheet parameters. 1 space like parameter , and 1 time like parameter, ⌧ which describe the free string. The 2d string is embedded in a D-dimensional space-time with coordinates X µ (⌧, ). We will keep D arbitrary until we arrive at the anomaly free result for the bosonic case of D = 26.
46 Classically the free bosonic string is described by the flat Minkowski action, S =
1 T 2
Z
d2
p
hh↵ @↵ X µ (⌧, )@ Xµ (⌧, )
with d2 = d d⌧, 0 Here T =
1 2⇡↵0
⇡,
(2.1)
1 ⌧ 1.
is the string tension and ↵0 is the Regge slope trajectory parameter.
h↵ is the 2d world-sheet Minkowsiki metric with signature ( , +) and the indices µ, ⌫ = 0, . . . , D
1. We will repeatedly suppress the explicit world-sheet coordinate
dependence of the space-time fields X µ but keep in mind their implicit dependence on the ⌧ and
parameters, except when it adds clarity to make them explicit. We
will also freely interchange the terms diffeomorphism and reparametrization invariance as in what follows herein, they are considered synonymous. Though it may be technically incorrect to do so, we also use interchangeably the terms Weyl, conformal, and (re)scaling invariance, unless a distinction between them is explicitly required. We note that the action (2.1) respects local reparametrization invariances under arbitrary infinitesimal diffeomorphisms ⇠ ↵ (⌧, ) X µ = ⇠ ↵ @↵ X µ h ↵ = ⇠ h↵ p
@ ⇠↵h
@ ⇠ h↵
(2.2)
p h = @↵ (⇠ ↵ h)
and Weyl scaling invariance under an arbitrary scalar factor ⇤(⌧, ), h↵ = ⇤h↵
(2.3)
Along with these local symmetries, (2.1) also enjoys global space-time Poincaré invariance for arbitrary coordinate transformations. For flat Minkowski space these are
47 the usual, X µ = aµ⌫ X ⌫ + bµ h
↵
(2.4)
=0
where ↵⌫µ is antisymmetric and constant, bµ is constant, and h↵ is a space-time scalar. Taking the variational derivative of (2.1) with respect to the inverse worldsheet metric h↵ , yields at the energy-momentum tensor T↵ =
2 1 S p T h h↵
= @ ↵ X µ @ Xµ
1 0 0 h ↵ h ↵ @ ↵0 X µ @ 0 X µ 2
(2.5)
= 0, which is traceless due to Weyl invariance, i.e. h↵ T↵ = 0.
(2.6)
Using the gauge freedom at hand, we may restrict the analysis to the conformal gauge. The result of the gauge choice allows us to choose the form of the world-sheet metric, h↵ = e
✓
◆ 1 0 . 0 1
(2.7)
This will describe the general conformal gauge world-sheet metric that we will make use of when we quantize the theory. At the quantum level this gauge choice is anomalous and requires much care in order to fix all the gauge degrees of freedom. For the classical case at hand we simply note that we have 2 reparametrization degrees of freedom and 1 Weyl scaling degree of freedom that we may use to make the metric
48 conformally flat. With this in mind, we may gauge fix the world-sheet metric for our present considerations to be of the form, hcgf ↵
= ⌘↵ =
✓
◆ 1 0 , 0 1
(2.8)
where cgf above denotes the classical gauge fixed metric subject to the gauge constraint equation, h↵
(2.9)
= 0.
Using the gauge fixed metric, we now have the classical gauge fixed action of (2.1) such that, S
! S
cgf
=
1 T 2
Z
d2 ⌘ ↵ @ ↵ X µ @ Xµ .
(2.10)
The massless classical bosonic string coordinate (X µ equation of motion is the 2d Euler-Lagrange equation from (2.10) ✓
@2 @ 2
@2 @⌧ 2
◆
X µ = 0.
(2.11)
This is equivalent to the general variation of S cgf under X µ ! X 0µ = X µ + X µ
(2.12)
such that up to boundary terms, S cgf
! S 0cgf = S cgf (X µ + X µ ).
(2.13)
This variation reproduces the classical equations of motion (2.11) plus a surface term. It is necessary for the surface terms to vanish. Periodicity in
insures that they vanish
49
x0 = t
x0 = t ···
··· ··· · ·· ··· ··· ··· · · · xk+1 , . . . , x25 ···
xk+1 , . . . , x25
x1 , . . . , x k
x1 , . . . , x k
Figure 2.1: The world-sheet of an open string embedded in D = 10 dimensions
for closed strings, however open-strings must take on vanishing boundary conditions. Thus we have, cgf Ssurf ace
=
T
Z
⇥ d⌧ Xµ0 X µ |
=⇡
Xµ0 X µ |
=0
⇤
=0
(2.14)
where the primes denote the partial derivatives of X µ with respect to the world-sheet parameter . For closed strings, (2.11) and periodic boundary conditions of the coordinates X µ, X µ (⌧, ) = X µ (⌧, + ⇡) yields a stationary solution to the classical equations of motion.
(2.15)
50 2.2.1 Solutions to the classical wave equation We will now find the general solutions of (2.11) for both open and closed strings. Following [15, 16, 21], it is convenient to change to light-cone coordinates on the world-sheet. We now define, ±
(2.16)
⌘⌧±
such that the general solution to (2.11) is expressed in terms of (2.16), X µ (⌧, ) = XRµ (
) + XLµ (
+
),
(2.17)
where we define the right/left moving coordinates XRµ /XLµ . In light-cone coordinates, the partial derivative operator and Minkowski metric become 1 (@⌧ ± @ ) 2 1 =⌘ += 2
@± ⌘ ⌘+
(2.18)
⌘±± = 0. We also note that the constraint equation (2.5) must supplement the massless wave equation (2.11). The original world-sheet parametrizations gives components of the constraint equation (2.5) as T10 = T01 = @⌧ X µ @ Xµ = 0, T00 = T11
1 = @⌧ X 2 + @ X 2 = 0. 2
(2.19)
In light-cone coordinates these constraints read, T++ =
1 (T00 + T11 ) 2 µ
= @ + X @ + Xµ
(2.20)
51 and T
=
1 (T00 2
T11 )
(2.21)
= @ X µ @ Xµ respectively. The trace (or rather traceless) condition of the energy-momentum tensor given by h↵ T↵ = 0 in light-cone coordinates takes the form of, T+ = T
+
=0
(2.22)
(2.20) and (2.21) along with the previous constraint equation (2.19) equivalently state that T00 = T11 . Using these results, we find that the constraint equations T++ = T
=0
(2.23)
become constraints on the ⌧ derivatives of the left/right modes X˙ R2 = X˙ L2 = 0.
(2.24)
In the above, X˙ 2 = @⌧ X µ @⌧ Xµ . The constraint equation of (2.24) correspond to the conserved quantities of (2.20), (2.21) and as such we have the freedom to set them to zero as in (2.23). We will address the validity of this choice in more detail below. 2.2.1.1
Intermission, Comments on gauge symmetries
Before solving the massless wave equation, let us briefly discuss the transformation to light-cone coordinates and implications of our gauge choice. This short digression will be relevant when we discuss the classical operators which generate the residual symmetries that are preserved by the above gauge choice. The operators will
52 form an infinite dimensional Lie algebra known as the Witt algebra, or the Virasoro algebra without a central extension. Upon quantization, the Witt algebra acquires a c-number anomaly as a central extension, and is then called the Virasoro algebra. First note that the above results lead to general statements about 2-dimensional quantum field theories having an infinite number of conserved charges as we will see below. Although these results have been obtained from a classical theory and in general may be anomalous upon quantization, the symmetries must be maintained even at the quantum level in order to have a sensible theory. Let us press forward with a quick analysis on this intermittent topic. In 2-dimensional quantum field theories, the conservation of energy and momentum becomes, @ T++ + @+ T+ = 0 @+ T
+@ T
+
(2.25)
= 0.
Using the tracelessness of T+ = 0, the conservation of energy and momentum now yields @ T++ = 0 and @+ T
(2.26)
= 0.
Since T++ is a conserved quantity and we note that these indeed are the constraints of (2.24), we have the freedom to set them to zero and move on. Also, for any function f(
+
) which depends on the world-sheet parameter
+
such that @ f (
+
) = 0 leads to
a conserved Noether current. We see that this is true by observing that given (2.26),
53 then for any such f (
+
) we have
@ f(
+
)T++ = 0 =) f (
+
)T++ = constant,
(2.27)
and similarly there is a corresponding charge Qf such that, Qf =
Z
The same is true for the case of + !
d f(
+
(2.28)
)T++ .
. Since f (
+
) was arbitrary, we have an
infinite number of conserved quantities. This holds in general for any 2-dimensional conformally invariant theory field theory. Given an infinite set of conserved quantities, it follows that we require an algebra of generators consistent with the infinite number of conserved quantities given above. This will lead us to the construction of the Virasoro algebra. The infinite number of conserved quantities above is a result of residual gauge symmetries carried over from the gauge fixing of the metric [16]. We may then conclude that setting h↵ = ⌘↵ does not gauge away all of the degrees of freedom. To explore this further, [16] notes that given any combination of diffeomorphisms and Weyl scalings for which (as given in (2.2) and (2.3)) (2.29)
@ ↵ ⇠ + @ ⇠ ↵ = ⇤⌘ ↵
holds, must also respect same the symmetries preserved by the gauge choice. Then for ⇠ ± = (⇠ 0 ± ⇠ 1 ) implies that ⇠ + (⇠ ) can be an arbitrary function of
+
(
). To wrap
up our tangent discussion about the gauge choices, we may consider the world-sheet reparametrizations to be
↵
= ⇠ ↵ , then the set of generators of the residual gauge
54 degrees of freedom are given by, V = {⇠ +
@ @
+
,⇠
@ @
(2.30)
}
These operators can then be considered as the classical generators of the 2-dimensional conformal transformations described above. We will revisit this shortly after finalizing our solutions to the 2-dimensional massless wave equation. 2.2.1.2
Back to general solutions of the classical wave equation
For the closed string we have the boundary conditions stated previously in (2.15), which simply respect a periodicity condition in the
world-sheet parameter. With
this in mind, the closed string general solution is given by two independent right and left moving mode expansions as 1 1 XRµ = xµ + ls2 pµ (⌧ 2 2
i X1 µ ) + ls ↵ e 2 n6=0 n n
1 1 i X1 µ XLµ = xµ + ls2 pµ (⌧ + ) + ls ↵ ˜ e 2 2 2 n6=0 n n
)
(2.31)
2in(⌧ + )
(2.32)
2in(⌧
where ↵n are Fourier mode coefficients, and xµ and pµ denote the center of mass position and momentum of the string. For the open string solution, the left and right moving modes must combine in order to form a standing wave. We recall that the surface term which resulted from the variation of the classical action (2.14) must vanish at the string endpoints. The boundary conditions consistent with this constraint are X 0µ |
=⇡ =0
= 0.
(2.33)
55 The general open string solution is then X µ (⌧, ) = xµ + ls2 pµ ⌧ + ils
X 1 µ ↵m e m m6=0
im⌧
cos(m )
(2.34)
The closed and open solutions obey the following relations defined by the Poisson brackets at equal ⌧ , [P µ ( , ⌧ ), P ⌫ ( 0 , ⌧ )]P.B = [X µ ( , ⌧ ), X ⌫ ( 0 , ⌧ )]P.B = 0 [P µ ( , ⌧ ), X ⌫ ( 0 , ⌧ )]P.B =
1 µ⌫ ⌘ ( T
0
(2.35) (2.36)
)
where P µ( , ⌧ ) =
S = T X˙ µ , ˙ Xµ
(2.37)
with primed fields denoting derivatives with respect to . We also have (2.38)
[pµ , xµ ]P.B. = ⌘ µ⌫
It follows from the Poisson brackets, that the Fourier oscillator modes satisfy µ µ [↵m , ↵n⌫ ]P.B. = [˜ ↵m ,↵ ˜ n⌫ ]P.B. = im⌘ µ⌫
m+n,0
(2.39)
µ ,↵ ˜ n⌫ ]P.B. = 0, [↵m
where ↵µ n = (↵nµ )†
and ↵ ˜ µ n = (˜ ↵nµ )†
(2.40)
follow because the bosonic coordinates are real. We will also make the convention that the zero oscillator modes are defined by ↵0 ⌘ ↵˜0 ⌘ 12 lpµ for closed strings and correspondingly ↵0 ⌘ ↵˜0 ⌘ lpµ for open strings. Here l is the length of the string defined in terms of the string Regge parameter or equivalently the string tension l=
p
2↵0 =
p1 . ⇡T
56 2.2.2 The Witt Algebra and Global Symmetries Given the classical bosonic string solutions and the commutation relations defined by the Poisson brackets in the previous section, we wish to use them to further inform our understanding of the free string. Proceeding along these lines we will eventually arrive at the Witt algebra which is the algebra of infinitesimal diffeomorphisms of S 1 and contains the generators of the residual symmetries preserved by the conformal gauge. Before constructing the Witt algebra, we will begin by making good use of the Poisson brackets and generate the free string Hamiltonian. We will then use the conserved quantities from the constraint equations of energy-momentum tensor to construct the algebra and show that the Hamiltonian is contained in the infinite dimensional Witt algebra. Once we have the Witt algebra we will be in a position to quantize the theory and obtain the Virasoro algebra. Let us begin by constructing the Hamiltonian of the free string. We define the Hamiltonian [16] in terms of a Legendre transformation of the classical action string action ⌘◆ T ⇣ ˙2 02 H= d X +X 2 0 Z ⇣ ⌘ T ⇡ = d X˙ 2 + X 02 2 0 Z
⇡
✓
X˙ µ P⌧µ
(2.41)
where dP µ = @⌧ P µ + @ P µ d⌧ is defined to be the total momentum flowing across an arbitrary line segment of the world-sheet, from which we directly see @⌧ X µ = P⌧µ . We µ write the Hamiltonian in terms of the oscillator modes ↵m ’s using bosonic coordinate
mode expansions for X˙ µ and X 0µ with the proper boundary conditions for open and
57 closed strings 1 1 X H= ↵ 2 n= 1
1 1 X H= (↵ 2 n= 1
n
for open strings
· ↵n ,
n
· ↵n + ↵ ˜
for closed strings.
·↵ ˜n) ,
n
(2.42) (2.43)
We combine the closed string mode expansions of the energy-momentum tensor constraint equations (T↵ = 0) with the result of our analysis of T±± which gave X˙ L2 = X˙ R2 = 0, and find that only the Fourier components of the mode expansions X µ survive. We have also seen that in the light-cone coordinates that T++ and T
are
conserved currents, which imply that the following operators at ⌧ = 0 are conserved Noether charges, known as the Virasoro generators. The right modes are Z T ⇡ 2im Lm = e T d 2 0 Z T ⇡ 2im ˙ 2 = e XR d 2 0 1 T X = ↵m n · ↵n , 2 n= 1
(2.44)
and similarly the left modes are
˜m = T L 2 = =
T 2 T 2
Z Z
⇡
e2im T++ d 0 ⇡
e
0 1 X
n= 1
2im
↵ ˜m
X˙ L2 d n
(2.45)
·↵ ˜n.
If we repeat the above to accommodate open string boundary conditions, we must then note that the Fourier components of the open string are not orthogonal functions. This simply implies that the open string modes are 2⇡ periodic. This can be seen by applying the periodicity relations for the left and right moving modes such that
58 !
+ ⇡ and impose the constraint equation T++ = 0 for the interval
⇡
⇡;
the result of which yields the open string Virasoro generators Lm = T = =
T 4 T 2
Z
⇡
e 0
Z
⇡
eim
0 1 X
im
⇣
↵m
n= 1
T
+ eim T++ d
X˙ + X 0 n
⌘2
(2.46)
d
· ↵n
Momentarily we will construct the algebra associated with Poisson brackets of the Virasoro generators. First, we rewrite with the Hamiltonian found in (2.42) in terms of the Virasoro generators. For open strings the Hamiltonian is given by (2.47)
H = L0 , while for the closed string Hamiltonian is ˜0 . H = L0 + L
(2.48)
We also note that the constraint equations of the energy-momentum tensor (2.23) require that L0
˜ 0 must also vanish. We note that L0 L
˜ 0 does not contain the L
space-time momentum pµ and is the generator of rigid rotations of the closed string along the
world-sheet parameter by an arbitrary angle.
The constraint equation (2.23) implies that the zero mode Virasoro operator satisfies L0 = 0. We take the invariant mass equation (M 2 =
pµ pµ ) subject to
the Virasoro constraints in order to write the invariant mass in terms of the internal
59 oscillator modes. At the classical level we then have the mass shell conditions, 1 1 X M = 0 ↵ ↵ n=1 2
M2 =
1 2 X (↵ ↵0 n=1
n
(for open strings)
· ↵n ,
n
· ↵n + ↵ ˜
n
·↵ ˜n) ,
(for closed strings)
(2.49) (2.50)
We take the Poisson brackets of the Virasoro generators and make use of the results from the oscillator mode brackets to arrive at the Witt algebra [Lm , Ln ]P.B. = (m
n)Lm+n .
(2.51)
To wrap up this section, we make contact with the statement of (2.30) being the classical generators of residual symmetries preserved by our gauge choice. If we consider an angular variable ✓ : 0 ✓ 2⇡ as the parametrization of the circle S 1 , then an infinitesimal coordinate transformation of ✓ ! ✓ + c(✓) may be considered as the action on ✓ by the diffeomorphism algebra of S 1 or dif f S 1 [16]. The dif f S 1 algebra is generated by the complete basis of circle diffeomorphisms operators defined by, Dn = ein✓ d Thus if we let ein✓ ! ⇠ ± and i d✓ !
@ @
±
d d✓
(2.52)
we would recover the generators of (2.30).
In this way, we see that the Witt algebra is isomorphic to dif f S 1 . This is only true "on shell", in the sense that we must apply the equations of motion in order to obtain the Fourier oscillator mode expansions subject to the corresponding boundary conditions. We may then consider
±
as an angular variable.
60 2.2.3 Free Bosonic String Quantization Let us briefly outline the procedure used to quantize the bosonic string. The results are summarized here, however the full details are given in (C). We first examen the quantized the Witt algebra from the "old covariant" method, where we take [ , ]P.B. !
i[ , ] and promote the fields to operators. Upon quantization the
algebra is referred to as the Virasoro algebra. The quantization process exposes a c-number anomaly or central extension, which prevents the closure of the Virasoro algebra due to a normal ordering ambiguity of the zero modes. The Virasoro anomaly is obtained at the quantum level from the non-zero expectation value of the Virasoro generators on vacuum states which are defined in terms of the energy-momentum tensor. The non-zero trace of the energy-momentum tensor signifies broken scale invariance and therefore is also referred to as the trace anomaly. The c-number anomaly is removed by the "modern covariant quantization" method discussed in [16], where the Euclidean path integral is quantized with the Faddeev-Popov prescription in the conformal gauge. Classically there are no repercussions from gauge fixing the metric. At the quantum level the path integral is subject to Faddeev-Popov ghost fields which are a consequence of gauge fixing the metric. When evaluated correctly the Faddeev-Popov prescription modifies the Virasoro algebra with the addition of the ghost Virasoro algebra. The ghost algebra also has a c-number anomaly, however the direct sum of the Virasoro and ghost Virasoro algebras is anomaly free. At this point the BRST formalism is implemented which
61 restricts the Fock space of states to only the physical states of the theory. The motivations to use this particular quantization prescription are at least 3fold; i) proper inclusion of the ghost states restores the quantum level scale invariance by removing the trace anomaly, ii) it is a self-consistent method where one has enough information to deduce the space-time dimension D and the normal ordering constant which appears in the zero mode Virasoro generator constraint condition (L0 = 0), and iii) this method is manifestly covariant. The complete anomaly free Virasoro generators are defined to be the sum of the bosonic matter and ghost Lm ’s such that, (c) Lm = L(↵) m + Lm
a
m,0 .
(2.53)
The contribution from the ghost sector maintains the 0th constraint condition, L0 = 0. The full ghost and matter anomaly contributions is given by the sum of (C.39) and (C.45), with the fully fixed anomaly constants from (C.49), (C.46), (C.48), A(m) =
D 3 (m 12
1 m) + (m 6
13m3 ) + 2am.
(2.54)
This is the final form of the anomaly obtained by quantizing the gauge fixed free bosonic string path integral. Conformal invariance is maintained when the corresponding anomalous terms vanish. Thus it can be seen directly from (C.50) that this is case if and only if D = 26 and a = 1. At this point we come to a crossroads and we pause to make a few brief comments of the results up to this point. We have seen that the bosonic formulation of string theory has anomalies which arose upon quantization of the theory. Specifi-
62 cally, cancelation of the conformal anomaly is needed to be addressed order to restore conformal invariance. There are other anomalies which also arise which have not been addressed. For instance, it is known that there exists a gravitational anomaly associated with world sheet chiral fermions (actually this is a general property of conformally invariant theories in 2 dimensions [16]). In this case it seems that one cannot couple the energy-momentum tensor to fermions in a consistent manner. The two-point functions of the left moving and right moving EMT’s in general do not respect energy-momentum conservation unless the background is flat. This is remedied if charges cL = cR , however this constraint breaks Weyl invariance unless D = 26 and cL = cR = 0. Under those conditions, then Weyl invariance is restored along with energy-momentum conservation. It is in this manner that the D integration can be ignored in (C.16). Secondly, though we have successfully quantized the free string, there is an issue with respect to ghosts, or negative norm states. The bosonic spectrum admits a tachyon, even at the quantum level. This requires us to restrict the spectrum to physical states which remove the tachyonic states. What normally follows after one quantizes covariantly with the Faddeev-Popov prescription is to follow it with BRST quantization formalism which eliminates the non-physical states from the spectrum [15, 16, 22]. To be rigorously consistent this should indeed be employed here. However, we only mention that this is technically prudent at the moment and postpone further discussion about the BRST formalism until after we quantize the superstring, which is the topic that we will begin now.
63 2.3
The RNS Superstring
The RNS superstring is a supersymmetric formulation of bosonic string theory which extends the world-sheet to a superspace formalism where fermionic fields are defined by anticommuting Grassmann variables which act as fermion coordinates in superspace. The notion of a field is generalized in the superspace formalism to a superfield [15,16,22]. In the RNS construction, world-sheet supersymmetry is a global symmetry and here we consider N = 1 supersymmetry. Superspace is parametrized by the super-world-sheet coordinates (
↵
, ✓A ), where
↵
= (⌧, ) and ✓A are two-
component Majorana spinors ✓A =
✓
✓ ✓+
◆
(2.55)
: {✓A , ✓B } = 0,
where A is the world-sheet spinor index, which we will suppress unless it suffices to add clarity and the super/sub-script placements are immaterial. We introduce a the general notion of a superfield defined by, Y µ( where B µ (
↵
↵
, ✓A ) = X µ (
) + ✓¯A
µ ↵ ) A(
1 + ✓¯A ✓A B µ ( 2
↵
),
(2.56)
) is an auxiliary field which allows the supersymmetry algebra to close
off-shell. The fermion field content { which form a D SO(D
↵
µ A,
µ A
⌫ B }P B
satisfy = ⇡⌘ µ⌫
AB
(
0
).
(2.57)
plet of Majorana fermions and the µ index transforms in the
1, 1) vector representation of the Lorentz group. We note that like the
negative mass states which result from the ⌘00 component of the metric in the bosonic
64 theory, the
0 A
spinor component carries the wrong metric statistics. This is resolved
with the addition of supersymmetry. The free supersymmetric string action is taken to be i S= 4⇡
Z
¯ µ DYµ , d2 d2 ✓DY
(2.58)
with i¯ ↵ i⇢↵ ✓@↵ X µ + ✓✓⇢ @↵ µ 2 i¯ DY¯ µ = ¯µ + B µ ✓¯ + i@↵ X µ ⇢↵ ✓¯ ✓✓@↵ µ ⇢↵ 2 DY µ =
µ
+ ✓B µ
(2.59) (2.60)
The above definitions give the component form of the action, S=
1 2⇡
Z
d2
i ¯µ ⇢al @↵
@ ↵ X µ @ ↵ Xµ
µ
B µ Bµ ,
(2.61)
and adopt the conventions 0
⇢ =
✓
0 i
◆ i , 0
1
⇢ =
✓
◆ 0 i , i 0
(2.62)
for the 2-dimensional Dirac representations which obey, {⇢↵ , ⇢ } = The components of
µ A
2⌘ ↵ .
(2.63)
◆
(2.64)
in this basis are, µ
=
✓
+
.
Supersymmetry is made manifest on the world-sheet with the use of the supersymmetry generators QA =
@ + i (⇢↵ ✓)A @↵ , A ¯ @✓
(2.65)
65 and a constant infinitesimal anticommuting supersymmetry generating parameter ✏A , such that we use ✏¯Q rather than Q itself. This generates the supersymmetric transformations of the super-world-sheet coordinates,
↵
⇥ ⇤ ✓A = ✏¯Q, ✓A ,
= [¯✏Q,
↵
(2.66)
] = i¯✏⇢↵ ✓.
With the above we define the supersymmetric transformations on the superfields (2.56) by, Y µ = [¯✏Q, Y µ ]
(2.67)
= ✏¯QY µ . The commutator of two transformations does indeed yield the expected off-shell spacetime translation, [ 1, 2] Y µ =
2i¯✏1 ⇢↵ ✏2 @↵ .
With the aid of the Fierz identity ✓A ✓¯B =
1 ✓¯ ✓ , 2 AB C C
(2.68) (2.68) can be expanded in
component form to give, X µ = ✏¯ µ
=
Bµ =
µ
i⇢↵ ✏@↵ X µ + B µ ✏ i¯✏⇢↵ @↵
(2.69)
µ
It is easily seen that since ✏¯Q is a first order differential operator, it obeys Leibniz’s rule for products of superfields. We should also note that in the Lagrangian defined
66 in (2.58) we have made implicit use of the superspace covariant derivative D=
@ @ ✓¯
(2.70)
i⇢↵ ✓@↵ ,
and obeys the anticommutation algebra, {DA , QB } = 0 ¯ B } = 2i (⇢↵ ) @↵ {DA , D AB {DA , DB } = 2i ⇢↵ ⇢0
AB
(2.71) @↵
2.3.1 RNS Constraint Equations We now briefly look at the constraint equations analogous to the bosonic case. We know that the spinors
µ A
obey the Dirac equation ⇢↵ @↵
µ A
= 0, this decouples
the right and left moving modes, (@ ± @⌧ )
µ ⌥
= 0.
(2.72)
We make this explicit, as in the bosonic case, by moving to world-sheet light-cone coordinates
±
= (⌧ ± ) and which, @± = 12 (@⌧ ± @ ) follows directly. We see that
given (2.72) and the bosonic equations of motion @ 2↵ X µ = 0 that @±
µ ⌥
= @± (@⌥ X µ ) = 0,
(2.73)
makes clear why supersymmetry can exist between bosons and fermions i.e., supersymmetry is the symmetry between
µ ±
and @± X µ [16].
Similar to the bosonic theory in D = 26-dimensions where the "Noether Method" gave constraints on the Noether current an energy-momentum tensor known
67 as the Virasoro constraints, we must also find constraints on the RN S superstring. These will be known as the super-Virasoro constraints. Technically these should be derived by gauge-fixing a gauge invariant action and then taking the proper variations. In the RN S superstring, this is tantamount to promoting supersymmetry from a global to a local symmetry where the action must incorporate a "zweibein" ea↵ (⌧, ) and a 2-component Majorana spinor with a world-sheet vector index known as a Rarita-Schwinger field,
A↵
as well as the physical coordinates X µ (⌧, ) and
µ
(⌧, ).
These additions naturally gives a different action than (2.58), which would also include the world-sheet metric in order to make manifest the reparametrization and local Lorentz invariance along with the extra terms involving the Rarita-Schwinger field, i.e. the gravitino and zweibeins needed to accommodate the full local supersymmetry transformations invariantly. The action discussed in the previous paragraph is really a supergravity action in D = 2 dimensions. By construction it also carries local Weyl invariance and an arbitrary Majorana fermionic symmetry, ⌘. Collectively, these symmetries define a superconformal theory on the world-sheet. We will not explicitly give the full action and derivation here, but the details are given in chapter 4.3.5 of [16]. Instead we simply note that there are 4 local bosonic symmetries; 2-world-sheet reparametrizations, 1-local Lorentz, and 1-Weyl scaling. Locally these can be used to gauge away the 4-components of the zweibein: ea↵ =
a ↵.
The 2-supersymmetries (✏)
and 2-superconformal symmetries (⌘) gauge away the 4-components of the gravitino: A↵
= 0. This simplifies the supergravity action discussed here to the globally super-
68 symmetric action given in (2.58) and (2.61) with equations of motion, @ 2↵ X µ = 0 and ⇢↵ @ ↵
µ A
= 0 stated above. Supplementing the equations of motion with the gauge
conditions on the gravitino and zweibein, gives the constraints on the supercurrent and energy momentum tensor, J↵A ⌘
⇡ 2✏
S A↵
1 = ⇢ ⇢↵ 2
T↵ = @ ↵ X µ @ Xµ +
µ A@
i ¯µ ⇢(↵ @ 2 A
Xµ = 0 )
A µ
(trace) = 0.
(2.74) (2.75)
If we transform the super-Virasoro constraints into light-cone coordinates as above, then blissful simplifications follow. Note that in the light-cone basis, J+A and J
A
are each two-component spinors where only the positive-chirality spinor component of J+A or the negative-chirality component of J
A
are non-zero due to the restriction
of ⇢↵ J↵A = 0, which is a consequence of the 2-D identity ⇢↵ ⇢ ⇢↵ = 0. For simplicity, we can suppress the spinor index and write the non-zero spinor components of the supercurrent and energy-momentum tensor as, J+ =
µ + @ + Xµ
J =
µ
(2.76)
@ Xµ ,
and i 2 i = @ X µ @ Xµ + 2
T++ = @+ X µ @+ Xµ +
µ + @+
T
µ
@+
µ+
µ
(2.77)
69 From the equal ⌧ (anti)commutators {
µ ±(
),
⌫ 0 ± ( )}P B
= ⇡⌘ µ⌫ (
0
⇡ [@± X µ ( ), @± X ⌫ ( 0 )]P B = ±i ⌘ µ⌫ 0 ( 2 {
µ +,
⌫
) 0
)
(2.78)
} = [@+ X µ , @ X ⌫ ]P B = 0,
we find that the supercurrent and energy-momentum tensor satisfy the classical algebra {J± ( ), J± ( 0 )}P B = ⇡ (
0
)T±± ( )
(2.79)
{J+ ( ), J ( 0 )}P B = 0, and are subject to the super-Virasoro constraints J± = 0
(2.80)
T±± = 0. We note that at the quantum level the super-algebra (2.79) of course has an anomaly term, and also the super-Virasoro constraints eliminate pesky timelike components which yield the wrong statistics in the (anti)commutators. 2.3.2 RNS Boundary Conditions and Mode Expansions Now that we have equations of motion and constraints for the RNS superstring, we must discuss the boundary conditions and mode expansions relevant to the fields. The space-time coordinates X µ (⌧, ) satisfy the same boundary conditions and constraints as in the bosonic case and therefore we omit them from the present
70 discussion. The same method for determining the bosonic boundary conditions and modes apply to the fermionic fields as well. We begin by analyzing the open superstring. By varying the fermionic piece of the Lagrangian, a surface term arises µ +
µ +
µ
µ
(2.81)
,
which is required to vanish for the open string sector. We choose the positive sign convention for the
= 0 boundary in the +,
Ramond (R) and ii) N eveu
modes, which will yield two cases, i)
Schwarz (NS), for the
= ⇡ boundary in the +,
modes. The positive sign convention applicable to both (R) and (NS) cases is µ + (⌧, 0)
=
µ
It follows that we have 8 > > µ µ > > ( = ⇡, ⌧ ), + ( = ⇡, ⌧ ) = > > > > < P µ µ in(⌧ i) (R) = p1 ( , ⌧ ) = n2Z dn e > 2 > > > > > > P > : µ ( , ⌧ ) = p1 dµ e in(⌧ + +
2
n2Z
n
B.C. )
Right Moving Modes
)
Left Moving Modes,
where the sums extend over all the integers and, 8 > > µ µ > > ( = ⇡, ⌧ ), + ( = ⇡, ⌧ ) = > > > > < P µ ii) (NS) = ( , ⌧ ) = p12 r2Z+ 1 bµr e ir(⌧ > 2 > > > > > > P > µ ir(⌧ + : µ ( , ⌧ ) = p1 1 b e +
2
r2Z+ 2
(2.82)
(⌧, 0)
r
(2.83)
B.C. )
Right Moving Modes
)
Left Moving Modes,
(2.84)
where the sums extend over half-integer modes. We establish the convention that we reserve the indices m or n for sums which run over integer modes, and the indices r or s for sums which run over half-integer modes.
71 For closed strings, the boundary terms vanish for (anti)periodic modes of each component of
and
µ
separately. We therefore have 8 > P µ 2in(⌧ ) > > < µ = dn e , n 2 Z, or µ = > > P µ 2ir(⌧ ) > : µ = br e , r 2 Z + 12 µ +
=
8 > > > < > > > :
µ + µ +
=
P ˜µ dn e
P = ˜bµr e
2in(⌧ + )
2ir(⌧ + )
, n 2 Z, or
,
r 2Z+
(2.85)
(2.86)
1 2
There exist four independent pairings of left moving and right moving modes which correspond to the NS-NS, NS-R, R-NS, R-R sectors. NS-NS and R-R sectors form closed string bosons and NS-R and R-NS form closed string fermions. The super-Vierasoro operators are given by the modes of the supercurrent and energy-momentum tensor. For the open superstring there is only one independent set of operators 1 Lm = ⇡
Z
⇡
d 0
e
im
T++ + e
im
T
1 = ⇡
Z
The R boundary conditions have the fermionic generators p Z ⇡ p Z 2 2 im im Fm = d e J+ + e J = ⇡ 0 ⇡ and the NS boundary conditions have the generators p Z ⇡ p Z 2 2 Gr = d eir J+ + e ir J = ⇡ 0 ⇡
⇡
d eim T++ .
(2.87)
⇡
⇡
d eim J+ ,
(2.88)
d eir J+ .
(2.89)
⇡
⇡ ⇡
For the closed superstrings there are two independent sets of generators (left and right moving modes), the modes analogous to the above generated by J+ and T++ , and the corresponding modes generated by J and T
.
72 2.3.3 RNS Superstring Quantization In this section we will quantize the RNS superstring in the same method applied to the bosonic case earlier. As the method is the same we will be more brief with the process, citing the important results when possible. We begin by the old-covariant method where we let { , }P B ! { , }. We will limit the discussion to the single set of modes ↵nµ and bµr or ↵nµ and dµn when working with the open superstring and the right moving sector for the closed superstring. The canonical anticommutation relations for the fermionic coordinates {
µ A(
, ⌧ ),
⌫
( 0 , ⌧ )} = ⇡ (
0
µ A(
)⌘ µ⌫
, ⌧ ) are
AB
(2.90)
It follows from (2.90) that the oscillator modes bµr and dµn satisfy, {bµr , b⌫s } = ⌘ µ⌫ {dµn , d⌫m } = ⌘ µ⌫
r+s,0
(2.91)
n+m,0
The zero oscillator modes of the super-Virasoro constraints give the mass-shell condition with a normal ordering constant aN.O. which we will ignore for now. The mass-shell condition is ↵0 M 2 = N + aN.O. ,
(2.92)
73 where N = N ↵ + N d or N = N ↵ + N b depending on the boundary conditions, and N↵ = Nd = Nb =
1 X
m=1 1 X m=1 1 X
↵
md rb
r= 12
· ↵m
m
r
m
· dm
(2.93)
· br .
The lowest mass state corresponds to the Fock-space ground state, 8 > > µ > 0 > > > µ :↵m |0i = bµr |0i = 0,
(2.94)
m, r > 0,
where the eigenvalues of the raising operators increase the value of ↵0 M 2 by one unit of the oscillator mode index of the corresponding operator. For the half integer modes, there is a unique non-degenerate ground state which we identify as the spin zero state. The integer modes satisfy {dµ0 , d⌫0 } = ⌘ µ⌫
(2.95)
and commute with the M 2 operator and thus it is not possible to identify them with a unique non-degenerate ground state. Actually dµ0 operators furnish an irreducible spinor representation of the Dirac algebra in the space-time background (which we will see is D=10) and thus a representation of SO(1, 9), which requires them to obey {
µ
,
⌫
}= µ
2⌘ µ⌫
p = i 2dµ0 ,
(2.96) (2.97)
where the last equation denotes the sign, normalization and metric conventions and are those of [16]. Since it is known that irreducible representations of the space-time
74 parametrizations correspond to spinors (here of SO(1, 9)), it follows that the integrally moded d oscillators must give fermionic representations. This is true because for every mass level (2.95) must admit a corresponding representation, and in particular for the Fock-space ground state which must be irreducible. Sectors with integrally moded world-sheet spinors are referred to as fermionic (or R) sectors and half-integral moded sectors are referred to as bosonic (or NS) sectors. The momentum and angular momentum densities along the string are Noether currents associated with the global symmetries X µ ! aµ⌫ X ⌫ + b⌫ and
µ
! aµ⌫
⌫
are
found to be 1 @↵ X µ ⇡ 1 = X µ @↵ X ⌫ ⇡
(2.98)
P↵µ = J↵µ⌫
X ⌫ @↵ X µ + i ¯µ ⇢↵
where the string tension has been set to T =
1 . ⇡
⌫
(2.99)
,
The form of Lorentz generators
associated with conserved Noether charges are given below J
µ⌫
where we have the
classical bosonic modes,
fermionic modes, K
µ⌫
=
=
Z
⇡ 0
8 > > > > > :E µ⌫
= xµ p⌫ =
i
x⌫ pµ
P1
1 n=1 n
8 > P > µ ⌫ > < i 1 r= 1 b r br i µ ⌫ [d , d0 ] 2 0
i
(2.101) ↵µ n ↵n⌫
b⌫ r bµr ,
2
> > > :
(2.100)
J⌧µ⌫ d = lµ⌫ + E µ⌫ + K µ⌫ ,
P1
n=1
dµ n d⌫n
↵⌫ n ↵nµ
(NS) d⌫ n dµn . (R)
(2.102)
75 The above satisfy the Lorentz algebra (C.12) and reduce to the purely bosonic string theory charges if the spinors 2.3.3.1
µ
in (2.98) are deleted.
The Quantum Super-Virasoro Algebra
Using the super-Virasoro constraints and mode expansions for the coordinates X µ and
µ A
fields, the Virasoro operators written in terms of the oscillator modes are L(↵) m L(b) m L(d) m
1 1 X = : ↵ n · ↵n : 2 n= 1 ◆ 1 ✓ 1 X 1 = r + m : b r · bm+r : 2 r= 1 2 ✓ ◆ 1 1 X 1 = n + m : d n · dm+n : . 2 n= 1 2
These are then combined to give the (NS) and (R) sector operators 8 > > (b) > > > (↵) (d) : Lm + Lm , (R),
(2.103)
(2.104)
where normal ordering is only required at the m = 0 level. The fermionic generators are correspondingly given as
Sf ermionic =
8 > P > > < Gr = 1 n=
> > P > :Fm = 1 n=
1
1
(NS)
↵
n
· br+n ,
↵
n
· dm+n , (R).
(2.105)
The bosonic (NS) super-Virasoro algebra is then [Lm , Ln ] = (m n)Lm+n + A(↵,b) (m) ✓ ◆ 1 [Lm , Gr ] = m r Gm+r 2 {Gr , Gs } = 2Lr+s + B (↵,b) (r)
r+s,0 ,
m+n,0
(2.106) (2.107) (2.108)
76 and have the c-number anomaly terms 1 A(↵,b) (m) = D m3 8 ✓ 1 (↵,b) B (r) = D r2 2
m ◆ 1 + 2aN.O.(N S) . 4
(2.109) (2.110)
The fermionic (R) sector super-Virasoro algebra is [Lm , Ln ] = (m n)Lm+n + A(↵,d) (m) ✓ ◆ 1 [Lm , Fn ] = m n Fm+n 2 {Fm , Fn } = 2Lm+n + B (↵,d) (m)
m+n,0 ,
m+n,0
(2.111) (2.112) (2.113)
with corresponding c-number anomaly terms 1 A(↵,d) (m) = Dm3 8 1 B (↵,d) (m) = Dm2 + 2aN.O.(R) . 2 2.3.3.2
(2.114) (2.115)
The Faddeev-Popov Prescription
We proceed with the Faddeev-Popov prescription, as we did in the bosonic theory, in order to cancel the anomalous terms present in the super-Virasoro algebra (2.106) and (2.111) above. We will only present the relevant aspects of the Faddeev-Popov ghost quantization, which amounts to the cancellation of the B (↵,b) (r) and B (↵,d) (m) anomalous terms above. A summary of the Faddeev-Popov method is given in Appendix C.2. We will then arrive at the values for critical superstring dimension and zero-mode operator constant. We will not present the cancellation of the anomalous terms A(↵,b) (m) and A(↵,d) (m), but we note that their cancellation may
77 be achieved by applying the Jacobi identity to [Lm , Ln ] in the (NS) and (R) sectors restricted to the specified critical dimension and zero-mode constant values [16]. The prescription follows the general principal that a change in variables of a field in an action which respects a specific symmetry (such as diffeomorphisms and Weyl invariances), induces a nontrivial Jacobian and ghost ghost fields. For our purposes the change of variables will correspond to symmetry parameters under which the action is invariant, and thus we do not need to integrate over them. We do not need to evaluate the path-integral in order to quantize the superstring theory and cancel the anomalous terms. For definiteness, we specify that we are working with the Euclidian world-sheet isometry group of SO(2) which describes the spinor or fermionic contribution to the path-integral. We introduce the arbitrary infinitesimal Grassmann variables ⌘ and ✏, which we will use gauge away the Rarita-Schwinger gravitino field
↵A
through a suitable change of variables and appropriate boundary
conditions. ↵A
has an index structure where ↵ is a vector or spin ±1 index, and A = ± 12
is a spinor index. Thus
↵A
has 4-components of spin ( 32 , 12 ,
1 , 2
3 ). 2
The gauge
symmetry parameters ⌘ and ✏ are each 2-component spinors with values ± 12 . We also introduce the temporary convention that the index of an operator/coordinate are written as the explicit spin state. Thus the covariant derivative operator carries a vector index ↵, which we identify as a vector with spin components of ±1 and denote it as r±1 . From this we define the infinitesimal transformations of
↵A
with respect
78 to the gauge parameters ⌘ and ✏ as, 3 2
= r1 ✏ 1
1 2
= r1 ✏
1 2
= r 1✏ 1 + ⌘
3 2
= r 1✏
2
+ ⌘1
1 2
2
2
1 2
(2.116) 1 2
.
We will focus on the nontrivial Jacobians which arise in the first and last transformations. The second and third transformation have no derivative on ⌘ and thus have only trivial Jacobians for
± 12
! ⌘± 1 . The other two Jacobians written in determinant 2
form as, 1
J 3 = det 1 r12
! 32
2
J
3 2
= det r 1
1 ! 2
1
3 2
(2.117) ,
where the notation is suggestively written to imply that r1 is an operator which maps a state of spin ghost fields
1 2
± 12
to a state of spin 32 . We introduce the commuting superconformal and
± 32
corresponding the insertion of the operators in the the
path-integral, J3 = 2
J
3 2
=
Z Z
D
1 2
D
D
1 2
D
3 2
3 2
exp exp
✓ Z 1 d2 ⇡ ✓ Z 1 d2 ⇡
3 2
3 2
r1
r
1
1 2
◆ 1 2
◆
(2.118) .
(2.119)
We may write the covariant superconformal ghost action as SF
P
=
i 2⇡
Z
d2 eh↵ ¯ @↵
.
(2.120)
79 where
and
± 12
± 32
are the components of the spinor
spectively and are constrained by ⇢↵AB
↵B
and vector-spinor
A
3 2
,
1 2
are right moving and
3 2
µ
(x) = eµm (x)
m
.
splits the superconformal ghosts such that
↵
,
re-
= 0. The field e is the determinant of the
frame fields eµm corresponding to the curved space gamma matrices The gauge choice of h↵ =
↵A
1 2
are left moving. By taking the variations
of (2.120) with respect to the world-sheet metric we find that the energy-momentum tensor and superconformal ghost current are T++ = J+gh =
i 2
1 2
3 2
@+ 1 2
3 2
+
3i 2
3 2
@+
(2.122)
,
with analogous left-moving energy-momentum tensor and current where 3 2
,
1 2
(2.121)
1 2
and @+ ! @ . The mode expansions for
and
3 2
,
1 2
!
in the fermionic (R) sector
are 1 1 X ⌥ 12 (⌧ ) = p 2 n= 1 1 1 X ± 32 (⌧ ) = p 2 n= 1
with
m,
m
!
r,
r
ne
(2.123)
⌥2in⌧
ne
⌥2in⌧
,
(2.124)
for modes in the half-integral bosonic (NS) sector. The super-
conformal ghosts also enjoy the commutation relations [
m(r) ,
n(s) ]
=
[
m(r) ,
n(s) ]
=[
m(r)+m(s),0 , m(r) ,
n(s) ]
m(r) =) R(N S) sectors = 0.
(2.125) (2.126)
The super-Virasoro generators receive ghost coordinate contributions from (2.121)
80 in the (R) sector Lgh m
=
X✓
1 n cn : +( + n) : 2
(m + n) : bm
8n
Fmgh
=
X✓ 8n
2b
n
1 m+n + ( n 2
m)c
m n n
n m+n
◆
:
◆
(2.127) (2.128)
,
and corresponding generators in the (NS) sector with m, n ! r, s and Fmgh ! Ggh r . In the (R) sector the ghost fermionic anticommutator is gh {Fmgh , Fngh } = 2Lgh m+n + B (m)
(2.129)
m+n,0 ,
gh gh and the (NS) sector is analogous with Fmgh ! Ggh r and B (m) ! B (r).
The (R) sector ghost anomaly is given by B gh (m) =
(2.130)
5m2 .
When we combine (2.130) above with the corresponding term in (2.111), we see that 1 B (↵,d) (m) + B gh (m) = Dm2 + 2aN.O.(R) 2
5m2
(2.131)
= 0 iff D = 10, aN.O.(R) = 0. For the (NS) sector, the ghost anomaly is B gh (r) =
1 4
(2.132)
5r2 .
The combination of (2.132) with the corresponding term in (2.106) gives B
(↵,b)
✓ 1 (r) + B (r) = D r2 2 gh
1 4
◆
+ 2aN.O.(N S) +
1 = 0 iff D = 10, aN.O.(N S) = . 2
1 4
5r2
(2.133) (2.134)
81 Thus we have seen that the critical dimension for the RNS open superstring is D = 10 and the normal ordering constant aN.O.(R) = 0 and aN.O.(N S) =
1 . 2
The open
superstring operators involve the combination of left and right moving modes and was not explicitly shown here. The derivation is analogous to the above, modulo proper open string mode boundary conditions and yields the same results as above. We also note that we have not derived the superstring with manifest space-time supersymmetry. The Green-Schwarz formalism begins with a gauge invariant action which is indeed space-time supersymmetric [15, 16, 22], however at the moment the Green-Schwarz formalism in most easily quantized in the light-cone gauge and it is not known if it is possible to quantize it covariantly as in Faddeev-Popov prescription. We note that manifest space-time supersymmetry is possible to achieve from the RNS formalism [22, 71], but requires BRST quantization and the GSO projection for this process. A mapping from the spectrum of Green-Schwarz construction to the spectrum of the RNS formalism in [16]. At this point we simply note that under the proper conditions, the two formalisms are equivalent. We will briefly address the BRST quantization in order to restrict the spectrum to physical states next, and a more detailed account of it is given in Appendix D. 2.3.4 BRST Quantization The BRST quantization formalism creates equivalence classes between states with equivalent number of ghosts. We employ the BRST formalism in order to eliminate non-physical negative norm states from the RNS superstring spectrum by
82 identifying states of zero ghosts. The formalism is outlined in the second section of the appendix, but we will briefly apply the formalism to the superconformal algebra developed in the previous section. We will limit the presentation to that of the right moving sector of closed superstring, as the left moving are constructed in the same manner, and open superstring case is analogous modulo proper boundary conditions. In the (R) sector using the definition of the Lie algebra cohomology operator given in the appendix and using the superconformal algebra (2.127) in order to obtain the structure constants fijk , the BRST operator Q can be shown to be 1 k i j f c c bk , 2 i,j
Q := ci Ki =
X⇣
L
(↵,d) n cn
+ F (↵,d)
n
8 n
X1 8 n
+
2
(m
n) : c
X 3 ( n + m) : c 2 8 n X
m
n bm+n
⌘
m c n bm+n
:
n
:
m m+n
(2.135)
ac0 ,
8 n
where in the first line ci , bk is meant to be (anti)ghosts for the commuting ghost fields
n,
m
or anticommuting ghost fields cn , bm as a appropriate to the canonical
(anti)commutation relations, {ci , bj } = [ i,
j]
=
i j
(2.136)
i+j,0
(2.137)
The equivalence class of states with ghost number of zero can be observed in
83 the following way [22] by noting that for the (R) sector {Q, bm } = Lm = L(↵,d) + Lgh m m [Q,
m]
=
Fm(↵,d)
=
Fm(↵,d)
+
(2.138)
Fmgh .
The (anti)commutation relations of the BRST charge operator Q and antighost coordinates bm ,
m
can be used to verify that Q nilpotent (Q2 = 0) provided that D = 10
and the normal ordering constant a = 0 in (2.135). With (2.138) we write L0 and F0 as L0 = {Q, b0 } F0 = [Q,
(2.139)
0] .
We recast the zero mode generators in the form of number operators which count the total number of oscillator modes including ghost contributions of a given state (↵,d,gh)
Ltot 0 = Ntot F0tot
=
1 + ↵02 2
(↵,d,gh) Nftot
0 0
aN.O.
(2.140)
+ aN.O. ,
where we have the more general expression with aN.O. 6= 0 for the (R) sector, and (↵,d,gh)
Ntot
(↵,d,gh)
and Nftot
(↵,d,gh) Ntot
(↵,d,gh) Nftot
= =
given by [15, 22]
1 ✓ X n=1 1 X
(
↵
n
· ↵n + (c
n n
n bn
1 b n cn ) + (c0 b0 2
b0 c 0 )
◆
(2.141)
n n) .
n=1
Thus for a state of zero ghost number |
0i
:|
0i
(↵,d,gh)
= |(Ntot
(↵,d,gh)
+ Nftot
= Ntot ), pµ i
which is an eigenstate of Ntot with implied appropriate eigenvalues and satisfies
84 Q |Ntot , pµ i = 0, we may write µ
(L0 + F0 ) |Ntot , p i =
✓
1 Ntot + p2 2
= Q(b0 If we assume that Ntot + 12 p2
◆
aN.O. |Ntot , pµ i
0 ) |Ntot , p
µ
(2.142)
i.
aN.O. 6= 0 then we may write
|Ntot , pµ i =
Q (b0 Ntot + 12 p2
0)
aN.O.
|Ntot , pµ i ,
(2.143)
which shows that (2.143) is in the same equivalence class as the zero ghost states, and thus solutions of the above generates the equivalence classes of states for zero ghost number. We note that that the solutions of (2.143) are trivial solutions. For non-trivial solutions on states, one defines the DDF operator which commutes with the entire superconformal algebra for physical states and generates a new physical state. We do not show this here, however detailed accounts can be found in [16, 22]. The (NS) sector is generated by the construction above with Fm ! Gr , r,
m
!
r,
m
!
a = 0 ! a = 1/2, and in the (R) sector, there is an infinite degeneracy
of the zero modes of the
and
oscillators. We do not further address this compli-
cation further here. We also note that the full closed superstring BRST quantization requires the inclusion of the left moving modes and is completely analogous to the above. The open superstring solutions are also analogous with the open superstring boundary conditions. We will now take a moment to briefly summarize the five consistent formulations of superstring theory and then move on to more modern topics such as D-branes.
85 2.4
5 Consistent Superstring Theories
The formalism developed in the previous sections has lead to five consistent formulations of the superstring. The type IA, types IIA and IIB, and the two heterotic theories (E8 ⇥ E8 and Spin(32)/Z2 . 2.4.1 Type IA The type IA superstring is a theory which has both open and closed strings. The closed string sector of the theory can have at most N = 2 space-time supersymmetry, and thus for the fermionic coordinates ✓A , (A = 1, 2) and periodic boundary conditions in , there is a choice to have ✓1 , ✓2 be of the same handedness or opposite handedness (i.e moving modes in the same direction or opposite directions) [16, 22]. In the case of open strings, the boundary conditions require that ✓1 , and ✓2 be equated at the endpoints. This requirement implies that the fermionic coordinates must be of the same handedness, since a left handed spinor cannot be equal to a right handed spinor. This acts as a constraint which restricts the space-time supersymmetry to N = 1 [16]. Chan-Paton factors corresponding to any classical gauge group are associated to the endpoints of the open superstrings and are used to construct a tree-level Yang-Mills theory. At the quantum level, the only group possible consistent gauge group is SO(32). This restricts the representation to be real and cannot correspond to an oriented string. This can be most easily seen of the one considers the gauge group to be gauge charges at the ends of the open superstrings, where a real representation cannot make a distinction between fundamental and antifunda-
86 mental representations, which would require a U (N ) or SU (N ) gauge group. Thus a consistent quantum interacting superstring consisting of open and closed strings must be unoriented theory with SO(32) gauge group in ten dimensions, where the closed strings form Yang-Mills singlets. 2.4.2 Type IIA and IIB The type IIA and type IIB superstring theories are theories based on closed superstrings only in D = 10 dimensions. The distinction between the type IIA and IIB theories lies in the chirality of the super-coordinates ✓1 and ✓2 . The type IIA theory are left-right symmetric or nonchiral. The ✓1 coordinate forms modes which move in one direction around the string and the ✓2 coordinate forms modes which move in the opposite direction. Similarly the type IIB theory is defined as a chiral theory, where the ✓1 and ✓2 coordinates move in the same direction, and thus are leftright asymmetric [16]. Since neither of the type II theories contain open strings they do not possess gauge degrees of freedom and therefore are unable to accommodate a Yang-Mills theory. This is true at the level of the basic distinct superstring theories discussed here. We will discover that a type IIA or type IIB theory in the presence of D-branes allows the inclusion of open superstrings [8, 10, 14, 72, 73]. 2.4.3 Spin32/Z2 and E8 ⇥ E8 Heterotic Superstrings The final two superstring theories are the heterotic theories, Spin32/Z2 or E8 ⇥ E8 . The two theories differ only in the lattice gauge groups used in their construction of consistent theories at the quantum level. The heterotic string is formulated by
87 combining the bosonic string with fermionic degrees of freedom for left moving modes in D = 26 dimensions while the right moving modes are supersymmetric incorporating one supersymmetric coordinate ✓ in D = 10 dimensions. For the left moving nonsupersymmetric modes the fermionic degrees of freedom cancel 16 of the bosonic fields contributions which are not considered as coordinate degrees of freedom, and leave behind 10 bosonic fields which are interpreted as space-time coordinates. Thus the left moving modes have D = 10 bosonic coordinates and the right moving modes have D = 10 from the combination of supersymmetric bosonic and fermionic degrees of freedom [14, 16, 72]. Thus the dimensionality of the heterotic string matches for left and right moving modes. 2.5
D-Branes
One of the greets tools to emerge from the second superstring revolution circa 1995, was Polchinski’s discovery of extended objects known as D-branes. D-branes may exist in dimensions less or equal to ten in which open superstrings with Dirichlet boundary conditions may terminate. D-branes are soliton-like "membrane" solutions in string theory that can be thought of as a space-time topological defect [58], where an incident closed string can transition to an open string upon interaction with the D-brane. They were first discovered for the bosonic string in [22,74,75], however their significance was fully realized when they were found to be the sources corresponding to (R-R) fluxes in the case of superstrings [22, 72, 76]. The incorporation of D-branes allowed theories that admitted only closed
88 string solutions without gauge degrees of freedom a way to include open strings with gauge degrees of freedom in a consistent manner. D-branes laid the foundation for the Maldacena’s seminal work on the AdS/CFT (or gauge/gravity) correspondence [8, 22], which implemented the holographic principle (proposed earlier by t’Hooft and Suskind) in order to make the first rigorous identification between gravitational theories in N-dimensions to gauge theories in (N-1)-dimensions. This shall be the topic of discussion in the following chapter.
Dirichlet: X m ( = 0) = X m ( = ) Xm Nuemann: = 0, for = 0, Xm
=0
$
Xm
=
Figure 2.2: Open strings with endpoints on D-branes which allow Yang-Mills degrees of freedom, where the open string endpoints act as Chan-Paton color sources.
89 2.5.1 Kaluza-Klein Compactification on S 1 In order to get at theory which at least has solutions which resemble our 4dimensional universe, we must find a way in which the extra dimensions imposed by our consistent string theory are not noticed. One particularly interesting case is when the extra dimensions are compactified in way that is essentially equivalent to a Kaluza-Klein compactification scheme. This method is the most straight forward in order to demonstrate the existence of D-branes, however orientifold planes and orbifold compactification [21, 72] schemes are also some of the earliest alternative methods but will not be discussed here. The general procedure in compactification is to identify the compactified spacetime as coordinates to take on periodic values. We demonstrate this in the unique bosonic string case with the 25th space-time coordinate as the compactified coordinate on an S 1 or radius R25 , X 25 ⇠ X 25 + 2⇡R25 w,
(2.144)
where w 2 Z is the winding number which counts the number of times the string wraps around the compact dimension, and X µ noncompact for µ = 0, . . . , 24. We momentarily adopt the conventions that the indices M, N run over the entire space-time dimensions D = 0, . . . , d = 25 and µ, ⌫ run over the noncompact dimensions, 0, . . . , d
1 = 24. Thus the full D = 26-dimensional theory has indices
0, . . . , d = 25. The metric splits into Gµ⌫ , Gµ25 , and G2525 sectors. From the d = 24 perspective these correspond to the 24-dimensional metric, a vector, and a scalar.
90 The metric can then be written as ds2 = GM N dX M dX N
(2.145)
= Gµ⌫ dX µ dX ⌫ + G2525 dX 25 + Aµ dX
µ 2
.
d-dimensional Reparmetrizations are the expected, 0
X µ (X ⌫ )
(2.146)
0d
X = X d + (X µ ), however the last case is the form of a gauge transformation on the noncompact dimensions µ, i.e. A0µ = Aµ If we consider a massless scalar
(2.147)
@µ .
in D-dimensions and let the metric com-
ponent for the compactified dimension be G2525 = 1 for simplicity, we find that the momentum in the d = 25 dimension is now quantized n . R25
p25 =
(2.148)
Here n 2 Z counts the number of times the momentum circles the X 25 coordinate. If we expand the space-time coordinates in a complete Fourier basis set we have, XM =
1 X
µ n (X ) exp
n= 1
inX d . R25
(2.149)
It follows that the D-dimensional wave equation @M @ M = 0 becomes @µ @ µ
n
(X µ ) =
n2 2 R25
n
(X µ ) .
(2.150)
91 We interpret the
n
modes in (2.150) of the D-dimensional fields as infinite tower of d-
dimensional fields which carry the label n. The invariant d-dimensional mass-squared of the compactfied theory is pµ pµ = For energies E
> 64M %2 sin2 ✓ > > 2 (cos(2✓)(2M %)+6M +%)3 > ` > > > < R= > > > > > > 2 3 2 2 > > : 64M sin ✓(` (% 2M )+2M (2M `2 (2`2 +M
2M %+%2 )) cos(2✓)(2M %)+M (2M +3%))3
N HEKS (5.48) N HN EKS
We note that both scalar curvatures in (5.48) are identical in the extremal
limit when `2 = 2M (M
%/2). However, the finite horizon and temperature in terms
of ✏, for the near-extremal case, allow for interesting and non trivial additions to the quantum geometric analysis of (5.47) and its thermodynamics. 5.4
Quantum Fields in N HN EKS
We now wish to perform a semi-classical analysis of the two dimensional dilaton field with respect to the Kaluza-Klein field content of (5.47). We know the quantum
176 effective action of a minimally coupled scalar has the form p ¯ Z 1 Sef f ⇠ d2 x g (2) R(2) R(2) + · · · , 16⇡ ⇤g(2) where ¯ =
const G
(5.49)
is the Weyl anomaly coefficient [111, 112]. We will determine the
full effective theory and value of ¯ to s-wave. This is a sensible approximation since is gravitational in origin and as such, should be real and unitless. Furthermore, most of the gravitational dynamics seem to be well approximated by this term of the non-local expansion [113]. In [55] it was shown that 'lm decays exponentially fast in time by analyzing the asymptotic behavior of its field equation for higher orders in l and m. To single out the relevant two dimensional near horizon theory, we begin by considering a four dimensional massless free scalar field in the background of (5.47). Z p 1 Sf ree = d4 x gg µ⌫ @µ '@⌫ ' 2 Z ⇥ ⇤ p 1 = d4 x ' @µ gg µ⌫ @⌫ ' 2 ✓ ◆ Z 1 `2 4 2 2 = d x' ` sin ✓ 2 @ + 2 U ✏2 t ✓ 2 ◆ U ✏2 2 ` sin ✓@U @U + @✓ (sin ✓@✓ ) + `2 ( ) ! ✓ ◆2 2 2 2 U ` (2` + M (2M + 3%) + M (2M %) cos(2✓)) `2 sin ✓ 2 + @2 + ` U 2 ✏2 64M 4 sin ✓ ✓ ◆ U `2 2 2 ` sin ✓ 2 2 @t @ '. ` U ✏2 (5.50) This calculation is essentially a scattering amplitude of a scalar field incident on the black hole. It was shown in [32], that such such a calculation is equivalent to the near horizon field theory.
177 In order to reduce the above to a two dimensional theory, we must integrate away angular degrees of freedom. Therefore we expand ' in terms of spherical harmonics '(t, U, ✓, ) =
X
(5.51)
'lm (t, U )Yl m (✓, ),
lm
and transform to tortoise coordinates defined by
dU ⇤ dU
=
`2 . U 2 ✏2
The resulting two
dimensional theory is further simplified by considering the region U ⇠ ✏, since mixing ⇤
and potential terms (⇠ l(l + 1) . . .) are weighted by a factor of f (U (U ⇤ )) ⇠ e2U , which vanish exponentially fast as U ! ✏. These considerations reduce to the functional, " # ✓ ◆2 Z 2 2 2 `2 ` U U ✏ Sf ree = d2 x '⇤lm @t im 2 + @U @U 'lm 2 U 2 ✏2 ` `2 Z `2 1 2 ⇤ = d x 'lm (@t imAt )2 + @U f (U )@U 'lm 2 f (U ) Z h i p `2 µ⌫ 2 ⇤ (2) = d x 'lm Dµ g g(2) D⌫ 'lm . 2
Dµ = @µ
(5.52)
imAµ is the gauge covariant derivative with a U (1) 2d complex scalar
charge e = m and we have introduced the Robinson-Wilczek two dimensional analogue (RW2DA) fields (2) gµ⌫
=
f (U ) =
✓
f (U ) 0
U2
✏2 `2
0 1 f (U )
◆
At =
U . `2
The quantum effective functional of (5.52), to s-wave '00 = . The factor of
p
(5.53)
A =At dt
(5.54) q
6 G
, for unitless
6 is chosen to coincide with the normalization of (5.49) for the
gravitational sector of the effective action. The wave function is obtained via path
178 integrating over , which amounts to a zeta-function regularization of the functional hp i µ⌫ (2) determinant of Dµ g g(2) D⌫ . The functional determinant is comprised of the two parts [114, 115]
=
grav
+
(5.55)
U (1) ,
where Z
grav
`2 = 16⇡
U (1)
3e2 `2 = ⇡
d2 x Z
in concurrence with (5.49) for ¯ =
p
g (2) R(2)
1 F F ⇤g(2) `2 . G
(5.56)
We restore locality in (5.56) with the intro-
duction of auxiliary (dilation and axion) scalars ⇤g(2)
1 R(2) and ⇤g(2)
and B such that
= R(2) and ⇤g(2) B = ✏µ⌫ @µ A⌫ ,
(5.57)
which in terms of general f (U ) and At (U ) are 1 2 @ + @U f (U )@U f (U ) t
=R(2)
1 2 @ B + @U f (U )@U B =F. f (U ) t
(5.58)
(5.59) have the general solutions (t, U ) =↵1 t + B(t, U ) = 1 t + where ↵i and
i
Z
Z
dU dU
↵2 2
f 0 (U ) f (U ) At (U ) , f (U )
(5.59)
are integration constants. In order to obtain a local action we make
use of (5.57) to transform (5.55) into our final Liouville type near horizon CF T given
179 by SN HCF T
`2 = 16⇡
Z
d2 x
3e2 `2 + ⇡
Z
p
g (2)
2
dx
p
g (2)
⇤g(2) (
+ 2 R(2)
B⇤g(2) B + 2B
✏µ⌫ p g (2)
!
@ µ A⌫
)
(5.60) .
For use in Section 5.5, we define the asymptotic (large U ) boundary fields (as in Section 1.3 and Section 5.1) as (0) gµ⌫
A
(0) t
U2 `2
✏2 `2
+O = 0 ✓ ◆3 U 1 = 2 +O , ` U +
1 3 U
`2 U2
0 +O
1 3 U
!
(5.61) (5.62)
which form an AdS2 ASG with scalar curvature R(2) =
2 `2
+O
1 U
.
Substitution of (5.61) and (5.62) into (5.59) allow us to define the boundary auxiliary scalars (0)
(0)
= (g (0) , A(0) ) and B (0) = B(g (0) , A(0) ) by
✓ ◆ 1 (t, U ) =↵1 t + 2 ln U
(0)
B (t, U ) = 1 t +
1
2`
2
U
undetermined constants ↵i and 5.5
↵2 `2 ✏2 + 2 U U ✏2 ( 2 ` 2 3U 3
↵ 2 ` 2 ✏2 +O 3U 3 ✓ ◆4 1) 1 +O , U
✓ ◆4 1 U
(5.63)
i.
Asymptotic Symmetries
We now calculate the quantum asymptotic symmetry group of the RW2DA fields. From the boundary fields (5.61) and (5.62), we impose the following metric, gauge field and scalar field fall of conditions gµ⌫ =
O O
1 3 U 1 0 U
0
O U1 O (U )
!
✓ ◆0 1 , A=O , U
✓ ◆0 ✓ ◆0 1 1 =O and B = O . U U (5.64)
180 A set of diffeomorphisms which preserve the above asymptotic boundary conditions is given by =
C1
U 2 ⇠(t) @t + C2 U ⇠ 0 (t)@U , U 2 ✏2
(5.65)
where ⇠(t) is an arbitrary function and Ci are arbitrary normalization constants. The variation of the gauge field under the above diffeomorphism is given by and thus,
A=O
1 0 U
is trivially elevated to a total symmetry !
(5.66)
+⇤
of the action. We now transform to conformal light cone coordinates (5.67)
x± = t ± U ⇤ in the above diffeomorphism (5.66), we obtain the components ±
=
U (U ⇤ ) ( C1 U (U ⇤ )⇠(x+ , x ) ± C2 `2 ⇠ 0 (x+ , x )) , U (U ⇤ )2 ✏2
(5.68)
which are well behaved on the asymptotic boundary. The corresponding quantum generator is defined by the conserved charge Q( ) =
Z
dxµ hTµ⌫ i
⌫
,
(5.69)
181 where hTµ⌫ i is the EMT of (5.60) defined by hTµ⌫ i = p
`2 = 8⇡
2 g (2)
⇢
SN HCF T µ⌫ g(2)
@µ @⌫
6e2 `2 + ⇡
⇢
2rµ @⌫
+
(2) gµ⌫
2R(2)
1 r↵ r↵ 2
1 gµ⌫ @↵ B@ ↵ B 2
@µ B@⌫ B
(5.70)
and hJ µ i = p
1 g (2)
SN HCF T 6e2 `2 1 p = ✏µ⌫ @⌫ B Aµ ⇡ g (2)
is the U (1) current, listed for completeness.
We now substitute the general solutions (5.59) into (5.70) and adopt asymptotic boundary conditions 8 > > > > > :hT
,
(5.71)
i = hJ i = 0 U ! ✏
from which we obtain the general behavior for the fields 8 > > > > > :f (U ) = At (U ) = 0 ` ! 1
.
(5.72)
These conditions allow us to determine the integration constants ↵i and ↵1 = 1
=
1 ↵2 = f 0 (✏) 2 1 2 = At (✏), 2
i
(5.73)
and thus specify both the EMT and U (1) current. The EMT exhibits a Weyl (trace) anomaly given by ⌦
↵ Tµ µ =
¯ 4⇡
R(2) ,
(5.74)
182 which determines the central charge via [19, 24, 25, 111, 113] ⇣ ⌘ p ¯ c 2 3/2 = ) c = 6` = 12 J + 4J E + O(✏2 ). 24⇡ 4⇡
(5.75)
The advantage to (5.71) is that at the asymptotic boundary of interest (and to O( 1` )2 , which will be denoted by the single limit x+ ! 1) the EMT is dominated by one holomorphic component hT
i. We may expand this component in terms of
the boundary fields (5.61) and (5.62) (or (5.63)) and compute the response to a total symmetry of the theory 8 > > > < +⇤ hT i = hT > > > :
+⇤
hJ i = O
⇣
1 3 U
while keeping in mind that f @± ⇠ ± and
⇠ ± +⇤ A±
0
i + 2 hT ⌘ ⇠ ± +⇤ g±⌥
i(
0
) +
c 24⇡
000
) +O
(
⇣
= ⇠ ± @± g±⌥ + g±⌥ @± ⇠ ± )
1 3 U
⌘
⇠ ± +⇤ f
(5.76)
,
= ⇠ ± @± f +
= ⇠ ± @± A± + A± @± ⇠ ± + @± ⇤.
The result is the trivial response for gauge variations, and the usual CF T response to gravitational variations of the EMT, which includes the anomalous Schwarzian derivative term [19, 22] and corresponds to one copy of the Virasoro algebra. This shows that hT
i transforms asymptotically as the EMT of a one dimensional CF T .
We compute the asymptotic charge algebra by compactifying the x coordinate to a circle from 0 ! 2⇡`2 /✏ and introducing the asymptotic conserved charge Qn = +lim
x !1
Z
dxµ hTµ⌫ i
(5.77)
⌫ n,
where ⇠(x+ , x ) has been replaced by circle diffieomorphisms
e
in(`2 /✏)x±
`2 /✏
in (5.68),
which is just the Fourier basis used in Section 1.3 and Section 5.1. The central
183 charges are initially set to zero as in [26] and we impose the initial constraint that the eigen-modes
n
form an asymptotic Witt or Dif f (S 1 ) subalgebra, which fixes
the arbitrary normalization constants Ci in (5.65) i
m,
= (m
n
n)
(5.78)
m+n .
The central charge is then found by the variation of Qn under a total asymptotic symmetry
m +⇤
Qn = [Qm , Qn ] = (m
n)Qn +
c m m2 12
1
m+n,0 .
(5.79)
This shows that the asymptotic quantum generators form a centrally extended Virasoro algebra with center (5.75) and computable non-zero lowest eigen-mode ⇣ ⌘ p 3/2 J + 4J E Q0 = + O(✏2 ). 2 5.6
(5.80)
AdS2 /CF T1 and Near-Exremal Kerr-Sen Thermodynamics
We have shown that the N HN EKS spacetime has the global symmetry group of AdS 2 ⇥ S2 (5.47). The spherical harmonic decomposition (5.51) in the near horizon limit of U ⇠ ✏, breaks the S2 section of the symmetry group. This reduces the free scalar theory (5.52) and hence the RW2DA fields (5.53) symmetry group to the local ASG AdS 2 ⇥ S1 , which has SL(2, R) ⇥ U (1) as its’ isometry group. Under these constraints, we find that N HN EKS is holographically dual to a CF T with central charge ⇣
c = 12 J +
p
4J 3/2 E
⌘
+ O(✏2 )
(5.81)
184 and lowest Virasoro eigen-mode
⇣
Q0 =
J+
p
4J 3/2 E
2
⌘
(5.82)
+ O(✏2 ).
We are now in a position to analyze the above results within the statistical Cardy formula. This yields the near-extremal Kerr-Sen entropy r ⇣ ⌘ p cQ0 Snear ext = 2⇡ = 2⇡ J + 4J 3/2 E + O(✏2 ). 6
(5.83)
We note that the near-etremal entropy has a peculiar form Snear
ext
= Sground + S1st
(5.84)
excited state .
Here Sground = 2⇡J is the extremal Kerr-Sen entropy and S1st
excited state
= 2⇡
q
12Jh , 6
p where h = 2 JE, is the excitation entropy just above extremality (the above result and h will be discussed more in the conclusion). The next concern is the N HN EKS horizon temperature and if it may be derived from our constructed correspondence. To answer this question, we will focus on the gravitational part of (5.60), given by Sgrav
`2 = 16⇡
Z
d2 x
p
⇤g(2)
g (2)
(5.85)
+ 2 R(2) .
The gravitational EMT calculated from (5.85) is hTµ⌫ i = p
2
SN HCF T g (2)µ⌫
g (2) ⇢ `2 = @µ @⌫ 8⇡
2rµ @⌫
+ g (2)µ⌫ 2R(2)
1 r↵ r↵ 2
(5.86) .
We now repeat the steps (5.70) and (5.71), but focus on the horizon limit U ! ✏ rather than the x+ ! 1 boundary, we obtain the single holomorphic component hT++ i =
`2 0 2 f (✏) , 32⇡
(5.87)
185 which is precisely the value of the Hawking Flux of the N HN EKS black hole weighted by the central charge (5.75). This allows us to extract the Hawking temperature [25, 35, 45, 58] via the standard identifications hT++ i = cHF =
c
⇡ f 0 (✏) (TH )2 ) TH = . 12 4⇡
(5.88)
This result tells us that the AdS2 /CF T1 correspondence constructed here intrinsically contains information about both the Kerr-Sen black hole entropy and temperature. However we are only able to obtain the temperature given prior knowledge of the central extension of the ASG.
186 CHAPTER 6 SUMMARY, CONCLUSIONS, AND FUTURE WORK
Q to the E to the D!...
Futurama, "The Prisoner of Benda" (2010) -Sweet Clyde (of the Globetrotter Homeworld)
6.1
Summary of Chapters 1,2,3
The goal of this work was to review string theory and holography in terms of the AdS/CF T (or more broadly the gauge/gravity) correspondence, and examine some of the applications of the correspondence(s). The review of (super)string theory began in Chapter 2 from first principles up through the modern developments of D-branes and the AdS/CF T correspondence. In Chapter 3 we reviewed Maldacena’s conjecture [8] in ’tHooft’s large N limit. The result was that an N = 4, d = 4, SU (N ) SYM gauge theory was dual to type IIB string theory in AdS 5 ⇥ S5 in the presence of D3-branes. The string theory interpretation is that a flat SUGRA background with parallel D3-branes decouple from the bulk geometry (in the field theory limit ↵0 ! 0), where the large N limit is then taken. The parallel D3-branes become extremely massive in this limit [58] and are no longer perturbed by bulk modes. This is equivalent a strong gravitational coupling and the parallel branes then collapse into a single black D3brane, analogous to a black hole. Here the strongly couple gravity theory is protected
187 from perturbations of the weakly coupled gauge theory. In this limit the AdS 5 portion of the AdS 5 ⇥ S5 background would look like a 4-dimensional Minkowski geometry on the boundary of AdS 5 . The calculational significance is realized in Section 3.3 when we equate the quantum partition functions of the boundary CFT theory to the classical partition function for the string (SUGRA) theory (3.15), ⌧
exp
Z
@AdS
0O
= ZS ( 0 ),
(6.1)
CF T
where Zs ( 0 ) = exp ( IS ( )), and IS ( ) is the classical string action. In this way the string partition function is treated as the generating functional for the boundary gauge theory. We gave several examples, mainly from [10], which showed how the correspondence functions with various field content. We also discussed how the infrared divergences of gravitational actions are made finite with boundary (surface integral) counter terms in Section 3.4. The gravity action counter terms, have gauge dual interpretations via the AdS/CF T correspondence, as local CFT counter terms which renormalize the gauge theory. We concluded Chapter 3 with the work of [78, 79] in Section 3.5. Here we discussed how various D3(D5)-brane configurations charged under the U (N ) gauge group, admit irreducible representations of SU (N ) that have a geometric interpretation of D-brane configurations. In this way, evaluating the classical D-brane action functionals is equivalent to evaluating the gauge theory partition functions via the AdS/CF T correspondence.
188 6.2
k-String Summary and Conclusions
6.2.1 Holographic k-String Conclusions In Chapter 4 we examined k-string configurations in strongly coupled, confining field theories by studying their dual D5-brane configurations. We embedded bosonic probe D5-branes with U (N ) gauge charge in the large N limit of the MNa [65], and MN [93] SUGRA backgrounds. The classical Born-Infeld plus topological ChernSimons action governed the probe D5-brane dynamics in the near horizon regime. In this limit, the gauge theory calculations yielded a positive string tension upon integration over the D5-brane world volume. The positive valued tension was identified as the QCD k-string tension between quark/anit-quark pairs [27]. We have also found supporting evidence towards the existence of a universal holographic form of the Lüscher term found in [83, 97–99]. We have borrowed intuition from developments based on the AdS/CF T correspondence in its conformal realization [78, 79] in order to address representations of D-brane configurations. Though our backgrounds here deviate slightly from conformality, it is suspected that the correspondence still exists under the proper conditions [8, 12]. Our tension results are summarized in table (4.1). At this point, we have only obtained qualitative agreement with the other approaches listed therein, however the analyses presented here have raised interesting questions in terms of representations of gauge theory and geometric D-brane configurations. These considerations are valuable in string theoretic contexts, as well as the analytically obtained tension values in d = 2 + 1 and d = 3 + 1. Though the results presented here are independent
189 of the dimensionality. This underscores the fact that the gauge/gravity method awaits further comparisons with other methods. We emphasize that the holographic approach requires a large N limit, namely, N ! 1 with k/N held fixed. A criticism of the approach here may be that the k-string configurations might not be reliably calculated with such small values of N . However, the results at hand lend themselves to the validity of the method in the regime in question. It seems that the gauge/gravity correspondence may have much to contribute to QCD discussions of k-strings as the technology becomes more sophisticated over time. Regardless the tensions of k-strings seems to offer ample ground for cross-field study. We highlight the theoretic support for the existence of a universal formula for the Lüscher term in holographic models of k-strings suggested in [97] and [83]. Future comparisons of our formula to those of the lattice calculations are eagerly anticipated given the ever increasing sophistication of computational methods in the lattice communities. 6.2.2 Open Questions and Future Work in k-strings Some open questions remain and are the aim of future efforts. One question in particular is, are the k-string representations of D-branes in a larger universality class of which the lattice gauge theory SU (N ) results are a subset? As we have seen in the tension analysis, there seems to be a robustness present in the theory with respect to scalar curvature and tension formulas of the probe D5-
190 branes irrespective of the SUGRA background. A possible relation is pointed out in [10] with an analogous example of N = 4, d = 4 on S4 SYM in contrast to the same field theory on R4 . The moduli space of classical vaccua on R4 is parametrized by the expectation value of six scalars
a
in the adjoint representation. In that case,
vacuum state of vanishing energy requires
⇥
a
,
b
⇤
= 0. The degenerate correlation
functions on R4 are not unique, but do depend on the choice of vacuum. The vacuum degeneracy and non-uniqueness are avoided on S4 , but as Witten points out, one should be more concerned with convergence of the path integral. The flat space directions in the potential diverge on R4 , but are finite on S4 due to the conformal coupling of the scalars to the S4 given by (R/6)T r
2
(where R is the S4 scalar
curvature). Witten’s analysis seems to suggest that for curvatures which are sufficiently close to the conformally round S4 , convergence in the path integral is protected. If it is generally true, that conformal curvature coupling for geometries sufficiently close to round geometries does imply path integral convergence as described by Witten’s example, then the scalar curvature of (4.31) seems to be fully robust against small deformations. However, it seems that the choice of embedding is intimately related to the these results. Investigation of this connection is a point of future interest. A full analysis of fermionic fluctuations of the solutions presented here with respect to the MN/MNa backgrounds is planned. Possible corrections to the Lüscher term may arise from this investigation though it seems unlikely that they will be able to contribute any additional low order corrections. This is a reasonable expectation
191 because the general form of the potential is Yukawa: VY ukawa (r) = g 2
e
kmr
r
,
where |r| ⇠ Lk
string ,
(6.2)
where g is the coupling, and m is the mass. Therefore we fully expect massive modes from the fermionic contributions to be exponentially suppressed for large Lk
string
as
compared to the fundamental scale. Other future considerations of interest are D-brane decay mechanisms. In particular, what are the explicit mechanisms at work in the decay of metastable Dpbrane configurations and how might some these mechanisms explicitly interact with the solutions presented here? One of the main efforts for future work is to investigate the existence of possible mixed representations of symmetric and antisymmetric D-brane configurations and their gauge theory dual descriptions. We note that this question is distinct from the configurations discussed in [78,79] where the Young diagrams constructed therein consider general configurations but do not actually correspond to any mixed representation. Rather they represent fermionic charged D5-branes or D3-branes charged by bosonized fermions where the corresponding Young diagrams are simply transposed. If a mixed representation is possible it is unclear of what the field content of such a representation should consist. 6.3
Kerr-Sen Summary and Conclusions
We have analyzed black hole thermodynamics of the N HN EKS space-time by constructing a near horizon effective action of its RW2DA fields, from which we
192 compute the resulting quantum ASG. This implies a AdS2 /CF T1 correspondence in the near horizon limit of the near-extremal Kerr-Sen metric with
Snear (5.71)
⇣ ⌘ p c =12 J + 4J 3/2 E + O(✏2 ), ⇣ ⌘ p 3/2 J + 4J E Q0 = + O(✏2 ) and 2 ⇣ ⌘ p 3/2 4J E + O(✏2 ). ext =2⇡ J +
(6.3)
Comparison between the results of Section 5.6 with those of the traditional Kerr/CF T correspondence of the Kerr-Sen black hole, shows (in the Kerr/CF T picture) the extremal Kerr-Sen black hole to be dual to a chiral CF T with centers and temperatures given as [52], cL =12J, TL =
1 , 2⇡
cR =12J
(6.4)
TR =0.
(6.5)
The extremal entropy is given by a thermal Cardy formula Sext =
⇡2 (cL TL + cR TR ) = 2⇡J. 3
(6.6)
The chiral structure seems to be missing in our results of (6.3), which is rooted in our choice and implementation of the boundary conditions (5.71), which singles out a specific homlomorphic component of the quantum EMT (5.70) at the asymptotic boundary. One possible interpretation is that the choice of (anti)holomorphic EMT definitions, forces both chiral contributions into one central charge c, and generator Q0 .
193 Similar results and analysis were found for the Kerr/CF T correspondence in [33]. There Strominger et al. attribute the result of a single half of a chiral CFT to the narrowing of the light cone for the vanishing chiral sector as they approach the near horizon limit. Indeed, as mentioned in Section 5.6, looking at the near-extremal entropy we see it has the form Snear
ext
=2⇡J + 2⇡
r
cR h R + O(✏2 ) 6
=Sground + S1st
excited state
+ ··· ,
(6.7) (6.8)
p where hR = 2 JE. In other words, the ground state entropy is completely specified by the left sector and the first excited state is specified by the right central charge and weight hR within a more traditional Cardy formula. This scenario, though plausible and consistent with [33], requires further investigations before a definitive statement can be made. However, departure from extremality should be accounted for in the entropy by contributions of the right sector of the chiral CF T similar to what was found by [116] with respect to extremal Kerr-N ewman AdS-dS black holes (and generalizations therein) and by [110] for Kerr/CF T in 4D and 5D, and [34] for the general Kerr/CF T correspondence. This was shown in the non-extremal Kerr-Sen case by [60], where a non-geometric analysis was employed and incorporated results derived separately at extremality. Another interesting feature is the dependence of the near extremality parameter ✏ in our construction. Taking the limit back to extremality, ✏ ! 0, would result in several divergences during intermediary computations. Though the final results of
194 the ASG are well defined, it is not clear how to extend the full calculation to this limit. However, there are still ample spacetime testing beds beyond Kerr which lack a near-extremality analysis of their asymptotic symmetry groups. Perhaps applications of the techniques developed in this article to a greater diversity of such spacetimes could help furnish the correspondence and better answer some of the questions posed above. 6.3.1 Kerr-Sen Future Considerations The heterotic string is an interesting and rich version of string theory as the left moving modes are not supersymmetric, though they do have both bosonic and fermionic excitations. The right moving modes however are supersymmetric. It was demonstrated in [109, 117] that the Kerr-Sen solution preserved only one half of the supersymmetries present in the heterotic string. In [109] the twisting solution generating method applied to the neutral string only acted on the holomorphic fields. The generating solutions therefore left the anti-holomorphic components intact and thus commute with supersymmetry transformations. In [117], the loss of half of the supersymmetries was shown to be a general property of stable heterotic string configurations from the supercharge generators action on the right moving fermionic modes in the sting ground state. In light of the field content contained in each chiral sector of the heterotic string, it poses an intriguing avenue for future considerations if non-trivial dynamics related to the chiral asymmetry described above in the N HN EKS solutions can
195 be found. An initial investigation could begin with the work of [118–120], where Carter showed that the Kerr family of geometries admits the existence of a conserved quantity called the Killing-Yano tensor (KY T -though it was not referred to by this name). Carter’s earlier investigation [118] revealed a fourth constant of motion in the Kerr solution related to conserved geodesics. The KYT is analogous to, or an extension of the notion Killing vectors. The rank two KYT fµ⌫ of [118–121], satisfies the following properties [35, 121] i) Antisymmetry : fµ⌫ =
f⌫µ
(6.9)
ii) Conservation : f r( fµ)⌫ = 0. Carter’s explicit construction of fµ⌫ in Boyer-Lindquist coordinates is given as f = a cos ✓dr ^ dt
a sin2 ✓d
2
r sin ✓d✓ ^
adt + r2 + a2 d
.
(6.10)
From (6.10), Carter constructed the symmetric conserved tensor Kµ⌫ related to the conservation of geodesics in the Kerr space-time [118] Kµ⌫ = fµ f ⌫ ,
which satisfies r(µ K⌫⇢) = 0.
(6.11)
The corresponding conserved charge for geodesics is [35, 118] Q = x˙ µ x˙ ⌫ Kµ⌫ ,
(6.12)
which consequently also commutes with the Laplace operator allowing for the use of separation of variables in the Kerr metric [35, 118]. Later Carter used these results to solve solve the Dirac equation in curved space-times (specifically Kerr family geometries in D = 4) [119, 120]
196 Gibbons et al. [121] employed Carter’s work [118–120] to introduce the possible existence of a new type of supersymmetry which is the square of the conserved Carter constant. The type of supersymmetry proposed in [121] is in some sense a generalization of supersymmetry to tensor fields and total supersymmetry admitted by a theory is schematically illustrated by the statement Qtotal = Qusual
Qnew
susy
susy .
However, we note that the existence of new supersymmetries are not general for all theories, furthermore they are generally dependent on the specific space-time geometry. For the rank 2 example discussed in (6.9),(6.10), and (6.12) with respect to the D = 4 Kerr geometry, the supercharge defined for a fermionic field Qi = fi µa ⇧µ
a
+
i ci 3!
a
abc
b
c
a
is (6.13)
,
where (6.13) is also generalized in [121] to higher rank tensors as well. The notational convention is that the index i = 0, . . . , N represents the total supersymmetries of the theory, i = 0 then represents the usual supersymmetry supercharge, and i = 1, . . . , N is the number of new supersymmetry supercharges found in the theory. The fields xµ ,
a
have susy variations under the infinitesimal Grassmann-odd parameter ✏ as xµ = i✏ {Q, xµ }DB = a
= i✏ {Q,
a
i✏eµa
(6.14)
a
}DB = ✏eµa x˙ µ + xµ !µ ab
b
,
(6.15)
where eµa are the frame fields or enbeins. The canonical momentum defined as ⇧µ = pµ + !µ = gµ⌫ x˙ ⌫ , and spin connection !µ = canonical Hamiltonian is H = 1/2g µ⌫ ⇧µ ⇧⌫ .
i/2!µab
a
b
. With these definitions the
197 The total supersymmetries and algebra are respectively given by µ
{J , H}DB = ⇧ {Qi , Qj }DB =
✓
@J Dµ J + Rµ⌫ @⇧⌫
◆
=0
(6.16) (6.17)
2iZij = Zji ,
where Z00 = H is the usual supersymmetry Hamiltonian, and Zij i, j = 1, . . . , N is the new bosonic symmetries, unless the Killing tensor Kijµ⌫ =
(ij) g
µ⌫
(i.e. is proportional
to the metric). If the latter is true, then this represents an extended supersymmetry. The Zij are Grassmann-even phase space functions, and since they satisfy the generalized Killing equations, their Dirac Brackets vanish with the Hamiltonian by construction. Therefore the Zij are constants of motion. It would be interesting to apply this formalism to the D = 10 heterotic string. Though the solutions of [104, 108, 109, 117, 122, 123] preserve one half of the original supersymmetries, the solution was not analyzed for any new supersymmetries in the KYT formalism1 of [121]. Generalized conformal symmetries were analyzed in [60, 122, 123], where a generalized antisymmetric three form was shown to exits, however generalized superconformal and supersymmetric symmetries were not specifically mentioned. In either case one may investigate the N HN EKS theory by probing the geometry with supersymmetric fields or with uncompactified fundamental length2 heterotic 1 To
the best of my knowledge, this possibility was not explored in the Kerr-Sen solution.
2 Here
I am specifically referring to microscopic sized (i.e. small but not necessarily of the fundamental Planck sacle) heterotic superstrings acting as incident particles on a KerrSen black hole target, as opposed to the macroscopic heterotic superstring from which the Kerr-Sen black hole solution is derived.
198 strings, in conjunction with the possibility of including interactions of the massless Kaluza-Klein modes of the compactified moduli (possibly by slightly enlarging one or more of the compactified radii, or considering modes above the ground state). Under these relaxed conditions, interesting non-trivial supersymmetric interactions could possibly be uncovered from the vast richness of the heterotic theory. 6.4
Recent Developments and Limitations of the AdS/CF T Correspondence 6.4.1 Black Hole Complimentarity
In this final section we briefly mention recent developments related to the application of the AdS/CF T correspondence to extremal black hole solutions. In the early to middle 1990’s Susskind, Thorlacous, and Uglum, posited the existence of black hole complimentarity [124]. In it they proposed that a black hole horizon was a membrane like surface (analogous to D-branes) which had a thickness on the order of a Planck length, known as a stretched horizon. From their supposition the argue that this type of description would result in tangible predictions of events at the horizon. Among their claims were that due to local Lorentz and diffeomorphism invariance, there would be no noticeable effects as one crosses the horizon. In more concrete terms, their arguments for consistency of physics at the horizon of a black hole requires the following to be true. 1. The formation and evaporation of a black hole viewed by a distant observer can be described within the context of quantum field theory. Thus the claim that
199 there exists a unitary S-matrix which describes the evolution of the black hole from infalling matter to outgoing Hawking radiation. 2. Outside of the stretched horizon, physics could be described in terms of a (semiclassical) low-energy effective theory. 3. To a distant observer, the black hole would appear as a quantum mechanical system with discrete entropic, and energetic states. 4. Due to the low energy effective Lorentz invariance, the probability of an infalling observer to observe a quanta of energy E / 1/rBH is exponentially suppressed by a decreasing adiabatic factor. In other words, there is no "drama" at the horizon. 6.4.2 Firewalls, Not Black Hole Complimentarity Recently in 2012, Polchinski et al. [125–127] claimed that items 1,2, and 4 cannot hold simultaneously for sufficiently old (i.e. eternal) black holes, and thus black hole complimentarity is flawed. Therefore there must exist a "firewall" very near to the horizon. We will not present rigorous point by point counter arguments as presented in [125]. Instead we present an overview of the AdS/CF T version of the firewall argument [127]. Consider an a vacuum state a, very near the horizon of a black hole. In the process of hawking radiation, a virtual particle b and virtual anti-particle ˜b are then created as in Figure 6.1. In the semi-classical, low energy description of quantum 1. a -is smooth adiabatic mode across horizon as seen by in falling observer and is
200
Ini$al'Picture' Horizon
˜b
b ˆb a time
Singularity
Boundary CFT
Figure 6.1: The semi-classical description of Hawking radiation occurring at the black hole horizon.
a "ground state". 2. b -is in a mixed excited thermal state defined by a superposition of states of a: b = Ba + Ca† . We assume that there is a Hilbert space H, with which we can write the ˆb , in the creation and annihilation operator basis of ˆb and ˆb† number operator N ˆb = ˆb†ˆb is diagonal. The operator N ˆb , is thermal in the a vacuum. For such that N eigenstates
bi
ˆb , it follows that of N h
ˆ |
b i | Na
bi i
O(1), 8
ˆa ) T r(N i =) T r(1) ˆa 6= 0) We also note that the projection P (N
bi ,
(6.18)
O(1). O(1). If all states are bounded from
201 below by unity, it must be that every state near the horizon is populated. This implies that the near horizon states have very large temperatures (highly energetic). Therefore there must be a firewall, and either unitarity is lost or a low energy effective action cannot describe the Hilbert space of states [127]. In order to see what the firewall implications are for the AdS/CF T correspondence, we ask the question "Does there exist a boundary CFT energy momentum ˜ operator Tˆµ⌫ , which is dual to T˜µ⌫ ? If we consider the CFT boundary operators which correspond to the infalling Hawking modes and the Hamiltonian H which governs the internal black hole dynamics, they the following commutation relations h i ˆ˜b† , ˆ˜b =
1 (6.19)
and
h
i H, ˆ˜b† =
!ˆ˜b† .
These interactions are the processes which allow for black hole evaporation. The commutator of the anti-virtual particles with the Hamiltonian must lower the energy of the black hole. This is because the timelike Killing vectors which are related to the Hamiltonian become spacelike beyond the horizon, where the energy becomes a momentum which can take on negative values. If we assume some initial state |ii of the black hole, and that there is a finite density of states which can describe the Hilbert space, the energy of the black hole is then bounded from above and below (for some small energy level ) such that 8 |ii : M < E < M + ,
(6.20)
202 where M is the black hole mass and E is the energy. If we now act on the state |ii with the operator ˆ˜b ˆ˜b† |ii
=)
M
! 0.
(C.7)
⇥ ⇤ Then for a state given by h0| a0m , a0† m |0i =
m, is of negative mass.
This new definition for L0 effects the mass-shell condition derived in the clas-
sical algebra. We now have (with ↵0 = 12 ) 2
M =
2a + 2
1 X
↵
n
n=1
(for open strings)
· ↵n ,
(C.8) (C.9)
and (C.5) implies that (L0
⇣
⌘
˜0 L
a | i = 0 must also hold for closed ⇣ ⌘ ˜ 0 | i = 0 gives the case for strings. Equivalently, this can be written as L L a) | i =
closed strings, M2 =
8a + 8
1 X
(↵
n=1
n · ↵n ) =
8a + 8
1 X
(˜ ↵
n=1
n
·↵ ˜ n ) , (for closed strings). (C.10)
For the closed string, the left and right modes couple for L0 such that 1 X n=1
↵
n
· ↵n =
1 X
↵ ˜
n=1
n
·↵ ˜ n , (for closed strings).
In regards to the operator equation (L0 may generalize this to (Lm
a
m,0 ) |
(C.11)
a) | i = 0 for physical states , we
i = 0 , m
0. This corresponds to semi-
positive definite conditions for the physical states and implies that the Fock space of physical states is actually a subspace of the complete Fock space.
215 We quickly note that there is no normal ordering ambiguity in the Poincaré algebra, and thus the Lorentz generators can interpreted directly as quantum operators and yield the expected commutation relations, (C.12)
[pµ , p⌫ ] = 0 [pµ , J ⌫⇢ ] = [J µ⌫ , J ⇢ ] = C.2
(C.13)
i⌘ µ⌫ p⌫ + i⌘ µ⇢ p⌫ i⌘ ⌫⇢ J µ + i⌘ µ⇢ J ⌫ + i⌘ ⌫ J µ⇢
i⌘ µ J ⌫⇢
(C.14)
The Faddeev-Popov Prescription
In this section of the appendix, we review the Faddeev-Popov prescription for quantizing the path integral as is discussed in [16]. The classical action in the path integral includes elements from the group of reparametrizations on the worldsheet metric in order to make the measure well defined. Essentially this inserts ”1” into the path integral, giving a Jacobian times a delta function for each coordinate diffeomorphism transformation. This also requires a reparametrization group action change of variables on the world-sheet metric such that h ! hg for the metric h and group element g. We leave the group unspecified here, but generally to tree level or 1 loop in ↵0 , g⇠ = {g 0 : g 0 2 SL(2, C), SL(2, R), or SL(2, Z)}
(C.15)
and depends on the coordinate chart used, whether the theory is open vs. closed, and the loop expansion order. The inclusion of the 2 Jacobians gives anticommuting Faddeev-Popov ghosts and antighosts, whose action we must also quantize. We give the final form of the
216 gauge fixed path integral below. Z=
Z
Z
D
DX( )Dc( )Db( ) e
S(X( ),b( ),c( ))
(C.16)
In the world sheet description there are two ghost fields for c( ) and therefore it is regarded as a two-component vector c↵ ( ) = (c+ , c ) in light-cone coordinates. Analogously, b( ) is regarded as a rank-two symmetric and traceless tensor b↵ ( ) with components b++ and b
. The forms of the ghost (antighost) actions in the
path integral can also be derived from an action similar to the Polyakov string action 1 2⇡
Sghost =
Z
d2
p
(C.17)
hh↵ c r↵ b ,
which in the conformal gauge simplifies to the form, SFcg P
=
1 ⇡
Z
d2
c+ @ b++ + c @+ b
.
(C.18)
The Faddeev-Popov ghost coordinates c and antighosts b are also conjugate variables and enjoy equal ⌧ anticommutation relations, {b++ ( ), c+ ( 0 )} = 2⇡ ( {b
0
( ), c ( )} = 2⇡ (
0
)
0
).
(C.19)
In the conformal gauge, their equations of motion are @ c+ = @ b++ = 0 @+ c = @+ b
(C.20)
= 0.
For open string boundary conditions, c+ = c , b++ = b
and have the Fourier
217 mode expansions, ±
c = b±± =
1 X
n= 1 1 X
cn e
in(⌧ ± )
bn e
in(⌧ ± )
(open strings)
(C.21) (C.22)
n= 1
The closed string periodicity condition requires two independent left and right propagating modes, c+ =
p
p
c =
1 X
2
cn e
n= 1 1 X
2in(⌧ + )
c˜n e
2in(⌧
bn e
2in(⌧ + )
2
(right moving closed strings)
(C.23)
,
(left moving closed strings),
(C.24)
,
(right moving closed strings)
(C.25)
(left moving closed strings).
(C.26)
, )
n= 1
and similarly for the b’s, b++ = b
p
=
2
p
1 X
n= 1 1 X
˜bn e
2
2in(⌧
)
,
n= 1
Given the anticommutation relations for c and b, the Fourier mode coefficients enjoy the anticommutation relations, {cm , bn } =
m+n,0
(C.27)
{cm , cn } = {bm , bn } = 0. We now calculate the ghost Virasoro operators. First we examine the energy momentum tensor from the ghost action. We move to the complex plane by letting z = ⌧ + i and z¯ = ⌧
i , thus the world-sheet Minkowski metric now takes on the
form h↵ ! e dzd¯ z . We do this here, so that the conformal dimension of the Virasoro
218 operators is manifest in the results. In doing so, the ghost action (C.18) now has the form, (c) SF P
i 4⇡
=
Z
d
c+ @ b++ + c @+ b
(C.28)
,
from which we vary the ghost action to obtain, (c) T++
T
(c)
= =
i i
1 + c @+ b++ + @+ c+ b++ 2 1 c @+ b 2
+@ c b
(C.29) (C.30)
.
Inserting the above ghost mode expansions into the definitions of the Virasoro operators at equal ⌧ (which for the open strings is Lm = 1 2⇡↵0
1 ⇡
R⇡
⇡
d eim T++ and T =
has been used for the fundamental string tension). This brings about L(c) m
=
1 X
[m(J
1)
n] bm+n c
(open strings).
n,
(C.31)
n= 1
˜m = For closed strings, we have L ˜ m with + $ Lm = L L(c) m
=
˜ (c) L m =
1 2⇡
R⇡ 0
d e
T++ for left moving modes and
2im
for the right moving modes, 1 X
n= 1 1 X
n,
(right closed strings)
[m(J
1)
n] bm+n c
[m(J
1)
n] ˜bm+n c˜ n , (left closed strings).
(C.32) (C.33)
n= 1
Here J is the conformal dimension of the ghost (antighost) operator, which is left arbitrary for the moment. For the case at hand J = 2 for b±± and J =
1 for c± .
Therefor the product of the ghosts and antighosts as in c± @± b±± and @± c± b±± has conformal dimension of h = J
1.
219 Similar to the Virasoro algebra, here there is also a normal ordering ambiguity h i (c) (c) which leads to an anomalous central extension term for Lm , Ln when m + n = 0. We will now investigate the anomaly terms for both the Virasoro algebra and the ghost Virasoro algebra. We normal order both algebras and explicitly verify that the quantized Faddeev-Popov (anti)ghost anomaly conspires to exactly cancel the quantum anomaly from the Virasoro algebra. C.3
Virasoro Anomaly of the Bosonic Matter
The matter field Virasoro anomaly term is discussed in [16,22,23]. We augment (↵)
the notation of the bosonic generators (2.44)-(2.46) to Lm (consistent with [16]), which specifies the quantum operators ↵nµ with respect to which the generators are defined. We will compute the open string Virasoro anomaly, where the resolution of the normal ordering ambiguity is more technically simple. The right and left moving oscillator operators, ↵nµ and ↵ ˜ nµ commute, therefore there is no loss of generality to consider the open string oscillators as ’fundamental’ operators for the process of quantization [22]. The result is unchanged for closed strings, and produces only a copy of the anomaly term for each independent mode. Note that the only terms which will contribute to the central extension of the Virasoro algebra are those of the form,[↵m · ↵m , ↵
m
·↵
m ].
We write the normal
220 (↵)
ordered Lm ’s by splitting sum such that, m 1
L(↵) m
X 1 ⌫ = ⌘µ⌫ ↵nµ ↵m 2 n=1
⌫ ⌫ + ↵0µ ↵m + ↵µ 1 ↵m+1 + ... ,
n
⇣
k 1
(↵) L m
X µ 1 = ⌘µ⌫ ↵ k ↵⌫ (m 2 k=1
k)
+ ↵
µ
⌫ m ↵0
+↵
µ
⌫ (m+1) ↵1
(C.34)
⌘
+ ... .
The central extension anomaly term is computed by only taking the commutator of terms outside of the parenthesis in (C.34), all other cross terms commute. A(↵) (m) =
m 1 1 X ⇥ : ↵n · ↵m 4 n,k=1
n, ↵ k
·↵
⇤
(m k) cen
(C.35)
:
We apply the operator identity [AB, CD] = CA [B, D] + C [A, D] B + A [B, C] D + [A, C] BD and make the following identifications (with Lorentz indices µ, ⌫, etc... suppressed), A = ↵n ,
B = ↵m
n,
C = ↵ k,
D=↵
(C.36)
(m k) .
We drop the terms which do not require normal ordering and normal order the remaining terms by making use of the canonical commotion relations (C.3). The result simplifies A
(↵)
(m) =
h
(↵) L(↵) m ,L m
i
m 1 1 X = ([A, D] [B, C] + [A, C] [B, D]) 4 n,k=1 m 1 1 X µ⌫ ⌘ ⌘µ⌫ (n = 4 n,k=1 m 1 D X = n(m 4 n,k=1
D = 2
m
m X1 n=1
n
n,m k (m
n) ( m X1 n=1
n,m k
n
2
!
n)
+
m n,k
n,k )
(C.37) +n
n,k (m
n)
n,k )
221 We employ the sum identities given below and extend the sum to include the n = m term since it is zero. m X
1 n2 = (2m3 + 3m2 + m) 6 n=1
(C.38)
m X
1 n = m(m + 1), 2 n=1
The result is the quantum Virasoro anomaly term A(↵) (m) =
D 3 (m 12
(C.39)
m),
where D is the space-time dimension. C.4
Virasoro Anomaly from (Anti)Ghost Matter
The (anti)ghost Virasoro anomaly is calculated analogously to that of the matter fields. This requires us to calculate calculate the commutator of (C.31) for values m and
m. It is useful to first normal order (C.31) such that the sum over
all n 2 Z is broken up over n
0 and n < 0, and express the desired commutator
(c)
in terms of Lm and the normal ordered sum over all n where the ghost/antighost Fourier coefficients are explicit. To proceed, it simplifies the entire calculation to have on hand the normal h i h i (c) (c) ordered expressions for Lm , bn and Lm , cn . The normal ordered sum of (C.31) is L(c) m
=:
1 X
(mJ 0
n) bm+n c
n
:
n= 1
=:
X
(mJ 0
n) bm+n c
n
:+:
X n 0
(mJ 0
n) bm+n c
n
n m, n > 0. > > : cn b m := b m cn ;
Using (C.31), the operator identity [AB, C] = A{B, C} {A, C}B, and the ghost/antighost Fourier coefficient anticommutation relations (C.27), we find that ⇥ ⇤ : L(c) m , bn := =
X
(mJ 0
k 0
X
X
(mJ 0
(mJ 0
k)[bm+k c k , bn ]
:0 X ⇠⇠ ⇠⇠ k)c k⇠ {b⇠ , bn } + (mJ 0 m+k
k){c k , bn }bm+k +
k 0
(mJ 0
k)bm+k {c k , bn }
k
