Topological Quantum Field Theory: Overview

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are maybe the most significant examples in the last years). In particular fluid dynamics, a topological macroscopic field theory, provides a powerful frame-.
278 Topological Quantum Field Theory: Overview

are maybe the most significant examples in the last years). In particular fluid dynamics, a topological macroscopic field theory, provides a powerful framework for modern theory of knots and links in 3-manifolds. Moreover, as we saw here, it provides a physical interpretation of the link, self-linking, and writhing number of knots and links. The present article was essentially aimed to illustrate such a relationship. Thus, the most fundamental result we reported here is the relation (formula) connecting the helicity of vector (magnetic) fields to the writhing number of knots: H(V) = Flux(V)2 Wr(K). So, writhing number for knots is the analog of helicity for vector fields. Both expressions of these invariants are variants of the (Gaussian) integral formula for the linking number of two disjoint closed space curves. Further investigations of these invariants and their mathematical properties might throw new light on the interfaces between many different areas of macroscopic and quantum physics. See also: The Jones Polynomial; Knot Theory and Physics; Magnetohydrodynamics; Mathematical Knot Theory; Stability of Flows; Superfluids; Topological Quantum Field Theory: Overview; Vortex Dynamics; Yang–Baxter Equations.

Further Reading Arnol’d V and Khesin B (1998) Topological Methods in Hydrodynamics. Heidelberg: Springer. Berger MA and Field GB (1984) The topological properties of magnetic helicity. Journal of Fluid Mechanics 147: 133–148. Berry MV and Dennis MR (2001) Knotted and linked phase singularities in nonchromatic waves. Proceedings of the Royal Society A 457: 2251–2263. Boi L (2005) Topological knots’ models in physics and biology. In: Boi L (ed.) Geometries of Nature, Living Systems and Human Cognition. New Interactions of Mathematics with Natural Sciences and Humanities, pp. 211–294. Singapore: World Scientific.

Boyland P (2001) Fluid mechanics and mathematical structures. In: Ricca RL (ed.) An Introduction to the Geometry and Topology of Fluid Flows, pp. 105–134. NATO-ASI Series: Mathematics. Dordrecht: Kluwer. Cantarella J, De Turk D, and Gkuck H (2001) The Biot–Savart operator for application to knot theory fluid dynamics, and plasma physics. Journal of Mathematical Physics 42: 876–905. Freedman MH and Zheng-Xu He (1991) Divergence free fields: energy and asymptotic crossing number. Annals of Mathematics 134: 189–229. Fuller FB (1978) Decomposition of the linking number of a closed ribbon: a problem from molecular biology. Proceedings of the National Academy of Sciences, USA 75: 3557–3561. Ghrist RW, Holmes PhJ, and Sullivan MC (1997) Knots and Links in Three-Dimensional Flows. Heidelberg: Springer. Hornig G (2002) Topological Methods in Fluid Dynamics. Preprint. Ruhr-Universita¨t-Bochum. Kauffman LH (1995) Knots and Applications. Series on Knots and Everything, vol. 6, Singapore: World Scientific. Lomonaco SJ (1995) The modern legacies of Thomson’s atomic vortex theory in classical electrodynamics. In: Kauffman LH (ed.) The Interface of Knots and Physics, Proc. Symp. Appl. Math., vol. 51, pp. 145–166. American Mathematical Society. Moffatt HK (1969) The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics 35: 117–129. Moffat HK (1990) The energy spectrum of knots and links. Nature 347: 367–369. Moffatt HK, Zaslavsky GM, Comte P, and Tabor M (1992) Topological Aspects of the Dynamics of Fluids and Plasmas, NATO ASI Series, Series E: Applied Sciences, vol. 218. Dordrecht: Kluwer Academic. Ricca RL (1998) New developments in topological fluid mechanics: from Kelvin’s vortex knots to magnetic knots. In: Stasiak A, Katritch V, and Kauffman LH (eds.) Ideal Knots. Singapore: World Scientific. Ricca RL and Moffat HK (1992) The helicity of a knotted vortex filament. In: Moffat HK (ed.) Topological Aspects of Dynamics of Fluids and Plasmas, pp. 225–236. Dordrecht: Kluwer. Tait PG (1900) On Knots I, II, III. In: Scientific Papers. Cambridge: Cambridge University Press. Thomson JJ (1883) A Treatise on the Motion of Vortex Rings. London: Macmillan. Trueba JL and Ran˜ada AF (2000) Helicity in classical electrodynamics and its topological quantization. Apeiron 7: 83–88. Woltjer L (1958) A theorem on force-free magnetic fields. Proceedings of the National Academy Sciences, USA 44: 489–491.

Topological Quantum Field Theory: Overview J M F Labastida, CSIC, Madrid, Spain C Lozano, INTA, Torrejo´n de Ardoz, Spain ª 2006 Elsevier Ltd. All rights reserved.

Introduction Topological quantum field theory (TQFT) constitutes one of the most successful fields of mathematical physics since it originated in the 1980s. It possesses an inherent property which makes it unique: TQFT provides predictions in mathematics which open

new fields of research. A well-known example is the prediction of Seiberg–Witten invariants as building blocks of Donaldson invariants. However, there are others such as the recent proposal for the coefficients of the HOMFLY polynomial invariants for knots as quantities related to enumerative geometry. These developments have drawn the attention of mathematicians and physicists into TQFT since the 1980s, a very fruitful period in which both communities have benefited from each other. Topology has always been present in mathematical physics, in particular when dealing with aspects of

Topological Quantum Field Theory: Overview

quantum physics. Global effects play an important role in quantum-mechanical models and topology becomes an essential ingredient in their description. TQFT itself appeared in the winter of 1987 after Witten’s work (Witten 1988a) on Donaldson theory (Donaldson 1990), but a series of papers during the 1980s which dealt with topological aspects of field and string theory anticipated its existence. Two of these correspond to Witten’s works on supersymmetric quantum mechanics and supersymmetric sigma models (Witten 1982) that led to a generalization of Morse theory. This generalization was considered by Floer (1987) in a new context that constituted the key element in Witten’s construction of TQFT. These developments were certainly influenced by Atiyah (1988). TQFT was born as a result of the interplay between physics and mathematics. This has been a constant feature all along its development. Soon after the formulation of the TQFT addressing Donaldson theory, now known as Donaldson–Witten theory, Witten formulated a new TQFT which focuses on knot invariants such as the Jones polynomial and its generalizations (Jones 1985). Witten (1989) constructed Chern–Simons gauge theory and proved its relation to the theory of knot and link invariants. This theory possesses different features than Donaldson–Witten theory, and in fact it turns out that these two theories fall into two different general types of TQFTs as will be explained in the following section. Anyhow, despite their formal differences, both Donaldson– Witten and Chern–Simons gauge theory emerged as a novel way to express topological invariants in terms of quantum field theory quantities as well as to generalize their previous formulation. But there was much more to them than it seemed in their beginnings. Once these topological invariants were formulated in field theory language, one had a huge machinery to study them from different points of view. Theoretical physicists have developed many useful tools to study quantum field theory. The use of these tools led to new frameworks for these topological invariants. In this overview we are going to provide the basics of TQFT and briefly describe two examples – Donaldson–Witten theory and Chern–Simons gauge theory – to explain how the general features are implemented. Some excellent reviews on the subject (Birmingham et al. 1991, Cordes et al. 1996, Labastida and Marin˜o 2004) are available. The organization of this work is as follows. In the following section we present a general introduction to TQFT from a functional integral point of view. Next, we touch upon the twisting of extended supersymmetry as a general constructive approach to TQFT. This is followed by a section on

279

Donaldson–Witten theory where we discuss the computation of its observables from a perturbative approach, showing their relation to the Donaldson invariants. Next, we introduce Chern–Simons gauge theory as a theory of knot and link invariants. The penultimate section deals with advanced developments in TQFT. Finally, we end up with some concluding remarks.

Topological Quantum Field Theory We will start our overview by presenting the most general structure of a TQFT from a functional integral point of view which, though not rigorously defined, is the approach that has led to the most important developments. As in conventional quantum field theory, axiomatic approaches to TQFT do exist, but we will not follow that route here. Let us consider an n-dimensional Riemannian manifold X endowed with a metric g and a quantum field theory on it. We will say that this theory is ‘‘topological’’ if there exist operators in the theory such that their correlation functions do not depend on the metric. If we denote these operators by Oi (where i is a generic label), then  hOi1    Oin i ¼ 0 g

½1

where h  i denotes a vacuum expectation value. The operators that satisfy this equation are called ‘‘topological observables.’’ The simplest way to achieve metric independence is to consider a theory whose action and operators do not depend on the metric. In this situation, if no anomalous metric dependence is generated upon quantization, the correlation functions of these operators satisfy [1] and lead to topological invariants on X. Theories of this sort are collectively referred to as Schwarz-type TQFTs, and well-known examples are Chern–Simons gauge theory and BF theories. However, Schwarz-type theories are too restrictive. One would like to have a theory satisfying property [1] with a weaker condition on the action. This can be achieved with the help of a symmetry. The resulting TQFTs are called of Witten or cohomological type, the main examples being Donaldson–Witten theory and topological sigma models (Witten 1988b). For TQFTs of Witten type, the action may depend on the metric. However, the theory has an underlying scalar symmetry  acting on the fields i . Since  is a symmetry, the action of the theory satisfies S(i ) = 0. In these theories, metric independence of the correlation functions is achieved as follows. Let T = (=g  )S(i ) be the energy–momentum tensor of

280 Topological Quantum Field Theory: Overview

the theory. It turns out that the energy–momentum tensor is -exact: T ¼ iG

½2

G being some tensor. Indeed, if [2] is satisfied, it follows that for any set of operators Oi which are -invariant,  hOi1 Oi2    Oin i ¼ hOi1 Oi2    Oin T i g ¼ ihOi1 Oi2    Oin G i ¼ ihðOi1 Oi2    Oin G Þi ¼0

½3

In this computation we have assumed that the symmetry  is not anomalous and that there are no contributions coming from boundary terms since we have integrated by parts in field space. This is not always the case and in fact the situations in which one of these two properties fails lead to rich phenomena. In those cases, for example, in Donaldson–Witten theory on manifolds with bþ 2 = 1, the correlation functions fail to be topological invariants in a controlled manner which unveils many interesting properties. We will now describe Witten-type theories in a general context. The general structure of Schwarz-type theories is much simpler and will be illustrated in the example presented below. In Witten-type theories the observables are the -invariant operators. It is simple to prove that -exact operators decouple from the theory. Indeed, if Oa is -exact, Oa = O^a , then hOa Oi1 Oi2    Oin i ¼ hO^a Oi1 Oi2    Oin i ¼ hðO^a Oi Oi    Oi Þi ¼ 0 1

2

n

½4

Thus, one can restrict the set of observables to the cohomology of : O2

Ker  Im 

½5

There is no reason a priori why the -symmetry should be a scalar Grassmannian symmetry, but in all known models of Witten-type TQFTs this turns out to be the case. Thus, these theories violate the spin-statistics theorem. In all these models the algebra of the  symmetry has the form 2 ¼ Z

½6

where Z is a symmetry transformation (typically a gauge symmetry of some sort). This property forces to consider Z-invariant observables and to work in the context of ‘‘equivariant cohomology.’’ The observables of Witten-type theories fit into a general pattern that we describe now. The key ingredient is a map between the homology of X and

the equivariant cohomology of . Given an operator (0) in the equivariant cohomology of , let us consider the following set of equations: dðnÞ ¼ ðnþ1Þ ;

n0

½7

where the operators (n) (n = 1, . . . , dim X) are differential forms of degree n on X and d is the de Rham differential. These differential equations are called ‘‘descent equations’’ and their solutions (n) (n  0) ‘‘topological descendants’’ of (0) . We will show how to construct a solution to these equations on general grounds. The topological descendants lead to the construction of a set of elements of the equivariant cohomology of . Let n be an n-cycle on X, n 2 Hn (X), and let us consider the following operator: Z ðn Þ Wð0Þ ¼ ðnÞ ½8 n

This operator is -invariant, Z Z Z ð Þ Wð0Þn ¼ ðnÞ ¼ dðn1Þ ¼ n

n

ðn1Þ ¼ 0

½9

@ n

since @ n = 0. On the other hand, if n were trivial in homology, that is, if n = @ nþ1 , we would have that W((0)n ) is -exact: Z Z Z ðn Þ ðnÞ ðnÞ Wð0Þ ¼  ¼ d ¼  ðnþ1Þ ½10 @ nþ1

nþ1

nþ1

(0)

Thus, given the operator  , we have constructed a map between the homology of X and the equivariant cohomology of . There are as many maps as basic operators (0) one finds in the theory. To actually construct these maps, we need to find a solution of the descent equations [7]. As announced before, there is a general solution to those equations in Witten-type theories. Since in this type of theories [2] holds, there exists an operator G  G0

½11

P ¼ T0 ¼ iG

½12

that satisfies

Notice that G is an anticommuting operator and a 1-form in spacetime. With the aid of this operator, one constructs the following solution to the descent equations [7]: ðnÞ ¼

1 ðnÞ  dx1 ^    ^ dxn n! 1 2 ...n

½13

where ð0Þ ðnÞ 1 2 ...n ðxÞ ¼ G1 G2    Gn  ðxÞ;

n ¼ 1; . . . ; dim X

½14

Topological Quantum Field Theory: Overview

One can easily check using [12] and the -invariance of (0) that the operators [13] do satisfy the descent equations [7]. We have seen that Witten-type TQFTs are characterized by property [2]. It would be desirable to have at hand a systematic procedure to build theories satisfying that property. It has been found that extended supersymmetry provides a very helpful starting point to build those theories. Although supersymmetry guarantees from first principles only the weaker condition [12] instead of [2], all TQFTs that have been constructed from extended supersymmetry actually satisfy [2]. To build a TQFT from a theory with extended supersymmetry, one needs to go through the twisting procedure that we now describe.

Twisting of Extended Supersymmetry All known Witten-type theories are related to an underlying extended supersymmetric quantum field theory. The topological theory is a modified version of the supersymmetric theory in which the Lorentz transformation properties (spins) of some of the fields have been modified. This modification of spin assignments is known as twisting, and it can be carried out on any theory with extended supersymmetry in any spacetime dimension. We will not consider the procedure in such a general setting but instead we will illustrate it by considering the case of N = 2 supersymmetry in four dimensions. We will begin with a general description and then we will apply it to a specific example: Donaldson–Witten theory. Let us consider the Euclidean version of the N = 2 supersymmetry algebra with no central charges. Central charges can be included without much ado but we will not consider them for simplicity. The total symmetry group of the theory is H = SU(2)þ  SU(2)  SU(2)R  U(1)R , K = SU(2)þ  SU(2) being the rotation group, and SU(2)R  U(1)R the internal symmetry group of the N = 2 supersymmetry algebra. The generator algebra takes the following form:  fQv ; Qw _ g ¼ 2 vw _ P ;

½P ; Qv _ ¼ 0

½P ; Qv  ¼ 0; ½M ; Qv  ¼ ð QÞv ;

½M ; Qv _  ¼0

½M_ _ ; Qv  ¼ 0;

½M_ _ ; Qv _  ¼ ð _ _ QÞv _

½Bvw ; Qu  ¼ uðv QwÞ  ;

½Bvw ; Q_  ¼  uðv Q_

½Qv ; R ¼ Qv ;

½Qv _ ; R ¼ Qv _

u

1 Qv and Qv ˙ transform under H as (0, 2, 2) and 1 (2, 0, 2) , respectively. M˙ ˙ and M are the generators of SU(2)þ and SU(2) , respectively, while Bvw and R generate SU(2)R and U(1)R , respectively. The twisting of a supersymmetric theory involves a modification of the couplings of the theory to a background metric on the space where the theory is defined. This modification is carried out redefining the Lorentz transformation properties of the different fields making use of the internal symmetry SU(2)R . In particular, we will redefine the couplings of the fields to the SU(2)þ spin connection according to the way they transform under SU(2)R . This is easily done by identifying the SU(2)R indices v with the SU(2)þ indices . ˙ The procedure involves a redefinition of the rotation group into K0 = SU0 (2)þ SU(2) , where SU0 (2)þ is generated by

M0_ _ ¼ M_ _  B_ _

½16

The supersymmetry generators Qv and Qv get ˙ transformed in the following way: Qv _ ! Q_ _

½17

Qv ! Q_ which allows supercharge’’:

us

to

define

the

‘‘topological

_

Q  _  Q_ _

½18

It is simple to prove using [15] and [16] that this quantity is a scalar under the new rotation group K0 : [M , Q] = 0 and [M0˙ ˙ , Q] = 0. In addition, from [15], it follows that Q is nilpotent (in the absence of central charges): 2

Q ¼0

½19

The scalar generator Q leads to the topological symmetry  of the previous section. Actually, the twisting procedure provides also the operator G in [12]. Defining i _ G ¼ ð  Þ Q_ 4

fQv ; Qw g ¼ 0

281

½20

one easily finds, after using [15] and [18],



½15 In these relations v, w 2 {1, 2} are SU(2)R indices and  and ˙ denote spinorial indices of SU(2) and SU(2)þ , respectively. The supersymmetry generators

fQ; G g ¼ @

½21

which is indeed equivalent to [12]. On general grounds we cannot prove that twisted supersymmetric theories lead to theories which satisfy [12]. However relation [12], which is weaker, is guaranteed. It turns out that in all the models originated from extended supersymmetry which have been studied, [2] is satisfied and thus the resulting theories are TQFTs of Witten type.

282 Topological Quantum Field Theory: Overview

Donaldson–Witten Theory One of the greatest successes of TQFT has been the discovery of Seiberg–Witten invariants as building blocks of Donaldson invariants. This was achieved in two main steps. First, Donaldson theory was reformulated in field-theoretical terms, using perturbative methods. Second, the resulting TQFT was solved using nonperturbative methods. In this section we are going to describe in some detail the first step. The second one will be briefly addressed later and is the main object of a separate article in the encyclopedia (see Seiberg–Witten Theory). Let us consider N = 2 supersymmetric Yang–Mills theory in four dimensions. The field content of the theory is the following: a gauge field A , two spinors v , and a complex scalar , all of them in the adjoint representation of a gauge group G. In addition, the theory possesses the auxiliary fields Dvw in the 3 of the internal SU(2)R . The theory has the following action:  Z 1 v d4 x tr r y r   i v  r  F F 4 1 1 i þ Dvw Dvw  ½; y 2  pffiffiffi vw v  ½y ; w  4 2 2  i v w_  pffiffiffi vw _ ½ ;  ½22 2 This action is invariant under the following N = 2 supersymmetric transformations: pffiffiffi  ¼ 2 vw v w v

A ¼ i v   i v 

v

 v ¼ Dv w w  i v ½; y   i    v F pffiffiffi w_ þ i 2 vw _ r  ðv

½23

ðv

v being spinorial N = 2 supersymmetric parameters. We can now twist the above theory following the procedure explained in the previous section. Upon twisting, the fields of the theory change their spin content as follows:

1 _ ð2; 2Þ 1

ð0; 0; 0Þ2 ! ð0; 0Þ2 y ð0; 0; 0Þ2 ! y ð0; 0Þ2 Dvw ð0; 0; 3Þ0 ! D_ _ ð0; 0Þ0

_ r

_ 

½Q;  ¼ 0

½24



fQ; g ¼ ½; y  pffiffiffi fQ;  g ¼ 2 2r  pffiffiffi ½Q; y  ¼ 2 2i

½26

fQ; _ _ g ¼ iðFþ_ _  D_ _ Þ pffiffiffi ½Q; D ¼ ð2r Þþ þ 2 2½;  ˙

A ð2; 2; 0Þ0 ! A ð2; 2Þ0 1 v ! ð0; 0Þ ; _ _ ð3; 0Þ1 _ ð0; 2; 2Þ

1 1 _

 F F þ D_ _ D_  4 4 1 i _  ½; y 2  pffiffiffi _  ½; _ _  2 2  pffiffiffi i _ y ;  ½25 þ i 2 ½;   pffiffiffi _ ½ 2 pffiffiffi where g = (det(g ))1=2 . To obtain the transformations of the fields under the topological symmetry, we need to compute the Q-transformations. These are easily obtained using [18] and [23]. They turn out to be i

½Q; A  ¼

Dvw ¼ 2i  r wÞ þ 2ir  wÞ pffiffiffi pffiffiffi ðv wÞ þ 2i 2 ðv ½ wÞ ; y  þ 2i 2 ½ ; 

v ð2; 0; 2Þ1 !

In this table the representations of the respective rotation groups carried by the fields have been indicated. The superindices refer to the U(1)R charge which is also called ‘‘ghost number’’ in the context of TQFT. The fields and  are given by the antisymmetric and symmetric pieces of ˙ ˙ : ˙ ˙ = ˙ (˙ )˙ and = (1=2) ˙  ˙ ˙ . Notice that the twisted fields in [24] are differential forms on X; therefore, the twisted theory makes sense globally on any arbitrary Riemannian 4-manifold. This is not the case with the original N = 2 supersymmetric Yang–Mills, which contains fermionic fields. Making global sense of those on arbitrary Riemannian 4-manifolds requires the manifold to be Spin. The dynamics of the twisted theory is governed by an action which can be obtained by twisting the action [22]. On an arbitrary Riemannian 4-manifold endowed with a metric g , the twisted action becomes  Z _ 4 pffiffiffi _ S ¼ d x g tr r r y  i    r _ _

F is the self-dual where  = ˙  and Fþ˙ ˙ =  ˙ ˙  part of F . Using these transformations, one easily 2 finds that Q is a gauge transformation. This is not unexpected since the N = 2 supersymmetric transformations [23] are in the Wess–Zumino gauge and they close only up to gauge transformations. This property implies that one must consider the equivariant cohomology of Q defined on the set of gaugeinvariant operators.

Topological Quantum Field Theory: Overview

The action [25] is Q-exact up to a topological term: Z 1 F^F ½27 S ¼ fQ; Vg  2

Using G we can now construct the map between the homology of X and the equivariant cohomology of Q. Let us consider the simple case SU(2). There exists only one independent Casimir and, correspondingly, only one basic operator:

where  i pffiffiffi _ _ V ¼ d4 x g tr _ _ ðF_  þ D_  Þ 4  1 1 _  ½; y  þ pffiffiffi _ r y 2 2 2

O ¼ trð2 Þ

Z

½28

Actually, it turns out that in all the theories obtained after twisting extended supersymmetry, the resulting actions are Q-exact up to topological terms. In the case of N = 2 theories, topological (theta) terms R F ^ F are generically not observable (due to a chiral anomaly), so it is customary to pick SDW ¼ fQ; Vg

½29

as the action of the theory, which immediately implies [2] and therefore the topological character of the theory. Notice, however, that [29] is stronger than [2]. As we described in the previous section, the observables of the theory can be constructed using the operator G in [20]. Its action on the twisted fields is easily obtained using [23]: 1 ½G ;  ¼ pffiffiffi  2 2 i ½G ; A  ¼ g  i 2pffiffiffi i 2 r ½G;  ¼  4  þ Dþ fG ;  g ¼ ðF  Þ ½G;  ¼ 0 3i ½G; Fþ  ¼ ir þ r pffiffiffi2 3i 2 fG; g ¼ 

r 8 3i 3i ½G; D ¼  r þ r 4 2 We now need to fix the basic operator (0) in [14]. The starting point must be a set of gauge-invariant, Q-closed operators which are not Q-trivial. Since [Q, ] = 0, these operators are the gauge-invariant polynomials in the field . For a simple gauge group of rank r the algebra of these polynomials is generated by r elements, and we shall denote this basis by On , n = 1, . . . , r. A simple choice for SU(N) consists of the following Casimirs: On ¼ trðnþ1 Þ;

n ¼ 1; . . . ; N

½31

½32

for which one finds the following set of descendants:   1 Oð1Þ ¼ tr pffiffiffi   dx 2  1 1 ð2Þ  O ¼  tr pffiffiffi ðF þ D Þ 2 2 ½33  1      dx ^ dx 4 .. . The map from the homology of X to the equivariant cohomology of Q can now be constructed very easily. Let i be an element of the homology group Hi (X). We associate to it the following observable: Z i ! Ii ði Þ ¼ OðiÞ ½34 i (i)

where O is given in [33]. The construction assures that Ii (i ) is invariant under Q and gauge transformations. Furthermore, it is also assured that Ii (i ) is not Q-exact. Let us consider the computation of correlation functions. The discussion will be presented for a generic gauge group. We will consider the topological theory defined by the Donaldson–Witten action SDW ¼ fQ; Vg

½30

283

½35

where V is defined in [28]. The property [35] has a very important consequence. The action SDW shows up in the correlation functions as exp(SDW =e2 ), where e is a free parameter which corresponds to the coupling constant of the N = 2 theory. Since the term involving the coupling constant is Q-exact, the correlation functions of Q-invariant operators are independent of e. Let us explain this in some detail. The (unnormalized) correlation functions of the theory are defined by Z 2 h1    n i ¼ D1    n eð1=e ÞSDW ½36 where 1 , . . . , n are invariant under Q transformations. Using the fact that SDW is Q-exact, one obtains @ 2 h1    n i ¼ 3 h1    n SDW i @e e 2 ¼ 3 hfQ; 1    n Vgi ¼ 0 e

½37

284 Topological Quantum Field Theory: Overview

where we have used the fact that Q is a symmetry of the theory, and therefore as in [3] the last functional integral gives zero. This result implies that one can compute these correlation functions in different limits of e. In the weak-coupling limit (semiclassical or saddle point approximation), one establishes the connection with Donaldson theory. In the strongcoupling limit, Seiberg–Witten invariants appear and one finds the connection between these two types of invariants. We will briefly explore the weakcoupling limit e ! 0. The functional integral [36] can be evaluated exactly in two steps: first one analyzes the zero modes or classical configurations that minimize the action, then one expands around them considering only quadratic fluctuations. The integration over these quadratic fluctuations involves ratios of determinants of kinetic operators that because of the Q-symmetry of the theory (which in fact is a Bose–Fermi symmetry) are 1. One is then left with an integral over the bosonic zero modes which leads to a finite-dimensional integral over the space of bosonic collective coordinates, and a finite Grassmannian integral over the zero modes of the fermionic fields. A careful analysis of the zero modes, first carried out by Witten, reveals that the infinite-dimensional functional integral is replaced by a finite-dimensional integral over the moduli space of anti-self-dual (ASD) connections MASD , þ that is, the space of connections satisfying F = 0. Therefore, the correlation functions [36] have the form h1    n i ¼

Z

We finish this section by pointing out that many features of the evaluation of the functional integral of the Donaldson–Witten theory developed here are common to most topological field theories of the Witten type. These features can be studied in the context of the Mathai–Quillen formalism which is the object of a separate article in the encyclopedia (see Mathai–Quillen Formalism).

Chern–Simons Gauge Theory for Knots and Links Chern–Simons gauge theory is the most important example of Schwarz-type TQFTs. Let us begin by introducing its basic elements. Chern–Simons gauge theory is a quantum field theory whose action is based on the Chern–Simons form associated to a nonabelian gauge group. The theory is defined by the following data: a smooth 3-manifold M which will be taken to be compact, a gauge group G which will be taken semisimple and compact, and an integer parameter k. The action of the theory is   Z k 2 SCS ðAÞ ¼ ½39 tr A ^ dA þ A ^ A ^ A 4 M 3 where A is a gauge connection and the trace is taken in the fundamental representation. The exponential of i times this action is invariant under gauge transformations, A ! A þ g1 dg

^1 ^    ^ ^n

½38

MASD

where the fields in 1    n are mapped to differential forms ^1    ^n on MASD – the degree of each form being given by the ghost number of its partner. Notice that the integral on the right-hand side vanishes unless the form has top degree. From the field-theoretical point of view, this is the requirement that the overall ghost number of the correlation function must be equal to dim MASD . The quantities on the right-hand side of [38] are – for gauge group SU(2) – precisely the Donaldson invariants. Thus, Witten’s work provided a new point of view on these invariants by reformulating them in a quantum field theory language. This is a very important contribution since quantum field theory is a very rich framework and a wide variety of methods can be used to analyze the correlation functions. This opened an entirely new strategy to investigate the Donaldson invariants. The emergence of Seiberg–Witten invariants is perhaps the greatest achievement of the implementation of this strategy.

½40

where g is a map g : M ! G. Notice that the action [39] is independent of the metric on the 3-manifold M. In this theory, appropriate observables lead to correlation functions which correspond to topological invariants. Candidates to be observables of this type must be metric independent and gauge invariant. Wilson loops satisfy these properties. They correspond to the holonomy of the gauge connection A along a loop. Given a representation R of the gauge group G and a 1-cycle  on M, it is defined as Z   WR ðAÞ ¼ trR Hol ðAÞ ¼ trR P exp A ½41 

Products of these operators are the natural candidates to obtain topological invariants after computing their correlation functions. These correlation functions are formally written as hWR11 WR22    WRnn i Z ¼ ½DAWR11 ðAÞWR22 ðAÞ    WRnn ðAÞeiSCS ðAÞ

½42

Topological Quantum Field Theory: Overview

where 1 , 2 , . . . , n are 1-cycles on M and R1 , R2 , and Rn are representations of G. In [42], the quantity [DA] denotes the functional integral measure and it is assumed that an integration over connections modulo gauge transformations is carried out. As usual in quantum field theory, this integration is not well defined. Field theorists have developed methods to assign a meaning to the righthand side of [42]. These methods mainly fall into two categories – perturbative and nonperturbative – and their degree of success mostly depends on the quantum field theory under consideration. For gauge theories, it is also possible to take an alternative approach, the large-N expansion, which in general provides further insights into the theory. In Chern– Simons gauge theory all these three methods have proved of great value. Witten (1989) showed, using nonperturbative methods, that when one considers nonintersecting cycles 1 , 2 , . . . , n without self-intersections, the correlation functions [42] lead to the polynomial invariants of knot theory discovered a few years earlier starting with the work of Jones (1985). Knot theory studies embeddings  : S1 ! M. Any two of such embeddings are considered equivalent if the image of one of them can be deformed into the image of the other by a homeomorphism on M. The main goal of knot theory is to classify the resulting equivalence classes. Each of these classes is a knot. Most of the work on knot theory has been carried out for the simple case M = S3 . Chern–Simons gauge theory, however, being a formulation intrinsically three dimensional, provides a framework to study the case of more general 3-manifolds M. A powerful approach to classify knots is based on the construction of knot invariants. These are quantities which can be computed for a representative of a class and are invariant within the class, that is, under continuous deformations of the chosen representative. At present, it is not known if there exist enough knot invariants to classify knots. Vassiliev invariants (Vassiliev 1990) are the most promising candidates, but it is already known that if they do provide such a classification, infinitely many of them are needed. The problem of the classification of knots in S3 can be reformulated in a two-dimensional framework using regular knot projections. Given a representative of a knot in S3, deform it continuously in such a way that the projection on a plane has simple crossings. Draw the projection on the plane, and at each crossing use the convention that the line that goes under the crossing is erased in a neighborhood of the crossing. The resulting diagram is a set of segments on the plane, containing the relevant

285

information at the crossings. The problem of classifying knots is equivalent to the problem of classifying knot projections modulo a series of relations among them. These relations are known as Reidemeister moves. Invariance of a quantity under the three Reidemeister moves is called invariance under ambient isotopy. If a quantity is invariant under all but the first move, it is said to possess invariance under regular isotopy. The formalism described for knots generalizes to the case of links. For a link of n components, one considers n embeddings, i : S1 ! M (i = 1, . . . , n), with no intersections among them. Again, the main problem that link theory faces is the problem of their classification modulo homeomorphisms on M. In this case one can also define regular projections and reformulate the problem in terms of their classification modulo the Reidemeister moves. The study of knot and link invariants experimented important progress in the 1980s. Jones (1985) discovered a new invariant which carries his name. The Jones polynomial can be defined very simply in terms of skein relations. These are a set of rules that can be applied to the diagram of a regular knot projection to construct the polynomial invariant. They establish a relation between the invariants associated to three links which only differ in a region as shown in Figure 1 where arrows have been introduced to take into account that the Jones polynomial is defined for oriented links. If one denotes by VL (t) the Jones polynomial corresponding to a link L, t being the argument of the polynomial, it must satisfy the skein relation:   pffiffi 1 1 VLþ  tVL ¼ t  pffiffi VL0 ½43 t t where Lþ , L , and L0 are the links shown in Figure 1. This relation plus a choice of normalization for the unknot (U) are enough to compute the Jones polynomial for any link. The standard choice for the unknot is VU ¼ 1

½44

though it is not the most natural one from the point of view of Chern–Simons gauge theory. After Jones

L+ Figure 1 Skein relations.

L–

L0

286 Topological Quantum Field Theory: Overview

work in 1984, many other polynomial invariants were discovered, as the HOMFLY and the Kauffman polynomial invariants. The pioneering work of Witten in 1988 showed that the correlation functions of products of Wilson loops [42] correspond to the Jones polynomial when one considers SU(2) as gauge group and all the Wilson loops entering in the correlation function are taken in the fundamental representation F. For example, if one considers a knot K, Witten showed that   VK ðtÞ ¼ WKF ½45 provided that one performs the identification   2i t ¼ exp kþh

½46

where h = 2 is the dual Coxeter number of the gauge group SU(2). Witten also showed that if instead of SU(2) one considers SU(N) and the Wilson loop carries the fundamental representation, the resulting invariant is the HOMFLY polynomial. The second variable of this polynomial originates in this context from the N dependence. However, these cases are just a sample of the general framework intrinsic to Chern–Simons gauge theory. Taking other groups and other representations, one possesses an enormous set of knot and link invariants. These invariants can also be obtained in the context of quantum groups. Many nonperturbative studies of Chern–Simons gauge theory have been carried out. The quantization of the theory has been studied from the point of view of the operator formalism as well as other more geometrical methods. Also, its connection to two-dimensional conformal field theory has been further elucidated, and a powerful method for the general computation of knot and link invariants has been developed by Kaul and collaborators. Chern–Simons theory is also amenable to perturbative analysis, which has provided important representations of the Vassiliev invariants. These invariants, proposed by Vassiliev in 1990, turned out to be the coefficients of the perturbative series expansion of the correlators of Chern–Simons gauge theory. Perturbative studies can be carried out in different gauges, originating a variety of new representations of Vassiliev invariants. Among the most relevant results related to these topics are the integral expressions for Vassiliev invariants by Kontsevich and by Bott and Taubes, as well as the recent combinatorial ones. These developments are not described here but the interested reader is referred to the recent review (Labastida 1999).

Advanced Developments Topological sigma models are another important type of (Witten-type) TQFTs. These theories are obtained after twisting 2D N = 2 supersymmetric sigma models. The twisting can be done in two different ways leading to two types of models, A and B. Their existence is related to mirror symmetry. Only type-A models will be described in what follows. These models can be defined on an arbitrary almost-complex manifold, though typically they are considered on Ka¨hler manifolds. The theory involves maps from two-dimensional Riemann surfaces  to target spaces X, together with fermionic degrees of freedom on  which are mapped to tangent vectors on X. The functional integral of the resulting theory is localized on holomorphic maps, defining the corresponding moduli space. The corresponding Q-cohomology provides the set of physical observables, which can be mapped to cohomology classes on the moduli space and integrated to produce topological invariants. Topological sigma models keep fixed the complex structure of the Riemann surface . Motivated by string theory, one also considers the situation in which one integrates over complex structures. In this case, one ends up working with holomorphic maps in the entire moduli space of curves. The resulting theories are called topological strings. We will review now a particular example of topological string theory which, besides being very interesting from the point of view of physics and mathematics, will be very useful in establishing a relation with Chern–Simons gauge theory. Let us consider topological strings with target manifold X a Calabi–Yau 3-fold. In this case, the virtual dimension of the moduli space of holomorphic maps turns out to be zero. Two situations can occur: either the space is given by a number of points (the real dimension is zero) or the moduli space is finite dimensional and possesses a bundle of the same dimension as the tangent bundle. In the first case, topological strings count the number of points weighted by the exponential of the area of the holomorphic map (the pullback of the Ka¨hler form integrated over the surface) times x2g2 , where x is the string-coupling constant and g is the genus of . In the second case, one computes the top Chern class of the appropriate bundles (properly defined), again weighted by the same factor. In both cases one can classify the contributions according to the cohomology class  on X in which the image of the holomorphic map is contained. The sum of the numbers obtained for each  and fixed g are known as Gromov–Witten

Topological Quantum Field Theory: Overview

invariants, Ng . The topological string contribution takes the form 0 1 R X X ! ½47 x2g2 @ Ng e  A g0

2H2 ðX;ZÞ

where ! is the Ka¨hler class of the Calabi–Yau manifold. In general, the quantities Ng are rational numbers. The precedent discussion has shown how Gromov– Witten invariants can be interpreted in terms of string theory. One could think that this is just a fancy observation and that no further insight on these invariants can be gained from this formulation. The situation turns out to be quite the opposite. Once a string formulation has been obtained, the whole machinery of string theory is at our disposal. One should look to new ways to compute the quantity [47], where Gromov– Witten invariants are packed. The hope is that, if this is possible, the new emerging picture will provide new insights on these invariants. This is indeed what occurred recently. It turns out that the quantity [47] can be obtained from an alternative point of view in which the embedded Riemann surfaces are regarded as D-branes. The outcome of this approach is that the Gromov–Witten invariants can be written in terms of other invariants which are integers and that possess a geometrical interpretation. To be more specific, the quantity [47] takes the form  2g2 R X X 1 dx d ! 2 sin ng e  ½48 d 2 g0 d>0 2H2 ðX;ZÞ

where ng are the new ‘‘integer’’ invariants. This prediction has been verified in all the cases in which it has been tested. A similar structure will be found in the next section in the context of knot theory in the large-N limit. Let us now consider also Donaldson–Witten theory from a new perspective. To be more specific, let us consider the case in which the gauge group is SU(2), and the 4-manifold X is simply connected and has þ bþ 2 > 1 (the case b2 = 1 is anomalous). In this situation there are 1 þ b2 physical observables [34], O = I1 and I(a ) = I2 (a ) (a = 1, . . . , b2 ), where a is a basis of H2 (X). These can be packed in a generating functional: * !+ X exp a Iða Þ þ O ½49 a

where and a (a = 1, . . . , b2 ) are parameters. In computing this quantity one can argue that the contribution is localized on the moduli space of instantons configurations and one ends up, after taking into account the selection rule dictated by the dimensionality of the moduli space, with integrations

287

over the moduli space of the selected forms. The resulting quantities are Donaldson invariants. As in the case of topological sigma models one could be tempted to argue that the observation leading to a field-theoretical interpretation of Donaldson invariants does not provide any new insight. Quite on the contrary, once a field theory formulation is available, one has at his disposal a huge machinery which could lead, on the one hand, to further generalizations of the theory and, on the other hand, to new ways to compute quantities such as [49], obtaining new insights on these invariants. This is indeed what happened in the 1990s, leading to an important breakthrough in 1994 when Seiberg and Witten calculated [49] in a different way and pointed out the relation of Donaldson invariants to new integer invariants that nowadays bear their names. The localization argument that led to the interpretation of [49] as Donaldson invariants is valid because the theory under consideration is exact in the weakcoupling limit. Actually, the topological theory under consideration is independent of the coupling constant and thus calculations in the strong-coupling limit are also exact. These types of calculations were out of reach before 1994. The situation changed dramatically after the work of Seiberg and Witten in which N = 2 super Yang–Mills theory was solved in the strongcoupling limit. Its application to the corresponding twisted version was immediate and it turned out that Donaldson invariants can be written in terms of new integer invariants now known as Seiberg–Witten invariants (Witten 1994). The development has a strong resemblance with the one described above for topological strings: certain noninteger invariants can be expressed in terms of new integer invariants. The Seiberg–Witten invariants are actually simpler to compute than Donaldson invariants. They correspond to partition functions of topological Yang–Mills theories where the gauge group is abelian. These contributions can be grouped into classes labeled by x = 2c1 (L), where c1 (L) is the first Chern class of the corresponding line bundle. The sum of contributions, each being 1, for a given class x is the integer Seiberg–Witten invariant nx . The strong-coupling analysis of topological Yang–Mills theory leads to the following expression for [49]: 2 X 21þð1=4Þð7þ11 Þ eððv =2Þþ2 Þ nx evx þi

þ =4

e

ððv2 =2Þ2 Þ

X x

x

nx eivx



½50

P where v = a a a , and  and are the Euler number and the signature of the manifold X. This result matches the known structure of [49] (structure theorem of Kronheimer and Mrowka) and provides

288 Topological Quantum Field Theory: Overview

a meaning to its unknown quantities in terms of the new Seiberg–Witten invariants. Equation [50] is a rather remarkable prediction that has been tested in many cases, and for which a general proof has been recently proposed. For a review of the subject, see Labastida and Lozano (1998). The situation for manifolds with bþ 2 = 1 involves a metric dependence and has been worked out in detail (Moore and Witten 1998). The formulation of Donaldson invariants in field-theoretical terms has also provided a generalization of these invariants. This generalization has been carried out in several directions: (1) the consideration of higher-rank groups, (2) the coupling to matter fields after twisting N = 2 hypermultiplets, and (3) the twist of theories involving N = 4 supersymmetry. We will now look at Chern–Simons gauge theory from the perspective that emerges from its treatment in the context of the large-N expansion. We will restrict the discussion to the case of knots on S3 with gauge group SU(N). Gauge theories with gauge group SU(N) admit, besides the perturbative expansion, a large-N expansion. In this expansion correlators are expanded in powers of 1/N while keeping the ’t Hooft coupling t = Nx fixed, x being the coupling constant of the gauge theory. For example, for the free energy of the theory, one has the general form F¼

1 X

Cg;h N22g t2g2þh

½51

g0 h1

FðVÞ ¼ log ZðVÞ ¼

In the case of Chern–Simons gauge theory, the coupling constant is x = 2i=(k þ N) after taking into account the shift in k. The large-N expansion [51] resembles a string-theory expansion and indeed the quantities Cg, h can be identified with the partition function of a topological open string with g handles and h boundaries, with N D-branes on S3 in an ambient sixdimensional target space T S3 . This was pointed out by Witten in 1992. The result makes a connection between a topological three-dimensional field theory and the topological strings described in the previous section. In 1998 an important breakthrough took place which provided a new approach to compute quantities such as [51]. Using arguments inspired by the AdS/CFT correspondence, Gopakumar and Vafa (1999) provided a closed-string-theory interpretation of the partition function [51]. They conjectured that the free energy F can be expressed as F¼

O(1) O(1) ! P 1 , t being the flux of the B-field through P 1 . The quantities Fg (t) have been computed using both physical and mathematical arguments, thus proving the conjecture. Once a new picture for the partition function of Chern–Simons gauge theory is available, one should ask about the form that the expectation values of Wilson loops could take in the new context. The question was faced by Ooguri and Vafa and they provided the answer, later refined by Labastida, Marin˜o, and Vafa. The outcome is an entirely new point of view in the theory of knot and link invariants. The new picture provides a geometrical interpretation of the integer coefficients of the quantum group invariants, an issue that has been investigated during many years. To present an account of these developments, one needs to review first some basic facts of large-N expansions. To consider the presence of Wilson loops, it is convenient to introduce a particular generating functional. First, one performs a change of basis from representations R to conjugacy classes C(k) of the symmetric group, labeled byP vectors k = (k1 , k2 , . . . ) with ki  0, and Pjkj = j kj > 0. The change of basis is Wk = R R (C(k))WR , where R P are characters of the permutation group S‘ of ‘ = j jkj elements (‘ is also the number of boxes of the Young tableau associated to R). Second, one introduces the generating functional:

1 X

N 22g Fg ðtÞ

½52

g0

where Fg (t) corresponds to the partition function of a topological closed-string theory on the noncompact Calabi–Yau manifold X called the resolved conifold,

X jCðkÞj ‘!

k

ðcÞ

Wk k ðVÞ

½53

where ZðVÞ ¼

X jCðkÞj k

k ðVÞ ¼

Y

‘!

Wk k ðVÞ

ðtr V j Þkj

j

In these expressions jC(k)j denotes the number of elements of the class C(k) in S‘ . The reason behind the introduction of this generating functional is that the large-N structure of the connected Wilson loops, Wk(c) , turns out to be very simple: 1 jCðkÞj ðcÞ X Wk ¼ x2g2þjkj Fg; k ð Þ ‘! g¼0

½54

where = et and t = Nx is the ’t Hooft coupling. Writing x = t=N, it corresponds to a power series expansion in 1/N. As before, the expansion looks like a perturbative series in string theory where g is the genus and jkj is the number of holes. Ooguri and Vafa conjectured in 1999 the appropriate stringtheory description of [54]. It corresponds to an open topological string theory (notice that the ones

Topological Quantum Field Theory: Overview

described in the previous section were closed), whose target space is the resolved conifold X. The contribution from this theory will lead to openstring analogs of Gromov–Witten invariants. In order to describe in more detail the fact that one is dealing with open strings, some new data need to be introduced. Here is where the knot description intrinsic to the Wilson loop enters. Given a knot K on S3 , let us associate to it a Lagrangian submanifold CK with b1 = 1 in the resolved conifold X and consider a topological open string on it. The contributions in this open topological string are localized on holomorphic maps f : g, h ! X with h = jkj which satisfy f [g, h ] = Q, and f [C] = j[] for kj oriented circles C. In these expressions  2 H1 (CK , Z), and Q 2 H2 (X, CK , Z), that is, the map is such that kj boundaries of g, h wrap the knot j times, and g, h itself gets mapped to a relative two-homology class characterized by the Lagrangian submanifold CK . The number of such maps (in the sense described in the previous section) is the open-string analog of Gromov–Witten invariants. They will be denoted by Q Ng, k . Comparing to the situation that led to [47] in the closed-string case, one concludes that in this case the quantities Fg, k ( ) in [54] must take the form Z R X ! Q Q ; Fg; k ð Þ ¼ Ng; e t ¼ ! ½55 k Q

P1

where ! is the Ka¨hler class of the Calabi–Yau t manifold R X and = e . For any Q, one can always write Q ! = Qt, where Q is in general a half-integer number. Therefore, Fg, k ( ) is a polynomial in 1=2 with rational coefficients. The result [55] is very impressive but still does not provide a representation where one can assign a geometrical interpretation to the integer coefficients of the quantum-group invariants. Notice that to match a polynomial invariant to [55], after obtaining its connected part, one must expand it in x after setting q = ex keeping fixed. One would like to have a refined version of [55], in the spirit of what was described in the previous section leading from the Gromov–Witten invariants Ng of [47] to the new integer invariants ng of [48]. It turns out that, indeed, F(V) can be expressed in terms of integer invariants in complete analogy with the description presented in the previous section for topological strings. A good review on the subject can be found in Marin˜o (2005).

Concluding Remarks In this overview we have introduced key features of TQFTs and we have described some of the most relevant results emerged from them. We have

289

described how the many faces of TQFT provide a variety of important insights in a selected set of problems in topology. Among these outstand the reformulation of Donaldson theory and the discovery of the Seiberg–Witten invariants, and the string-theory description of the large-N expansion of Chern–Simons gauge theory, which provides an entirely new point of view in the study of knot and link invariants and points to an underlying fascinating interplay between string theory, knot theory, and enumerative geometry which opens new fields of study. In addition to their intrinsic mathematical interest, TQFTs have been found relevant to important questions in physics as well. This is so because, in a sense, TQFTs are easier to solve than conventional quantum field theories. For example, topological sigma models are relevant to the computation of certain couplings in string theory. Also, Witten-type gauge TQFTs such as Donaldson–Witten theories and its generalizations play a role in string theory as effective world-volume theories of extended string states (branes) wrapping curved spaces, and TQFTs arising from N = 4 gauge theories in four dimensions have shed light on field- (and string-) theory dualities. Most of these developments, and others that we have not touched upon or only mentioned in passing have their own entries in the encyclopedia, to which we refer the interested reader for further details. See also: Axiomatic Approach to Topological Quantum Field Theory; BF Theories; Chern–Simons Models: Rigorous Results; Donaldson–Witten Theory; Gauge Theoretic Invariants of 4-Manifolds; Gauge Theory: Mathematical Applications; Hamiltonian Fluid Dynamics; The Jones Polynomial; Knot Theory and Physics; Mathai–Quillen Formalism; Mathematical Knot Theory; Schwarz-Type Topological Quantum Field Theory; Seiberg–Witten Theory; Stationary Phase Approximation; Topological Sigma Models.

Further Reading Atiyah MF (1988) New invariants of three and four dimensional manifolds. In: The Mathematical Heritage of Herman Weyl, Proc. Symp. Pure Math., vol. 48. American Math. Soc. pp. 285–299. Birmingham D, Blau M, Rakowski M, and Thompson G (1991) Topological field theory. Physics Reports 209: 129–340. Cordes S, Moore G, and Rangoolam S (1996) Lectures on 2D Yang–Mills theory, equivariant cohomology and topological field theories. In: David F, Ginsparg P, and Zinn-Justin J (eds.) Fluctuating Geometries in Statistical Mechanics and Field Theory, Les Houches Sesion LXII, p. 505 (hep-th/9411210). Elsevier. Donaldson SK (1990) Polynomial invariants for smooth fourmanifolds. Topology 29: 257–315. Floer A (1987) Morse theory for fixed points of symplectic diffeomorphisms. Bulletin of the American Mathematical Society 16: 279.

290 Topological Sigma Models Freyd P, Yetter D, Hoste J, Lickorish WBR, Millet K, and Ocneanu A (1985) A new polynomial invariant of knots and links. Bulletin of the American Mathematical Society 12(2): 239–246. Gopakumar R and Vafa C (1999) On the gauge theory/geometry correspondence. Advances in Theoretical and Mathematical Physics 3: 1415 (hep-th/9811131). Jones VFR (1985) A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society 12: 103–112. Jones VFR (1987) Hecke algebra representations of braid groups and link polynomials. Annals of Mathematics 126(2): 335–388. Kauffman LH (1990) An invariant of regular isotopy. Transactions of American Mathematical Society 318(2): 417–471. Labastida JMF (1999) Chern-Simons gauge theory: ten years after. In: Falomir H, Gamboa R, and Schaposnik F (eds.) Trends in Theoretical Physics II, ch. 484, 1. New York: AIP (hep-th/9905057). Labastida JMF and Lozano C (1998) Lectures in topological quantum field theory. In: Falomir H, Gamboa R, and Schaposnik F (eds.) Trends in Theoretical Physics, ch. 419, 54. New York: AIP (hep-th/9709192).

Labastida JMF and Marin˜o M (2005) Topological Quantum Field Theory and Four Manifolds. Dordrecht: Elsevier; Norwell, MA: Springer. Marin˜o M (2005) Chern–Simons theory and topological strings. Reviews of Modern Physics 77: 675–720. Moore G and Witten E (1998) Integrating over the u-plane in Donaldson theory. Advances in Theoretical and Mathematic Physics 1: 298–387. Vassiliev VA (1990) Cohomology of knot spaces. In: Theory of Singularities and Its Applications, Advances in Soviet Mathematics, vol. 1, pp. 23–69. American Mathematical Society. Witten E (1982) Supersymmetry and Morse theory. Journal of Differential Geometry 17: 661–692. Witten E (1988a) Topological quantum field theory. Communications in Mathematical Physics 117: 353. Witten E (1988b) Topological sigma models. Communications in Mathematical Physics 118: 411. Witten E (1989) Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121: 351. Witten E (1994) Monopoles and four-manifolds. Mathematical Research Letters 1: 769–796.

Topological Sigma Models D Birmingham, University of the Pacific, Stockton, CA, USA ª 2006 Elsevier Ltd. All rights reserved.

Introduction Topological sigma models govern the quantum mechanics of maps from a Riemann surface  to a target space M. In contrast to the standard supersymmetric sigma model, the topological version has a special local shift symmetry. This symmetry takes the form ui = i , where i is an arbitrary local function of the coordinates on the base manifold . In essence, this topological shift symmetry ensures that all local degrees of freedom of the model can be gauged away. As a result, the dynamics of such a model resides in a finite number of global topological degrees of freedom. This feature is generic to all topological field theories of Witten type, also known as cohomological field theories (see Topological Quantum Field Theory: Overview). The topological shift symmetry is responsible for the special topological nature of the model, which is seen most readily by BRST quantizing the local shift symmetry. This gives rise to a nilpotent BRST operator Q. The properties of this BRST operator are crucial for establishing the topological nature of the model. The key point in the construction of any cohomological field theory is the fact that the full quantum action Sq can be written as a BRST commutator Sq = {Q, V}, where V is a function of the

fields needed to define the path integral. In particular, one can show that the partition function and all correlation functions are independent of the metric on both the base manifold  and the target space M. For example, let us define the path integral by Z ½1 Z ¼ d efQ;Vg where  denotes the full set of fields required at the quantum level. In general, the function V depends on geometric data of both  and M. Nevertheless, one can easily establish that the partition function is independent of this data by noting the following. Variation of Z with respect to the metric of the target space g (for example) gives Z g Z ¼  d efQ;Vg fQ; g Vg ½2 The right-hand side of this equation is nothing but the vacuum expectation value of a BRST commutator, and this vanishes by BRST invariance of the vacuum. It is important to note here that the BRST operator Q can be constructed to be independent of g. Apart from the necessity of introducing the metric tensor, these models also require additional geometric data for their construction. The complex structure of , and at least an almost-complex structure on M, is required. By a similar argument, one can show that the partition function and correlation functions are independent of this extra