on string theory and axionic strings & instantons - SLAC - Stanford

I

XX-PUB-5659 September. 0’)

ON STRING

THEORY AND AXIONIC INSTANTONS*

STRINGS

1991

&

Soo- Jong Rey + Stanford Linear Stanford University,

Accelerator Stanford,

Center CA 94309

ABSTRACT Progress in understanding -

dimensional axionic Superstring

exact superconformal

instantons and axionic strings is reported.

theory is the first promising

haps unified with strong and electroweak standing

is largely based on weak coupling

candidate of quantum

interactions.

physics of superstring

tive understandings:

to name. a few, dynamical

Therefore,

just

constant

problem

in this talk,

topics in superstring tons. Recently,

and spacetime

analysis. On the other

seem to require nonperturbasupersymmetry

singularity

theory to the above direction:

considerable

suitably

under-

breaking,

are such examples.

I will discuss what I find one of the most ‘interesting stringy

progress in nonperturbative

strings has been made. Here, I will restrict dimensions,

gravity, per-

Still, our current

perturbative

hand, most of interesting cosmological

field theories of four-

compactified

solitons and instanaspects of noncritical

myself to critical

to four dimensions

superstrings

in ten-

keeping N = 1 spacetime

supersymmetry. At scales well below that may be described

of compactification,

by a low-energy

supergravity

massless string

theory. It is observed that, in

all known string theories, there arises ubiquitously tiplet:

graviton,

excitations .- _ -.* Work supported

dilaton

and Kalb-Ramond

depends largely

a massless gravity

(axion)

in part by the Department

of Energy, contract

supermul-

fields. The other massless

on the way a specific compactification

7perm&ent address: Center for Theoretical Haven CT 06511 USA

Presented Vancouver,

excitations

scheme is

DE-ACO3-76SF00515

Physics, Sloane Laboratory,

at the Particle and BC, Canada, August

Fields 18-22,

Yale University,

‘91, 1991

New

--

chosen. Since I would like to understand superstrings,

..

I will only keep the gravity

intrinsic multiplet

nonperturbative

aspects of

in the low-energy

whose dy-

namics is described by an effective acti0n.l S eff = ~3!~xdi?[ -R( Q) + (higher J Here, Q2, denotes a general affine connection

orders)].

(1)

with nonzero dilaton

and torsion

Cl,iab = L@ + $=Vbl@ + Hpab,

1.

Axionic

Instanton

After Wick rotation

to Euclidean

tions to self-dual Einstein

spacetime, we find axionic instantons

as solu-

equation2

.- - _

7yyb(Q)

= uz,““b(

This turns out to be equivalent to d@ = f”H solution G,,(x)

i-2). and R,,

(2) = 0. We find N-instanton

as = S,,,

IQ01 - xcJ2'

e-‘($) = 5 .,127+

which is stabilized

N &a H,,,A=~T~pvA~(x

by a conserved topological

An exact superconformal be N = 1 supersymmetric

- xa)D,

(x - x,)4

(3)

charge Q = & &,‘aEH = Z.

field theory of the axionic instanton

5214(2) x u;( 1) = u(2) WZNW

Q denotes level of su(2) WZNW theory. The self-duality

a=1

is found’ to

model of ir = 4. Here,

model and 5 a background

charge of u(l)

equation of Eq.(2) b ecomes an algebraic self-duality 7=*&i-

In this case, the supersymmetric

u(2) WZNW

(4) model can be written

Sw=~{JtT(~~t~~)+~ Jtr(E’dx)3}+i J&(2) ln(det where C = exp($X”

$ ia - X)

-possesses a larger symmetry:

as3

C+det Ct)+(ferm;ons)

(5) E U(2). It can be shown”y3 that the instanton

doubly extended N = 4 SUQ-~(~)

2

x

sur(2)

x

ut( 1)

superconformal u(2) currents,

invariance.

Denoting

we find the generators T = - -& K”K” Go=

K” E $8X0, K s J $ i@ A g out of the of doubly extended

N = 4 SCA to be

+ K . K) - (IE”d%I!” •t %‘q)

•t +K”

-(~“K”+JI.K-~.~r\8)+ld~” & 7

(6)

-

u = &x0. If the background

N = 4 SCA with

bly extended central

charge is c = B

i contributes duality

is satisfied,

symmetry

central the total

axionic instantons

gauge connection

tended (4,4)

KM subalgebra

worldsheet

guarantees

is identified

a difficult

charge $. central

SUQ-i(2)

the original5

x sui(2)

U( 1) background

Therefore,

with

The charge

if the algebraic

self-

charge adds up to 6. Wonderfully,

spin connection

u(1) alstring in

possess doubly

ex-

-1,3 This hidden N = 4 superconformal

a nonrenormalization

of the axionic

expansions. Patching

sectors into a modular

dou-

x u(l).

in type II string and in heterotic

supersymmetry.

all orders in worldsheet perturbation tiholomorphic

produces

N = 4 SCA remains to hold even after we twist

extended

gebra! Therefore,

off, Eq.(6)

= 6 - $. In our model,

an additional

Eq.(4)

the doubly which

charge $ is dropped

invariant

step to analyze even though

partition

significant

instanton

solution

holomorphic function

to

and an-

seems to be

progress has been achieved

recently.sv7 In the axionic

instanton

background,

it was shown2 that

there exist only

two chiral fermion

zero modes (as opposed, e.g., to four nonchiral

in Eguchi-Hanson

instantons)

1 expansions.

to the lowest order in worldsheet

The exact SCFT of axionic instanton

orders as well. The spacetime supersymmetry holomorphic

(1,0) primary

fields. Denoting

3

zero modes perturbation

enables us to prove this to all

is generated by contour integrals of zero modes of N = 4 supercurrents

I -

asG~~JT~,G,~$T~,N=l if they annihilate

..

{G;,Gb,}l0~~=

sp acetime supersymmetry

the Ramond

vacuum

SQb(Lo- &)\O>R=

that G0 annihilate

Since the Ramond

a background

only

it is easy to show that

0, in which contribution

model cancels that from u(1) with imposing

10 >R. First,

is guaranteed

from su(2) WZNW

charge. On the other hand,

(O>R, we have

ground state is GSO projected,

I’ii(O >R=

$10 >R, Eq.(7)

implies that (1 f I’,) 10>R= 0. Therefore, there arises only two chiral fermionic zero modes if and only if the algebraic self-duality this is yet another proof of the aforementioned also entails a very significant metry .

breaking’

low-energy

y = kIJ&

is satisfied. Indeed,

nonrenormalization

implications2

theorem. It

to dynamical

since only when there exist precisely

supersym-

two chiral fermion

zero

modes, gravitino

can acquire nonzero mass, thus breaks local supersymmetry.

2.

Strings

Axionic

It is also possible to find a superconformal that

the axionic

instanton

was a source of quantized

field. That is why compact, instanton.

Furthermore,

rectly interpreted

field theory of axionic strings. Recall

sum

WZNW

integer-valued

model was relevant for the axionic

level Q of SUQ(~) WZNW

model was corfield. Funda-

mental string is an axionic string, thus a source of Kalb-Ramond

electric field.

we expect a relevant

model with metric

signature

ous value, and interpreted exact SCFTs. radial direction

(+ + -).

to be noncompact,

1) x u(l)

and au-~(1,

1) x u(l)i

WZNW

models as

modei twisted

by 4 along

and a compact U( 1) model, while the second a su+( u( 1) model with a background

signature

scribes correct dilaton

(-

+ ++)

and (- - ++)

and Kalb-Ramond

order solution ‘” but aparently

1) WZNW

electric charge. Indeed, for axionic

1,l) WZNW

charge $ which may be

as a wrong sign u(1) model with a background

spacetime

su+(l,

In this case, the level Q takes a continu-

The first consists of a su(1, 1) WZNW

model and a Feign-Fuks -- - have

SCFT

as a Kalb-Ramond

string, we find3vg ~%-~,+(l,

interpreted

magnetic

magnetic

charge of Kalb-Ramond

Therefore,

as quantized

Kalb-Ramond

charge t. Thus, they

respectively.

field backgrounds

The first3 de-

of a known lowest

different global geometry. The second is intimately

4

-

related to the axionic instanton and u+(l)

--+ u:(l)

The total

central

critical

through tra.nsformations”:

suck

3 su+(l,

1)

(The reader is warned that this is not equivalent to ~(1, 1)). charge is ?tot = (3 + 4/Q)

six if an algebraic self-duality

+ (1 - 4/y2).

condition

7 = &fl

This equals to the is satisfied.

In fact,

the doubly extended N = 4 algebra Eq.(5) remains to hold in the axionic string as well, and the axionic strings are spacetime supersymmetric worldsheet

perturbation

is that the manifold . tion with nontrivial

expansions!

Furthermore,

is a solution of (+ $ --)

a generalized connection,

to all orders in the

its spacetime interpretation

signature self-dual Einstein

Therefore,

ths axionic string

constitutes

soliton solution of N = 2 gauged string theoryl’ at critical

My motivations

of studying

(1) Recently,

black hole in critical

was shown to be described by su(l, 1)/u(l)

dimension!

connected to one-dimensional

N = 1 supersymmetry

bosonic string

gauged WZNW

can generalize it to an N = 1 supersymmetric by a manifestly

a

the axionic string are two-fold.

two-dimensional

pears to be intimately

equa-

theory

model,12 which ap-

noncritical

strings.

One

black hole, which is described

su(1, 1)/u(l)

gauged WZNW

model.

If-one choose Landau gauge, the black hole SCFT consists of su(1, 1) WZNW model, negative metric free field theory and (b, c) ghosts. They contribute tral charges ir = (3 + 4/Q), 1 and -2 a SCFT of su-o( 1,l) Furthermore,

x u(1) WZNW

respectively.

symmetry

The first two combines to

model with central charge 2 = 4 + 4/Q.

it can be shown that the WZNW

N = 4 superconformal

model possess a doubly extended

as well! This is almost identical

SCFTs of axionic strings. The N = 4 superconformal related to Harmonic

cen-

superspace and twistor

to the above

theories being intimately

theory, the above unexpected

con-

nections between the 2-D black hole and the axionic strings might shed light on uncovering

hidden VV, symmetry

(2) Even though conceptually gate topological

in c = 1 quantum

gravity.

simple, I found it technically

sectors of N = 1 superstrings.

hard to investi-

The gauged N = 2 string theory

is very simple since string field consists only of one scalar field, the Ktihler potential, yet rich enough to allow soliton sectors. Therefore, technical

questions such as collective

coordinates,

I do expect that many

moduli

space, definitions

of

aspects of string theory are -_-. topological charge etc. and other nonperturbative easier- to investigate and to understand in gauged N = 2 string theory. The author is grateful

to C. Callan, J.A. Harvey, G.T. Horowitz,

5

E. Mar-

tinec, A. Stromingeer

and C. Vafa for useful conversations

This work was supported

in part

by the U.S. DOE

on related subjects.

under

Grant

DE-AC-OS-

76SF00515.

References 1. J. Scherk and J.H. Schwarz, Phys. Lett. 52B

(1974) 347.

2. S.-J. Rey, Phys. Rev. D41 (1991); A xionic String Instantons Energy

Implications,

and L. Clavelli,

in ‘Particle

World

Theory

Scientific

tons, SLAC preprint 4. C.G. Callan,

(August,

and Superstrings’,

Pub. (November,

3. S.-J. Rey, Ezact Superconformal

eds. B. Harms

1989).

Field Theory of Azionic

Strings and Instan-

1991).

J.A. Harvey and A. Strominger,

erotic Instantons

and Their Low-

and Solitons,

EFI-91-03,

Worldsheet

PUPT-1233

Approach

preprint

5. A. Sevrin, W. Troost and A. van Proeyen,

Phys. Lett. 208B

6. J.L.

Properties

Petersen

and -A. Taormina,

N = 4 Superconformal Models’,

EFI-90-61

7. H. Ooguri,

Algebras

preprint

Modular and Their

Connection

to Het-

(1991).

(1988) 447.

of Doubly

Extended

to Rational

Torus

(1990).

S.-J. Rey and C. Vafa, unpublished

8. J. Louis, Status of Supersymmetry

Breaking

work (1991). in String

Theory, in this Pro-

ceeding and references therein. 9. S.-J. Rey, unpublished N = 2 String

(March,

Self-Dual (August,

Cosmic Strings 1991).

HUTP-91/01

preprint

(1990)

1389: Geometry

(1991).

On String Theory and Black Holes, IASSNS-HEP-91/12

1991).

in Gauged

and J.A. Harvey, Phys. Rev. Lett. 63 (1989) 719.

and C. Vafa, Mod. Phys. Lett. A5

N = 2 Strings, 12. E. Witten,

(1990);

Theory, SLAC preprint

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work

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