Correlation functions in super liouville theory

(1)

VOLUME 68, NUMBER 11

PH

YSICAL REVI

EW

LETTERS

Correlation Functions

in

Super

Liouville

Theory

16MARCH 1992

E.

Abdalla, ' M.

C. B.

Abdalla, D. Dalmazi, and Koji Harada ' '"' "'Instituto deFisica, Unit ersidade deSaoPaulo, CP 20516,SaoPaulo, Brazil -'Instituto de Fisica Teorica, UNESP, Rua Pamplona 145,CEP01405,SaoPaulo, Brazil

(Received 27August 1991)

We calculate three- and four-point functions in super Liouville theory coupled toa super Coulomb gas

on world sheets with spherical topology. We first integrate over the zero mode and assume that a pa-rameter takes an integer value. We find the amplitudes, give plausibility arguments in favor ofthe

re-sult, and formally continue the parameter to an arbitrary real number. Remarkably the result is

com-pletely parallel to the bosonic case.

PACS numbers: 1I.17.+y,04.60.+n, 1l.3G.Pb

measure, the total action is given by

S

=SSL+S~,

The matrix model definition

of

2D gravity is proving to

be very powerful in calculating correlation functions

[1],

although it seems difficult to generalize the results to su-persymmetric theories. On the other hand, in the

contin-uum approach (Liouville theory)

[2-5]

it is difficult to calculate correlation functions, while its supersymmetric generalization (super Liouville theory)

[6-8]

is well known. Recently, however, several authors

[9-13]

have

exactly calculated correlation functions in the continuum

approach to conformal matter fields coupled to 2D

gravi-ty (see also Ref.

[141).

They have used a technique based

on the integration over the Liouville zero mode, and their results agree with those obtained earlier in the discrete approach (matrix models). It is thus very urgent to ex-tend the continuum method

[15]

to the supersymmetric

case, i.e., superconformal matter fields coupled to 2D su-pergravity.

The aim

of

this Letter is to calculate the three- and

four-point functions in super Liouville theory coupled to superconformal matter with the centra1 charge ~& 1, rep-resented as a super Coulomb gas

[16].

Our approach is

close to that of Di Francesco and Kutasov

[10].

The

re-sult is remarkable and is very parallel to the bosonic case; it amounts to a redefinition

of

the cosmological constant

and ofthe primary superfields, resulting in the same

am-plitudes as those

of

the bosonic theory. The strategy is to

first calculate the amplitudes in terms

of

a conformally invariant integral. We find the result of the integral

as-suming analyticity (see also

[17]),

using its symmetries

and asymptotic behavior. We give plausibility arguments

in favor

of

the result.

The relevant framework has been given by Distler, Hlousek, and Kawai

[8].

With a translation invariant

1

5'sL=

d

zE(

—, D& sLD'4 _sL

4m~

—

_QYesL

—

_«ue

'

'")

Ssr

=

d z

E(

2D

@~D

@sr+2iapY@sr),

(2)

4~~

where @sL and

4~

are super Liouville and matter

superfields, respectively (see Refs. [8,

18]).

The matter

sector has the central charge

c„,

=1

—

8a0.

The parame-ters Qand

a

~

are given by

Q

=2(i+

a')

'"

a

= —

—,'

g+

—,' (Q

—

4)'

= —

—,'

g~

Iapl.

The (gravitationally dressed) primary superfield +Ns is

given by

Eeike~(z) P(/)e~„(z)

Nsz,

z e p

P(k)

= —

.

'

Q+1k

—

~pl.

(4)

(5)

Note that the expressions for

a+

and

p(k)

are the same

in terms ofQ and ap as those

of

the bosonic theory. Screening charges in the matter sector are

of

the form

d ze' —

"

',

where d+ are the two solut&ons

of

—,'

d(d

—

2ap)

=

—,. ln this Letter, however, _{we will}

con-centrate on the case without screening charges. The case with screening charges, N-point (N

~

4)

functions, and

the inclusion of the Ramond sector will be discussed else-where.

We shall calculate three-point functions

of

the primary

field +Ns on world sheets with spherical topology hout screening charges), that is,

(6)

(wtt

rI„+

(

.

k)

=J

[&

+

][

+

]II+

i=I

i=1

Our first step isto integrate over the zero modes,

(

3 3

P

J

4'

(z;,

k;)—

:

2n'h

g

k;

—

2a

_A(k,

,

k,

k

),

i l

i=I

1 ₃

A(k

i,kp,

k3)

=1

(

—s)

5

1P

~

~2- ik;W~(Z.) Pi+SL~~i~ ' &2 +++SL~Z~

dze

'

'ei

'

dze

i

J

₀

(7)

(2)

VOLUME 68, NUMBER 11 PH

YSICAL REVIEW

LETTERS

16 MARCH 1992

where ( . .)0 denotes the expectation value evaluated in the free theory (p

=0)

and we have absorbed the factor

[a+(

—

x/2) ] ' _into _{the normalization} _of_{the path} _integral. _{The parameter} _s_is

defined as

1 3

s=

—

_g+g

_p;

₍₈₎

a+

In general, s can take any real value and there is no obvious way ofcalculating the path integral. However, ifwe

as-sume that s is a non-negative integer

[8-13],

this isjust a free-field correlator. Under this assumption, we evaluate the path integral, and formally continue s to noninteger values. For sas anon-negative integer,

a(k,

,

k,

,

k,

)

=r(

—

s)

r ₃ r

3 .c S

'"

.

_rrd"-,

_nd",

_rtl*-„I""

-"'n

_rr

_I.

-,

—;-~,

_~,

l-"rr

_I.

„I-"

i=1

i=l

i&j

i I

j

I

i&j

3

=r(-s)

2

rtd";d'&ll

I;+&&

I

"'ll

Il

—

_'I

"'ll

_I;,

I

"

F

i=I

i&j

(9)

We have divided by the SL2volume by setting zI

=0, z2=1,

z3

=~,

02=03=0,

and OI=—0. The integral is a

supersym-metric generalization offormula

(B.

9)

of Ref.

[19].

Alternatively, using @si

=p+OIv+Oy,

we can write

A(k,

,

k,

,

k,

)

=r(

—

s)

3

p,

'„Hd",

III,

I

"'ll

—,

I

"'

r

i=l

A(k

i,kz,

ki)

= —

i

where

nl

I"'(,

p;

)=,

'„,

„d'

Hd'g;d';I

I'

'll-I=I

2Q

x

_rt

_lz; _z&l +(tyy(0)lyly( z)

i'

' _lpga(vz))p.

(10)

i&j

This isnonvanishing only for sodd; we thus write s

=2m+

l.

One may evaluate

_(y

_y)o and

(y

y)0 independent-ly. Since the rest

of

the integrand is symmetric, one may write the result in asimple form by relabeling coordinates:

r 2

A

—

s) As

+

I)

I"'(a,

p;p),

(ii)

a+

z

II

I

—

g;I

"I

—

~,I

"I&,

n,I

'U

—

I&,

l'ln,

I'

ll

—

(,

I

"Il

i=I

i=]

m ni

II

I

(;

—

~,I

"II

I&;

—

(,

I

"I

n;

—

~,

I"'

(i

2)

(i3)

i=0

x

_rt

g(

—,'

+ a+

(2i

—

I)

p)

5(

—'

+

p+

(2i

—

I)

p)

4(

—

—,'

—

a

—

p+

(2i

—

4m

—

I

)

p),

i I

where C„,

(a,

p;p) has the same symmetries as

I"'(a,

p;p),

and where h(x)=—I

(x)/I (I

—

x).

By looking at the large-a behavior,

yin'

ia,

_p,'p~. i

j

—

a

—2«I—2(2«I+I)p—4pnI(2ns+I)

(is)

one can confirm that C„,

(a,

p;p) is, as a function of a, bounded as lal

—

~

and assumed analytic. This means that

C„,

(a,

P;p) is independent

of a,

and by symmetry, ofP as well; C„,

=C„,

(p).

It is hard to calculate C„,

(p).

Forthis purpose, it isuseful to consider the simpler integral

(i

4)

nI nl nI nt nI

J"'(,

P;)',

p)

=„

II

d'(,

d'n; III&;I

'ln,

l

'll

—

_(,

I

"Il

—

n,l

"

III&;

—

g,

l"

III(;

(,

I

"In;

—

n,l"—

U

Ii;;

ij

i&j

i I

(16)

By using similar arguments, one may obtain

ni—I

J"'(a,

P;

yp)

=C„,

()',p)

+

A(I

+a+

2ip)A(l

+P+

2ip)A(

—

I

—

a

—

_P

—

2)'+

(2i

—

4m+

2)p)

i=0

and

a=

—

a+p~,

p=

a+pz, and

p=

—

—.

a+.

We now have to calculate

I"'(a,

p;p).

Using the residual

SL(2, C)

transformations it is actually not diII'tcult to show

that

I"'

has the following symmetries:

I"'(,

P;p)

=I"'(P,

a;p)

=I"'(—

I

—

a

—

_P

—

_mp,_P;p).

Thus we may write

I"'(a,

_P;p) in the following way:

nI

I"'(a,

_P;p)

=C„,

(a,

P;p)

Q

h(I

+

a+2ip)h(I

+P+2ip)h(

—

a

—

P+(2i

—

4m)p)

(3)

VOLUME 68, NUMBER 11

PH

YSICAL REVI

EW

LETTERS

16MARCH 1992

(2o)

Notice that after Eq.

(15)

we have assumed that C„,

(a,

P;p) is an analytic function of

a

and P. Although a full proof

ofanalyticity

of

C„,

(a,

P;p) isoutside the scope

of

the present paper, we give three arguments of plausibility which give a strong basis to the result. First, C„,can be derived from

C

(y=

—

—,;p) using

(20).

The proof

of

the analyticity

of

the latter relies on the results of Ref.

[18].

Thus one can prove analyticity in that particular case. Second, we suppose

that nothing new should happen for particular values ofm (or

s).

At

m=1,

the calculation can be done explicitly. At last, one can compare the pole structure with that

of

string amplitude in a sphere with punctures (we thank A. Schwim-mer for this comment).

Now we are ready to write down the amplitude. Without loss

of

generality, we can choose

kl,

k3~

a0, k2

~

a0. By

using

(5), (8),

and

_g;-~k;

=2ao,

one gets

'p(1

—

s)

(ao&

0),

p=&

_—

—,'

—

ps

(ao&o).

It is easily seen that, for

ao&

0,

A

=0

identically, as in the bosonic theory. For ap &0,there are many cancellations,

leading to

m m

I"'(a,

_P;p)

=C„,

(p)

+

6(

—,'

—

(2i+1)p)

Q

6( —

2ip)A(l+a+2mp)h(

—,

—a+

p)

i 0

x

+

₆₍

I

+ a+

y+

(2i

—

I

)

p)

6(

I

+

P+

y+

(2i

—

I

)

p)h(

—

I

—

a

—

_P

—

y+

(2i

—

4m+

2)

p)

.

(i7)

Again, it is very difficult to calculate C„,

(y;p).

Unfortunately we could not get it in a rigorous way. A series of trials

and errors, however, led us to the following form:

&2mml

C. {r

p)=

_„,

[~{—

(r+p))]'"'Q~{I+2{y+ip)»{I+y+(2i

—l)p).

(i

8)

i=l

This isconsistent with

C~(yp)

=

l

z

[/3{

—

(y+p))]

h(I+y+p)d(I+2(y+p))

and with the two other (calculable) cases p

=0

and

y=o

(up to symmetry factors). It is very difficult to get anything

else consistent with these constraints. And

a

posteriori it seems to be correct since it gives a physically reasonable re-sult. Let us assume that

(18)

iscorrect and see its consequences. The two integrals are related by

I"'(e,

_P

_p)

= —

(x/2"'m!)A(1+a)A(l+P)A(

—

e

—

P)J

(2p,P;

—

2 ',

p)

.

Therefore C„,

(p)

= —

(~/2"'m!)C„,

(

—

2;p)h(

—,'

—

p)A(

—,'

+(2m+1)p).

If

we substitute

(18)

we get

2n~+1 nl IP1

C,

(p)

= —

[h(-'

—p)l'"'+'QA(2ip)P

&(2

+(2i+1)p)

.

22nl i I

i=0

2nt+I

=

₍

—

_I

₎

"'+'

_[a(-.

—

_{p)] '"'+}

_'(4p)

'"'A(1

₊

_a+2mp)a(

—,'

—a+

p)

.

(m!)'

Wefinally obtain the three-point function:

3

(2i)

A(k

i, kg,k3)

=

1 p

2 2 A(I

+a+2mp)a(-,

'

—a+

p)

2 2

3

=

—

p

_~

—

1

—

_p

_Q

—

_~(-,

_{[I+P, —}

_k,

_]).

2 2

_j

l 2

By redefining the cosmological constant and the primary superfield +Ns as

2 1

u-,

u. +~s(k,

)-

_.

.

+Ns(k,

),

6(

—,'

—

p)

(

—

ix/2)a(-,

'

[I

+P

—

_k

'])

(22)

(24)

(23)

we get our main result

A(k

),kp,

k3)

p'.

Remarkably, this amplitude isofthe same form as the bosonic one

[10].

It is natural toexpect that this feature continues tobe true for N-point (N

~

4)

functions. In fact, for k~,k2, k3

~

ao, k4

~

ao &

0

(and without screening charges), the four-point function turns out tobe

A(ki,

k2,k3,

k4)

=(s+

l)p',

₍₂₅₎

with the same redefinition

of

the cosmological constant and the primary superfields. In order to get the amplitude for general k;, one may argue, as in Ref.

[10],

that nonanalyticity comes entirely from massless intermediate states and one

(4)

VOLUME 68, NUMBER 11 PH

YSICAL

R

EVI

EW

LETTERS

16 MARCH 1992

may calculate the amplitude by using the analyticity of the one-particle irreducible

(I PI)

correlators. After setting p

=1,

we obtain the four-point function for all

k;:

&(&i,

&2,&s, &4)

=

&—

[I&i+&2

ttol+ I&

i+&

i &ol+I&

i+&4

&ol]+~it't

(26)

with A~pt

=

(I

+a:).

Compare this with Eq.

(37)

in Ref.

[10].

The analogy to the bosonic case is obvious. A de-tailed account will appear elsewhere.

The close connection to the bosonic amplitudes was suggested from super KP systems

[20],

and more recently, from

supermatrix models

[21].

We think, however, that our demonstration ismore direct.

In conclusion, we calculated the three- and four-point functions ofsuper Liouville theory coupled to a super Coulomb gas (without screening charges) on a sphere and found that they are essentially the same as those ofthe usual Liouville

theory, obtained in Ref.

[11].

As a by-product we get the supersymmetric generalization offormula

(B.9)

of Ref.

[19]

(s

=2m+ I),

5 5 5 5

—

_„H

d";

d'&H

I;+

019;

I'

ll

II

—;I

"II

I;,

I"

j=l

i=1

i=l

j&j

—

₍ 1)nr 2m+I 2m/ ₍&

)2m+I

I)I f)I t)1

g

d(2'p)

+

6(

—,

+(2'+

l)p)g

d(1+

+2'p)h(l+P+2'p)A(

— —

P+(2' —

4

)p)

i=I

j=0

I)I

x

+

6(

—,'

+a+(2i —

I

)p)A(

—,

+P+(2i —

1)p)h(

—

—,

—

a

—

P+(2i

—

4m

—

I

)p)

.

i=1

(27)

After completing this work, we found Refs.

[17]

and

[22],

which deal with the same issue, and, in fact, complement the present work.

K.H. would like to thank the members of Instituto de

Fisica Teorica, UNESP, for their hospitality extended to

him and

FAPESP

(Grant No.

90/1799-9)

for the

finan-cial support. He also acknowledges communications on

2D gravity with

Y.

Tanii and with N. Ishibashi. D. D. thanks

FAPESP

(Grant No. 90/2246-3) for financial

support. The work of

E.

A. and M.

C.B.

A. is partially

supported by CNPq.

"

Present address: Department ofPhysics, Kyushu Univer-sity, Fukuoka 812,Japan.

[I]

E. Brezin and V. A. Kazakov, Phys. Lett. B 236, 144 (1990);M. R. Douglas and S. H. Shenker, Nucl. Phys.

B335, 635 (1990);D.

J.

Gross and A. A. Migdal, Phys.

Rev. Lett.64, 127 (1990).

[2]A. M.Polyakov, Phys. Lett. 103B,207

(1981).

[3]T. L. Curtright and C. B.Thorn, Phys. Rev. Lett. 4$,

1309 (1982); E. Braaten, T. L. Curtright, and C. B.

Thorn, Phys. Lett.

IISB,

115(1982);Ann. Phys. (N.Y.)

147, 365 (1983);E. Braaten, T. L. Curtright, G.

Gan-dour, and C. B.Thorn, Phys. Rev. Lett.

Sl,

19 (1983); Ann. Phys. (N.Y.) 153,147 (1984);

J.

L.Gervais and A.

Neveu, Nucl. Phys. B199,50 (1982);B209, 125 {1982); B224, 329 (1983);B238,123 (1984);B238, 396 {1984);

E. D'Hoker and R. Jackiw, Phys. Rev. D 26, 3517

(1982); T.Yoneya, Phys. Lett. 148B,1 11 (1984).

[4] F. David, Mod. Phys. Lett. A 3, 1651 (1988);

J.

Distler and H. Kawai, Nucl. Phys. B321, 509

(1989).

[5]N. Seiberg, in Proceedings ofthe 1990Yukawa

Interna-tional Seminar on Common Trends in Mathematics and Quantum Field Theory, and Cargese Meeting on Random

Surfaces, Quantum Gravity and Strings, 1990

(unpub-lished);

J.

Polchinski, Slrings '90 Conference, College Station, Texas, /990 [Nucl. Phys. B357,241

(1991)].

[6] A. M.Polyakov, Phys. Lett. 103B,211

(1981).

[7]

J.

F. Arvis, Nucl. Phys. B212, 151 (1983); B2IS, 309 (1983);O.Babelon, Nucl. Phys. B258, 680(1985);Phys.

Lett. 141B,353 (1984); T. Curtright and G. Ghandour,

Phys. Lett. l36B,50(1984).

[8]

J.

Distler, Z. Hlousek, and H. Kawai, Int.

J.

Mod. Phys. A5, 391 (1990).

[9]M.Goulian and M. Li, Phys. Rev. Lett. 66,2051

(1991).

[10]P. Di Francesco and D. Kutasov, Phys. Lett. B 261,385

(1991).

[I I]Vl.S.Dotsenko, Mod. Phys. Lett. A6, 3601

(1991).

[12] Y.Kitazawa, Phys. Lett. B265, 262

(1991).

[13]N. Sakai and Y. Tanii, Prog. Theor. Phys. 86, 547

(1991).

[14]A. Gupta, S. P. Trivedi, and M. B. Wise, Nucl. Phys.

B340,475 (1990).

[15]See also M. Bershadsky and I. R. Klebanov, Phys. Rev. Lett. 65,3088 (1990);Nucl. Phys. B360,559

(1991);

A. M. Polyakov, Mod. Phys. Lett. A6,635

(1991).

[16]M. A. Bershadsky, V.G.Knizhnik, and M. G.Teitelman,

Phys. Lett. 15IB,131 (1985);D. Friedan, Z.Qiu, and S.

Shenker, Phys. Lett. 151B,37(1985).

[17]P. Di Francesco and D. Kutasov, Report No. PUTP-1276

(unpublished).

[18]E.Martinec, Phys. Rev. D28, 2604 (1983).

[19]Vl. S. Dotsenko and V. Fateev, Nucl. Phys. B240, 312

(1984);B251,691 (1985).

[20]P. Di Francesco,

J.

Distler, and D. Kutasov, Mod. Phys.

Lett. A5,2135(1990).

[21]L. Alvarez-Gaume and

J.

L. Manez, Mod. Phys. Lett. A

6, 2039

(1991).

[22] L. Alvarez-Gaume and Ph. Zaugg, Phys. Lett. B273,81

(1991);

K. Aoki and E. D'Hoker, Report No. UCLA/91/TEP/33, 1991 (unpublished).