2D string theory coupled to quantum gravity

Physics Letters B 312 (1993) 411-416 North-Holland

PHYSICS LETTERS B

2D string theory coupled to quantum gravity Nobuyukl Ishlbashl Theory Group, KEK, Tsukuba, Ibarakt 305, Japan Received 25 May 1993, revised manuscript received 22 June 1993 Editor M Dine

We consider self-avoiding Nambu-Goto open strings on a random surface We have shown that the partition function of such a string theory can be calculated exactly The string susceptibility for the disk is evaluated to be - ½ We also consider modifications of the Nambu-Goto action which are exactly soluble on a random surface

1. Introduction String theory appears in many aspects of physics It ~s an old ~dea that QCD might be represented as a string theory [ 1 ] Also, the three dimensional Islng model was argued to be described as a three dimensional superstrmg theory [2] Therefore, it is important to search for consistent string theories m each dimension and classify the umversahty classes of them Such string theories wall be useful m describing various physical systems as point particle field theories was

However, constructing consistent string theories appropriate for such apphcatxons ~s not an easy task It is always possible to construct low dlmens~onal string theories by cornpactifylng crmcal string theories, but such theories have features hke the existence of a massless spin two pamcle, which are not desirable m most cases Hence we should look for n o n c r m cal string theories However, constructing a consistent string theory preserving Lorentz or rotation l n v a n anee ~s not so easy The noncrmcal Polyakov string was quantlzed by the authors of [ 3 ], but their quantlzation ~s not consistent m d d~menslonal space-hme w~th 1 < d < 25 The string susceptlbd~ty becomes complex, if one naively applies the KPZ formula to these cases Therefore one should overcome this "c = 1 barrier" in order to construct Polyakov string theories of physical interest There can be another approach to noncrmcal string theory, namely the d~rect quant~zatlon of the N a m b u -

Goto action Usually the N a m b u - G o t o action is quantlzed by transforming it Into the Polyakov action In 26 dimensional space-time, we can quantize the N a m b u - G o t o string directly and It gives the same results as that of the quantlzatlon a la Polyakov However, there is no reason to believe in their equivalence in the noncritical cases Therefore, it is possible that the N a m b u - G o t o string has no "c = 1 barrier" when directly quantlzed The N a m b u - G o t o string IS mamfestly Lorentz or rotation m v a n a n t and ~t can be useful m describing various systems Unfortunately the N a m b u - G o t o string has a highly n o n h n e a r action and is retractable in general Here let us discuss the two dimensional self-avoiding N a m b u - G o t o string as the simplest case xn th~s direction In two d~menslonal stung theones, the embeddrag of the worldsheet into the space-time is generically singular, revolving folds If one requires selfavoldmgness, such folds are forbidden and the N a m b u - G o t o action becomes t n v m l for closed strings [4] However for open strings the theory is not so m v l a l For example, the partition function corresponding to the d~sk graph of such an open string can be written as

Z(Iz) = Z

e -uAw)

(1)

F

Here F denotes the self-avoiding loop m the spacet~me corresponding to the boundary of the open string graph and the sum ~s over such loops A ( F ) is the area of the space-time F encloses For the self-avoiding

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411

Volume 312, number 4

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19 August 1993

We will show that the string susceptlblhty for the disk IS l We also discuss modifications of the N a m b u Goto action Section 4 contains concluding remarks

open strings, the N a m b u - G o t o actaon depends only on the boundary of the graph, which coincides with A(F) This is a n o n t n v l a l sum and it is not sure ff this open string theory is consistent contrary to the usual noncritical string case Th~s self-avoiding open string theory may be apphed to two dimensional systems of interest, e g two &menslonal QCD [5 ], which have been actively stud~ed recently [6] The expectation value of the Wilson loop for two &mens~onal pure Yang-Mllls theory IS known to be

The techmques needed for performing the sum m eq ( 1 ) over random walks on a random surface was developed by Duplantler and Kostov They considered self-avoiding walks on a random surface

( T r R P e x p ( f AudxU)) =exp[-g2C2(R)A],

Z(R,m) =

2 Self-avoiding random walks on a random surface

e-ml-aV'

Z

(3)

F, metrlc C

(2) where g is the gauge coupling, C2(R) the quadratic Caslmtr operator for representation R of the gauge group, and A is the area enclosed by the loop If one tries to incorporate dynamical quarks into the theory, one should sum the above expectation value over the fluctuations of the loop C Thus we encounter a sum over loops slmdar to eq ( 1 ) Eq (1) may correspond to two &menslonal QCD with self-avoiding quarks Anyway, it seems difficult to calculate the sum in eq ( 1 ) In this paper, we wall discuss this self-avoiding N a m b u - G o t o open string theory on a random surface In other words, we will consider the string theory coupled to q u a n t u m gravity m the target space On a random surface, ~t is possible to calculate the partition functions of the open string theory exactly We wall show that the string susceptibility is not complex contrary to the usual noncritical string case This fact implies the consistency of this string theory, at least m the presence of q u a n t u m gravity m the target space In a sense, what we are deahng w~th ~s related to two dimensional QCD w~th self-avoiding quarks coupled to q u a n t u m gravity The orgamzat~on of this paper is as follows In section 2, we consider self-avoiding random walks on a random surface as a warm-up We wall first review the techniques developed by Duplantler and Kostov [7] to calculate the partition function of self-avoiding random walks on a random surface Using their results, we express the configuration sum of a loop on a random surface in terms of the wave function of quantum grawty In section 3, we go on to the case of selfavoiding strings and evaluate the partition function 412

where the sum is over self-avoiding walks F and the space-time metrics with the Boltzmann weight revolving the total length l of F and the volume V of the space-time Let us first review their results [7] The ensemble of random surfaces was defined by the c o n t i n u u m limit of the dynamical triangulation of the surfaces If we restrict the topology of the surface to that of the sphere, the partition function Z (2, m ) m eq (3) is dlscretlzed as

Z(fl, K) = Z

e-BI°IS(G)~-~Klrl

G

(4)

F

Here G denotes a planar ~b3 graph, IG r is the n u m b e r of vertices of the graph G and S(G) is the symmetry factor The dual of G describes a triangulated surface F is a self-avo~dlng random walk on the graph G fl and K correspond to the parameters 2 and rn respectively, m the c o n t i n u u m hmlt Duplantxer and Kostov rewrote eq (4) in terms of the partition function G~ of random graphs w~th n external legs, G.(fl) =

Z

e-BI°l

(5)

n leg planar G

Here, let us concentrate on the case where F in eq (4) is a loop Then F divides the surface into two parts Z (fl, K ) can be expressed by two Gn's representing these two parts as 1

Z(fl, K) = .--. m + n

(m + n ) '

m' n'

m,n

× (e-~K)m+nGm(fl)Gn(fl)

(6)

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PHYSICS LETTERS B

Gn (fl) was calculated using the large-N hmlt of the matrix model and it has an integral expression [8]

19 August 1993

in this llmlt to define the sum in eq (3) with m and 2 as above As 8 ~ 0,

2b

/ ,

Gn(fl) = /

(7)

d2p (2)2 n

i #

2a

0

Here p (2) is the density of elgenvalues of the N × N matrix given in [8] Substituting this expression, we obtain 2b

× gX( l ) ~ (l)e -ml) ,

(12)

where

2b 2b

Z(fl, K) = - f d 2 1 f d 2 2 p ( 2 1 ) p ( 2 2 ) 2a

x ln(1 - e-#K(21 + 22))

Kc = e&14bc ,

(9)

where bc is the value of b at fl = tic When K approaches K~ simultaneously as fl approaches tic#l, the integral in eq (8) diverges because the point 21 = 22 = 2b in the integration region approaches the singularity of the logarithm in the lntegrand Loops with infinite length dominate at this critical point Eq (8) is the main result of Duplantler and Kostov, which we will use in this paper In the rest of this section, we will calculate Z (2, m ) in eq (3) using eq (8) In order to extract the singular part and take the c o n t i n u u m limit, it is convenient to rewrite eq (8) as

z(...,= S 2a

2b

S

<-0

(?-(~)l

2a

O~

×idt(-iexp{-ttl-e-'K(2'+Jz)]})'

(10)

0

(

d#p(/x)exp-8

2a

(8)

The singular behaviour of Z (fl, K ) can be seen from this integral representation There is a crmcal point fl~ for fl which corresponds to the pure gravity critical point The critical point for K is at

2b

S

~P(/) =llmS-S/26~0

2a

4bc

} (13)

The right hand side of eq (12) has a natural interpretatlon as follows ~P(l) is the partition function of two dimensional gravity for the disk with the boundary length l, which can be regarded as the wave function of q u a n t u m gravity [9] The right hand side of eq (12) can be considered as a regularized form of

i ~l~(l)~U(l)e-mt

(14)

0

When m = 0, this formally coincides with the inner product of the wave functions The above result shows that such an inner product gives the n u m b e r of configurations of self-avoiding loops on a random surface Since the wave function in q u a n t u m gravity does not depend on time, but rather is something integrated over the time variable, it is easy to understand intuitively that the inner product would give the n u m b e r of ways of taking a time slice on a random surface The time slice we are considering here is a self-avoiding random walk in the space-time Using p(/~) in [8], we can evaluate ~ ( l ) up to an overall constant factor as

and change the variables as ~U(l) = l-S/2(1 + v ~ / ) e - v 7 / , e - a = e - & ( 1 - 282),

(15)

K = Kc(1 - m 6 ) ,

t = 118

(11)

In the c o n t m u u m l l m l t , 8, which is supposed to be thelattlce spacing, approaches zero We use Z (fl, K) #1 Here we consider only the ddutephase m [7]

after a rescahng of the variable 2 Substituting this expression, it is easy to calculate the right hand side of eq (12) ~ ( l ) ~ ( l ) e -mr in the lntegrand can be expressed as a finite sum of terms of the form ln e - (m+2,/i)t with n an integer The following formula is useful in evaluating the right hand side o f e q (12) 413

Volume 312, number 4

h m O, ~0

(-j

PHYSICS LETTERS B

dl F - l ln e -(m+2v~)l

)

0

= -(lne-(m+2v%l)oln(m + 2x/Z) + (terms analytic in m)

(16)

Here ( f ( l ) ) o denotes the coefficient a0 of l ° in the Laurent expansion f ( l ) = y~anl" .2 Thus we obtain as Z (2, m)

19 August 1993

by two Gn's each of which corresponds to the outside and the inside of the loop F*3 However, this time the cosmological constant in the Gn corresponding to the inside is shifted because of the term /*A in the Boltzmann weight Therefore, after the same procedure as in section 2, we can derive

Z(Z,/*)

= llmO.

I'-j

dll'

~,+a(/)~(l)

0

(20)

-(~U(l)g-'(l)e-ml)oln(m + 2x/2) + (terms analytic in m )

(17)

The result includes the nonunxversal part which is analytic in m and vanishes when differentiated sufficiently many times It is the contrlbutmn o f very short loops Therefore, the first term in eq (17) is the universal part which should be taken as the continuum hmlt We can show that it is really universal and does not change if one changes the way of dlscretlzatlon by making use o f quadrangles etc instead o f triangles Discarding the nonuniversal part, we obtain as the continuum limit Z ( 2 , m) = ( ~1( r n

+ 2if2) 5

_

2 x / ~ ( m + 2x/-2) 4

Here ~ (l) denotes the disk partition function of two dimensional quantum gravity with the cosmological constant 2 The right hand side o f e q (20) can be consldered as the regularized inner product o f two wave functions with different cosmologmal constants The calculation of eq (20) goes exactly as in section 2 Substituting eq (15), one can again use the formula eq (16) with a slight modification In order to select the universal part, we should Insert e -mr into the lntegrand of eq (20), discard the terms analytic in m and take m to be 0 in the end Eventually we obtain

zI2,u) = +

+ ~(m

+ 2 v ~ ) 3 l n ( m + 2x/2)

(18)

3. Self-avoiding open strings on a random surface

It is straightforward to apply the techniques in the previous section to self-avoiding open strings on a random surface The configuration sum we have to d e a l w i t h is a s follows

v/277) ' xA-77(v

x ln(x/2 + ¢ 2 + / * )

e-UA-~V

+

x/277)31 (21)

W h e n / * >> 2, the typical area of the worldsheet of the open string is much smaller than the typical area of the s p a c e - t i m e Then the compactness o f the s p a c e time becomes irrelevant to the fluctuations o f the open string and we expect that the partitmn function scales as

Z (2,/*) ~/*-rdlsk+2

Z(2,/*) = E

1

(22)

(19)

F metric

This F~tr should be taken as the definition o f the string susceptibility for the disk When/* >> 2, eq (21 ) gives

In this case, the Boltzmann weight involves the area A enclosed by the self-avoiding walk F Let us restrict ourselves to the case where the topology of the s p a c e time is that o f the sphere If one dlscretizes t h i s sum as in the previous section, it can again be rewritten

and Fd,sk = --½ Therefore the string susceptibility o f this two dimensional string theory is not complex and

*2 ( f (l))0 appeared in the definmon of an tuner product of the wave functmns m [9]

#3 We c o n s i d e r t ha t F is o r i e n t e d a n d we can de fi ne the i ns i de a n d the o u t s i d e

414

Z (2,/*) ~/*~ In(/*),

(23)

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PHYSICS LETTERS B

we do not encounter the inconsistency contrary to the case of the usual noncritical string It is intriguing to observe the exponent 5 in eq (23) originates from the scaling dimension of the disk partition function ~ ( l ) (cf eq (15)) The above result indicates that the fluctuations of the N a m b u - G o t o string are essentially the same as those of the disk partltion function of pure q u a n t u m gravity, but for the log correction The In/t term appears when we take the universal part of eq (20), which includes a logarithmic divergence The ultraviolet behaviour (l ~ 0 in the integral in eq (20)) of the boundary of the string is responsible for the divergence Therefore it seems that the string susceptibility for the self-avoiding N a m b u Goto string is real because the string may be describable by pure gravity if we take proper care of the stnngs with short boundaries Further study is needed to make the above statement more preose It is possible to generalize our self-avoiding N a m b u - G o t o string on a random surface as follows The action of the N a m b u - G o t o string is the area of the worldsheet However any reparametrization mvariant quantity is conceivable as a string action Let us perturb the N a m b u - G o t o action by reparametrizatlon lnvarlant operators O, Then the sum m eq (1) will be modified to be

Zg, O,),

Z(2,#,g,) = Z e x p ( - / t A -

(24)

F

and we obtain instead o f e q

(20)

(\ ~~--~7---7, Z ( A , / t , g , ) = , ~lim0~ 0 [e)

d l l C-l

0

× ~ +~,g,(l) % (l))

(25)

Here ~"//t+2,g, (l) is the wave function of q u a n t u m gravIty with the action perturbed by ~ g, O, We will show that such perturbations to the N a m b u - G o t o action can change the critical behaviours of the theory By choosing O, to be the reparametrizanon invarlant observables of 2D q u a n t u m gravity in [10], and fine tuning g,, ~uu+xg, (l) becomes the wave function of q u a n t u m gravity in the multlCrltlCal phase [ 11] The explicit form of such a wave function in the m-th

19 August 1993

multlcrItlcal point lS ~:4

~J,u+A,g,(l) = l - l ( V@ + "~)m-l/2 gm-1/2( V/fl +

2l), (26)

where Kin- 1/2 IS the Bessel function [9 ] m = 2 corresponds to the pure gravity and the critical points with m > 2 can be reached starting from the pure gravity If one perturbs the N a m b u - G o t o action so that the wave function ~vu+,~,g' (l) which appears in eq (25) is the multlCrltlcal wave function eq (26), one can obtain a new class of string theories We may call such a string the "multIcritical string" The string susceptibihty for the disk of the multlcritical string theory can be calculated using eq (25) and eq (26) as

~dlsk =

4

- m +______~1 if m = 2, 4 or odd, 2 -m + 4 otherwise 2

(27)

Conclusions

We have shown that the two dimensional selfavoiding N a m b u - G o t o string is exactly solvable when it is coupled to q u a n t u m gravity The string susceptlblhty can be calculated and we obtain a real value Usually we quantlze string theory ~ la Polyakov If one tries to quanUze the two dimensional Polyakov string, one either spoils Lorentz or rotation lnvarlance, or encounters a complex string susceptibility One has to overcome the difficulty of 2D q u a n t u m gravity coupled to c > 1 matter The N a m b u - G o t o string theory considered here possesses a real string susceptibility because it seems that the fluctuations of this two dimensional string is quite similar to those of the pure gravity (c = 0) The reduction of the central charge may be attributed to the self-avoldlngness We have also constructed new string models on a random surface by perturbing the N a m b u - G o t o action We have shown that multlcrltical phases can be reached by such perturbations It is possible to calculate multlloop amplitudes and obtain the mass spectrum of such string theories We will report on this problem elsewhere ~:4 Here we take the conformal background m [9], because we want # to couple to the area of the worldsheet 415

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Acknowledgement We would hke to thank K Hlgashlj~ma, Y Okada, N Tsuda, Y Yamada, T Yukawa and other members of K E K theory group for useful d~scusslons and encouragements

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[4] K Fujlkawa, J Kubo and H Terao, Phys Lett B 263 (1991) 371 [5] A Mlgdal, Sov Plays JETP 42 (1975) 413 [6] D Gross, preprmt PUPT 1356, D Gross and W Taylor, preprmts PUPT 1376, PUPT 1382, J Mmahan, prepnnt UVA-HET-92-10, M R Douglas, preprmt RU-93-13, M R Douglas and V A Kazakov, preprmt LPTENS93/20 [7]B Duplantmr and I K Kostov, Phys Rev Lett 61 (1988) 1433, Nucl Phys B340 (1990) 491 [8] E Brezm, C Itzykson, G Pans1 and J B Zuber, Commun Math Plays 59 (1978)35 [9] G Moore, N Smberg and M Staudacher, Nucl Plays B 362(1991) 665 [10] E Brezln and V Kazakov, Phys Lett B 236 (1990) 144, M Douglas and S Shenker, Nucl Plays B 335 (1990) 635, D Gross and A Mlgdal, Plays Rev Lett 64 (1990) 127, Nucl Plays B 340 (1990) 333 [ I I ] V Kazakov, Mod Phys Lett A 4 (1989)2125