JUCLEAR PHYSICS Nuclear Physics B (Proc. Suppl.) 25A (:992) 203 210 North-Holland

PROCEEDINGS SUPPLEMENTS

UNCONVENTIONAL METHODS IN 2D QUANTUM GRAVITY: SOLVING (llq) MODELS

David Montano

California Ins~itete of Technology, Pasadena, CA 911~5, USA

In this talk 1 will discuss the (l, q)-models of d < 1 quanttun gravity. These are the first critical points of the models with q - J primary fields. I will show how these models may be understood as topological theories whose correlation functions arise from contact terms and surface degenerations. I will then provide explicit solutions for these models using the generalized KdV hlermx Ly. These solutions are espescially useful because they are valid for arbitrary values of q.

The title for this lecture requires some words

of explanation. The methods I will discuss are

unconventional in that. they are not part of the

s tandard formalism for studying quantum grav-

ity in higher dimeosions; i.e., quantum field the-

ory. The m~st useful tools for understanding 2-D

quantum gravity coupled to c < 1 matter h a w been based on topological arguments [1,2] and

the KdV equations [3] (motivated by the ma-

trix models [4]). People have also relied on the

more s tandard ideas of Liouv~lle theory [5], but

this has been less fruitful in providing exact solu-

tions. Thus, in this lecture I will concentrate on

the fact tha t ~opology and the KdV equations

provide a complete solution to gravity for e < 1

matter.

In part icular, I will be interested i , the "pure"

topological theories. These are the models de-

fined by a BRST trivial Lagrangian. "Per turba- tions" of these Lagrangians with BRST invari-

ant, but nontrivial, operators lead to tim other

e < l theories, often referred to as the higher

multicritical points [6,7]. I will come back to this

point later.

The rest of this talk will then be organ:zed as follows: I will first discuss "pure" 2-D gray-

ity and how it can be completely solved through

self-consistency [8]. I will then show how gravity

with many primary fields may be investigated

and solved, first using self-consistency and then

the KdV equations. I will also show the close

relation between the KdV and topological ap-

proaches, in particular, emphasizing the role of ghost numbc: conservation [9]. Precisely whal

I mean by "pu:e" gravity, pr imary fields, and

the relation with the KdV equations will be ex-

pbined shortly.

1. Pure Gravity

"Pure" 2-D gravity can he interpreted vari-

ously as any of the following,

a) e = - 2 Liouville theory [6]; h) SI,(2,R) topological gauge theory [10]', c) intersection theory on the moduli space

of Riemann surfaces [1].

The topological field theory approach begins

with a BILST trivial Lagrangian,

£0 = {Q,^} , (1)

where Q is the BRST operator which arises from

gauging the topological symmetry. It can also

0920-5632/92/$ 05.00 ~ 1992- Elsevier Science Publisher s B.V. All rights reserved.

204 D. Montano o I [Ynconventio~al ~ t h c d s it, 2D Quantum Gravity"

be interpreted as the exterior derivative on the

corresponding moduli space. For pure 2-D g~av- ity the relevant moduli space is that of complex structures of Riemann surfaces. Associated with this BRST operxtor there is an anomalous U(1) ghost number symmetry; the anomaly is ~qual to the dimension of the corresponding moduli space. The observables in the theor.: art given by the eohomol~,gy of Q. There is a hierarchy of BILST invatiants given hy zero, one- and two-

forms on the surface:

0 = { Q , ~ ) } , da~)+, - {Q,a~ )}

da~}+l = {Q,a~)}, d a ~ ) = O,

where #~) corresponds to the i-form invariant

and m refers to the ghost charge. Correlation functions are constructed from combinations of the following,

~0~, I ~ . ) £ ~ , 12) f ~ ,

with a net ghost charge canceling that of the anomaly. The correlation functions may also be

interpreted as integrals over the moduli space of the wedge-product of forms whose degree is given by the ghost charge.

In "pure" gravity we have the following ghost number assignments,

ghost number/4[a(m °)] = 2rt,, (3)

anomalous dimension: 6 9 - 6 + 2N, (4)

for N marked points on a Riemann surface, E.

1.1. Con tac t Algebra

In a topological quantum field theory (TQFT)

correlation functions cannot depend on distance, a metrical property. #1 Thus, contributions to

# ! Note that an integration over metrks does not neces- sarily imply that correlators are independent of dis- tahoe. Clearly, ther~ ~ could be preferred metrics which minimize the action.

the correlation functions ~,rise only from contact terms and surface degenerations as originally ad- vocated by [2]. I will show that the c o n s i s t e n c y

of this p r o c e d u r e is sufficient to completely solve pure gravity.

The ghost number of the two-forms is given

by,

U [ ~ ~1 = 2m - 2. (5)

The contact term can then be written ~s follows,

I~C. °!) = Am" ~+._~. (el

From ghost number conservation we know that there can only be one ficid on the right hand side of eq. 6. The ghost number uniquely determines

the operator. This is an important property and is maintained in the theories with many primary fields.

Since the order of the operators in a correla- tion fimction is irrelevant, one can construct the

following consistency condition,

which ~sing eq. 6 leads to the following,

A n k +m - i A m k _ Amt~+n-lan k -.

(A,~ m - Arn'~)Am+,~-I ~. (8)

We proved in [8] that the most general polyno

mial solution to eq. 8 is given by,

A , ~ " = a n + b. (9)

The coefficients a and b caq be doermined by either normalizing the co-relators,

{o'0a0o'0) and (al) , (10)

or by fitting to the corresponding KdV result. The linear solution for Am n implies a type of

D. Montano / Unconventional Me'hods in 2D Quantum Gravity 205

Virasoro algebra,

(71 -- m ) I f i re+n- l ) - (11)

Recursion relations can now be terived which

completely solve the model in term: of a and b. These recursion relations can. also I-~e derived us- ing the V.casoro constraints frorr, the KdV (ma- trix mo iel) point of view. ! will come back to this later.

Tbe multieritical points of pure gravity arise from considering the following Lagrangian,

, / L;= ~ : o + z / ~ o + t l #l, (12)

where £o is the BRST trivial Lagrangian. By

choosing t) = &..I one gets the k th multicritical point, the (k, 2)-models [6,7].

By expanding in powers of • and tk one can derive the multicritical correlation functions in te r~s of the correlation functions of £0 alone, which conserve ghost number. It should be noted

'ha t the multicritical points no longer conserve ghost number, but they are solvable in tert~.~ of

the ghost number conserving correlators of the first critical point of pure gravity. Expanding the

correlation functions in powers of z and t (t is set to 1), we get,

(o,+~ H ) g . ' = rn¢S

, v - (z#o)* (#kY

I,p na~$

One can also derive the following general recur- sion relation using that for pure gravity,

(%+k H cr")k = x(on H #")*+ m($ rues

E(2mj 4-1)(On+mj+k-i H o n , ) ~ r j i~j

n

I=1 rn¢S

S=XuY icX

(-,+~-, H (~,,)1. (141 i(Y

The scaling behavior of the specific he~t can be seen t . given by,

(aoo'o " .7: I l k . (15)

This is one of the famous consequences of the one-matrix model [4].

2. (l,q)- Models

I will uow discuss the models with more than one primary field: these will lead to a variety of complications. The (l,q)-models have q - 1

primary fields. They are tlle (1,q)-models of the generalized KdV hierarchy (to be discussed later). Ttley have the following observables

(BILST ,ontrivial operators):

where f c ~ = 0 , . . . , q - 2 ore(Go), t m = 0, 1,2,...

These models are also described by the twisted N = 2 super-conformal field theories [I 1], and

s.lso by Hie q - 1 matrix models. They have the following ghost number assign-

Iqlell| s~

/~'[~,,,(o~)l = (m - l)q + o. (16)

The primary fields are #o(O~) = O~. Tile om(Oo) are referred to as the mta descendants

of O~. Note that all the primary fields carry a

negative ghost charge, and as before, the ghost charge mliquely determines the c erator.

Let me digress briefly to point out that the ghost nnlnber is related to the area scaling ex-

ponents~

206 D. Montano / Unconventional Methods in 2D Quantum Gravity

(~,~(O,~)) ~ • ~zl(m-1)q+~]. (17)

Schematically, one can write the contact alge-

bra as follows,

o,,o, , = ~ c:,,o.,, (18) -f

where,

~, , (O~) - O=~+~, and

L/rOll -- ~ - q, c~ ~ - l m o d q.

l~om ghost number conservation we see that

there can be only one field on the right-hand side

ofeq. 18,

OaOo = A~BOa+~-~,

c "1 = A,,B~o+~--f-q. (19)

The consistency condition is the same as for pure gravity (eq. 8). So, again we have that ,

c~ O = (aB + b)6o+~_~_q.

But is all this still true? Not comFletely. Re-

call that the operators, O,~, only exist for c~

- 1 mod q. Thus, the contact algebra must be of

the form,

f ( ~ + l ) O o + ~ _ q , i f ~ + B ~ - I Oo O# 0 , i f a + B = - I

where the above condition is mod q.

This algebra does not, in general, satisfy the

consistency condition, eq. 8, for q > 4. Opera-

tors with a + B = - 1 rood q imply that c~o = 0, but if the other o's don' t vanish, the consistency

condition cannot be satisfied. This is simple to

verify. In order to resolve this issue, we can re-

strict to contacts bctwe.r. ~ O ~ where a = 0

mod q, in which case ~ + B would never equal

- 1 mod q. One can now derive the following re- curs!on relation [8],

(~,(Oo) [I ~,(o~))g= (n,O)ES (n,O)ES

(nq ~- ,~ -.[- l)(o'rn+n_l(Ot~ ) H O"({~*t))g (.,~)¢(n,~)

m - !

J= l B,7

(V'-t(OB)~"'-!-'(O') I'[ vn(O,)),-t (n,S)ES

S=Xul',gl+ga=#

((if,- ! (O~) 1-[ ~.(O6)),. (. ,~)eX

( " . , - t ( O , ) 1"[ ~ . ( 0 , ) ) . ) } . (20) (.,6)EY

These are known as the "Virasoro constraints."

We see that these recursion relations are a conse-

quence of two-point contacts with descendants of

the identity, and earlier it was shown tha t these obey a Virasoro algebra, eq. 11.

Unfortunately, to completely solve the ( l ,q) - models we need more information. There has

been speculation tha t W-constraints and mul-

lipoint contacts are the solution. We have been

unable as of yet to derive a complete and consis-

tent contact algebra to solve all the (1, q) models

and agree with the KdV. We are currently work- lug in that direction. #o At any rate, in this talk

I will show how a complete and explicit solution

for the (I ,q)-models can be arrived at from the

KdV equations. The important point is tha t I

will provide explicit expre.~sions for the correla-

tion functions of an~l (1,q) model [9].

3. KdV Gravity

Ill this approach the correlation functions gen-

erated by thv t~o-point function of the lowest di- mension operators (also referred to as the specific

#2 This is work in progress with Gil Rivlis and Jacob Sonnenscheht.

D. Montano / Unconventional Methods in 2D Qu,mtum Gravity 207

heat} can be computed systematically using the

flows of the .,eneralized KdV hierarchy [12]. This was originally motivated by the work of Michael Douglas I3] on the continuum limit of the matrix models. Before proceeding I would like to review some concepts necessary for understanding KdV

gravity. First, let me define a pseudo-differential oper-

ator as follows,

Do O (2') O = ~ - " . n t ( z . J " - L D = 0~

i=0

and

~-~( 1~/( r e + i - l ) ! f ( 1 ) D - a - i D - " l ( x ) = z . . ~ . - - , g . ~ g - - i ~ . "

i=o " "

D n f ( z ) = i i=0

It will be useful to define the order and scaling dimension of a pseudo-differential operator,

order(O) - n(highest derivative). (22)

If a0(x) = 1, then we can define the sealing di-

mensions,

dim(O) = ord(O) = n ('2.3)

dim(D) = t (24)

dim(a/) = i. ~25)

Scaling information will play an important role

in solving the (I ,q) KdV gravity. It has been taken for granted since [3] that

(p, q)-KdV gravity is equivalent to the (p: q) min-

imal matter coupled to gravity or the (q - 1)- matrix model with a potential of orde, p. Let us proceed with the (p,q)-KdV by defining the

following differential operator and its roots,

Q = D q + uq_rzD q-2 + uq_3 Dq-3 + ... + us (26)

and,

Do Qplq = D r + ~ vtD p-/. (27)

The vl are uniquely determined by requiring that (QP/q)¢ = QP. We now define the differential op-

erator,

p - IQpl~)+, (28)

where by (O)+ it is understood that only the differential part of 0 is kept. In terms of the

matrix models , ~ t, perators Q and P can be

roughly interpreted as,

d P ~ - ~ ,

where A is a matrix eigenvalue. Then, following Douglas [3] it is natural to impose the so-called

"string eqnation,"

[P, Q] = 1. (29)

For each "primary field" in the corresponding

eonformal field theory or matrix in the original

lattice, there is a corresponding operator,

p~, -. Q"/q , where a -7- !, ...,q - l (30)

and "descendants."

Pi = P,,+-,q = a,,,(P~a = Q"~+olq (31)

Note that,

ord(a,,(/~,)) --- dim[am( tJo)] -- c, + qm. (32)

CorrelalimL ftLnctions are now determined

by the following rules. We identify uq-z with tile two-point function of lowest dimension

primaries. The lowest dimension operator is the

puncture operator, and its flow parameter,a:, is called tile cosmological constant. This terminol- ogy is a ¢ou~quenee of the fact that ae is related to tile inver~ of the area. It Ls interesting to note how in this way a topological theory can

208 D. Montano / Unconventional Methods in 2D Quantum Gravity

provide sealing informat;on with respect to the area. Thus, we see that uv_z is the specific heat,

1 • = (P ,P , ) = ResQ '1`, (33) qUq-2

where the Res(O) is the eooflieient of D - I . To understand how correlation fun=tions are

computed we can imagine that the Lagrangian for this theory takes the form,

£ = £0 + E t,~,~o'n(O~), (34)

• and to,o = x. Thus,

O~ ( P , P , ) = "O'~x 2 Z ,

where Z is the partition function. All the corre- lation functions can he determined by differenti- ating with respect to the parameters, tm.~. The

KdY hierarchy tells us precisely what happens upon differentiation,

0 QIIq = [Q,~+oI~,Q,I~]. (35) ~rn,c~

We now have a systematic procedure for comput-

ing all the correlation functions. We will investi- gate this in some detail for the ( t ,q) models.

3.1. Solving the ( l ,q) models

The (1, q)-modeis are vastly simpler than the higher critical points. But as m~ntie.~ed earlier, perturbations of these models should lead to the higher critical points, (p, q)-models. This has

been shown explicitly by Distler [6] when he com- puted (p, 2) correlation functions by perturbing the (1,2) pure gravity. For the ( l ,q) models this procedure is more subtle and has not yet be~n implemented in the literature.

In the ( l ,q) models there is a major simplifi- cation because

P = Q ~ ' = D.

The string equation,

[P, QI = l,

theu leads to,

u~ = ~to ~ ua = z6to.

The integration constants were set to zero in order to have well-defined scaling dimensions.

Note that,

d i m ( x ) = q ==~ d i m ( ~ z z ) = q + l , (36)

which is inconsistent with the scaling behavior of the constants in ut. We thus have that,

Q = D ~ + z ,

P = O. (37)

One should not be deceived by t lis simple result.

The dependence on the other KdV parameters, l~, has been dropped, hut it will bc necessary to i:~cludL it in order to compute correlation func- t'ons.

3.2. Correlation .functions and selection rules

The starting point for the computation of the

correlation functions is the specific heat,

(PIP1) = 1 qn~_~_ = aesQ 11~. (38)

The higher point functions follow from the KdV flows, eq. 35.

(PIP, Pi) = ~ ( P I P I )

= Res[Pi+,Q IIq]

= (ResPd'. (39)

Correlation functions involving fewer than one

pu~lcture require an integration,

(PIP,) =- ResPi, (40)

D. Montano / Unconventional Methods in 2D Quantum Gravity 209

(P~P, Pi) = ~s[e,+,P~], (40

(ele, GPk) = aes{[[A+, P + q+, G] (42)

+ [Pi+, [Pt+, Pj]]}. (43)

Before proceeding into an explicit computa-

tion, it is useful to recall the ghost number con- servation of the topCogi.-ai theories in Section 2.

We expect a similar conservation law to hold in

the KdV gravity; we will see that tha t is indeed

the case [9]. It should be observed tha t any cor-

relation function is of the form,

(PIPi,Pi~.. .Pi.) = ResO. (44)

One can then show that ,

n - I o rdO = i . - - E ( q + l - i j )

.i=1 n

-- q + 1 - y ~ g h ( e , , ) , (4S) j=l

where the ghost number is defined to be,

gh(Pi , ) ~ q + 1 - ij. (46)

Now in order that O has a residue we require

that ,

n I - Egh(Pi,) _> - I , (47) q +

or equivalently,

n E gh(PG) _< 2 ( q + 1), (48) j=o

where gh(Pio) =gh(P l ) - q. The second se-

lection rule arises only in the topological limit,

z ~ O ,

n E / . / = 0 mud (q + 1 ). (49) j=0

This follows from the requirement tha t ResO ~ 0

as x ~ 0. Thus, ResO must be a constant which

as seen earlier (eq. 36) has a scaling dimension,

0 mud q + I. Combining eqs. 48 and 49, we get

the following impor tant selection rule which cor-

responds to ghost-number conservation, n

~] g h ( & ) = 2(I - ~)(q + t), (50) j=O

where g is a half-integer. Thus, for a correlation

function not to vanish we require eq. 50 to be

satisfied. It should be noted tha t g can be in-

terpreted as the genus, and thus we know from

which genera each correlation roeeives n contri-

bution. It should also be noted tha t the con-

served charge in eq. 50 corresponds to the di-

mension of the moduli space of flat connections

of S t (q, R). Using the above, we are now able to write an

explicit expression for Pi,

Pi = D i + i:rDi-q - i (q . - i)D~_V_ 1 q 2q

_ i ( q - i)r2Di_2q 2q 2

+ i ( q - i ) ( 2 q - i ) z D i _ 2 q - 1 2q 2

_ i(q - i)(2q - i ) ( S q - 3i + 4)Di_~.,_2 24q 2

+ i(q - i)(2q - i ) t3Di_3 , + ... (.51) 6q 3

This expression may be verified by no,.ing tha t

PIPj = Pi+j and Pq = Dq + z = Q . i can now provide some explict correlatioh

functions for ( I ,q ) models,

(PIP/) = q - lz61,q_ 1 q

(P' P'Pa) = ~ 6 , + j , . (52)

Tile higher point functions of the primaries van-

ish even when they obey the selection rule,

eq..50. Some correlation functions of desccndants

are,

210 D. Montane / Unconventional Methods it; 21) Quantum Gravity

1 I'41 ( # + ~ ¢ k ( P , - l ) = ~ ] ' [ ( " q - 1)

(P~l(Pi)~k(."q-i+=) = ( t + ] ) ( - l ) k - I 24qk+2

k4-2 i ( i - q ) ( i - 2q) ] ' ] ( i - n~ - 2) .

Note ~ow t.~e above equations appear to arise

from contact terms, l~urther details and a more

extensive list of cv~-ielation functions can be

found in references [9,13].

4. Concluding remarks

We have now seen how the ( l ,q) models of 2D

gravity may be solved without resorting to a field

theory interpretation. We have seen the power

of self-consist, ency and the KdV equations; how-

ever, there are still many open questions. Can

the arguments of self-consistency be used to solve

topological gravity with more than one primary

field? We arc currently working on this prob-

lem and the results seem promising. #3 There has

been speculation that the missing recursion rela-

tions correspond to W, constraints. We will soon

be able to answer that question with certainty by

fxplicitly providing the recursioq relations which

match the KdV. There is also the question of how

exactly to perturb the (l ,q) models in order to

arrive at the gelleral (p,q) models. We have, un-

fortunately, nrthing to say on this question. It

appears to be a difficult prob!~w. Overall, one

can ~ly that there has been much progress but

many interesting questions remain.

Acknowledgements

1 am grateful to Gil W.~ lis and Helen Tuck for

proof-reading this manuscript.

Re fe r ences

[1] E. Witten, Nucl. Phys. B340(1990) 281. [2] F. Vedinde and H V*rlinde, " A .qol~tion to ~D

Gravltl/," PUPT-1176 (1990). [3] M. Douglas, PAys. Lett B238 (1990) 176. [4] E. Brezln, V. Kazakov, PAls. Left. B23e (1990) 144;

M. Douglas, S. Shenker, NRcL Phys. B33fi {1990) 635; D. Gross, A. Migdsd, PArs. Re~. Left. 64 (1990) 127.

[5] F. D,:.vld, Mod. PArs. Lett. A3 (1958) 1651; J. Distler, H. Kawal, NacL PAys. B321 (1989) 599.

[G] J. Distler, N~el. Phys. B342 (1990) 523. [7] Ft. Dijkgraaf, E. Witten, NIcl. PAys. B342 (l~JO)

486. [8] K. Aokl, D. Monlano, J. So~lenschein, "The role o/

the contact algebra in mwltiraatrix models," CALT- 68-1677, to appear in the Inter. Jol t . ~] Mod. PAys.

A (t~1). [91 D. Montano, G. Rivl~s, Nzcl. Phys. 13360 (1991) 524. [10] D. Montane, J. Sonnenschein, N~cl. Phys. B324

(1989) 348. [11] K. Li, N~cl. Phys. B354 (1991) 711. [12] P. Di Francesco, D. Kuts~ov, ~Correlation Fune.

tiuns in ~-D String Theor~h" PUPT-1237 (1991). [1~] G. Hivlis, "Two Topics in ~D Qvant~m Field The-

ory," PhD. thesis, Caltech, (1991)

#a Work in progre~ ~ .h Gil Rivlir and Jacob Sonnen. schein.