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.
Top Related