Two dimensional gravity in genus one in Matrix Models, Topological and Liouville approaches
aa r X i v : . [ h e p - t h ] J un Two dimensional gravity in genus one in Matrix Models,Topological and Liouville approaches.
A.Belavin , G.Tarnopolsky
L.D. Landau Institute for Theoretical PhysicsChernogolovka, 142432, Russia
Abstract
One-matrix model in p -critical point on torus is considered. The generating function ofcorrelation numbers in genus one is evaluated and used for computation correlation numbersin KdV and CFT frames. It is shown that the correlation numbers in KdV frame in genus onesatisfy the Witten topological gravity recurrence relations. There exist several different approaches to the 2d Quantum gravity. One of them is the con-tinuous approach. In this approach the theory is determined by the functional integral overall metrics [1]. Calculation of this integral in the conformal gauge leads to the Liouville fieldtheory. Therefore this approach is called the Liouville gravity.The other way to describe the sum over 2d surfaces is the discrete approach. It is basedon the idea of approximation of two-dimensional geometry by an ensemble of planar graphs ofbig size. Technically the ensemble of graphs is usually defined by expansion into a series ofperturbation theory of the integral over N × N matrixes. That is why this approach is calledthe Matrix Models (further MM). References to the both approaches can be found in the review[2].After [3, 4, 2] the coincidence of the gravitational dimensions was found the conjecture thatboth of these approaches describe the same variant of the 2d Quantum gravity appeared. There-fore it was naturally to expect that the correlation numbers will are also the same. Howeverthe attempt of a naive identification of the correlation numbers doesn’t lead to the agreementin a general case.In [5, 6] a conjecture was proposed and checked that there exists a “resonance” transformationof coupling constants in Matrix Models, from the standard definition (the so-called KdV frame)to another (the so-called Liouville or CFT frame), such that new defined correlation numbersof MM coincide with naturally defined ones in the Liouville Gravity.The form of the transformation was conjectured in [6] for the particular case of the p -criticalOne-Matrix Model (OMM), which corresponds to the Minimal Liouville gravity MG / p +1 . Theconjectured identity of the correlation numbers was checked up to five-point case in genus zero[6, 7].At last the third approach – 2d Topological gravity was invented by Witten in [8], who builtaxiomatics of this theory along the lines of intersection theory . It was conjectured and checked for genus zero) in [8] that correlation numbers in Topological gravity and in Matrix modelscoincide. It should be mentioned that this fact takes place if correlation numbers in OMM arecalculated in KdV frame.The article is organized in the following way. At first we review the method of orthogonalpolynomials for the solution of Matrix model, the double scaling limit and Douglas sting equa-tion. Then we use these tools to compute the torus partition function in p -critical One-matrixmodel .We use the explicit expression for the partition function in genus one to compute the corre-lation numbers in KdV, as well as in CFT frames.The results in CFT frame should be compared with the correlation numbers in the MinimalLiouville gravity, which have not been computed yet. In genus one we expect a coincidencesimilar to that observed in genus zero [5, 6].After that, we evaluate explicitely the first three correlation numbers in KdV frame. After-wards we derive the recursion relation in genus one and observe that it coincides with the oneof Topological gravity. Then we compute two first correlation numbers in Conformal frame. Atthe end we discuss some open problems. Here we give some well-known aspects of the method of orthogonal polynomials [2]. Thepartition function which includes surfaces of all genera is Z ( v k , N ) = log Z dM e − tr V ( M ) , (1)where M is hermitian N × N matrix and V ( M ) = N P p +1 k =1 v k M k is the polynomial potentialwith different coupling constants v k , and in further calculation we fix v p +1 = ( p +1)! p !(2 p +2)! for sim-plicity. On the other hand, it is known [2] that partition function (1) is expressed through thesum Z = ∞ X h =0 N − h Z h , (2)where h is genus of surface and Z h is the partition function of all surfaces with genus h . In thispaper we evaluate torus partition function Z , expanding the integral (1) in /N series.In this section we carry on evaluation using the method of orthogonal polynomials.Since the integrand in (1) depends only on the eigenvalues of the matrix M , we can factorizethe integration measure into the product of the Haar measure for unitary matrices and anintegration measure for eigenvalues.Thus we have Z ( v k , N ) = log Z N Y i =1 dλ i ∆ ( λ ) e − P i V ( λ i ) , (3)where λ i ’s are the N eigenvalues of the hermitian matrix M and ∆( λ ) = Q i 0) = √− t . The correlation numbers are expressed through the formula h O k ...O k n i = ∂ n Z ∂t k ...∂t k n (cid:12)(cid:12)(cid:12)(cid:12) t = ... = t p − =0 , (54)where the index h i denotes the correlation numbers on torus. The first three correlationnumbers are (see Appendix D) h O k i = p + k u − k − c , h O k O k i = ( p + 2 + k + k )( k + k ) + 2 p − k k u − k − k − c , h O k O k O k i = 196 (cid:18) k k i p + 4) k + 2 k i + (6 p + 8) k + 8 p − k k k (cid:19) u − k − k − k − c , (55)where k = k + k + k , k i = k + k + k , and k i = k + k + k . Comparison with Topological Gravity In paper [8] E.Witten gave the definition of 2d Topological gravity. The recursion relationbetween correlation numbers has been derived in [8] by studying intersection theory. Usingthis relation Witten computed correlation numbers in genus zero and checked their coincidencewith expressions for correlation numbers in One-matrix model. This resemblance lead Wittento the conjecture about the equivalence between Topological gravity and One-matrix model,what was proved later by M.Kontsevich [13].In this section we show that the same recursion relation in genus one as well as in genuszero holds also in One-matrix model. Our apprach uses the explicit expression for the partionfunction of One-matrix model and differs from that used by Kontsevich.Let h i and h i denote the genus zero and genus one correlation numbers. The recursionrelations between correlation numbers from [8] look as follows h σ k σ k ...σ k s i = k X S = X ∪ Y h σ k − Y i ∈ X σ k i σ i h σ Y j ∈ Y σ k j σ k s − σ k s i , (56) h σ k σ k ...σ k s i = 112 k h σ k − σ k ...σ k s σ σ i + k X S = X ∪ Y h σ k − Y i ∈ X σ k i σ i h σ Y j ∈ Y σ k j i , (57)where σ k ↔ O p − k − and the symbol P S = X ∪ Y represent a sum over all decomposition of S = { , , ..., s } as a union of the two sets X and Y .Witten has shown that the relations (56) and (57) are fulfilled for correlation numbers h σ k ...σ k s i , of the more general theory which depends on relevant parameters { a k } ( a k ↔ t p − k − ) in such way that expectation values of any observable N obey ∂∂a k h N i = h σ k N i , (58)if the following relations h σ k σ k σ k i = k h σ k − σ i h σ σ k σ k i , (59) h σ k i = 112 k h σ k − σ σ i + k h σ k − σ i h σ i (60)hold.In One-matrix model the correlation numbers with arbitrary { t k } are given by the formula h O k ...O k n i , = ∂ n Z , ∂t k ...∂t k n . (61)Therefore the assumption (58) is holds automatically. The fulfillment of (59) for correlationnumbers in genus zero was checked by A.B. Zamolodchikov [11].Below we check (59) and (60) using explicit expressions for Z and Z from (49) and (50).In terms of observables O k in One-matrix model, the expressions (59) and (60) writes ( σ k ↔ O p − k − ) h O p − k − O p − k − O p − k − i = k h O p − k O p − i h O p − O p − k − O p − k − i , (62) h O p − k − i = 112 k h O p − k O p − O p − i + k h O p − k O p − i h O p − i . (63)From formula (49) at arbitrary { t k } one can get h O k O k i = ∂ Z ∂t k ∂t k = ( u ∗ ) p − k − k − p − k − k − , h O k O k O k i = ∂ Z ∂t k ∂t k ∂t k = − ( u ∗ ) p − k − k − k − P ′ ( u ∗ ) . (64) hus it is easy to see from (64) that equality (62) is fulfilled. From the torus partition function(50) also at arbitrary { t k } we get h O k i = ∂Z ∂t k = − p − k − P ′ ( u ∗ ) ( u ∗ ) p − k − + P ′′ ( u ∗ )12( P ′ ( u ∗ )) ( u ∗ ) p − k − . (65)The expressions (64) and (65) indeed satisfy the equation (63). Consequently we have provedthat the correlation numbers in One-matrix model in KdV frame satisfy the recurrence relation(56) and (57), assuming replacement σ k → O p − k − . The CFT frame is defined by a different set of parameters { λ k } , which are associated with { t k } by "resonance" transformation [6]. As it was shown in [6] after "resonance" transformation thepolynomial P ( u, { t k } ) from (52) up to the factor ( p +1)!(2 p − u p +1 c takes the form Q ( x, { λ k } ) = ∞ X n =0 p − X k ...k n =1 λ k ...λ k n n ! d n − dx n − L p − P k i − n ( x ) , (66)where x = u/u c , u c is u ∗ at λ , ..., λ p − = 0 and L n ( x ) are the Legendre polynomials. We alsoassume that (cid:0) ddx (cid:1) − L p = R L p dx = L p +1 − L p − p +1 . Below we use the notation Q k ...k n ( x ) = d n − dx n − L p − P k i − n ( x ) , Q ( x ) = L p +1 − L p − p + 1 . (67)The correlation numbers are expressed through the formula hO k ... O k n i = ∂ n Z ∂λ k ...∂λ k n (cid:12)(cid:12)(cid:12)(cid:12) λ = ... = λ p − =0 . (68)If we calculated correlation numbers for the partition function in genus zero, inserting thepolynomial Q ( x, { λ k } ) instead of P ( x, t k ) in (49) and (50) we would obtained results in [6].Let us compute the partition function in genus one. Thus taking into account the commonformulas for correlation numbers (97) and (98) in Appendix D and using some values forLegendre polynomials and consequently for polynomial Q ( x, { λ k } ) in critical point ( x = 1 ) Q ′ (1) = 1 , Q ′′ (1) = p ( p + 1)2 , Q ′′′ (1) = ( p + 2)( p + 1) p ( p − , Q k i = 1 ,Q ′ k i (1) = ( p − k i )( p − k i − , Q ′′ k i (1) = 18 Y r =1 ( p − r − k i + 2) ,Q ′ k i k j (1) = 18 Y r =1 ( p − r − k i − k j + 1) , (69)one can obtain from (68) the first two correlation numbers of the partition function in genusone in CFT frame hO k i = (2 p − k )( k + 1)24 , hO k O k i = − 124 (1 + k )(1 + k )(( k + k − p + 2)( k + k ) − k k − p ) . (70) Conclusion In this paper we have derived the torus partition function Z in p -critical One-matrix model.Using the explicit expression for the partition function in genus one we compute the correlationnumbers in KdV, as well as in CFT frames.We show the fulfillment of recurrence relation for correlation numbers in OMM in KdV framein genus one, which are the same as that in 2d Topological gravity .The results in CFT frame should be compared against the correlation numbers in the Min-imal Liouville gravity, which have not been computed yet. We expect the coincidence in genusone similarly that was observed on sphere [5, 6, 14] and on disk [15]. 10 Acknowledgements We are grateful to V. Belavin, M. Bershtein, M. Lashkevich, Ya. Pugai and A. Zamolodchikovfor useful discussions.This work was supported by Federal Program Scientific-Pedagogical Personnel of InnovationRussia (contract No. 02.740.11.5165) and by grant Scientific Schools 6501.2010.2. A.B. wassupported also by RFBR initiative interdisciplinary project grant 09-02-12446-ofi-m and RBRF-CNRS grant PICS-09-02-91064. AppendixA For our aim in section 3 we use Euler-Maclorein formula [12]. It helps express summation ofdiscrete function through integration of this function and some other terms.Let function f ( x ) is considered in section [ a, b ] . Let h = b − an , where n is natural number,then n X k =1 f ( a + ( k − h ) = 1 h Z ba f ( x ) dx − 12 ( f ( b ) − f ( a )) + ∞ X m =1 h m − B m (2 m )! ( f (2 m − ( b ) − f (2 m − ( a )) , (71)where B m are Bernoulli numbers ( B = 1 , B = − , B = ). In our analysis of torus partitionfunction we only use terms in Euler-Maclorein formula up to m = 1 . B In this section we deal with part of the sum from (10) ˜ W n = 2 nv n X { σ n − } R k + m · ... · R k + m n , (72)where { σ n − } denotes all paths of n − steps ( n − steps up and n steps down) starting at k and ending at k − . Each step down from m to m − receives a factor of R m and each stepup receives a factor of unity.We assume the existence of smooth function R ( ξ, N ) of variable ξ ∈ [0 , , such that ( kN , N ) = R k . Thus R ( ξ + m/N, N ) and ˜ W n have the Taylor expansion R ( ξ + m/N, N ) = R ( ξ, N ) + mN R ξ ( ξ, N ) + m N R ξξ ( ξ, N ) + O (cid:0) N (cid:1) , (73) ˜ W n = W n + 1 N W n + 1 N W n + O (cid:0) N (cid:1) . (74)Since R ( ξ + m N , N ) · ... · R ( ξ + m n N , N ) == R n + R n − R ξ N n X i =1 m i + R n − R ξξ N n X i =1 m i + R n − R ξ N n X i 1) = 1 . (90)Solution for this equation is a k = , thus we have A k = k ( k − (cid:16) C k k k +1 (cid:17) u k − . For the secondequation from (88) we put B k = b k k ( k − k − (cid:16) C k k k +1 (cid:17) u k − , then we derive equation for theconstants b k : b k +1 ( k + 1 / − b k ( k − / 2) = 1 , (91)and obtain b k = , therefore B k = k ( k − k − (cid:16) C k k k +1 (cid:17) u k − . We can see that these solutions for A k and B k satisfy the third equation in (88).Summarizing all results, we have P k = (cid:18) C k k k +1 (cid:19) u k , M k = 0 ,A k = k ( k − (cid:18) C k k k +1 (cid:19) u k − = 16 P ′′ k B k = k ( k − k − (cid:18) C k k k +1 (cid:19) u k − = 112 P ′′′ k , (92)therefore one can writes S k as follows S k = P k + ε (cid:18) P ′′ k u xx + 112 P ′′′ k u x (cid:19) + O ( ε ) . (93) D The singular part of the partition function on torus Z ( t , t , ...t p − ) is Z = − log P ′ ( u ∗ )12 , (94)where u ∗ = u ∗ ( t , t , ..., t p − ) is the suitably chosen root of the polynomial P ( u ) = u p +1 + t u p − + p − X k =1 t k u p − k − . (95) he correlation numbers are expressed through the formula h O k ...O k n i = ∂ n Z ∂t k ...∂t k n (cid:12)(cid:12)(cid:12)(cid:12) t = ... = t p − =0 . (96)Thus in common form first two correlation numbers are (denote P k = ∂ P /∂t k ) h O k i = − (cid:18) P ′ k P ′ − P ′′ P k ( P ′ ) (cid:19) , (97) h O k O k i = − (cid:18) P ′ k k P ′ − P ′′ k P k + P ′′ k P k + P ′ k P ′ k + P ′′ P k k ( P ′ ) ++ 2 P ′′ P ′ k P k + 2 P ′′ P ′ k P k + P ′′′ P k P k ( P ′ ) − P ′′ ) P k P k ( P ′ ) (cid:19) . (98)And the third correlation number is (denote P i = P k i = ∂ P /∂k i ) h O k O k O k i = − P ′ P ′ − P ′′ (12 P + P ′ (12 P + P ′′ (1 P + P ′′ P ( P ′ ) ++ 1( P ′ ) (cid:0) P ′′ P ′ (12 P + P ′′′ (1 P P + 2 P ′′ (1 P ′ P + P ′′′ P (12 P + 2 P ′′ P ′ (1 P + 2 P ′ P ′ P ′ (cid:1) −− P ′ ) (cid:0) P ′′ P ′′ (1 P P + 6 P ′′ P ′ (1 P ′ P + 2( P ′′ ) P (12 P + 3 P ′′′ P ′ (1 P P + P ′′′′ P P P (cid:1) ++ 1( P ′ ) (cid:0) P ′′ ) P ′ (1 P P + 7 P ′′′ P ′′ P P P (cid:1) − P ′′ ) P P P ( P ′ ) (cid:19) , (99)where parentheses denote symmetrization (for instance P ′′ (12 P = P ′′ P + P ′′ P + P ′′ P ).In KdV critical point i.e. t = ... = t p − = 0 , we have u c = u ∗ ( t , , ..., 0) = √− t , and fordifferen derivatives of polynomial P ( u ) from (95) one can get P ′ ( u c ) = 2 u pc , P ′′ ( u c ) = 2(2 p − u p − c , P ′′′ ( u c ) = 6( p − u p − c P k i ( u c ) = u p − k i − c , P ′ k i ( u c ) = ( p − k i − u p − k i − c , P ′′ k i ( u c ) = ( p − k i − p − k i − u p − k i − c , P k i k j ( u c ) = 0 . 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