't Hooft expansion of multi-boundary correlators in 2d topological gravity
PPrepared for submission to JHEP ’t Hooft expansion of multi-boundary correlators in 2dtopological gravity
Kazumi Okuyama a and Kazuhiro Sakai b a Department of Physics, Shinshu University,3-1-1 Asahi, Matsumoto 390-8621, Japan b Institute of Physics, Meiji Gakuin University,1518 Kamikurata-cho, Totsuka-ku, Yokohama 244-8539, Japan
E-mail: [email protected], [email protected]
Abstract:
We study multi-boundary correlators of Witten-Kontsevich topological grav-ity in two dimensions. We present a method of computing an open string like expansion,which we call the ’t Hooft expansion, of the n -boundary correlator for any n up to anyorder by directly solving the KdV equation. We first explain how to compute the ’t Hooftexpansion of the one-boundary correlator. The algorithm is very similar to that for thegenus expansion of the open free energy. We next show that the ’t Hooft expansion ofcorrelators with more than one boundaries can be computed algebraically from the corre-lators with a lower number of boundaries. We explicitly compute the ’t Hooft expansionof the n -boundary correlators for n = 1 , ,
3. Our results reproduce previously obtainedresults for Jackiw-Teitelboim gravity and also the ’t Hooft expansion of the exact result ofthe three-boundary correlator which we calculate independently in the Airy case. a r X i v : . [ h e p - t h ] J a n ontents In a recent paper [1] it was shown that the path integral of Jackiw-Teitelboim (JT) gravity[2, 3] is equivalent to a certain double-scaled random matrix model. The genus expan-sion of this random matrix model describes the splitting/joining of the baby universes[1]. In this correspondence we can consider the average (cid:104) Z ( β ) (cid:105) of the partition function Z ( β ) = Tr e − βH where the average is defined by the integral over the random matrix H .More generally, we can consider the multi-point function (cid:104) (cid:81) ni =1 Z ( β i ) (cid:105) of the partition func-tions Z ( β i ) ( i = 1 , . . . , n ). On the bulk gravity side it corresponds to the multi-boundarycorrelator, i.e. the gravitational path integral on the spacetime with n boundaries withfixed lengths β i . As argued in [4, 5], the connected part (cid:104) (cid:81) ni =1 Z ( β i ) (cid:105) conn of this correlatorcomes from the contribution of the Euclidean wormhole connecting the n boundaries.Of particular interest is the two-point function (cid:104) Z ( β ) Z ( β ) (cid:105) or its analytic continua-tion (cid:104) Z ( β + i t ) Z ( β − i t ) (cid:105) , known as the spectral form factor. The spectral form factor is auseful diagnostic of the quantum chaos [6, 7] and exhibits the characteristic behavior calledramp and plateau, as a function of time. The ramp comes from the eigenvalue correlations[8] while the plateau arises from the pair-creation of eigenvalue instantons [9]. The transi-tion from the ramp to plateau occurs at what is called Heisenberg time t H ∼ g − , where g s is the genus-counting parameter. Around this time scale, the operator insertion Z ( β ± i t )into the matrix integral backreacts to the eigenvalue distribution and two eigenvalues arepulled out from the dominant support (or cut) of the eigenvalue distribution.– 1 –his reminds us of the “giant Wilson loop” in 4d N = 4 SU ( N ) super Yang-Millstheory. In that case, path integral of the expectation value of the 1 / k , when k is of order N the dual object is a fundamental string on AdS × S ,but when k becomes of order N the bulk dual object morphs into a D3-brane [12]. In theGaussian matrix model picture, what is happening for k ∼ O ( N ) is that one eigenvalueis pulled out from the cut due to the insertion of the large Wilson loop operator into thematrix integral [13, 14]. This is exactly the same mechanism as the ramp-plateau transitionin the spectral form factor [9]. The different bulk dual pictures of the winding Wilson loopfor k ∼ O ( N ) and k ∼ O ( N ) are reflected in the different forms of the genus expansion:closed string like expansion for k ∼ O ( N ) and open string like expansion for k ∼ O ( N ).The above discussion suggests that one can study the open string like expansion of thecorrelators (cid:104) (cid:81) i Z ( β i ) (cid:105) by taking the following scaling limit g s (cid:28) , β i (cid:29) s i = g s β i , (1.1)which we call the ’t Hooft limit. Indeed, in our previous papers [15, 16] we studied the’t Hooft limit of the multi-boundary correlators in the JT gravity matrix model. In [15]we pointed out that JT gravity is a special case of general Witten-Kontsevich topologicalgravity, where infinitely many couplings t k are turned on with a specific value t k = γ k with γ = γ = 0 , γ k = ( − k ( k − k ≥ . (1.2)In this paper we consider the ’t Hooft expansion of the correlators (cid:104) (cid:81) i Z ( β i ) (cid:105) for Witten-Kontsevich topological gravity with general couplings t k . We find that the leading term ofthis expansion is closely related to the open free energy defined via the Laplace transformof Baker-Akhiezer function [17, 18]. We also show that the higher order corrections to the’t Hooft expansion of the correlators (cid:104) (cid:81) i Z ( β i ) (cid:105) can be systematically obtained from theKdV equation (2.10) or (5.15). It turns out that the ’t Hooft expansion of the one-pointfunction is obtained by using a similar algorithm for the computation of open free energydeveloped in [19], while the ’t Hooft expansion of n -point function with n ≥ n -point functioncan also be computed algebraically. In section 6, we use this method to obtain the ’t Hooftexpansion of the three-point function. We also calculate the exact result in the Airy case.Finally we conclude in section 7. In appendix A, we present a proof of the relation (5.15). We would like thank Shota Komatsu for discussion on the analogy with giant Wilson loops. – 2 –
Multi-boundary correlators in topological gravity
In Witten-Kontsevich topological gravity [20, 21] (see e.g. [18] for a recent review) observ-ables are made up of the intersection numbers (cid:104) τ d · · · τ d n (cid:105) g,n = (cid:90) M g,n ψ d · · · ψ d n n , d , . . . , d n ∈ Z ≥ . (2.1)They are defined on a closed Riemann surface Σ of genus g with n marked points p , . . . , p n .We let M g,n denote the moduli space of Σ and M g,n the Deligne-Mumford compactificationof M g,n . Here τ d i = ψ d i i and ψ i is the first Chern class of the complex line bundle whosefiber is the cotangent space to p i . The generating function for the intersection numbers isdefined as F ( { t k } ) := ∞ (cid:88) g =0 g g − F g ( { t k } ) , F g ( { t k } ) := (cid:68) e (cid:80) ∞ d =0 t d τ d (cid:69) g . (2.2)In this paper we consider the n -boundary connected correlator (which we also call the n -point function) [22] Z n ( { β i } , { t k } ) = (cid:42) n (cid:89) i =1 Z ( β i ) (cid:43) conn (cid:39) B ( β ) · · · B ( β n ) F ( { t k } ) , (2.3)where B ( β ) := g s (cid:114) β π ∞ (cid:88) d =0 β d ∂∂t d . (2.4) B ( β ) can be thought of as the “boundary creation operator.” The symbol “ (cid:39) ” in (2.3)means that the equality holds up to an additional non-universal part [22] when 3 g − n <
0. Such a deviation appears only in the genus-zero part of n = 1 , n ≥ Z n as well as F satisfy a set of simple differential equations, which allow us to computetheir genus expansion. To see this, let us first introduce the notation (cid:126) := g s √ , x := t (cid:126) τ := t (cid:126) (2.5)and ∂ k := ∂∂t k , (cid:48) := ∂ x = (cid:126) ∂ , ˙ := ∂ τ = (cid:126) ∂ . (2.6)The differential equations are simply written in terms of the derivatives W n := Z (cid:48) n , W := F (cid:48) , u := g ∂ F = 2 F (cid:48)(cid:48) . (2.7)– 3 –ecall that u satisfies the KdV equation [20, 21]˙ u = uu (cid:48) + 16 u (cid:48)(cid:48)(cid:48) . (2.8)Integrating this equation once in x we obtain˙ W = ( W (cid:48) ) + 16 W (cid:48)(cid:48)(cid:48) . (2.9)Since B ( β i ) commutes with ˙ = ∂ τ and (cid:48) = ∂ x , we immediately obtain a differential equationfor W n (cid:39) B ( β ) · · · B ( β n ) W by simply acting B ( β ) · · · B ( β n ) on both sides of the aboveequation. The result is [16]˙ W n ( β , . . . , β n ) = (cid:88) I ⊂ N W (cid:48)| I | W (cid:48)| N − I | + 16 W (cid:48)(cid:48)(cid:48) n ( β , . . . , β n ) . (2.10)Here N = { , , . . . , n } , W (cid:48)| I | = W (cid:48)| I | ( β i , . . . , β i | I | ) with I = { i , i , . . . , i | I | } and the sum istaken for all possible subset I of N including the empty set.As explained in [16] one can solve this equation and compute the genus expansion of W n up to any order. The genus expansion of Z n is then obtained by merely integrating W n once in x . This can be done without ambiguity. In [23] we demonstrated this computationin the (off-shell) JT gravity case t k = γ k ( k ≥ t k by merely replacing B n → ( − n +1 I n +1 ( n ≥ . (2.11)Here I n = I n ( u , { t k } ) = ∞ (cid:88) (cid:96) =0 t n + (cid:96) u (cid:96) (cid:96) ! ( n ≥
0) (2.12)are Itzykson-Zuber variables [24], u := ∂ F (2.13)is the genus-zero part of u and B n are Itzykson-Zuber variables restricted to the JT gravitycase B n = ( − n +1 I n +1 ( u , { t k = γ k ( k ≥ } ) ( n ≥ ∞ (cid:88) k =0 ( − k u k k !( k + n )! . (2.14)The key to solving (2.10) order by order is the change of variables ∂ = 1 t ( ∂ u − I ∂ t ) , ∂ = u ∂ − ∂ t . (2.15) This change of variables was originally introduced by Zograf (see e.g. [25]). – 4 –hat is, instead of t and t we take u and t := ( ∂ u ) − = 1 − I (2.16)as independent variables and regard t k ≥ as parameters. In the new variables the inte-gration constant is trivially fixed at every step of solving the differential equation. This isensured by F = − log t and by the fact that F g ( g ≥
2) are polynomials in the generators I n ≥ and t − [24, 26, 27].To summarize, we know that one can compute the small g s expansion of the n -boundarycorrelator Z n up to any order. This expansion can be thought of as a closed string likeexpansion. Interestingly, Z n also admits an open string like expansion. This is again asmall g s expansion, but is performed in the ’t Hooft regime (1.1). In the rest of the paperwe will show that one can also compute this expansion up to any order. In the scaling regime (1.1) the one-point function admits the ’t Hooft expansion F = log Z = ∞ (cid:88) ˜ g =0 g ˜ g − F ˜ g , (3.1)where F ˜ g is a function of s = g s β/ t k . In [15] we calculated F ˜ g with ˜ g = 0 , , t k = γ k ( k ≥
0) by saddle point method. In [16] we generalized thecalculation to the “off-shell” case t k = γ k ( k ≥
2) with t , t being unfixed. In what followswe will present a method of computing F ˜ g up to any ˜ g with general t k by solving thedifferential equation (2.10). This method is very similar to that of computing the genusexpansion of the open free energy [19] and is much more efficient than the saddle pointcalculation.Instead of directly dealing with (3.1), we first compute the genus expansion G = log W = ∞ (cid:88) ˜ g =0 g ˜ g − G ˜ g . (3.2)Since G and F are related by G = F + log ∂ x F , (3.3)the expansion (3.1) will immediately be obtained once (3.2) is computed. The differentialequation (2.10) for n = 1 is written as ∂ W = u∂ W + g ∂ W . (3.4) F ˜ g in this paper are related to those in our previous work [15, 16] by F here˜ g ↔ √ − ˜ g F there˜ g with theidentification λ = √ s . – 5 –his implies ∂ G = u∂ G + g (cid:16) ∂ G + 3 ∂ G∂ G + ( ∂ G ) (cid:17) , (3.5)which is rewritten as − ∂ t G = g
12 ( ∂ G ) + g ∂ G∂ G + g ∂ G + ( u − u ) ∂ G. (3.6)Recall that the genus expansion u = ∞ (cid:88) g =0 g g s u g (3.7)is computed by solving the KdV equation (2.8) (see e.g. [15, 19] for the results in ourconvention). By plugging (3.2) and (3.7) into (3.6) one obtains − ∂ t G = 112 ( ∂ G ) ,DG = 14 ∂ G ∂ G (3.8)for ˜ g = 0 , DG ˜ g = 112 (cid:88) ≤ i,j,k< ˜ gi + j + k =˜ g ∂ G i ∂ G j ∂ G k + 14 ˜ g − (cid:88) k =0 ∂ G ˜ g − k − ∂ G k + 112 ∂ G ˜ g − + (cid:98) ˜ g (cid:99) (cid:88) k =1 u k ∂ G ˜ g − k (3.9)for ˜ g ≥
2. Here we have introduced the differential operator D := − ∂ t −
14 ( ∂ G ) ∂ . (3.10)In what follows we will solve the above differential equations and compute G ˜ g . Firstof all, the explicit form of G is obtained as follows. Recall that W is related to theBaker-Akhiezer function ψ ( ξ ) as [15] W ( s ) = e G ( s ) = (cid:90) ∞−∞ dξe sξg s ψ ( ξ ) (3.11)and ψ ( ξ ) is expanded as [15, 19] ψ ( ξ ) = exp (cid:32) ∞ (cid:88) ˜ g =0 g ˜ g − A ˜ g (cid:33) (3.12)with A = − tz ∞ (cid:88) n =1 I n +1 (2 n + 3)!! z n +3 , z = (cid:112) ξ − u ) . (3.13)– 6 –he integral (3.11) can be evaluated by the saddle point method. G is given by G = 2 sξ ∗ + 2 A ( ξ ∗ ) , (3.14)where the saddle point ξ ∗ is determined by the condition ∂ ξ [2 sξ + 2 A ( ξ )] (cid:12)(cid:12)(cid:12) ξ = ξ ∗ = 0 . (3.15)This is equivalent to s = − ∂ ξ A (cid:12)(cid:12)(cid:12) ξ = ξ ∗ = tz ∗ − ∞ (cid:88) n =1 I n +1 (2 n + 1)!! z n +1 ∗ , z ∗ = (cid:112) ξ ∗ − u ) . (3.16)As we showed in [19], this relation is inverted as z ∗ = (cid:88) j a ≥ (cid:80) a j a = k (cid:80) a aj a = n (2 n + k )!(2 n + 1)! s n +1 t n + k +1 ∞ (cid:89) a =1 I j a a +1 j a !(2 a + 1)!! j a . (3.17)Plugging this back into (3.14) we obtain the explicit form of G . In fact, G is exactlytwice the genus zero part F o0 of the open free energy studied in [19], for which the followingsimple expression is available: G = 2 F o0 = 2 u s + 2 (cid:88) j a ≥ (cid:80) a j a = k (cid:80) a aj a = n (2 n + k + 1)!(2 n + 3)! s n +3 t n + k +2 ∞ (cid:89) a =1 I j a a +1 j a !(2 a + 1)!! j a . (3.18)It also follows that [19] ∂ G = 2 z ∗ , ∂ s G = 2 ξ ∗ . (3.19)In terms of z ∗ the differential equations (3.8) are written as − ∂ t G = 23 z ∗ ,DG = z ∗ ∂ z ∗ (3.20)and D in (3.10) becomes D = − ∂ t − z ∗ ∂ . (3.21)We saw in [19] that G = 2 F o0 indeed satisfies the first equation in (3.20).The operator D has interesting properties. (This is analogous to D in [19].) Forinstance, we see that Dz ∗ = − ∂ t z ∗ − z ∗ ∂ z ∗ = z ∗ ∂ s z ∗ − z ∗ ∂ ( ξ ∗ − u ) = z ∗ ∂ ξ ∗ − z ∗ ∂ ( ξ ∗ − u )= z ∗ t ,Dξ ∗ = − ∂ t ξ ∗ − z ∗ ∂ ξ ∗ = − z ∗ ∂ t z ∗ − z ∗ ∂ s z ∗ = 0 , (3.22)– 7 –here we have used [19] − ∂ t z ∗ = z ∗ ∂ s z ∗ . (3.23)We can also show that Dξ ( n ) ∗ = D∂ s ξ ( n − ∗ = − ∂ t ∂ s ξ ( n − ∗ − z ∗ ∂ ∂ s ξ ( n − ∗ = − ∂ s ∂ t ξ ( n − ∗ − ∂ s (cid:0) z ∗ ∂ ξ ( n − ∗ (cid:1) + ( ∂ s z ∗ ) ∂ ξ ( n − ∗ = ∂ s Dξ ( n − ∗ + 2 z ∗ ( ∂ s z ∗ ) ∂ n − s ∂ ξ ∗ = ∂ s Dξ ( n − ∗ − ∂ t z ∗ ) z ( n ) ∗ , (3.24)where ξ ( n ) ∗ := ∂ ns ξ ∗ , z ( n ) ∗ := ∂ ns z ∗ ( n ≥ . (3.25)From this we find Dξ ( n ) ∗ = − n − (cid:88) k =0 ∂ ks (cid:0) z ( n − k ) ∗ ∂ t z ∗ (cid:1) = − n − (cid:88) k =0 k (cid:88) (cid:96) =0 (cid:32) k(cid:96) (cid:33) (cid:0) ∂ k − (cid:96)s z ( n − k ) ∗ (cid:1) ∂ (cid:96)s ∂ t z ∗ = 2 n − (cid:88) k =0 k (cid:88) (cid:96) =0 (cid:32) k(cid:96) (cid:33) z ( n − (cid:96) ) ∗ ∂ (cid:96)s ( z ∗ ∂ s z ∗ )= 2 n − (cid:88) k =0 k (cid:88) (cid:96) =0 (cid:32) k(cid:96) (cid:33) z ( n − (cid:96) ) ∗ (cid:96) (cid:88) m =0 (cid:32) (cid:96)m (cid:33) z ( (cid:96) − m ) ∗ z ( m +1) ∗ = 2 n − (cid:88) k =0 k (cid:88) (cid:96) =0 (cid:96) (cid:88) m =0 k !( k − (cid:96) )!( (cid:96) − m )! m ! z ( n − (cid:96) ) ∗ z ( (cid:96) − m ) ∗ z ( m +1) ∗ . (3.26)As explained in [19] z ( n ≥ ∗ can be expressed in terms of ξ ( n ≥ ∗ and z ∗ : z (1) ∗ = ξ (1) ∗ z ∗ ,z (2) ∗ = ξ (2) ∗ z ∗ − (cid:0) ξ (1) ∗ (cid:1) z ∗ ,z (3) ∗ = ξ (3) ∗ z ∗ − ξ (1) ∗ ξ (2) ∗ z ∗ + 3 (cid:0) ξ (1) ∗ (cid:1) z ∗ . (3.27)– 8 –herefore Dξ ( n ≥ ∗ can also be expressed in terms of ξ ( n ≥ ∗ and z ∗ . For n = 1 , , Dξ (1) ∗ = 2 (cid:0) ξ (1) ∗ (cid:1) z ∗ ,Dξ (2) ∗ = 6 ξ (1) ∗ ξ (2) ∗ z ∗ − (cid:0) ξ (1) ∗ (cid:1) z ∗ ,Dξ (3) ∗ = 8 ξ (1) ∗ ξ (3) ∗ z ∗ + 6 (cid:0) ξ (2) ∗ (cid:1) z ∗ − (cid:0) ξ (1) ∗ (cid:1) ξ (2) ∗ z ∗ + 18 (cid:0) ξ (1) ∗ (cid:1) z ∗ . (3.28)On the other hand, as in [19] we evaluate the r.h.s. of (3.9) using ∂ z ∗ = z (1) ∗ z ∗ − tz ∗ ,∂ ξ ( n ) ∗ = z ( n +1) ∗ ( n ≥ . (3.29)In this way, one can express both sides of (3.9) as a polynomial in the variables t − , I k ≥ , z − ∗ , (cid:0) ξ (1) ∗ (cid:1) − and ξ ( n ≥ ∗ .Almost in the same way as in [19], we can formulate the following algorithm to solve(3.9) and obtain G ˜ g from the data of { G ˜ g (cid:48) } ˜ g (cid:48) < ˜ g .(i) Compute the r.h.s. of (3.9) and express it as a polynomial in the variables t − , I k ≥ , z − ∗ , (cid:0) ξ (1) ∗ (cid:1) − and ξ ( n ≥ ∗ .(ii) Let t − m f ( I k , z ∗ , ξ ( n ) ∗ ) denote the highest order part in t − of the obtained expression.This part can arise only from D (cid:32) − f ( I k , z ∗ , ξ ( n ) ∗ )( m − t m − z ∗ I (cid:33) . (3.30)Therefore subtract this from the obtained expression.(iii) Repeat the procedure (ii) down to m = 3. Then all the terms of order t − automat-ically disappear and the remaining terms are of order t − or t . Note also that theexpression does not contain any I k .(iv) In the result of (iii), collect all the terms of order t − and let t − z ∗ ∂ z ∗ g ( z ∗ , ξ ( n ) ∗ ) denotethe sum of them. This part arises from Dg ( z ∗ , ξ ( n ) ∗ ) . (3.31)Therefore subtract this from the result of (iii). The remainder turns out to be inde-pendent of t .(v) In the obtained expression, let h (cid:0) ξ ( n ≥ ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) m (3.32)– 9 –enote the part which is of order z − ∗ as well as of the lowest order in (cid:0) ξ (1) ∗ (cid:1) − . Thispart arises from D (cid:32) h (cid:0) ξ ( n ≥ ∗ (cid:1) m + 1) (cid:0) ξ (1) ∗ (cid:1) m +1 (cid:33) . (3.33)Therefore subtract this from the obtained expression.(vi) Repeat the procedure (v) until the resulting expression vanishes.(vii) By summing up all the above obtained primitive functions we obtain G ˜ g .Using this algorithm we can compute G ˜ g up to high order. ( G is also obtained by solving(3.20).) The first few of G ˜ g are G = 12 log ξ (1) ∗ − log z ∗ ,G = − I t z ∗ − tz ∗ + 3 ξ (1) ∗ z ∗ − ξ (2) ∗ z ∗ ξ (1) ∗ + ξ (3) ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) ,G = I t z ∗ + I t z ∗ + I t z ∗ + 58 t z ∗ − I ξ (1) ∗ t z ∗ + I ξ (2) ∗ t z ∗ ξ (1) ∗ + 5 ξ (2) ∗ tz ∗ ξ (1) ∗ − ξ (1) ∗ tz ∗ − ξ (2) ∗ z ∗ − ξ (4) ∗ z ∗ (cid:0) ξ (1) ∗ (cid:1) + 7 ξ (3) ∗ ξ (2) ∗ z ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) + 3 (cid:0) ξ (1) ∗ (cid:1) z ∗ + 3 ξ (3) ∗ z ∗ ξ (1) ∗ − (cid:0) ξ (2) ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) + ξ (5) ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (3) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) − ξ (4) ∗ ξ (2) ∗ (cid:0) ξ (1) ∗ (cid:1) + 3 ξ (3) ∗ (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) . (3.34)– 10 – ˜ g can easily be obtained by inverting the relation (3.3). We obtain F = G = 2 F o0 , F = G − log( (cid:126) ∂ F )= 12 log ξ (1) ∗ − z ∗ − log (cid:126) , F = G − ∂ F ∂ F = − I t z ∗ − tz ∗ + 2 ξ (1) ∗ z ∗ − ξ (2) ∗ z ∗ ξ (1) ∗ + ξ (3) ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) , F = G − ∂ F ∂ F + 12 (cid:18) ∂ F ∂ F (cid:19) = I t z ∗ + I t z ∗ + I t z ∗ + 134 t z ∗ − I ξ (1) ∗ t z ∗ + I ξ (2) ∗ t z ∗ ξ (1) ∗ + 17 ξ (2) ∗ tz ∗ ξ (1) ∗ − ξ (1) ∗ tz ∗ + 7 ξ (3) ∗ ξ (2) ∗ z ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) z ∗ (cid:0) ξ (1) ∗ (cid:1) + ξ (3) ∗ z ∗ ξ (1) ∗ − ξ (4) ∗ z ∗ (cid:0) ξ (1) ∗ (cid:1) + 3 ξ (3) ∗ (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) − ξ (4) ∗ ξ (2) ∗ (cid:0) ξ (1) ∗ (cid:1) + 10 (cid:0) ξ (1) ∗ (cid:1) z ∗ − ξ (2) ∗ z ∗ + ξ (5) ∗ (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (3) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) − (cid:0) ξ (2) ∗ (cid:1) (cid:0) ξ (1) ∗ (cid:1) . (3.35)We computed F ˜ g for ˜ g ≤
13. We have checked that the above F ˜ g with ˜ g = 0 , , F ) with √ − ˜ g F ˜ g in [16] under theidentification (2.11). In this section let us consider ’t Hooft expansion of the two-boundary correlator. While Z itself admits ’t Hooft expansion, for many purposes it is convenient to consider instead’t Hooft expansion of (cid:101) Z ( s , s ) = Tr (cid:104) e β Q Π e β Q Π (cid:105) = Z ( s + s ) − Z ( s , s ) , (4.1)where Q := ∂ x + u, Π := (cid:90) x −∞ dx (cid:48) | x (cid:48) (cid:105)(cid:104) x (cid:48) | . (4.2)Correspondingly, let us define the derivative (cid:102) W ( s , s ) = ∂ x (cid:101) Z ( s , s ) = W ( s + s ) − W ( s , s ) (4.3)and “free energies” K (2) = log (cid:101) Z , G (2) = log (cid:102) W . (4.4)– 11 – (2) and K (2) are related by G (2) = K (2) + log ∂ x K (2) . (4.5)We consider the expansions G (2) = ∞ (cid:88) ˜ g =0 g ˜ g − G (2)˜ g , K (2) = ∞ (cid:88) ˜ g =0 g ˜ g − K (2)˜ g , (4.6)where G (2)˜ g and K (2)˜ g are functions of s i = g s β i / i = 1 ,
2) and t k .The differential equation (2.10) for n = 2 is written as ∂ W = u∂ W + g ∂ W + √ g s ∂ W ( s ) ∂ W ( s ) . (4.7)Subtracting (4.7) from (3.4) with s = s + s we obtain the differential equation for (cid:102) W ( s , s ) − ∂ t (cid:102) W = ( u − u ) ∂ (cid:102) W + g ∂ (cid:102) W − √ g s ∂ W ( s ) ∂ W ( s ) . (4.8)This implies √ g s ∂ G ( s ) ∂ G ( s ) e G ( s )+ G ( s ) − G (2) = ∂ t G (2) + ( u − u ) ∂ G (2) + g (cid:16) ∂ G (2) + 3 ∂ G (2) ∂ G (2) + ( ∂ G (2) ) (cid:17) . (4.9)In [16] we derived that K (2)0 ( s , s ) = F ( s ) + F ( s ) . (4.10)From this and (4.5) we have G (2)0 ( s , s ) = G ( s ) + G ( s ) (4.11)as the initial condition.We observe that starting with (4.11) and comparing both sides of (4.9) order by orderin the small g s expansion one can algebraically determine G (2)˜ g from the data of (cid:8) G (2)˜ g (cid:48) (cid:9) ˜ g (cid:48) < ˜ g and { G ˜ g (cid:48) } ˜ g (cid:48) ≤ ˜ g . For instance, for ˜ g = 1 we obtain G (2)1 = 12 log ξ (1)1 ∗ + 12 log ξ (1)2 ∗ − log z ∗ − log z ∗ − log( z ∗ + z ∗ ) + 32 log 2 . (4.12)As in the case of one-point function, K (2)˜ g can also be algebraically determined by (4.5)from the data of (cid:8) G (2)˜ g (cid:48) (cid:9) ˜ g (cid:48) ≤ ˜ g . Therefore, given the data of { G ˜ g (cid:48) } ˜ g (cid:48) ≤ ˜ g we can compute K (2)˜ g without any integration procedure. For instance, we obtain K (2)1 = G (2)1 − log (cid:0) (cid:126) ∂ K (2)0 (cid:1) = 12 log ξ (1)1 ∗ + 12 log ξ (1)2 ∗ − log z ∗ − log z ∗ − z ∗ + z ∗ ) + 12 log 2 − log (cid:126) . (4.13)– 12 –sing the above method we computed K (2)˜ g for ˜ g ≤
8. We verified that K (2)˜ g with˜ g = 1 , K (2)1 ) with √ − ˜ g K (2)˜ g givenin (3.52) of [16].As a further nontrivial check, let us compare the above results with the low temperatureexpansion of the two-point function studied in [16]. As we mentioned, the results in [16]is trivially generalized to the case of general t k by the replacement (2.11). Recall that thelow temperature expansion of e K (2) is written as (see (4.32) of [16] and notations therein) e K (2) = Tr( e β Q Π e β Q Π)= Erfc( (cid:112) D ) e h t + yT √ πh ∞ (cid:88) (cid:96) =0 T (cid:96) (cid:96) ! z (cid:96) − B ∞ (cid:88) (cid:96) =0 T (cid:96) +1 (cid:96) ! g (cid:96) = 12 √ πh e u T + h t − D (cid:34) e D Erfc( (cid:112) D ) ∞ (cid:88) (cid:96) =0 T (cid:96) (cid:96) ! z (cid:96) − t (cid:114) D π ∞ (cid:88) (cid:96) =0 T (cid:96) (cid:96) ! g (cid:96) (cid:35) . (4.14)Using the data of z (cid:96) , g (cid:96) (0 ≤ (cid:96) ≤
7) we verified that this expression indeed reproducesthe above obtained K (2)˜ g in the form of small- s expansion. We performed the expansion of K (2)˜ g (2 ≤ ˜ g ≤
5) up to the order of s − ˜ g ) and observed perfect agreement. Since small- s expansion of K (2)˜ g (˜ g ≥
2) starts at the order of s − ˜ g ) , this serves as a rather nontrivialcheck. In this section let us consider ’t Hooft expansion of multi-boundary correlators. As in thecase of the two-boundary correlator, it is convenient to consider the ’t Hooft expansion K ( n ) = log (cid:101) Z n = ∞ (cid:88) ˜ g =0 g ˜ g − K ( n )˜ g (5.1)with (cid:101) Z n ( β , . . . , β n ) = Tr( e β Q Π · · · e β n Q Π) . (5.2)– 13 – ( n )˜ g in (5.1) is a function of s i = g s β i / i = 1 , . . . , n ) and t k . As we saw in [16] Z n and (cid:101) Z n are related as Z ( β ) = Tr (cid:104) e βQ Π (cid:105) = (cid:101) Z ( β ) ,Z ( β , β ) = Tr (cid:104) e ( β + β ) Q Π − e β Q Π e β Q Π (cid:105) = (cid:101) Z ( β + β ) − (cid:101) Z ( β , β ) ,Z ( β , β , β ) = Tr (cid:104) e ( β + β + β ) Q Π + e β Q Π e β Q Π e β Q Π + e β Q Π e β Q Π e β Q Π − e β Q Π e ( β + β ) Q Π − e β Q Π e ( β + β ) Q Π − e β Q Π e ( β + β ) Q Π (cid:105) = (cid:101) Z ( β + β + β ) + (cid:101) Z ( β , β , β ) + (cid:101) Z ( β , β , β ) − (cid:101) Z ( β , β + β ) − (cid:101) Z ( β , β + β ) − (cid:101) Z ( β , β + β ) (5.3)for n = 1 , ,
3. In general, the relation is given by the formula [28] Z n ( β , . . . , β n ) = Tr log (cid:32) (cid:34) − n (cid:89) i =1 (1 + z i e β i Q ) (cid:35) Π (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O ( z ··· z n ) = Tr log n (cid:88) k =1 (cid:88) i < ···
0) is in perfect agreement with 2 − / K (3)2 (cid:12)(cid:12)(cid:12) y =0 ,t =1 of [16]. Note that in– 16 –he case of t k = γ k ( k ≥
0) we have t = 1 , I = 1 ,ξ (1) i ∗ = z i ∗ cos (cid:0) √ z i ∗ (cid:1) , ξ (2) i ∗ = √ z i ∗ sin (cid:0) √ z i ∗ (cid:1) + cos (cid:0) √ z i ∗ (cid:1) cos (cid:0) √ z i ∗ (cid:1) ,ξ (3) i ∗ = − z i ∗ cos (cid:0) √ z i ∗ (cid:1) + 3 √ (cid:0) √ z i ∗ (cid:1) cos (cid:0) √ z i ∗ (cid:1) + 6 z i ∗ cos (cid:0) √ z i ∗ (cid:1) . (6.3)Note also that z i ∗ ’s here are related to those in [16] by z here i ∗ = √ z there i ∗ . In this subsection we consider the Airy case corresponding to a particular subspace ofcouplings { t n } t , t (cid:54) = 0 , t k = 0 ( k ≥ . (6.4)In this case the Itzykson-Zuber variables I n in (2.12) become I = t + u t , I = t , I k ≥ = 0 , (6.5)and from the genus-zero string equation u = I we find u = t t , t = 1 − t . (6.6)From (6.5) and (6.6), one can see that various quantities introduced in section 3 take verysimple form F o0 = u s + s t , z ∗ = st , ξ ∗ = u + s t ,ξ (1) ∗ = st = z ∗ t , ξ (2) ∗ = 1 t , ξ ( k ≥ ∗ = 0 . (6.7)Using these we obtain the ’t Hooft expansion (6.1) of K (3) in the Airy case. For instance, K (3)2 and K (3)3 are given by K (3)2 = − z ∗ + z ∗ + z ∗ + 3 z ∗ z ∗ z ∗ tz ∗ z ∗ z ∗ ( z ∗ + z ∗ )( z ∗ + z ∗ )( z ∗ + z ∗ ) , K (3)3 = 18 t [ z ∗ z ∗ z ∗ ( z ∗ + z ∗ )( z ∗ + z ∗ )( z ∗ + z ∗ )] × [5 (cid:0) z ∗ + z ∗ + z ∗ (cid:1) + 6 (cid:0) ( z ∗ + z ∗ ) z ∗ + ( z ∗ + z ∗ ) z ∗ + ( z ∗ + z ∗ ) z ∗ (cid:1) + 14 (cid:0) z ∗ z ∗ z ∗ + z ∗ z ∗ z ∗ + z ∗ z ∗ z ∗ (cid:1) − (cid:0) z ∗ z ∗ + z ∗ z ∗ + z ∗ z ∗ (cid:1) + 21 z ∗ z ∗ z ∗ ] . (6.8)In the Airy case, it is known that multi-point functions Z n ( β , · · · , β n ) can be writtenin the integral representation [29]. In particular, we can write down Z n for n = 1 , Z ( β ) = 1 − t g s (cid:112) πβ exp (cid:18) βt − t + g β − t ) (cid:19) ,Z ( β , β ) = Z ( β + β )Erf (cid:32) g s (cid:112) β β ( β + β )2 √ − t ) (cid:33) , (6.9)where Erf( z ) denotes the error functionErf( z ) = 2 √ π (cid:90) z dxe − x . (6.10)For n = 3, the closed form expression of Z ( β , β , β ) with β = β = β is obtained in[30] in terms of the Owen’s T -function T ( z, a ) = 12 π (cid:90) a dx e − z (1+ x ) x . (6.11)We can generalize the result of [30] for Z ( β , β , β ) with arbitrary β , β , β . As discussedin [31], one can determine Z by solving the KdV equation for W (2.10) on the subspace(6.4) using Z , in (6.9) as inputs. In this way we find Z ( β , β , β ) Z ( β + β + β ) = 1 − T (cid:32) g s (cid:112) β ( β + β )( β + β + β )2(1 − t ) , (cid:115) β β β ( β + β + β ) (cid:33) − T (cid:32) g s (cid:112) β ( β + β )( β + β + β )2(1 − t ) , (cid:115) β β β ( β + β + β ) (cid:33) − T (cid:32) g s (cid:112) β ( β + β )( β + β + β )2(1 − t ) , (cid:115) β β β ( β + β + β ) (cid:33) . (6.12)One can easily see that this reduces to the result of [30] when β = β = β .From this exact result (6.12) of Z , one can compute the ’t Hooft expansion (6.1) of K (3) = log (cid:101) Z in the Airy case. Using the relation between Z and (cid:101) Z in (5.3) and thefollowing property of the Owen’s T -function T ( z, ∞ ) = 14 −
14 Erf (cid:18) z √ (cid:19) , (6.13)we find (cid:101) Z ( β , β , β ) Z ( β + β + β ) =2 (cid:101) T (cid:32) g s (cid:112) β ( β + β )( β + β + β )2(1 − t ) , (cid:115) β β β ( β + β + β ) (cid:33) +2 (cid:101) T (cid:32) g s (cid:112) β ( β + β )( β + β + β )2(1 − t ) , (cid:115) β β β ( β + β + β ) (cid:33) +2 (cid:101) T (cid:32) g s (cid:112) β ( β + β )( β + β + β )2(1 − t ) , (cid:115) β β β ( β + β + β ) (cid:33) , (6.14) Strictly speaking, the closed form of Z , is obtained in [29] when t n = 0 ( n ≥ – 18 –here we defined (cid:101) T ( z, a ) = T ( z, ∞ ) − T ( z, a ) = 12 π (cid:90) ∞ a dx e − z (1+ x ) x . (6.15)In the large z regime with finite a , this is expanded as (cid:101) T ( z, a ) = e − (1+ a ) z π ∞ (cid:88) k =0 z − k +1) (cid:16) a ∂ a (cid:17) k a (1 + a )= e − (1+ a ) z π (cid:20) a (1 + a ) z − a a (1 + a ) z + 3 + 10 a + 15 a a (1 + a ) z + · · · (cid:21) . (6.16)Using this expansion, we have checked that the ’t Hooft expansion of the exact result of (cid:101) Z in (6.14) reproduces K (3)2 and K (3)3 in (6.8). This serves as a nontrivial consistency checkof our formalism. In this paper we developed a formalism to compute the ’t Hooft expansion of the multi-boundary correlators in topological gravity with general couplings t k . The ’t Hooft expan-sion of the n -point function can be obtained from the relation (5.15), which is equivalentto (2.10) for n = 1 ,
2. For the one-point function, we developed an algorithm for the com-putation of ’t Hooft expansion in section 3, which is almost parallel to the computation ofopen free energy studied in [19]. We find that the ’t Hooft expansion of n -point function( n ≥
2) is determined algebraically from the lower point functions by using (5.15). Ourcomputation reproduces the JT gravity case studied in [15, 16] and the ’t Hooft expansionof the exact result of correlators in the Airy case, as it should.There are several interesting open questions. It is suggested in [32] that we can definethe multi-point analogue of the spectral form factor in random matrix models and it exhibitsa similar behavior as the ramp and plateau. Using our formalism, it would be possible tostudy the ramp-plateau transition regime of the multi-point version of the spectral formfactor. We leave this as an interesting future problem.Recently, the quenched free energy (cid:104) log Z ( β ) (cid:105) of JT gravity is studied by the replicamethod [33–35]. It is shown in [31] that the quenched free energy is written as a certainintegral transformation of the generating function of multi-boundary correlators. The lowtemperature behavior of quenched free energy in JT gravity is quite interesting since it issuggested in [33] that JT gravity exhibits a spin glass phase at low temperature. In [31, 33]the quenched free energy is analyzed in the Airy regime β ∼ g − / . It would be interestingto study the quenched free energy of JT gravity in the ’t Hooft regime β ∼ g − using ourformalism. Acknowledgments
This work was supported in part by JSPS KAKENHI Grant Nos. 19K03845 and 19K03856,and JSPS Japan-Russia Research Cooperative Program.– 19 –
Proof of master differential equation
In this section we prove (5.15) for general n . Let us introduce the notation (cid:104) k ( p ) , l ( q ) (cid:105) := (cid:90) ∞−∞ dξ k (cid:90) ∞−∞ dξ k +1 · · · (cid:90) ∞−∞ dξ l e (cid:80) lj = k β j ξ j ∂ px ψ k K k,k +1 K k +1 ,k +2 · · · K l − ,l ∂ qx ψ l . (A.1)Derivatives of (cid:104) , n (cid:105) are calculated as (cid:104) , n (cid:105) (cid:48) = (cid:88) 23 ( ψ (cid:48)(cid:48)(cid:48) i ψ j + ψ i ψ (cid:48)(cid:48)(cid:48) j ) + u ( ψ (cid:48) i ψ j + ψ i ψ (cid:48) j ) + u (cid:48) ψ i ψ j (cid:21) = 23 ( ψ (cid:48)(cid:48) i ψ j + ψ i ψ (cid:48)(cid:48) j − ψ (cid:48) i ψ (cid:48) j ) + uψ i ψ j (A.4)and thus ( ∂ τ − u∂ x ) K ij = 23 ( ψ (cid:48)(cid:48) i ψ j + ψ i ψ (cid:48)(cid:48) j − ψ (cid:48) i ψ (cid:48) j ) . (A.5)Using (A.3) and (A.5) we obtain( ∂ τ − u∂ x ) (cid:104) , n (cid:105) = 23 (cid:88) B252 (1985) 343–356.[3] C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,”Phys. Lett. (1983) 41–45.[4] P. Saad, S. H. 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