't Hooft expansion of multi-boundary correlators in 2D topological gravity

We study multi-boundary correlators of Witten-Kontsevich topological gravity 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 any order by directly solving the Korteweg-De Vries equation. We first explain how to compute the 't Hooft expansion of the one-boundary correlator. The algorithm is very similar to that for the genus expansion of the open free energy. We next show that the 't Hooft expansion of correlators with more than one boundary can be computed algebraically from the correlators with a lower number of boundaries. We explicitly compute the 't Hooft expansion of the $n$-boundary correlators for $n=1,2,3$. Our results reproduce previously obtained results for Jackiw-Teitelboim gravity and also the 't Hooft expansion of the exact result of the three-boundary correlator which we calculate independently in the Airy case.


Introduction
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 expansion of this random matrix model describes the splitting/joining of the baby universes [1]. In this correspondence we can consider the average Z(β) 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 n i=1 Z(β i ) of the partition functions Z(β i ) (i = 1, . . . , n). On the bulk gravity side it corresponds to the multi-boundary correlator, i.e. the gravitational path integral on the spacetime with n boundaries with fixed lengths β i . As argued in [4,5], the connected part n i=1 Z(β i ) conn of this correlator comes from the contribution of the Euclidean wormhole connecting the n boundaries.
Of particular interest is the two-point function Z(β 1 )Z(β 2 ) or its analytic continuation Z(β + it)Z(β − it) , known as the spectral form factor. The spectral form factor is a useful diagnostic of the quantum chaos [6,7] and exhibits the characteristic behavior called ramp 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 transition from the ramp to plateau occurs at what is called Heisenberg time t H ∼ g −1 s , where g s is the genus-counting parameter. Around this time scale, the operator insertion Z(β ± it) into the matrix integral back-reacts to the eigenvalue distribution and two eigenvalues are pulled out from the dominant support (or cut) of the eigenvalue distribution. This reminds us of the "giant Wilson loop" in 4d N = 4 SU(N ) super Yang-Mills theory. 1 In that case, the path integral of the expectation value of the 1/2 Bogomol'nyi-Prasad-Sommerfield Wilson loops reduces to the Gaussian matrix integral [10,11]. For the winding Wilson loop with winding number k, when k is of the order of N 0 , the dual object is a fundamental string on AdS 5 × S 5 , but when k becomes of the order of N the bulk dual object morphs into a D3-brane [12]. In the Gaussian matrix model picture, what is happening for k ∼ O(N ) is that one eigenvalue is pulled out from the cut due to the insertion of the large Wilson loop operator into the matrix integral [13,14]. This is exactly the same mechanism as the ramp-plateau transition in the spectral form factor [9]. The above discussion suggests that one can study the open string like expansion of the correlators i Z(β i ) by taking the following scaling limit g s 1, β i 1 with s i = g s β i 2 fixed, (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 topological gravity, where infinitely many couplings t k are turned on with a specific value t k = γ k with In this paper we consider the 't Hooft expansion of the correlators i Z(β i ) for Witten-Kontsevich topological gravity with general couplings t k . We find that the leading term of this expansion is closely related to the open free energy defined via the Laplace transform of the Baker-Akhiezer function [17,18]. We also show that the higher-order corrections to the 't Hooft expansion of the correlators i Z(β i ) can be systematically obtained from the Korteweg-De Vries (KdV) equation (2.10) or (5.15). It turns out that the 't Hooft expansion of the one-point function is obtained by using a similar algorithm for the computation of open free energy developed in [19], while the 't Hooft expansion of an n-point function with n ≥ 2 is determined algebraically from the lower point functions.
This paper is organized as follows. In section 2, we briefly review how to compute the genus expansion of the multi-boundary correlators by solving the KdV equation. In section 3, we present a method of computing the 't Hooft expansion of the one-point function. In section 4, we explain that the 't Hooft expansion of the two-point function can be computed algebraically. In section 5, we show that the 't Hooft expansion of a general n-point function can also be computed algebraically. In section 6, we use this method to obtain the 't Hooft expansion 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).

Multi-boundary correlators in topological gravity
In Witten-Kontsevich topological gravity [20,21] (see e.g. [18] for a recent review) observables are made up of the intersection numbers They are associated with a closed Riemann surface Σ of genus g with n marked points p 1 , . . . , p n . We let M g,n denote the moduli space of Σ and M g,n the Deligne-Mumford compactification of M g,n . Here τ d i = ψ d i i and ψ i is the first Chern class of the complex line bundle whose fiber is the cotangent space to p i . The generating function for the intersection numbers is defined as In this paper we consider the n-boundary connected correlator (which we also call the n-point function) [22] B(β) can be thought of as the "boundary creation operator." The symbol " " in (2.3) means that the equality holds up to an additional non-universal part [22] when 3g −3+n < 0. Such a deviation appears only in the genus-zero part of n = 1-, 2-boundary correlators and does not affect their higher genus parts nor correlators with n ≥ 3 (see e.g. [16] for a more detailed explanation). Z n as well as F satisfy a set of simple differential equations, which allows us to compute their genus expansion. To see this, let us first introduce the notation and (2.6) The differential equations are simply written in terms of the derivatives Recall that u satisfies the KdV equation [20,21] u = uu + 1 6 u . (2.8) Integrating this equation once in x we obtaiṅ Since B(β i ) commutes with˙= ∂ τ and = ∂ x , we immediately obtain a differential equation for W n B(β 1 ) · · · B(β n )W 0 by simply applying B(β 1 ) · · · B(β n ) to both sides of the above equation. The result is [16] W n (β 1 , . . . , β n ) = I⊂N W |I| W |N −I| + 1 6 W n (β 1 , . . . , β n ). (2.10) . . , i |I| }, and the sum is taken for all possible subsets 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 [16] we demonstrated this computation in the (off-shell) JT gravity case t k = γ k (k ≥ 2), but as detailed in [23], all the results are immediately generalized to the case of general t k by merely replacing Here are Itzykson-Zuber variables [24], is the genus-zero part of u, and B n are Itzykson-Zuber variables restricted to the JT gravity case (2.14) The key to solving (2.10) order by order is the change of variables 2 That is, instead of t 0 and t 1 we take u 0 and as independent variables and regard t k≥2 as parameters. In the new variables the integration constant is trivially fixed at every step of solving the differential equation. This is ensured by F 1 = − 1 24 log t and by the fact that F g (g ≥ 2) are polynomials in the generators I n≥2 and t −1 [24,26,27].
To summarize, we know that one can compute the small g s expansion of the n-boundary correlator Z n up to any order. This expansion can be thought of as a closed string like expansion. Interestingly, Z n also admits an open string like expansion. This is again a small g s expansion, but is performed in the 't Hooft regime (1.1). In the rest of the paper we will show that one can also compute this expansion up to any order.

't Hooft expansion of the one-boundary correlator
In the scaling regime (1.1), the one-point function admits the 't Hooft expansion 3 where Fg is a function of s = g s β/2 and t k . In [15] we calculated Fg withg = 0, 1, 2 in the JT gravity case t k = γ k (k ≥ 0) by saddle point method. In [16] we generalized the calculation to the "off-shell" case t k = γ k (k ≥ 2) with t 0 , t 1 being unfixed. In what follows we will present a method of computing Fg up to anyg with general t k by solving the differential equation (2.10). The method is very similar to that of computing the genus expansion of the open free energy [19] and is much more efficient than the saddle point calculation.
Instead of directly dealing with (3.1), we first compute the genus expansion Since G and F are related by the expansion (3.1) will immediately be obtained once (3.2) is computed. The differential equation (2.10) for n = 1 is written as 3 Fg in this paper are related to those in our previous work [15,16] by F herẽ This implies which is rewritten as Recall that the genus expansion is computed by solving the KdV equation (2.8) (see e.g. [15,19] for the results in our convention). By plugging (3.2) and (3.7) into (3.6), one obtains forg ≥ 2. Here we have introduced the differential operator In what follows we will solve the above differential equations and compute Gg. First of all, the explicit form of G 0 is obtained as follows. Recall that W 1 is related to the Baker-Akhiezer function ψ(ξ) as [15] and ψ(ξ) is expanded as [15,19] The integral (3.11) can be evaluated by the saddle point method. G 0 is given by where the saddle point ξ * is determined by the condition This is equivalent to As we showed in [19], this relation is inverted as Plugging this back into (3.14), we obtain the explicit form of G 0 . In fact, G 0 is exactly twice the genus zero part F o 0 of the open free energy studied in [19], for which the following simple expression is available: It also follows that [19] In terms of z * , the differential equations (3.8) are written as and D in (3.10) becomes We saw in [19] that G 0 = 2F o 0 indeed satisfies the first equation in (3.20). The operator D has interesting properties. (This is analogous to D in [19].) For instance, we see that where we have used [19] −∂ t z * = z * ∂ s z * . (3.23) We can also show that From this we find As explained in [19], z (n≥1) * can be expressed in terms of ξ (n≥1) * and z * : can also be expressed in terms of ξ (n≥1) * and z * . For n = 1, 2, 3 we have On the other hand, as in [19] we evaluate the r.h.s. of (3.9) using In this way, one can express both sides of (3.9) as a polynomial in the variables t −1 , I k≥2 , . Almost in the same way as in [19], we can formulate the following algorithm to solve (3.9) and obtain Gg from the data of {Gg }g <g : (i) Compute the r.h.s. of (3.9) and express it as a polynomial in the variables t −1 , I k≥2 , denote the highest-order part in t −1 of the obtained expression. This part can arise only from (3.30) Therefore subtract this from the obtained expression.
(iii) Repeat procedure (ii) down to m = 3. Then all the terms of order t −2 automatically disappear and the remaining terms are of order t −1 or t 0 . Note also that the expression does not contain any I k .
(iv) In the result of (iii), collect all the terms of order t −1 and let t −1 z * ∂ z * g(z * , ξ (n) * ) denote the sum of them. This part arises from (3.31) Therefore subtract this from the result of (iii). The remainder turns out to be independent of t. This part arises from Therefore subtract this from the obtained expression.
(vi) Repeat procedure (v) until the resulting expression vanishes.
(vii) By summing up all the above-obtained primitive functions, we obtain Gg.
Using this algorithm we can compute Gg up to a high order. (G 1 is also obtained by solving (3.20).) The first few Gg terms are (3.34) Fg can easily be obtained by inverting the relation (3.3). We obtain (3.35) We computed Fg forg ≤ 13. We have checked that the above Fg withg = 0, 1, 2 are in perfect agreement (up to the constant part of F 1 ) with √ 2 1−g Fg in [16] under the identification in (2.11).

't Hooft expansion of the two-boundary correlator
In this section let us consider 't Hooft expansion of the two-boundary correlator. While Z 2 itself admits 't Hooft expansion, for many purposes it is convenient to consider instead the 't Hooft expansion of where Correspondingly, let us define the derivative and "free energies" G (2) and K (2) are related by We consider the expansions where G (2) g and K (2) g are functions of s i = g s β i /2 (i = 1, 2) and t k . The differential equation (2.10) for n = 2 is written as Subtracting (4.7) from (3.4) with s = s 1 + s 2 , we obtain the differential equation for W 2 (s 1 , s 2 ): (4.9) In [16] we derived that as the initial condition. We observe that starting with (4.11) and comparing both sides of (4.9) order-by-order in the small g s expansion one can algebraically determine G (2) g from the data of G can also be algebraically determined by (4.5) from the data of G (2) g g ≤g . Therefore, given the data of {Gg }g ≤g we can compute K (2) g without any integration procedure. For instance, we obtain 2 * − log z 1 * − log z 2 * − 2 log(z 1 * + z 2 * ) +  Using the above method we computed K (2) g forg ≤ 8. We verified that K (2) g with g = 1, 2 are in perfect agreement (up to the constant part of K g given in (3.52) of [16].
As a further nontrivial check, let us compare the above results with the low-temperature expansion of the two-point function studied in [16]. As we mentioned, the results in [16] are trivially generalized to the case of general t k by the replacement (2.11). Recall that the low-temperature expansion of e K (2) is written as (see (4.32) of [16] and notations therein) (4.14) Using the data of z , g (0 ≤ ≤ 7) we verified that this expression indeed reproduces the 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 3(6−g) and observed perfect agreement. Since small-s expansion of K (2) g (g ≥ 2) starts at the order of s 3(1−g) , this serves as a rather nontrivial check.

General formalism for multi-boundary correlator
In this section let us consider 't Hooft expansion of multi-boundary correlators. As in the case of the two-boundary correlator, it is convenient to consider the 't Hooft expansion with Z n (β 1 , . . . , β n ) = Tr(e β 1 Q Π · · · e βnQ Π).

(5.4)
Let us next introduce W n (β 1 , . . . , β n ) = ∂ x Z n (β 1 , . . . , β n ). (5.5) One may expect that (2.10) leads to a differential equation for W n similar to (4.8), which enables us to compute the 't Hooft expansion of log W n in the same way as in the last section. However, this is not the case. This is because W n (β 1 , . . . , β n ) are only cyclically symmetric with respect to the variables β i and for n ≥ 3 multiple W n with different orders of β i appear in the single differential equation (2.10). Consequently, the differential equation is not determinative for W n≥3 . On the other hand, by taking a different approach it is still possible to compute 't Hooft expansion of W n≥3 . Our new approach is based on the fact that Z n is expressed in terms of the Baker-Akhiezer function ψ i := ψ(ξ i ; {t k }), which satisfies where From (5.6) we are able to derive a new differential equation, which is not for W n itself, but for its constituents, as we will see below.
We first recall that (5.2) is rewritten as [16] Z n (β 1 , β 2 , . . . , β n ) = ∞ −∞ dξ 1 · · · ∞ −∞ dξ n e n j=1 β j ξ j K 12 K 23 · · · K n−1,n K n,1 , where K ij is the Darboux-Christoffel kernel. It is written in terms of the Baker-Akhiezer function ψ i as This means that Let us introduce the notation With a slight abuse of notation let us also introduce (5.13) Using (5.10) we see that W n (β 1 , β 2 , . . . , β n ) = n k=1 k, k − 1 . (5.14) We find that 1, n satisfies the simple differential equation The proof of this equation is not difficult but rather lengthy, so that we relegate it to Appendix A. One can easily check that (5.15) indeed reproduces the differential equations (3.4) and (4.8) for the n = 1, 2 cases by observing that To compute the 't Hooft expansion of log 1, n we need to know the initial condition, i.e. the genus zero part. This is derived as follows. Recall that K ij is written as (see e.g. [16]) Since the Baker-Akhiezer function admits the expansion (3.12), K ij is expanded as By solving the differential equation (5.15) with the initial condition (5.19) one can determine the 't Hooft expansion of 1, n purely algebraically, as in the case of W 2 . Then, using the relation (5.14), one immediately obtains the 't Hooft expansion of W n and Z n .

General results
In this section let us compute the 't Hooft expansion of the three-point function using the formalism developed in the last section. We consider the 't Hooft expansion As we mentioned, one can algebraically compute the above genus expansion. The results are 3 * − log [z 1 * z 2 * z 3 * (z 1 * + z 2 * )(z 2 * + z 3 * )(z 3 * + z 1 * )] + ln 2, Note also that the z i * terms here are related to those in [16] by z here
From this exact result (6.12) of Z 3 , one can compute the 't Hooft expansion (6.1) of K (3) = log Z 3 in the Airy case. Using the relation between Z 3 and Z 3 in (5.3) and the following property of the Owen's T -function where we defined In the large z regime with finite a, this is expanded as (6.16) Using this expansion, we checked that the 't Hooft expansion of the exact result of Z 3 in (6.14) reproduces K 2 and K 3 in (6.8). This serves as a nontrivial consistency check of our formalism.

Conclusions and outlook
In this paper we developed a formalism to compute the 't Hooft expansion of the multiboundary correlators in topological gravity with general couplings t k . The 't Hooft expansion of the n-point function can be obtained from the relation (5.15), which is equivalent to (2.10) for n = 1, 2. For the one-point function, we developed an algorithm for the computation of 't Hooft expansion in section 3, which is almost parallel to the computation of open free energy studied in [19]. We find that the 't Hooft expansion of an n-point function (n ≥ 2) is determined algebraically from the lower point functions by using (5.15). Our computation reproduces the JT gravity case studied in [15,16] and the 't Hooft expansion of 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 define the multi-point analogue of the spectral form factor in random matrix models and it exhibits a similar behavior as the ramp and plateau. Using our formalism, it would be possible to study the ramp-plateau transition regime of the multi-point version of the spectral form factor. We leave this as an interesting future problem.
Recently, the quenched free energy log Z(β) of JT gravity is studied by the replica method [33][34][35]. It is shown in [31] that the quenched free energy is written as a certain integral transformation of the generating function of multi-boundary correlators. The lowtemperature behavior of quenched free energy in JT gravity is quite interesting since it is suggested 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 −2/3 s . It would be interesting to study the quenched free energy of JT gravity in the 't Hooft regime β ∼ g −1 s using our formalism.

(A.2)
Next, from (5.6) we see that It also follows thaṫ and thus (∂ τ − u∂ x ) K ij = 2 3 (ψ i ψ j + ψ i ψ j − ψ i ψ j ). Note that (5.17) is rewritten as and thus Using this we obtain (ξ 1 − u) 1 , n = (ξ n − u) 1 , n + The first term of the r.h.s. can be rewritten by using the relations