JT gravity and the asymptotic Weil-Petersson volume

A path integral in Jackiw-Teitelboim (JT) gravity is given by integrating over the volume of the moduli of Riemann surfaces with boundaries, known as the"Weil-Petersson volume,"together with integrals over wiggles along the boundaries. The exact computation of the Weil-Petersson volume $V_{g,n}(b_1, \ldots, b_n)$ is difficult when the genus $g$ becomes large. We utilize two partial differential equations known to hold on the Weil-Petersson volumes to estimate asymptotic behaviors of the volume with two boundaries $V_{g,2}(b_1, b_2)$ and the volume with three boundaries $V_{g,3}(b_1, b_2, b_3)$ when the genus $g$ is large. Furthermore, we present a conjecture on the asymptotic expression for general $V_{g,n}(b_1, \ldots, b_n)$ with $n$ boundaries when $g$ is large.

The genus g partition function with n boundaries, Z g,n (β 1 , . . . , β n ), is given by an integral over a function involving the Weil-Petersson volume of the moduli of Riemann surfaces of genus g with n boundaries [3].
A Hermitian-matrix integral computes the JT gravity path integral [3]. This is owing to the fact that Mirzakhani's recursion relation [25], which holds for the Weil-Petersson volumes of the moduli of the Riemann surface on the JT gravity side, corresponds [26] to the topological recursion of Eynard and Orantin [27], which yields the genus expansion of a Hermitian-matrix integral as discussed in [28].
In JT gravity, connected correlators are given as the sums over genus g JT path integrals [3]: The genus g partition function with n boundaries, Z g (β 1 , . . . , β n ) is an integral over a function involving the Weil-Petersson volume V g,n [3] Z g (β 1 , . . . , β n ) = α n Therefore, connected correlators and path integrals in JT gravity can be computed when the Weil-Petersson volume V g,n (b 1 , . . . , b n ) is known (together with Schwarzian theory along the boundaries). For this reason, the evaluation of the Weil-Petersson volume V g,n (b 1 , . . . , b n ) has a physical importance.
In this work, we aim to estimate the asymptotic behavior of the Weil-Petersson volume V g,2 (b 1 , b 2 ) with genus g and two boundaries of geodesic lengths b 1 and b 2 when g is large.
We also estimate the asymptotic behavior of the Weil-Petersson volume with three boundaries, V g,3 (b 1 , b 2 , b 3 ), when g is large. Furthermore, we extend these results to propose a conjecture concerning an asymptotic expression for V g,n (b 1 , . . . , b n ). This study focuses on bosonic JT gravity.
In mathematics, Mirzakhani's recursion relation [25] is known for the Weil-Petersson volume V g,n (b 1 , . . . , b n ), which provides a method for computing the Weil-Petersson volume recursively. Although computational results are known up to some finite genus g and up to some finite number of boundaries n, the computation becomes difficult with increasing g. In this note, we study the asymptotic behavior of the Weil-Petersson volume V g,n (b 1 , . . . , b n ) when the genus g of Riemann surfaces becomes large 1 .
Mirzakhani's recursion relation expresses the derivative of b 1 V g,n in terms of V g ,n , where 2g +n is less than 2g + n. However, this expression necessarily involves V g−1,n+1 . For this reason, when g is large, it is considerably difficult to deduce the asymptotic expression for V g,n+1 from that of V g,n recursively. To resolve this difficulty, we utilize partial differential equations [30,31] that hold for the Weil-Petersson volume V g,n . This approach yields V g,n+1 from V g,n up to the leading order in g, when the genus g is sufficiently large. This approach might be useful in evaluating the JT path integral and connected correlators in bosonic JT gravity.
This report is structured as follows. Section 2 summarizes our strategy to estimate the asymptotic behavior of the Weil-Petersson volume V g,n (n ≥ 2) when the genus g is large. In Section 3.1, we provide an explicit calculation for V g,2 (b 1 , b 2 ) when the genus g is large. A leading term in g is obtained, and a subleading correction ∼ 1/g is also discussed. In Section 3.2, we estimate the asymptotic expression for V g,3 (b 1 , b 2 , b 3 ), namely the Weil-Petersson volume with genus g and three geodesic boundaries of lengths b 1 , b 2 , b 3 , when g is large. We also provide a conjecture on the asymptotic expression for V g,n (b 1 , . . . , b n ) when g is large. In Section 4 we comment on the asymptotic Weil-Petersson volume when the geodesic lengths b 1 , . . . , b n are large, and we mention a related question. When n = 1, the Weil-Petersson volume in this region was predicted in [3]. Section 5 closes with concluding remarks and an outlook on some open problems.
2 Summary of our strategy to estimate asymptotic V g,n (b 1 , . . . , b n ) As noted in the Introduction, we analyze the asymptotic behavior of the Weil-Petersson volume V g,n (b 1 , . . . , b n ) when the genus g is large. Here, V g,n (b 1 , . . . , b n ) denotes the Weil-Petersson volume of the moduli of genus g Riemann surfaces with n boundaries of geodesic lengths b 1 , . . . , b n . The evaluation of the Weil-Petersson volume enables the computation of the JT path integral and of the connected correlators [3].
In performing this analysis, if the asymptotic expression for the volume V g,n is known for a large enough g, can the asymptotic expression be deduced for the volume V g,n+1 ?
In principle, Mirzakhani's recursion formula [25] provides a method for computing V g,n for every g and n, starting from V 0,3 and V 1,1 , recursively. However, computing V g,n precisely and directly from the recursion formula is considerably difficult when g is large. When the asymptotic expression for V g,n is concerned, there is the additional difficulty of estimating V g,n+1 from V g,n from the recursion formula. Mirzakhani's recursion formula expresses ∂ b 1 b 1 V g,n+1 (b 1 , . . . , b n+1 ) as the sum of the integrals of V g,n times a function, the products V g 1 ,n 1 V g 2 ,n 2 times a function (where g 1 and g 2 add up to g, and n 1 and n 2 add to n + 2), and V g−1,n+2 times a function. Therefore, to estimate the asymptotic behavior of V g,n+1 from V g,n via the recursion formula, one needs to know the asymptotic behavior of V g−1,n+2 where g is large.
To avoid this difficulty, our approach utilizes partial differential equations that hold for V g,n (b 1 , . . . , b n ), as deduced in [30,31]. The Weil-Petersson volumes satisfy the following partial differential equations [30,31]: Here, ∂ j represents the derivative with respect to b j , where j = 1, . . . , n + 1. Because we focus on the asymptotic expressions for V g,n , we impose the following asymptotic conditions on the genus g and the geodesic lengths of the boundaries, b 1 , . . . , b n : Then, we may replace the second differential equation in (3) with the following reduced equation under the asymptotic conditions (4): The partial differential equations (3) impose highly nontrivial constraints on the asymptotic expressions for the Weil-Petersson volumes V g,n . For example, when n = 0, the first differential equation in (3) becomes [30,31] Asymptotic expressions for V g,0 and V g,1 (b) were predicted in [3] from the matrix-integral analysis using a contour integral for large g. One can verify that, when g >> 1 and g >> b, the formulas for V g,0 and V g,1 (b) in [3] satisfy the equation (6). This provides a consistency check of the asymptotic expressions for V g,0 and V g,1 (b) predicted in [3].
Here, we estimate V g,2 (b 1 , b 2 ) when g is large by applying (3), (5) to the asymptotic expression for V g,1 (b) obtained in [3]. The differential equations in (3) and (5) are highly effective for estimating V g,2 when this is asymptotic in g. Furthermore, applying (3), (5) to the deduced asymptotic The iteration of this process leads us to a conjecture on the asymptotic expression for V g,n (b 1 , . . . , b n ) when g is large.
One can compute large genus contributions to the JT path integral and the connected correlators from the deduced expressions.

Asymptotic behavior of the Weil-Petersson volume
We estimate the asymptotic expression for V g,2 (b 1 , b 2 ) when the genus g is large, using the partial differential equations (3) deduced in [30,31]. When the Riemann surface has two boundaries, the differential equations (3) become As stated in Section 2, we impose asymptotic conditions on the genus g and geodesic lengths b 1 , b 2 as follows: The second equation in (7) becomes reduced to the following equation under these conditions: From a physical argument on 2d topological gravity under the condition g >> 1 and g >> b 1 , the asymptotic form of From this V g,1 (b 1 ), we estimate the asymptotic expression for V g,2 (b 1 , b 2 ) when the conditions (8) are satisfied. It is well known in mathematics that any V g,n (b 1 , . . . , b n ) is symmetric about b 1 , . . . , b n . This implies in particular that V g,2 must be symmetric about b 1 and b 2 . Utilizing this symmetry property of the Weil-Petersson volume and the first equation in (7), one is naturally led to considering the following asymptotic expression 2 for V g,2 (b 1 , b 2 ): When g is large (g >> 1), one can replace Γ(2g as g tends toward infinity. Thus, a natural asymptotic expression for V g,2 (b 1 , b 2 ) under the conditions (8) is The coefficient is chosen to satisfy the first equation in (7).
Confirming whether the asymptotic expression (13) actually satisfies the reduced differential equation (9) yields a nontrivial check. One can confirm that (13) indeed satisfies equation (9) as follows: when our expression (13) is substituted into the left-hand side of (9), we obtain Substituting (10) into the right-hand side of (9) yields ). This confirms that the leftand right-hand sides of (9) are equal when V g,2 (b 1 , b 2 ) is expressed as (13). We thus confirmed that (13) satisfies the reduced equation (9).
The expression for V g,2 (b 1 , b 2 ) (13) is consistent with Conjecture 1 given by Zograf in [29]. This can be confirmed by comparing the Weil-Petersson volume with intersection numbers using [32] 3 then setting b 1 and b 2 to zero. (i, j on the right-hand side of (16) are non-negative integers, and their sum ranges from 0 to 3g − 1.) Zograf's conjectures in [29] for the Weil-Petersson volume when the genus g is large are partially proved rigorously in [35,36,37]. The fact that (13) is consistent with Conjecture 1 in [29] also supports our expression.
We compared the precise computations of Peter Zograf of V g,2 (b 1 , b 2 ) up to g = 18 [38] with our expression (13). 4 , we compared the coefficients of b 2i 2 b 2j 2 obtained by expanding the expression (13) in terms of b 2i 2 b 2j 2 , which we denote by c asymp.

gij
, with the precise coefficient computed by Zograf, which we denote by c precise gij .
For example, when one sets b 1 = b 2 = 0, one obtains the coefficient c g00 . This corresponds to We compared the constant term in (13) with the results by Zograf [38]. The error percentages obtained from 1 − c precise g00 c asymp. g00 (18) are within 6 % for 2 ≤ g ≤ 18. The precision of the expression (13) improves as genus g increases: for 6 ≤ g ≤ 18, the error percentages are less than 2%, and for 12 ≤ g ≤ 18, the error percentages are less than 1 %. We also compared the coefficients of b 2i 1 b 2j 2 for 0 ≤ i, j ≤ 2 (where i and j are not simultaneously zero) obtained from the asymptotic expression (13) with the results computed by Zograf [38] when g = 18. Because coefficients are symmetric under the exchange of i and j, we only consider the case i ≥ j here. For all the cases (i, j) = (1, 0), (1,1), (2,0), (2,1), (2,2), the error percentages obtained from gij (19) are less than 10%. Except for the case (i, j) = (2, 2), the error percentages are less than 6%. The comparison seems to suggest that genus g required for a precision of agreement, say error percentage of less than 2%, increases as i and j rise.
It is worth noting that the asymptotic expression (13) for V g,2 (b 1 , b 2 ) yields a leading term in g. There is a correction term of order ∼ 1/g. One can see this as follows: The Weil-Petersson volume V g,n (b 1 , . . . , b n ) also satisfies an integral equation [30,31]: When there are two boundaries, this equation takes the following particular form: Integrating the result for V g,1 (b 1 ) (10) in [3], we expect However, expression (13) for V g,2 (b 1 , b 2 ) vanishes when b 2 = 2πi 5 . This suggests that there is a subleading correction term ∼ 1/g to the leading term (13), Therefore, from the above argument, we deduce the following more precise asymptotic expression for V g,2 under (8): The form of the function f (b 1 , b 2 ) is left undetermined.

Asymptotic
Now, we would like to estimate the asymptotic expression for general V g,n (b 1 , . . . , b n ) when condition (4) is satisfied. First, we estimate the Weil-Petersson volume with three boundaries V g,3 for g >> 1, g >> b 1 , b 2 , b 3 . For this situation, V g,3 must satisfy the following two differential equations [30,31]: 5 With our asymptotic expression (13) for V g,2 (b 1 , b 2 ), we have 4g−3 . Additionally, g >> b 1 under (8). Therefore, we may set b1 0 b 1 V g,1 (b 1 )db 1 to zero to the leading order in g. Based on this reasoning, for leading order in g, the vanishing of (13) when b 2 assumes the value 2πi does not suggest an inconsistency here.
We used the reduced equation (5) for the second equation in (3) owing to the conditions g >> By a similar argument to that presented in Section 3.1, we exploit symmetry to estimate the asymptotic expression for V g ,3 (b 1 , b 2 , b 3 ): Assuming that V g,2 (b 1 , b 2 ) is given by (13) under (8), expression (25) satisfies the two equations in (24) (when g tends toward infinity). Furthermore, expression (25) is consistent with Conjecture 1 in [29], as can be confirmed by setting b 1 , b 2 , b 3 in (25) to zero. These results support our expression (25) to some degree. Similar to the Weil-Petersson volume with two boundaries V g,2 (b 1 , b 2 ), V g,3 (b 1 , b 2 , b 3 ) must satisfy the integral equation [30,31]: With our expression (25) , the left-hand side in (26) vanishes, while the righthand side does not vanish when (13) is substituted into V g,2 (b 1 , b 2 ). This suggests that there is a subleading correction ∼ 1/g, similar to the situation discussed in Section 3.1. Integrating (13) times b 1 and b 2 , we deduce that the subleading correction is of the form Iteration of this computation leads to the following conjecture on the asymptotic expression for V g,n (b 1 , . . . , b n ) under the conditions (4): As g tends toward infinity, this expression satisfies the two differential equations mentioned in Section 2: Here, we used the reduced equation for the second equation in (29) owing to the conditions (4), as previously explained in Section 2. Furthermore, setting b 1 , b 2 , . . . , b n to zero, the consistency of expression (28) with Conjecture 1 in [29] is confirmed.
4 Asymptotic behavior of the Weil-Petersson volume V g,n (b 1 , . . . , b n ) with large b Mirzakhani proved [25] that the Weil-Petersson volume V g,n (b 1 , . . . , b n ) is a polynomial in b given by where b on the right-hand side represents b = (b 1 , b 2 , . . . , b n ), α = (α 1 , α 2 , . . . , α n ), and α i , i = 1, . . . , n are non-negative integers. b 2α is defined as (30). 6 With regard to the theorem of Mirzakhani, it is worth making a remark about the Weil-Petersson volume in the limit where the boundary lengths b become large, i.e., b >> g >> 1. It follows from Mirzakhani's theorem that in the large b regime (b 1 , . . . , b n >> g >> 1), the highestorder terms in the polynomial (30) are dominant. Therefore, in the regime b 1 , . . . , b n >> g >> 1, the Weil-Petersson volume should be approximated by the sum of these highest-order terms in b: where the coefficients c g (α) with |α| = 3g − 3 + n are positive and are in Q. However, the method discussed in this report, which employs partial differential equations (3) to estimate the asymptotic Weil-Petersson volume, does not apply in the large b regime to determine the coefficients in (31). This is owing to the fact that the conditions b 1 , . . . , b n >> g >> 1 are imposed on b.
It might be interesting to consider if there is a method for estimating the asymptotic expression for the Weil-Petersson volume V g,n (b 1 , . . . , b n ) in the regime b 1 , . . . , b n >> g >> 1, from the volume V g,n−1 (b 1 , . . . , b n−1 ). This is equivalent to estimating the coefficients c g (α) in (31) for asymptotic V g,n , when the coefficients c g (α) are known for asymptotic V g,m , m < n, in this regime.

Concluding remarks and open problems
Herein, we utilized partial differential equations satisfied by the Weil-Petersson volume [30,31] to estimate the asymptotic Weil-Petersson volumes V g,2 (b 1 , b 2 ) and V g,3 (b 1 , b 2 , b 3 ) for large genus g (g >> 1 and g >> b 1 , b 2 , b 3 ). We also conjectured the asymptotic expression for the volume V g,n (b 1 , . . . , b n ) for general n, when the genus g is large (g >> 1 and g >> b 1 , . . . , b n ). The obtained asymptotic expressions satisfy the partial differential equations deduced in [30,31] to leading order in g. We also confirmed that, when the b i all vanish, our asymptotic expressions for the volumes (V g,2 (b 1 , b 2 ) with two boundaries, V g,3 (b 1 , b 2 , b 3 ) with three boundaries, and V g,n (b 1 , . . . , b n ) in general) are all consistent with Conjecture 1 for the asymptotic Weil-Petersson volumes specified by Zograf in [29]. We also compared our asymptotic expression (13) for V g,2 (b 1 , b 2 ) with the precise computational results by Zograf [38]. The comparison showed good agreement.
The Weil-Petersson volumes yield the intersection numbers of cohomology classes on the moduli of Riemann surfaces [32]. For example, Mirzakhani proved [32] Witten's conjecture [39] 7 by relating the intersection numbers of cohomology classes on the moduli of Riemann surfaces to the coefficients of the Weil-Petersson volume in (30). The asymptotic expressions for the Weil-Petersson volumes obtained in this note provide information on the intersection numbers of certain cohomology classes on the moduli of Riemann surfaces. It might be interesting to compare the known intersection numbers of cohomology classes with our results. Physically, they include the correlation functions of 2d topological gravity [43].
The asymptotic Weil-Petersson volumes V g,0 and V g,1 (b) were predicted in [3] in the context of bosonic JT gravity, by using the density of eigenvalues via topological recursion. As discussed in [3], their method of using topological recursion 8 to predict asymptotic Weil-Petersson volumes may extend to general V g,n (b 1 , . . . , b n ). If so, it might also be interesting to study whether our expressions agree with the results obtained from the approach of topological recursion.
If the method of topological recursion succeeds in evaluating the asymptotic Weil-Petersson volume V g,n (b 1 , . . . , b n ) in the regime b 1 , . . . , b n >> g >> 1, the saddle-point approximation should yield the approximation of the Weil-Petersson volume V g,n (b 1 , . . . , b n ) in the form (31) discussed in Section 4. This provides a non-trivial check of the result given by the topological recursion, if it is applied to evaluate the Weil-Petersson volume V g,n (b 1 , . . . , b n ) for general n in the regime b 1 , . . . , b n >> g >> 1.