Topological recursion of the Weil–Petersson volumes of hyperbolic surfaces with tight boundaries

Budd, Timothy; Zonneveld, Bart

doi:10.1063/5.0192711

The Weil–Petersson volumes of moduli spaces of hyperbolic surfaces with geodesic boundaries are known to be given by polynomials in the boundary lengths. These polynomials satisfy Mirzakhani’s recursion formula, which fits into the general framework of topological recursion. We generalize the recursion to hyperbolic surfaces with any number of special geodesic boundaries that are required to be tight. A special boundary is tight if it has minimal length among all curves that separate it from the other special boundaries. The Weil–Petersson volume of this restricted family of hyperbolic surfaces is shown again to be polynomial in the boundary lengths. This remains true when we allow conical defects in the surface with cone angles in (0, π) in addition to geodesic boundaries. Moreover, the generating function of Weil–Petersson volumes with fixed genus and a fixed number of special boundaries is polynomial as well, and satisfies a topological recursion that generalizes Mirzakhani’s formula. This work is largely inspired by recent works by Bouttier, Guitter, and Miermont [Ann. Henri Lebesgue 5, 1035–1110 (2022)] on the enumeration of planar maps with tight boundaries. Our proof relies on the equivalence of Mirzakhani’s recursion formula to a sequence of partial differential equations (known as the Virasoro constraints) on the generating function of intersection numbers. Finally, we discuss a connection with Jackiw–Teitelboim (JT) gravity. We show that the multi-boundary correlators of JT gravity with defects are expressible in the tight Weil–Petersson volume generating functions, using a tight generalization of the JT trumpet partition function.

I. INTRODUCTION

A. Topological recursion of Weil–Petersson volumes

In the celebrated work¹ Mirzakhani established a recursion formula for the Weil–Petersson volume V_g,n(L) of the moduli space of genus-g hyperbolic surfaces with n labeled boundaries of lengths

L = (L_{1}, \dots, L_{n}) \in R_{> 0}^{n}

⁠. Denoting [n] = {1, 2, …, n} and using the notation

L_{I} = {(L_{i})}_{i \in I}

⁠, I ⊂ [n], for a subsequence of L and

L_{\hat{I}} = L_{[n] \ I}

⁠, the recursion can be expressed for (g, n) ∉ {(0, 3), (1, 1)} as

\begin{align} V_{g, n} (L) & = \frac{1}{2 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \int_{0}^{\infty} d y x y K_{0} (x + y, t) [V_{g - 1, n + 1} (x, y, L_{\hat{{1}}}) \\ + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} V_{g_{1}, 1 + | I |} (x, L_{I}) V_{g_{2}, 1 + | J |} (y, L_{J})] \\ + \frac{1}{2 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \sum_{j = 2}^{n} x (K_{0} (x, t + L_{j}) + K_{0} (x, t - L_{j})) V_{g, n - 1} (x, L_{\hat{{1, j}}}), \end{align}

(1)

where

K_{0} (x, t) = \frac{1}{1 + \exp (\frac{x + t}{2})} + \frac{1}{1 + \exp (\frac{x - t}{2})} .

(2)

Together with V_0,3(L) = 1 and

V_{1, 1} (L) = \frac{L_{1}^{2}}{48} + \frac{π^{2}}{12}

this completely determines V_g,n as a symmetric polynomial in

L_{1}^{2}, \dots, L_{n}^{2}

of degree 3g − 3 + n.

This recursion formula remains valid^1,2 when we replace one or more of the boundaries by cone points with cone angle α_i ∈ (0, π) if we assign to it an imaginary boundary length L_i = iα_i. Cone points with angles in (0, π) are called sharp, as opposed to blunt cone points that have angle in (π, 2π). The Weil–Petersson volume of the moduli space of genus-g surfaces with n geodesic boundaries or sharp cone points is thus correctly computed by the polynomial V_g,n(L).

It was recognized by Eynard and Orantin³ that Mirzakhani’s recursion (in the case of geodesic boundaries) fits the general framework of topological recursion. To state this result explicitly one introduces for any g, n ≥ 0 satisfying 3g − 3 + n ≥ 0 the Laplace transformed⁴ Weil–Petersson volumes

ω_{g, n}^{(0)} (z) = \int_{0}^{\infty} [\prod_{i = 1}^{n} d L_{i} L_{i} e^{- z_{i} L_{i}}] V_{g, n} (L),

(3)

which are even polynomials in

z_{1}^{- 1}, \dots, z_{n}^{- 1}

of degree 6g − 4 + 2n, while setting

ω_{0, 1}^{(0)} (z) = 0, ω_{0, 2}^{(0)} (z) = \frac{1}{{(z_{1} - z_{2})}^{2}} .

(4)

Then Mirzakhani’s recursion translates into the recursion (Ref. 3, Theorem 2.1)

ω_{g, n}^{(0)} (z) = \underset{u \to 0}{R e s} \frac{π}{(z_{1}^{2} - u^{2}) \sin 2 π u} [ω_{g - 1, n + 1}^{(0)} (u, - u, z_{\hat{{1}}}) + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} ω_{g_{1}, 1 + | I |}^{(0)} (u, z_{I}) ω_{g_{2}, 1 + | J |}^{(0)} (- u, z_{J})]

(5)

valid when g, n ≥ 1 and 3g − 3 + n ≥ 0, which one may recognize as the recursion for the invariants

ω_{g, n}^{(0)} (z)

of the complex curve

\{\begin{cases} x (z) = z^{2} \\ y (z) = \frac{1}{π} \sin (2 π z) . \end{cases}

(6)

The main purpose of the current work is to generalize these recursion formulas to hyperbolic surfaces with so-called tight boundaries, which we introduce now.

B. Hyperbolic surfaces with tight boundaries

Let S_g,n be a fixed topological surface of genus g with n boundaries and

T_{g, n} (L)

the Teichmüller space of hyperbolic structures on S_g,n with geodesic boundaries of lengths

L = (L_{1}, \dots, L_{n}) \in R_{> 0}^{n}

⁠. Denote the boundary cycles⁵ by ∂₁, …, ∂_n and the free homotopy class of a cycle γ in S_g,n by

{[γ]}_{S_{g, n}}

⁠. For a hyperbolic surface

X \in T_{g, n} (L)

and a cycle γ, we denote by ℓ_γ(X) the length of γ, in particular

ℓ_{\partial_{i}} (X) = L_{i}

⁠. The mapping class group of S_g,n is denoted Mod_g,n and the quotient of

T_{g, n} (L)

by its action leads to the moduli space

M_{g, n} (L) = T_{g, n} (L) / {M o d}_{g, n} .

Let us denote by

{\overset{⌢}{S}}_{g, n, p} \supset S_{g, n + p}

the topological surface obtained from S_g,n+p by capping off the last p boundaries with disks (Fig. 1). Note that the free homotopy classes

{[\cdot]}_{S_{g, n + p}}

of S_g,n+p are naturally partitioned into the free homotopy classes

{[\cdot]}_{{\overset{⌢}{S}}_{g, n, p}}

of

{\overset{⌢}{S}}_{g, n, p}

⁠. In particular,

{[\partial_{j}]}_{S_{g, n + p}}

for j = n + 1, …, n + p are all contained in the null-homotopy class of

{\overset{⌢}{S}}_{g, n, p}

⁠. For i = 1, …, n the boundary ∂_i of

X \in M_{g, n + p} (L)

is said to be tight in

{\overset{⌢}{S}}_{g, n, p}

if ∂_i is the only simple cycle γ in

{[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}

of length ℓ_γ(X) ≤ L_i. Remark that both

{[\partial_{i}]}_{S_{g, n + p}}

and

{[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}

for i = 1, …, n are Mod_g,n+p-invariant, so these classes are well-defined at the level of the moduli space. This allows us to introduce the moduli space of tight hyperbolic surfaces

M_{g, n, p}^{tight} (L) = {X \in M_{g, n + p} (L) : \partial_{1}, \dots, \partial_{n} are tight in {\overset{⌢}{S}}_{g, n, p}} \subset M_{g, n + p} (L) .

(7)

Note that

M_{g, n, 0}^{tight} (L) = M_{g, 0, n}^{tight} (L) = M_{g, n} (L)

⁠, while

M_{0, 1, p}^{tight} (L) = \emptyset

because ∂₁ is null-homotopic and

M_{0, 2, p}^{tight} (L) = \emptyset

because

{[\partial_{1}]}_{{\overset{⌢}{S}}_{0, 2, p}} = {[\partial_{2}]}_{{\overset{⌢}{S}}_{0, 2, p}}

and therefore ∂₁ and ∂₂ can never both be the unique shortest cycle in their class. In general, it is an open subset of

M_{g, n + p} (L)

and therefore it inherits the Weil–Petersson symplectic structure and Weil–Petersson measure dμ_WP from

M_{g, n + p} (L)

⁠. The corresponding tight Weil–Petersson volumes are denoted

T_{g, n, p} (L) = \int_{M_{g, n, p}^{tight} (L)} d μ_{WP} \leq V_{g, n + p} (L),

(8)

such that T_g,n,0(L) = T_g,0,n(L) = V_g,n(L) and T_0,1,p(L) = T_0,2,p(L) = 0.

FIG. 1.

A surface Sg,n+p (left) and the result S⌢g,n,p (right) of capping off the last p boundaries, with in this case g = 1, n = 4 and p = 6. If the original surface was equipped with a hyperbolic structure, it is natural to think of the resulting surface as the Riemannian manifold obtained by attaching constantly curved hemispheres of appropriate diameter to the p geodesic boundaries.

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A surface S_g,n+p (left) and the result ${\overset{⌢}{S}}_{g, n, p}$ (right) of capping off the last p boundaries, with in this case g = 1, n = 4 and p = 6. If the original surface was equipped with a hyperbolic structure, it is natural to think of the resulting surface as the Riemannian manifold obtained by attaching constantly curved hemispheres of appropriate diameter to the p geodesic boundaries.

We can extend this definition to the case in which one or more of the boundaries ∂_n+1, …, ∂_n+p is replaced by a sharp cone point with cone angle α_i ∈ (0, π). In this case we make the usual identification L_i = iα_i, and still denote the corresponding Weil–Petersson volume by T_g,n,p(L). Our first result is the following.

Proposition 1.

For g, n, p ≥ 0 such that 2g − 3 + n ≥ 0, the tight Weil–Petersson volume T_g,n,p(L) of genus g surfaces with n tight boundaries and p geodesic boundaries or sharp cone points is a polynomial in $L_{1}^{2}, \dots, L_{n + p}^{2}$ of degree 3g − 3 + n + p that is symmetric in L₁, …, L_n and symmetric in L_n+1, …, L_n+p.

For most of the upcoming results we maintain the intuitive picture that the tight boundaries are the “real” boundaries of the surface, whose number and lengths we specify, while we allow for an arbitrary number of other boundaries or cone points that we treat as defects in the surface. To this end we would like to encode the volume polynomials in generating functions that sum over the number of defects with appropriate weights. A priori it is not entirely clear what is the best way to organize such generating functions, so to motivate our definition we take a detour to a natural application of Weil–Petersson volumes in random hyperbolic surfaces.

C. Intermezzo: Random (tight) hyperbolic surfaces

If we fix g, n and

L \in ([0, \infty) \cup i {(0, π))}^{n}

⁠, then upon normalization by 1/V_g,n(L) the Weil–Petersson measure dμ_WP provides a well-studied probability measure on

M_{g, n} (L)

defining the Weil–Petersson random hyperbolic surface, see e.g. Refs. 6–10. A natural way to extend the randomness to the boundary lengths or cone angles is by choosing a (Borel) measure μ on [0, ∞) ∪ i(0, π) and first sampling

L \in ([0, \infty) \cup i {(0, π))}^{n}

from the probability measure

\frac{1}{μ^{\otimes n} (V_{g, n})} V_{g, n} (L) d μ (L_{1}) \dots d μ (L_{n}), μ^{\otimes n} (V_{g, n}) : = \int V_{g, n} (L) d μ (L_{1}) \dots d μ (L_{n}),

(9)

and then sampling a Weil–Petersson random hyperbolic surface on

M_{g, n} (L)

⁠. If the genus-g partition function¹¹

F_{g} [μ] = \sum_{n \geq 0} \frac{μ^{\otimes n} (V_{g, n})}{n!}

(10)

converges, we can furthermore make the size n ≥ 0 random by sampling it with the probability μ^⊗n(V_g,n)/(n!F_g[μ]). The resulting random surface (of random size) is called the genus-g Boltzmann hyperbolic surface with weight μ. See the upcoming work¹² for some of its statistical properties.

A natural extension is to consider the genus-g Boltzmann hyperbolic surface with n tight boundaries of length L = (L₁, …, L_n), where the number p of defects and their boundary lengths/cone angles K = (K₁, …, K_p) are random. The corresponding partition function is

T_{g, n} (L; μ] = \sum_{p \geq 0} \frac{μ^{\otimes p} (T_{g, n, p}) (L)}{p!}, μ^{\otimes p} (T_{g, n, p}) (L) : = \int T_{g, n, p} (L, K) d μ (K_{1}) \dots d μ (K_{p}) .

(11)

If it is finite, we can sample p with probability μ^⊗p(T_g,n,p)(L)/(p!T_g,n(L; μ]) and then K from the probability measure

\frac{1}{μ^{\otimes p} (T_{g, n, p}) (L)} T_{g, n, p} (L, K) d μ (K_{1}) \dots d μ (K_{n})

(12)

and then finally a random tight hyperbolic surface from the probability measure dμ_WP/T_g,n,p(L, K) on

M_{g, n, p}^{tight} (L, K)

⁠. Note that for μ = 0 the genus-g Boltzmann hyperbolic surface with n tight boundaries reduces to the Weil–Petersson random hyperbolic surface we started with.

The important observation for the current work is that the partition functions F_g[μ] and T_g,n(L; μ] of these random surfaces can be thought of as (multivariate, exponential) generating functions of the volumes V_g,n(L) and T_g,n,p(L, K) if we treat μ as a formal generating variable. Since we will not be concerned with the details of the measures (9) and (12) and F_g[μ] and T_g,n(L; μ] only depend on the even moments ∫L^2kdμ(L), we can instead take these moments as the generating variables.

D. Generating functions

To be precise, we let a weight μ be a real linear function on the ring of even, real polynomials (i.e.,

μ \in R {[K^{2}]}^{*}

⁠). For an even real polynomial f we use the suggestive notation

μ (f) = \int d μ (K) f (K),

(13)

making it clear that the notion of weight generalizes the Borel measure described in the intermezzo above. For L ∈ [0, ∞) ∪ i(0, π), the Borel measure given by the delta measure δ_L at L gives a simple example of a weight μ = δ_L satisfying δ_L(f) = f(L). The choice of weight μ is clearly equivalent to the choice of a sequence of times

(t_{0}, t_{1}, \dots) \in R^{Z_{\geq 0}}

recording the evaluations of μ on the even monomials, up to a conventional normalization,

t_{k} [μ] = \frac{2}{4^{k} k!} μ (K^{2 k}) = \int d μ (K) \frac{2 K^{2 k}}{4^{k} k!} .

(14)

Naturally we can interpret

μ^{\otimes p} \in {(R {[K^{2}]}^{*})}^{\otimes p}

as an element of

{(R {[K^{2}]}^{\otimes p})}^{*} ≅ R {[K_{1}^{2}, \dots, K_{p}^{2}]}^{*}

by setting

μ^{\otimes p} (f_{1} (K_{1}) \dots f_{p} (K_{p})) = μ (f_{1}) \dots μ (f_{p})

for even polynomials f₁, …, f_p and extending by linearity. More generally, we can view μ^⊗p as a linear map

R [L_{1}^{2}, \dots, L_{n}^{2}, K_{1}^{2}, \dots, K_{p}^{2}] ≅ R [L_{1}^{2}, \dots, L_{n}^{2}] [K_{1}^{2}, \dots, K_{p}^{2}] \to R [L_{1}^{2}, \dots, L_{n}]

⁠. We use the notation

μ^{\otimes p} (f) = \int f (K) d μ (K_{1}) \dots d μ (K_{p}), μ^{\otimes p} (f) (L) = \int f (L, K) d μ (K_{1}) \dots d μ (K_{p}) .

(15)

One can then naturally introduce the generating function F[μ] of a collection of symmetric, even polynomials f₁(L₁), f₂(L₁, L₂), … via

F [μ] = \sum_{p \geq 0} \frac{μ^{\otimes p} (f_{p})}{p!}

⁠. Then the generating function of tight Weil–Petersson volumes is defined to be

T_{g, n} (L; μ] = \sum_{p = 0}^{\infty} \frac{μ^{\otimes p} (T_{g, n, p}) (L)}{p!} = \sum_{p = 0}^{\infty} \frac{1}{p!} \int d μ (K_{1}) \dots \int d μ (K_{p}) T_{g, n, p} (L, K),

(16)

which we interpret in the sense of a formal power series, so we do not have to worry about convergence. We could make this more precise by fixing a weight μ and considering

T_{g, n} (L; x μ] \in R [[x]]

as a univariate formal power series in x. Or we could view T_g,n(L; μ] as a multivariate formal power series in the times (t₀, t₁, …) defined in .

What is important is that we can make sense of the functional derivative

\frac{δ}{δ μ (L)}

on these types of series defined by

\frac{δ}{δ μ (L)} P [μ] = {\frac{\partial}{\partial x} P [μ + x δ_{L}]|}_{x = 0} .

(17)

In particular, if f(L, K), with L = (L₁, …, L_n) and K = (K₁, …, K_p), is an even polynomial that is symmetric in K₁, …, K_p then

\frac{δ}{δ μ (L)} μ^{\otimes p} (f) (L) = p μ^{\otimes p - 1} (f) (L, L) .

(18)

At the level of the generating function we thus have

\frac{δ}{δ μ (L)} T_{g, n} (L; μ] = \sum_{p = 0}^{\infty} \frac{1}{p!} \int d μ (K_{1}) \dots \int d μ (K_{p}) T_{g, n, p + 1} (L, L, K) .

(19)

In terms of formal power series in the times we may instead identify the functional derivative in terms of the formal partial derivatives as

\frac{δ}{δ μ (L)} = \sum_{k = 0}^{\infty} \frac{2 L^{2 k}}{4^{k} k!} \frac{\partial}{\partial t_{k}}, \frac{δ}{δ μ (0)} = 2 \frac{\partial}{\partial t_{0}} .

(20)

E. Main results

To state our main results about T_g,n(L; μ], we need to introduce the generating function R[μ] as the unique formal power series solution satisfying R[μ] = ∫dμ(L) + O(μ²) to

Z (R [μ]; μ] = 0, Z (r; μ] : = \frac{\sqrt{r}}{\sqrt{2} π} J_{1} (2 π \sqrt{2 r}) - \int d μ (L) I_{0} (L \sqrt{2 r}),

(21)

where I₀ and J₁ are (modified) Bessel functions. Let also the moments M_k[μ] be the defined recursively via

M_{0} [μ] = \frac{1}{\frac{δ R}{δ μ (0)}}, M_{k} [μ] = M_{0} [μ] \frac{δ M_{k - 1} [μ]}{δ μ (0)}, k \geq 1,

(22)

where the reciprocal in the first identity makes sense because

\frac{δ R}{δ μ (0)} = 1 + O (μ)

⁠. Alternatively, for k ≥ 0 we may express M_k as

M_{k} [μ] = Z^{(k + 1)} (R [μ]; μ] = {(\frac{- \sqrt{2} π}{\sqrt{R [μ]}})}^{k} J_{k} (2 π \sqrt{2 R [μ]}) - \int d μ (L) {(\frac{L}{\sqrt{2 R [μ]}})}^{k + 1} I_{k + 1} (L \sqrt{2 R [μ]}),

(23)

where Z^(k+1)(r; μ] denotes the (k + 1)th derivative of Z(r; μ] with respect to r.

We further consider the series

η (u; μ] = \sum_{p = 0}^{\infty} M_{p} [μ] \frac{u^{2 p}}{(2 p + 1) ‼}, \hat{X} (u; μ] = \frac{\sin (2 π u)}{2 π u η (u; μ]},

(24)

which we both interpret as formal power series in u with coefficients that are formal power series in μ. The reciprocal in the second definition is well-defined because η(0) = M₀[μ] = 1 + O(μ). We can now state our main result that generalizes Mirzakhani’s recursion formula.

Theorem 2.

The tight Weil–Petersson volume generating functions T_g,n(L; μ] satisfy

\begin{align} T_{g, n} (L; μ] & = \frac{1}{2 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \int_{0}^{\infty} d y x y K (x + y, t; μ] [T_{g - 1, n + 1} (x, y, L_{\hat{{1}}}; μ]) \\ + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} T_{g_{1}, 1 + | I |} (x, L_{I}; μ] T_{g_{2}, 1 + | J |} (y, L_{J}; μ] \\ + \frac{1}{2 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \sum_{j = 2}^{n} x (K (x, t + L_{j}; μ] + K (x, t - L_{j}; μ]) T_{g, n - 1} (x, L_{\hat{{1, j}}}; μ], \end{align}

(25)

which is the same recursion formula (1) as for the Weil–Petersson volumes V_g,n(L) except that the kernel K₀(x, t) is replaced by the “convolution”

K (x, t, μ] = \int_{- \infty}^{\infty} d X (z) K_{0} (x + z, t),

(26)

where X(z) = X(z; μ] is a measure on

R

determined by its two-sided Laplace transform

\int_{- \infty}^{\infty} d X (z) e^{- u z} = \hat{X} (u; μ] = \frac{\sin (2 π u)}{2 π u η (u; μ]} .

(27)

Furthermore, we have

T_{0, 3} (L_{1}, L_{2}, L_{3}; μ] = \frac{1}{M_{0} [μ]},

(28)

T_{1, 1} (L; μ] = - \frac{M_{1} [μ]}{24 M_{0} {[μ]}^{2}} + \frac{L^{2}}{48 M_{0} [μ]} .

(29)

We will not specify precisely what it means to have a measure X(z; μ] that itself is a formal power series in μ. Importantly its moments

\int_{- \infty}^{\infty} d X (z) z^{p} = {(- 1)}^{p} p! [u^{p}] \hat{X} (u; μ]

are formal power series in μ, so for any

x, t \in R

K (x, t; μ] = \sum_{p = 0}^{\infty} {(- 1)}^{p} \frac{\partial^{p}}{\partial x^{p}} K_{0} (x, t) [u^{p}] \hat{X} (u; μ]

(30)

is a formal power series in μ as well.

In the case μ = 0, it is easily verified that

M_{k} [0] = \frac{{(- 2 π^{2})}^{k}}{k!},

(31)

η (u, 0] = \frac{\sin (2 π u)}{2 π u},

(32)

so

\hat{X} (u; 0] = 1

and X(z; 0] = δ₀(z) and therefore one retrieves Mirzakhani’s kernel K(x, t) = K₀(x, t). Given that the form of Mirzakhani’s recursion is unchanged except for the kernel, this strongly suggests that the Laplace transforms

ω_{g, n} (z) = ω_{g, n} (z; μ] : = \int_{0}^{\infty} [\prod_{i = 1}^{n} d L_{i} L_{i} e^{- z_{i} L_{i}}] T_{g, n} (L; μ]

(33)

of the tight Weil–Petersson volumes can be obtained as invariants in the framework of topological recursion as well. When μ = 0 this reduces to the Laplace-transformed Weil–Petersson volumes

ω_{g, n} (z; 0] = ω_{g, n}^{(0)} (z)

defined in . The following theorem shows that this is the case in general.

Theorem 3.

Setting

ω_{0, 2} (z) = {(z_{1} - z_{2})}^{- 2}

and ω_0,0(z) = ω_0,1(z) = 0, the Laplace transforms (33) satisfy for every g ≥ 0, n ≥ 1 such that 3g − 3 + n ≥ 0 the recursion

ω_{g, n} (z) = \underset{u \to 0}{R e s} \frac{1}{2 u (z_{1}^{2} - u^{2}) η (u; μ]} [ω_{g - 1, n + 1} (u, - u, z_{\hat{{1}}}) + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} ω_{g_{1}, | I |} (u, z_{I}) ω_{g_{2}, | J |} (- u, z_{J})] .

(34)

These correspond precisely to the invariants of the curve

\{\begin{cases} x = z^{2} \\ y = 2 z η (z; μ] . \end{cases}

(35)

Another consequence of Theorem 2 is that for all g ≥ 0, n ≥ 1, such that 3g − 3 + n ≥ 0, the tight Weil–Petersson volume T_g,n(L; μ] is expressible as a rational polynomial in $L_{1}^{2}, \dots, L_{n}^{2}$ and $M_{0}^{- 1}, M_{1}, M_{2}, \dots$ ⁠.

Besides satisfying a recursion in the genus g and the number of tight boundaries n, these also satisfy a recurrence relation in n only.

Theorem 4.

For all g, n ≥ 0, such that n ≥ 3 for g = 0 and n ≥ 1 for g = 1, we have that

T_{g, n} (L; μ] = \frac{1}{M_{0}^{2 g - 2 + n}} P_{g, n} (L, \frac{M_{1}}{M_{0}}, \dots, \frac{M_{3 g - 3 + n}}{M_{0}}),

(36)

where

P_{g, n} (L, m)

is a rational polynomial in

L_{1}^{2}, \dots, L_{n}^{2}, m_{1}, \dots, m_{3 g - 3 + n}

. This polynomial is symmetric and of degree 3g − 3 + n in

L_{1}^{2}, \dots, L_{n}^{2}

, while

P_{g, n} (\sqrt{σ} L, σ m_{1}, σ^{2} m_{2}, σ^{3} m_{3}, \dots)

is homogeneous of degree 3g − 3 + n in σ. For all g ≥ 0, n ≥ 1 such that 2g − 3 + n > 0 the polynomial

P_{g, n} (L, m)

can be obtained from

P_{g, n - 1} (L, m)

via the recursion relation

\begin{align} P_{g, n} (L, m) & = \sum_{p = 1}^{3 g - 4 + n} (m_{p + 1} - \frac{L_{1}^{2 p + 2}}{2^{p + 1} (p + 1)!} - m_{1} m_{p} + \frac{1}{2} L_{1}^{2} m_{p}) \frac{\partial P_{g, n - 1}}{\partial m_{p}} (L_{\hat{{1}}}, m) \\ + (2 g - 3 + n) (- m_{1} + \frac{1}{2} L_{1}^{2}) P_{g, n - 1} (L_{\hat{{1}}}, m) + \sum_{i = 2}^{n} \int d L_{i} L_{i} P_{g, n - 1} (L_{\hat{{1}}}, m), \end{align}

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where we use the shorthand notation

\int d L L f (L, \dots) = \int_{0}^{L} d x x f (x, \dots)

. Furthermore, we have

P_{0, 3} (L) = 1

(38)

P_{1, 1} (L_{1}, m_{1}) = \frac{1}{24} (- m_{1} + \frac{1}{2} L_{1}^{2})

(39)

and

P_{g, 0}

for g ≥ 2 is given by

P_{g, 0} (m_{1}, \dots, m_{3 g - 3}) = \sum_{\begin{array}{c} d_{2}, d_{3}, \dots \geq 0 \\ \sum_{k \geq 2} (k - 1) d_{k} = 3 g - 3 \end{array}} {⟨ τ_{2}^{d_{2}} τ_{3}^{d_{3}} \dots ⟩}_{g} \prod_{k \geq 2} \frac{{(- m_{k - 1})}^{d_{k}}}{d_{k}!},

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where

{⟨ τ_{2}^{d_{2}} τ_{3}^{d_{3}} \dots ⟩}_{g}

are the ψ-class intersection numbers on Deligne–Mumford compactification of the moduli space of curves

{\bar{M}}_{g, n}

with n = ∑_kd_k ≤ 3g − 3 marked points.

For instance, the first few applications of the recursion yield

\begin{array}{l} P_{0, 4} (L, m) & = \frac{1}{2} (L_{1}^{2} + \dots + L_{4}^{2}) - m_{1}, \\ P_{0, 5} (L, m) & = \frac{1}{8} (L_{1}^{4} + \dots + L_{5}^{4}) + \frac{1}{2} (L_{1}^{2} L_{2}^{2} + \dots + L_{4}^{2} L_{5}^{2}) - \frac{3}{2} (L_{1}^{2} + \dots + L_{5}^{2}) m_{1} + 3 m_{1}^{2} - m_{2}, \\ P_{1, 2} (L, m) & = \frac{1}{192} (L_{1}^{4} + L_{2}^{4}) + \frac{1}{96} L_{1}^{2} L_{2}^{2} - \frac{1}{24} (L_{1}^{2} + L_{2}^{2}) m_{1} + \frac{1}{12} m_{1}^{2} - \frac{1}{24} m_{2}^{2} . \end{array}

Note that this provides a relatively efficient way of calculating the Weil–Petersson volumes V_g,n(L) from the polynomial

P_{g, 0}

⁠, since

V_{g, n} (L) = T_{g, n} (L; 0] = {P_{g, n} (L, m)|}_{m_{k} = {(- 2 π^{2})}^{k} / k!} .

A simple corollary of Theorem 4 is that the volumes satisfy string and dilaton equations generalizing those for the Weil–Petersson volumes derived by Do and Norbury in Ref. 13, Theorem 2.

Corollary 5.

For all g ≥ 0 and n ≥ 1, such that 2g − 3 + n > 0, we have the identities

\sum_{p = 0}^{\infty} 2^{p} p! M_{p} [μ] [L_{1}^{2 p}] T_{g, n} (L; μ] = \sum_{j = 2}^{n} \int d L_{j} L_{j} T_{g, n - 1} (L_{\hat{{1}}}; μ] + 1_{{g = 0, n = 3}},

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\sum_{p = 1}^{\infty} 2^{p} p! M_{p - 1} [μ] [L_{1}^{2 p}] T_{g, n} (L; μ] = (2 g - 3 + n) T_{g, n - 1} (L_{\hat{{1}}}; μ],

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where the notation

[L_{1}^{2 p}] T_{g, n} (L; μ]

refers to the coefficient of

L_{1}^{2 p}

in the polynomial T_g,n(L; μ].

As explained in Ref. 13, the string and dilaton equations for symmetric polynomials, in particular for the Weil–Petersson volumes, give rise to a recursion in n for genus 0 and 1. Using Theorem 4, we also get such a recursion for higher genera in the case of tight Weil–Petersson volumes.

F. Idea of the proofs

This work is largely inspired by the recent work¹⁴ of Bouttier, Guitter, and Miermont. There the authors consider the enumeration of planar maps with three boundaries, i.e., graphs embedded in the triply punctured sphere, see the left side of Fig. 2. Explicit expressions for the generating functions of such maps, also known as pairs of pants, with controlled face degrees were long known, but they show that these generating functions become even simpler when restricting to tight pairs of pants, in which the three boundaries are required to have minimal length (in the sense of graph distances) in their homotopy classes. They obtain their enumerative results on tight pairs of pants in a bijective manner by considering a canonical decomposition of a tight pair of pants into certain triangles and diangles, see Fig. 2.

FIG. 2.

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An example of a tight pair of pants, i.e., a planar map with three boundaries (in blue) of minimal length in their homotopy class in the triply punctured sphere. On the right the canonical decomposition described in Ref. 14. Figure adapted from Ref. 14, Fig. 1.1.

Our result (28) for the genus-0 tight Weil–Petersson volumes with three distinguished boundaries can be seen as the analogue of Ref. 14, Theorem 1.1, although less powerful because our proof is not bijective. Instead, we derive generating functions of tight Weil–Petersson volumes from known expressions in the case of ordinary Weil–Petersson volumes. The general idea is that a genus-0 hyperbolic surface with two distinguished (but not necessarily tight) boundaries can unambiguously be cut along a shortest geodesic separating those two boundaries, resulting in a pair of certain half-tight cylinders (Fig. 4). Also a genus-g surface with n distinguished (not necessarily tight) boundaries can be shown to decompose into a tight hyperbolic surface and n half-tight cylinders. The first decomposition uniquely determines the Weil–Petersson volumes of the moduli spaces of half-tight cylinders, while the second determines the tight Weil–Petersson volumes. This relation is at the basis of Proposition 1.

To arrive at the recursion formula of Theorem 2 we follow the line or reasoning of Mirzakhani’s proof¹⁵ of Witten’s conjecture¹⁶ (proved first by Kontsevich¹⁷). She observes that the recursion equation (1) implies that the generating function of certain intersection numbers satisfies an infinite family of partial differential equations, the Virasoro constraints. Mulase and Safnuk¹⁸ have observed that the reverse implication is true as well. We will demonstrate that the generating functions of tight Weil–Petersson volumes and ordinary Weil–Petersson volumes are related in a simple fashion when expressed in terms of the times (14) and that the former obey a modified family of Virasoro constraints. These constraints in turn are equivalent to the generalized recursion of Theorem 2.

G. Discussion

Mirzakhani’s recursion formula has a bijective interpretation.¹ Upon multiplication by 2L₁ the left-hand side 2L₁V_g,n(L) accounts for the volume of surfaces with a marked point on the first boundary. Tracing a geodesic ray from this point, perpendicularly to the boundary, until it self-intersects or hits another boundary allows one to canonically decompose the surface into a hyperbolic pair of pants (three-holed sphere) and one or two smaller hyperbolic surfaces. The terms on the right-hand side of Ref. 1 precisely take into account the Weil–Petersson volumes associated to these parts and the way they are glued.

It is natural to expect that Theorem 2 admits a similar bijective interpretation, in which the surface decomposes into a tight pair of pants (a sphere with 3 + p boundaries, three of which are tight) and one or two smaller tight hyperbolic surfaces. However, Mirzakhani’s ray shooting procedure does not generalize in an obvious way. Nevertheless, working under the assumption that a bijective decomposition exists, one is led to suspect that the generalized kernel K(x, t, μ] of Theorem 2 contains important information about the geometry of tight pairs of pants. Moreover, one would hope that this geometry can be further understood via a decomposition of the tight pairs of pants themselves analogous to the planar map case of Ref. 14 described above.

Since a genus-0 surface

X \in M_{0, 3 + p} (0, 0, 0, L)

with three distinguished cusps is always a tight pair of pants (since the zero length boundaries are obviously minimal), a consequence of a bijective interpretation of Theorem 2 is a conjectural interpretation of the series

\hat{X} (u; μ] = \sin (2 π u) / (2 π u η)

in in terms of the hyperbolic distances between the three cusps. To be precise, let c₁, c₂, c₃ be unit-length horocycles around the three cusps and

Δ (X) = d_{hyp} (c_{1}, c_{2}) - d_{hyp} (c_{1}, c_{3})

the difference in hyperbolic distance between two pairs, then it is plausible that

\hat{X} (u; μ] \overset{?}{=} \sum_{p \geq 0} \frac{1}{p!} \int (\int_{M_{0, 3 + p} (0, 0, 0, L)} e^{2 u Δ (X)} d μ_{WP} (X)) d μ (L_{1}) \dots d μ (L_{p}) .

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Or in the probabilistic terms of Sec. I C, the measure M₀[μ]X(z; μ] on

R

⁠, which integrates to 1 due to , is the probability distribution of the random variable

2 Δ (X)

in a genus-0 Boltzmann hyperbolic surface

X

with weight μ. In upcoming work we shall address this conjecture using very different methods.

Another natural question to ask is whether the generalization of the spectral curve (6) of Weil–Petersson volumes to the one of tight Weil–Petersson volumes in Theorem 3 can be understood in the general framework of deformations of spectral curves in topological recursion.¹⁹

H. Outline

The structure of the paper is as follows:

In Sec. II we introduce the half-tight cylinder, which allows us to do tight decomposition of surfaces, which relates the regular hyperbolic surfaces to the tight surfaces. Using the decomposition we prove Proposition 1.

In Sec. III we consider the generating functions of (tight) Weil–Petersson volumes and their relations. Furthermore, we use the Virasoro constraints to prove Theorem 2, Theorem 4, and Corollary 5.

In Sec. IV we take the Laplace transform of the tight Weil–Petersson volumes and prove Theorem 3. We also look at the relation between the disk function of the regular hyperbolic surfaces and the generating series of moment η.

Finally, in Sec. V we briefly discuss how our results may be of use in the study of Jackiw–Teitelboim (JT) gravity.

II. DECOMPOSITION OF TIGHT HYPERBOLIC SURFACES

A. Half-tight cylinder

Recall that a boundary ∂_i of

X \in M_{g, n + p} (L)

is said to be tight in

{\overset{⌢}{S}}_{g, n, p}

if ∂_i is the only simple cycle γ in

{[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}

of length ℓ_γ(X) ≤ L_i and we defined the moduli space of tight hyperbolic surfaces as

M_{g, n, p}^{tight} (L) = {X \in M_{g, n + p} (L) : \partial_{i} is tight in {\overset{⌢}{S}}_{g, n, p}} \subset M_{g, n + p} (L) .

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We noted before that when g = 0 and n = 2 we have

M_{0, 2, p}^{tight} (L) = \emptyset

because ∂₁ and ∂₂ belong to the same free homotopy class of

{\overset{⌢}{S}}_{0, 2, p}

and can therefore never both be the unique shortest cycle. Instead, it is useful for any p ≥ 1 to consider the moduli space of half-tight cylinders

H_{p} (L) = {X \in M_{0, 2 + p} (L) : \partial_{2} is tight in {\overset{⌢}{S}}_{0, 2, p}} \subset M_{0, 2 + p} (L),

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which is non-empty whenever L₁ > L₂ > 0. We will also consider

\bar{H_{p}} (L) = {X \in M_{0, 2 + p} (L) : \partial_{2} has minimal length in {[\partial_{2}]}_{{\overset{⌢}{S}}_{0, 2, p}}} \subset M_{0, 2 + p} (L)

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and denote its Weil–Petersson volume by H_p(L).

Lemma 6.

When L₁ ≥ L₂ > 0, $H_{p} (L)$ is an open subset of $M_{0, 2 + p} (L)$ , and when it is non-empty (L₁ > L₂) its closure is $\bar{H_{p}} (L)$ . In particular, both have the same finite Weil–Petersson volume H_p(L) ≤ V_0,2+p(L) when L₁ > L₂, but $H_{p} (L)$ has 0 volume and $\bar{H_{p}} (L)$ non-zero volume H_p(L) when L₁ = L₂.

Proof.

For L₁ > L₂, $H_{p} (L)$ is the intersection of the open sets {ℓ_γ(X) > L₂} indexed by the countable set of free homotopy classes γ in ${[\partial_{2}]}_{{\overset{⌢}{S}}_{0, 2, p}}$ ⁠. It is not hard to see that in a neighborhood of any $X \in H_{p} (L)$ only finitely many of these are important, so the intersection is open. Its closure is given by the countable intersection of closed sets {ℓ_γ(X) ≥ L₂}, which is precisely $\bar{H_{p}} (L)$ ⁠.

In the case L₁ = L₂, $H_{p} (L)$ is empty, because ∂₁ is a curve in ${[\partial_{2}]}_{{\overset{⌢}{S}}_{0, 2, p}}$ of the same length as ∂₂. On the other hand, $\bar{H_{p}} (L)$ is non-empty and of full dimension. To see this, observe, for instance, that there exists an ϵ > 0, depending only on L₁, such that $\bar{H_{p}} (L)$ includes all surfaces in $M_{0, 2 + p} (L)$ that contain a pair of pants bounded by ∂₁, ∂₂ and a simple closed geodesic of length less than ϵ. In such a surface, each simple closed geodesics other than ∂₁ and ∂₂ must exit the pair of pants and can thus be ensured to be of length at least L₁. Hence, $\bar{H_{p}} (L)$ has positive volume H_p(L) > 0.□

B. Tight decomposition

We are now ready to state the main result of this section.

Proposition 7.

The Weil–Petersson volumes T_g,n,p(L) and H_p(L) satisfy

\begin{array}{l} V_{g, n + p} (L) & = \sum_{I_{0} ⊔ \dots ⊔ I_{n} = {n + 1, \dots, n + p}} \int T_{g, n, | I_{0} |} (K, L_{I_{0}}) \prod_{\begin{array}{c} 1 \leq i \leq n \\ I_{i} \neq \emptyset \end{array}} H_{| I_{i} |} (L_{i}, K_{i}, L_{I_{i}}) K_{i} d K_{i}, (g \geq 1 or n \geq 3) \\ V_{0, 2 + p} (L) & = H_{p} (L) + \sum_{\begin{array}{c} I_{1} ⊔ I_{2} = {3, \dots, 2 + p} \\ I_{1}, I_{2} \neq \emptyset \end{array}} \int_{0}^{L_{2}} H_{| I_{1} |} (L_{1}, K, L_{I_{1}}) H_{| I_{2} |} (L_{2}, K, L_{I_{2}}) K d K, (L_{1} \geq L_{2}) \end{array}

where in the first equation it is understood that K_i = L_i whenever I_i = ∅.

The remainder of this section will be devoted to proving this result. But let us first see how it implies Proposition 1.

Proof of Proposition 1.

Clearly H₁(L) = V_0,3(L) = 1 for L₁ ≥ L₂ and T_g,n,0(L) = V_g,n(L). Rewriting the equations as

\begin{array}{l} T_{g, n, p} (L) & = V_{g, n + p} (L) - \sum_{\begin{array}{c} I_{0} ⊔ \dots ⊔ I_{n} = {n + 1, \dots, n + p} \\ | I_{0} | < p \end{array}} \int T_{g, n, | I_{0} |} (K, L_{I_{0}}) \prod_{\begin{array}{c} 1 \leq i \leq n \\ I_{i} \neq \emptyset \end{array}} H_{| I_{i} |} (L_{i}, K_{i}, L_{I_{i}}) K_{i} d K_{i}, \\ H_{p} (L) & = V_{0, 2 + p} (L) - \sum_{\begin{array}{c} I_{1} ⊔ I_{2} = {3, \dots, 2 + p} \\ I_{1}, I_{2} \neq \emptyset \end{array}} \int_{0}^{L_{2}} H_{| I_{1} |} (L_{1}, K, L_{I_{1}}) H_{| I_{2} |} (L_{2}, K, L_{I_{2}}) K d K, \end{array}

it is clear that they are uniquely determined recursively in terms of V_g,n(L). Moreover, by induction we easily verify that H_p(L) in the region L₁ ≥ L₂ is a polynomial in

L_{1}^{2}, \dots, L_{2 + p}^{2}

of degree p − 1 that is symmetric in L₃, …, L_2+p, and T_g,n,p is a polynomial in

L_{1}^{2}, \dots, L_{n + p}^{2}

of degree 3g − 3 + n + p that is symmetric in L₁, …, L_n and symmetric in L_n+1, …, L_n+p.□

C. Tight decomposition in the stable case

1. Shortest cycles

The following parallels the construction of shortest cycles in maps described in Ref. 14, Sec. 6.1.

Lemma 8.

Given a hyperbolic surface $X \in M_{g, n + p}$ for g ≥ 1 or n ≥ 2, then for each i = 1, …, n there exists a unique innermost shortest cycle $σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)$ on X, meaning that it has minimal length in ${[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ and such that all other cycles of minimal length (if they exist) are contained in the region of X delimited by ∂_i and $σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)$ . Moreover, if g ≥ 1 or n ≥ 3, the curves $σ_{{\overset{⌢}{S}}_{g, n, p}}^{1} (X), \dots, σ_{{\overset{⌢}{S}}_{g, n, p}}^{n} (X)$ are disjoint.

Proof.

First note that if a shortest cycle exists, it is a simple closed geodesic. As a consequence of Ref. 20, Theorem 1.6.11, there are only finitely many closed geodesics with length $\leq L_{i}$ in ${[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ ⁠. Since $\partial_{i} \in {[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ has length L_i, this proves the existence of at least one cycle in ${[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ with minimal length.

Regarding the existence and uniqueness of a well-defined innermost shortest cycle, suppose $α, β \in {[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ are two distinct simple closed geodesics with minimal length ℓ (see left side of Fig. 3). Since $α \in {[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ ⁠, cutting along α separates the surface in two disjoint parts. Therefore, α and β can only have an even number of intersections. If the number of intersections is greater than zero, we can choose two distinct intersections and combine α and β to get two distinct cycles γ₁ and γ₂ by switching between α and β at the chosen intersections, such that γ₁ and γ₂ are still in ${[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ ⁠. Since the total length is still 2ℓ, at least one of the new cycles has length $\leq ℓ$ ⁠. This cycle is not geodesic, so there will be a closed cycle in ${[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ with length $< ℓ$ ⁠, which contradicts that α and β have minimal length. We conclude that α and β are disjoint. Since all cycles in ${[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ with minimal length are disjoint and separating, the notion of being innermost is well-defined.

Consider $α_{i} = σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)$ and $α_{j} = σ_{{\overset{⌢}{S}}_{g, n, p}}^{j} (X)$ for i ≠ j (see right side of Fig. 3). Just as before, since α_i is separating and α_i and α_j are simple, the number of intersections is even. If α_i and α_j are not disjoint, we can choose two distinct intersections and construct two distinct cycles γ_i and γ_j by switching between α_i and α_j at the chosen intersections, such that γ_i and γ_j are in ${[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ and ${[\partial_{j}]}_{{\overset{⌢}{S}}_{g, n, p}}$ respectively. Since the total length of the cycles stays the same, there is at least one a ∈ {i, j} such that γ_a has length less or equal than α_a. Since γ_a is not geodesic, there is a closed cycle in ${[\partial_{a}]}_{{\overset{⌢}{S}}_{g, n, p}}$ with length strictly smaller than α_a, which is a contradiction, so the innermost shortest cycles are disjoint.□

In particular the proof implies the following criterions are equivalent:

A simple closed geodesic $α \in {[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ is the innermost shortest cycle $σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)$ ⁠;
For a simple closed geodesic $α \in {[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ we have ℓ(α) ≤ L_i and each simple closed geodesic $β \in σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)$ that is disjoint from α has length ℓ(β) ≥ ℓ(α) with equality only being allowed if β is contained in the region between α and ∂_i.

FIG. 3.

Illustrations of why two intersecting geodesics α and β cannot both have minimal length in their free-homotopy classes on S⌢g,n,p.

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Illustrations of why two intersecting geodesics α and β cannot both have minimal length in their free-homotopy classes on ${\overset{⌢}{S}}_{g, n, p}$ ⁠.

2. Integration on Moduli space

Let us recap Mirzakhani’s decomposition of moduli space integrals in the presence of distinguished cycles (Ref. 1, Sec. 8). A multicurve Γ = (γ₁, …, γ_k) is a collection of disjoint simple closed curves Γ = (γ₁, …, γ_k) in S_g,n which are pairwise non-freely-homotopic. Given a multicurve, in which each curve γ_i may or may not be freely homotopic to a boundary ∂_j of S_g,n, one can consider the stabilizer subgroup

Stab (Γ) = {h \in {M o d}_{g, n} : h \cdot γ_{i} = γ_{i}} \subset {M o d}_{g, n} .

Note that if

γ_{i} \in {[\partial_{j}]}_{S_{g, n}}

is freely homotopic to one of the boundaries ∂_j then h · γ_i = γ_i for any h ∈ Mod_g,n. The moduli space of hyperbolic surfaces with distinguished (free homotopy classes of) curves is the quotient

M_{g, n} {(L)}^{Γ} = T_{g, n} (L) / Stab (Γ) .

For a closed curve γ in S_g,n and

X \in M_{g, n}

⁠, let ℓ_γ(X) be the length of the geodesic representative in the free homotopy class of γ. For

K = (K_{1}, \dots, K_{k}) \subset R_{> 0}^{k}

we can restrict the lengths of the geodesic representatives of curves in Γ by setting

M_{g, n} {(L)}^{Γ} (K) = {X \in M_{g, n} {(L)}^{Γ} : ℓ_{γ_{i}} (X) = K_{i}, i = 1, \dots, k} \subset M_{g, n} {(L)}^{Γ} .

If

γ_{i} \in {[\partial_{j}]}_{S_{g, n}}

then this set is empty unless K_i = L_j. Denote by

π^{Γ} : M_{g, n} {(L)}^{Γ} \to M_{g, n} (L)

the projection. If there are exactly p cycles among Γ that are not freely homotopic to a boundary, then this space admits a natural action of the p-dimensional torus

{(S^{1})}^{p}

obtained by twisting along each of these p cycles proportional to their length. The quotient space is denoted

M_{g, n} {(L)}^{Γ *} (K) = M_{g, n} {(L)}^{Γ} (K) / {(S^{1})}^{p}

and is naturally equipped with a symplectic structure inherited from the Weil–Petersson symplectic structure on

M_{g, n} {(L)}^{Γ}

⁠. If we denote by S_g,n(Γ) the possibly disconnected surface obtained from S_g,n by cutting along all γ_i that are not freely homotopic to a boundary and by

M (S_{g, n} (Γ), L, K)

its moduli space, then according to Ref. 1, Lemma 8.3, the canonical mapping

M_{g, n} {(L)}^{Γ *} (K) \to M (S_{g, n} (Γ), L, K)

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is a symplectomorphism. Given an integrable function

F : M_{g, n} {(L)}^{Γ} \to R

that is invariant under the action of

{(S^{1})}^{p}

⁠, there exists a naturally associated function

\tilde{F} : M (S_{g, n} (Γ), L, K) \to R

such that (essentially Ref. 1, Lemma 8.4)

\int_{M_{g, n} {(L)}^{Γ}} F (X) d μ_{W P} (X) = \int \prod_{\begin{array}{c} 1 \leq i \leq n \\ γ_{i} \notin {[\partial_{j}]}_{S_{g, n}} \end{array}} K_{i} d K_{i} \int_{M (S_{g, n} (Γ), L, K)} \tilde{F} (X) d μ_{W P} (X) .

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3. Shortest multicurves

Suppose g ≥ 1 or n ≥ 3, meaning that we momentarily exclude the cylinder case (g = 0, n = 2). We consider now a special family of multicurves Γ = (γ₁, …, γ_n) on S_g,n+p for n ≥ 1, p ≥ 0. Namely, we require that $γ_{i} \in {[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}$ is freely homotopic to the boundary ∂_i in the capped-off surface ${\overset{⌢}{S}}_{g, n, p}$ for i = 1, …, n. Then there exists a partition I₀ ⊔⋯ ⊔ I_n = {n + 1, …, n + p} such that S_g,n+p(Γ) has n + 1 connected components s₀, …, s_n, where s₀ is of genus g and is adjacent to all curves Γ as well as the boundaries ${(\partial_{j})}_{j \in I_{0}}$ while for each i = 1, …, n, s_i is of genus 0 and contains the ith boundary ∂_i as well as ${(\partial_{j})}_{j \in I_{i}}$ and is adjacent to γ_i. Note that I_i = ∅ if and only if $γ_{i} \in {[\partial_{i}]}_{S_{g, n + p}}$ ⁠. Finally, we observe that mapping class group orbits ${{Mod}_{g, n + p} \cdot {[Γ]}_{S_{g, n + p}}}$ of these multicurves Γ are in bijection with the set of partitions {I₀ ⊔ ⋯ ⊔ I_n = {n + 1, …, n + p}}.

With the help of Lemma 8 we may introduce the restricted moduli space in which we require γ_i to be (freely homotopic to) the innermost shortest cycle in

{[\partial_{i}]}_{{\overset{⌢}{S}}_{g, n, p}}

⁠,

{\hat{M}}_{g, n, p} {(L)}^{Γ} = \{X \in M_{g, n + p} {(L)}^{Γ} : γ_{i} \in [σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)] for i = 1, \dots, n\} .

Lemma 9.

The natural projection

⨆_{Γ} {\hat{M}}_{g, n, p} {(L)}^{Γ} \to M_{g, n + p} (L),

where the disjoint union is over (representatives of) the mapping class group orbits of multicurves Γ, is a bijection.

Proof.

If $X, X^{'} \in T_{g, n + p} (L)$ are representatives of hyperbolic surfaces in ${\hat{M}}_{g, n, p} {(L)}^{Γ}$ and ${\hat{M}}_{g, n, p} {(L)}^{Γ^{'}}$ respectively, then by definition $[γ_{i}] = [σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)]$ and $[γ_{i}^{'}] = [σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X^{'})]$ ⁠. If X and X′ represent the same surface in $M_{g, n + p} (L)$ ⁠, they are related by an element h of the mapping class group, X′ = h · X, and therefore also $σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X) = h \cdot σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X^{'})$ and $[γ_{i}] = h \cdot [γ_{i}^{'}]$ ⁠. So Γ and Γ′ belong to the same mapping class group orbit and, if Γ and Γ′ are freely homotopic, we must have h ∈ Stab(Γ). Hence, X and X′ represent the same element in the set on the left-hand side, and we conclude that the projection is injective. It is also surjective since any $X \in T_{g, n + p} (L)$ is a representative of ${\hat{M}}_{g, n, p} {(L)}^{Γ}$ if we take $Γ = (σ_{{\overset{⌢}{S}}_{g, n, p}}^{1} (X), \dots, σ_{{\overset{⌢}{S}}_{g, n, p}}^{n} (X))$ ⁠, which is a valid multicurve due to Lemma 8.□

We can introduce the length-restricted version ${\hat{M}}_{g, n, p} {(L)}^{Γ} (K) \subset M_{g, n + p} {(L)}^{Γ} (K)$ as before.

Lemma 10.

The subset

{\hat{M}}_{g, n, p} {(L)}^{Γ} (K) \subset M_{g, n + p} {(L)}^{Γ} (K)

is invariant under twisting [the torus-action on

M_{g, n + p} {(L)}^{Γ} (K)

described above]. The image of the quotient

{\hat{M}}_{g, n, p} {(L)}^{Γ *} (K)

under the symplectomorphism (47) is precisely

M_{g, n, | I_{0} |}^{tight} (K, L_{I_{0}}) \times \prod_{\begin{array}{c} 1 \leq i \leq n \\ I_{i} \neq \emptyset \end{array}} \bar{H_{| I_{i} |}} (L_{i}, K_{i}, L_{I_{i}}) .

(49)

Proof.

Let $X \in M_{g, n + p} {(L)}^{Γ} (K)$ be a hyperbolic surface with distinguished multicurve Γ. The lengths of the geodesics associated to Γ as well as the lengths of the geodesics that are disjoint from those geodesics are invariant under twisting X along Γ. The criterion explained just below Lemma 8 for γ_i to be the innermost shortest cycle $σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)$ is thus also preserved under twisting, showing that the subset ${\hat{M}}_{g, n, p} {(L)}^{Γ} (K)$ is invariant.

Let $X_{0} \in M_{g, n, | I_{0} |} (K, L_{I_{0}})$ and $X_{i} \in M_{0, 2 + | I_{i} |} (L_{i}, K_{i}, L_{I_{i}})$ for those i = 1, …, n for which I_i ≠ ∅ be the hyperbolic structures on the connected components s₀, …, s_n of S_g,n+p(Γ) obtained by cutting X along the geodesics associated to Γ. For each i = 1, …, n the criterion for γ_i to be the innermost shortest cycle $σ_{{\overset{⌢}{S}}_{g, n, p}}^{i} (X)$ is equivalent to the following two conditions holding:

the ith boundary of X₀ is tight in the capped-off surface associated to s₀;
I_i = ∅ (meaning γ_i = ∂_i) or $X_{i} \in \bar{H_{| I_{i} |}} (L_{i}, K_{i}, L_{I_{i}})$ [recall the definition in (46)].

Hence, we have $X \in {\hat{M}}_{g, n, p} {(L)}^{Γ} (K)$ precisely when $X_{0} \in M_{g, n, | I_{0} |}^{tight} (K, L_{I_{0}})$ and $X_{i} \in \bar{H_{| I_{i} |}} (L_{i}, K_{i}, L_{I_{i}})$ when I_i ≠ ∅. This proves the second statement of the lemma.□

It follows that the Weil–Petersson volume of

{\hat{M}}_{g, n, p} {(L)}^{Γ *} (K)

is equal to the product of the volumes of the spaces appearing in . Combining with Lemma 9 and the integration formula this shows that

\begin{array}{l} V_{g, n + p} (L) & = \sum_{Γ} \int_{{\hat{M}}_{g, n, p} {(L)}^{Γ}} d μ_{W P} \\ = \sum_{I_{0} ⊔ \dots ⊔ I_{n} = {n + 1, \dots, n + p}} \int T_{g, n, | I_{0} |} (K, L_{I_{0}}) \prod_{\begin{array}{c} 1 \leq i \leq n \\ I_{i} \neq \emptyset \end{array}} H_{| I_{i} |} (L_{i}, K_{i}, L_{I_{i}}) K_{i} d K_{i}, \end{array}

where it is understood that K_i = L_i whenever I_i = ∅. This proves the first relation of Proposition 7.

D. Tight decomposition of the cylinder

The decomposition we have just described does not work well in the case g = 0 and n = 2, because ∂₁ and ∂₂ are in the same free homotopy class of the capped surface

{\overset{⌢}{S}}_{0, 2, p}

⁠. Instead, we should consider a multicurve Γ = (γ₁) consisting of a single curve γ₁ on S_0,2+p in the free homotopy class

{[\partial_{1}]}_{{\overset{⌢}{S}}_{0, 2, p}} = {[\partial_{2}]}_{{\overset{⌢}{S}}_{0, 2, p}}

⁠, see Fig. 4. In this case there exists a partition I₁ ⊔ I₂ = {3, …, p + 2} such that S_0,2+p(Γ) has two connected components s₁ and s₂, with s_i a genus-0 surface with 2 + |I_i| boundaries corresponding to ∂_i, γ₁ and

{(\partial_{j})}_{j \in I_{i}}

⁠. We consider the restricted moduli space

{\hat{M}}_{0, 2, p} {(L)}^{Γ} = \{X \in M_{0, 2 + p} {(L)}^{Γ} : γ_{1} \in [σ_{{\overset{⌢}{S}}_{0, 2, p}}^{1} (X)]\},

which thus treats the two boundaries ∂₁ and ∂₂ asymmetrically, by requiring that γ₁ is the shortest curve farthest from ∂₁. Lemma 9 goes through unchanged: the projection

⨆_{Γ} {\hat{M}}_{0, 2, p} {(L)}^{Γ} ⟶ M_{0, 2 + p} (L),

where the disjoint union is over the mapping class group orbits of Γ = (γ₁), is a bijection. Assuming L₁ ≥ L₂, we cannot have

γ_{1} \in {[\partial_{1}]}_{S_{0, 2 + p}}

so I₁ ≠ ∅. There are two cases to consider:

$γ_{1} \in {[\partial_{2}]}_{S_{0, 2 + p}}$ and therefore I₂ = ∅: this means that ∂₂ has minimal length in ${[\partial_{2}]}_{{\overset{⌢}{S}}_{0, 2, p}}$ ⁠, so ${\hat{M}}_{0, 2, p} {(L)}^{Γ} = \bar{H_{p}} (L)$ ⁠.
I₂ ≠ ∅: by reasoning analogous to that of Lemma 10 we have that ${\hat{M}}_{0, 2, p} {(L)}^{Γ *} (K)$ is symplectomorphic to
$\bar{H_{| I_{1} |}} (L_{1}, K, L_{I_{1}}) \times H_{| I_{2} |} (L_{2}, K, L_{I_{2}}) .$
Hence, when L₁ ≥ L₂ we have
$V_{0, 2 + p} (L) = H_{p} (L) + \sum_{\begin{array}{c} I_{1} ⊔ I_{2} = {3, \dots, 2 + p} \\ I_{1}, I_{2} \neq \emptyset \end{array}} \int_{0}^{L_{2}} H_{| I_{1} |} (L_{1}, K, L_{I_{1}}) H_{| I_{2} |} (L_{2}, K, L_{I_{2}}) K d K .$
This proves the second relation of Proposition 7.

FIG. 4.

A surface with two distinguished boundaries ∂1 and ∂2 naturally decomposes into a pair of half-tight cylinders.

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A surface with two distinguished boundaries ∂₁ and ∂₂ naturally decomposes into a pair of half-tight cylinders.

III. GENERATING FUNCTIONS OF TIGHT WEIL–PETERSSON VOLUMES

A. Definitions

Let us define the following generating functions of the Weil–Petersson volumes, half-tight cylinder volumes and tight Weil–Petersson volumes:

F_{g} [μ] = \sum_{n = 0}^{\infty} \frac{1}{n!} \int d μ (L_{1}) \dots \int d μ (L_{n}) V_{g, n} (L),

(50)

H (L_{1}, L_{2}; μ] = \sum_{p = 1}^{\infty} \frac{1}{p!} \int d μ (L_{3}) \dots \int d μ (L_{2 + p}) H_{p} (L),

(51)

{\tilde{F}}_{g} [ν, μ] = \sum_{n = 0}^{\infty} \frac{1}{n!} \int d ν (L_{1}) \dots \int d ν (L_{n}) T_{g, n} (L_{1}, \dots, L_{n}; μ] .

(52)

Furthermore, for g ≥ 2 we recall the polynomial

P_{g, 0} (m_{1}, \dots, m_{3 g - 3}) = \sum_{\begin{array}{c} d_{2}, d_{3}, \dots \geq 0 \\ \sum_{k \geq 2} (k - 1) d_{k} = 3 g - 3 \end{array}} {⟨ τ_{2}^{d_{2}} τ_{3}^{d_{3}} \dots ⟩}_{g} \prod_{k \geq 2} \frac{{(- m_{k - 1})}^{d_{k}}}{d_{k}!},

(53)

where

{⟨ τ_{2}^{d_{2}} τ_{3}^{d_{3}} \dots ⟩}_{g}

are the ψ-class intersection numbers on Deligne–Mumford compactification of the moduli space of curves

{\bar{M}}_{g, n}

with n = ∑_kd_k ≤ 3g − 3 marked points. Then according to Ref. 21, Theorem 3²²

F_{0} [μ] = \frac{1}{2} \int_{0}^{R} d r Z {(r; μ]}^{2},

(54)

F_{1} [μ] = - \frac{1}{24} \log M_{0} [μ],

(55)

F_{g} [μ] = \frac{1}{{(M_{0} [μ])}^{2 g - 2}} P_{g, 0} (\frac{M_{1} [μ]}{M_{0} [μ]}, \dots, \frac{M_{3 g - 3} [μ]}{M_{0} [μ]}) for g \geq 2,

(56)

where the moments M_k[μ] are defined in Eq. .

In the genus-0 case we can take successive derivatives to find useful formulas for one, two or three distinguished boundaries of prescribed lengths,

\frac{δ F_{0} [μ]}{δ μ (L_{1})} = - \int_{0}^{R [μ]} I_{0} (L_{1} \sqrt{2 r}) Z (r; μ] d r,

(57)

\frac{δ^{2} F_{0} [μ]}{δ μ (L_{1}) δ μ (L_{2})} = \int_{0}^{R [μ]} I_{0} (L_{1} \sqrt{2 r}) I_{0} (L_{2} \sqrt{2 r}) d r,

(58)

\frac{δ^{3} F_{0} [μ]}{δ μ (L_{1}) δ μ (L_{2}) δ μ (L_{3})} = \frac{1}{M_{0} [μ]} [\prod_{i = 1}^{3} I_{0} (L_{i} \sqrt{2 R [μ]})] .

(59)

B. Volume of half-tight cylinder

The equations of Proposition 7 turn into the equations

\frac{δ^{n} F_{g} [μ]}{δ μ (L_{1}) \dots δ μ (L_{n})} = \int T_{g, n} (K; μ] \prod_{i = 1}^{n} (K_{i} H (L_{i}, K_{i}; μ] + δ (K_{i} - L_{i})) d K_{i}, (g \geq 1 or n \geq 3)

(60)

\frac{δ^{2} F_{0} [μ]}{δ μ (L_{1}) δ μ (L_{2})} = H (L_{1}, L_{2}; μ] + \int_{0}^{L_{2}} H (L_{1}, K; μ] H (L_{2}, K; μ] K d K . (L_{1} \geq L_{2})

(61)

Let us focus on the last equation, which should determine H(L₁, L₂; μ] uniquely. The left-hand side depends on μ only through the quantity R[μ], and the dependence on R is analytic,

\frac{δ^{2} F_{0} [μ]}{δ μ (L_{1}) δ μ (L_{2})} = R + \frac{1}{4} (L_{1}^{2} + L_{2}^{2}) R^{2} + \frac{1}{48} (L_{1}^{4} + 4 L_{1}^{2} L_{2}^{2} + L_{2}^{4}) R^{3} + \dots .

Hence, the same is true for H(L₁, L₂; μ] and one may easily calculate order by order in R that

H (L_{1}, L_{2}; μ] = R + \frac{1}{4} (L_{1}^{2} - L_{2}^{2}) R^{2} + \frac{1}{48} {(L_{1}^{2} - L_{2}^{2})}^{2} R^{3} + \dots .

This suggests that H(L₁, L₂; μ] depends on L₁ and L₂ only through the combination

L_{1}^{2} - L_{2}^{2}

⁠. Let’s prove this.

Lemma 11.

The half-tight cylinder generating function satisfies

(\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{L_{2}} \frac{\partial}{\partial L_{2}}) H (L_{1}, L_{2}; μ] = 0,

(62)

and is therefore given by

H (L_{1}, L_{2}; μ] = \sum_{ℓ = 0}^{\infty} \frac{2^{- ℓ} R {[μ]}^{ℓ + 1}}{ℓ! (ℓ + 1)!} {(L_{1}^{2} - L_{2}^{2})}^{ℓ} = \sqrt{\frac{2 R [μ]}{L_{1}^{2} - L_{2}^{2}}} I_{1} (\sqrt{L_{1}^{2} - L_{2}^{2}} \sqrt{2 R [μ]}) (L_{1} \geq L_{2}) .

(63)

Proof.

By construction

H (L_{1}, 0; μ] = \frac{δ^{2} F_{0} [μ]}{δ μ (L_{1}) δ μ (0)}

and the integral with L₂ = 0 evaluates to

H (L_{1}, 0; μ] = \frac{δ^{2} F_{0} [μ]}{δ μ (L_{1}) δ μ (0)} = \frac{\sqrt{2 R}}{L_{1}} I_{1} (L_{1} \sqrt{2 R}) .

(64)

The identity

\frac{\partial}{\partial r} (\frac{\sqrt{2 r}}{L_{1}} I_{1} (L_{1} \sqrt{2 r}) \frac{\sqrt{2 r}}{L_{2}} I_{1} (L_{2} \sqrt{2 r})) = (\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{L_{2}} \frac{\partial}{\partial L_{2}}) I_{0} (L_{1} \sqrt{2 r}) I_{0} (L_{2} \sqrt{2 r}),

which can be easily checked by calculating the derivatives, implies that

H (L_{1}, 0; μ] H (L_{2}, 0; μ] = (\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{L_{2}} \frac{\partial}{\partial L_{2}}) \frac{δ^{2} F_{0} [μ]}{δ μ (L_{1}) δ μ (L_{2})} .

Hence, by we find that

\begin{array}{l} (\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{L_{2}} \frac{\partial}{\partial L_{2}}) H (L_{1}, L_{2}; μ] & = H (L_{1}, 0; μ] H (L_{2}, 0; μ] - (\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{L_{2}} \frac{\partial}{\partial L_{2}}) \int_{0}^{L_{2}} H (L_{1}, K; μ] H (L_{2}, K; μ] K d K \\ = H (L_{1}, 0; μ] H (L_{2}, 0; μ] - H (L_{1}, L_{2}; μ] H (L_{2}, L_{2}; μ] \\ - \int_{0}^{L_{2}} (\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{L_{2}} \frac{\partial}{\partial L_{2}}) H (L_{1}, K; μ] H (L_{2}, K; μ] K d K \\ = - \int_{0}^{L_{2}} \frac{\partial}{\partial K} (H (L_{1}, K; μ] H (L_{2}, K; μ]) d K \\ - \int_{0}^{L_{2}} (\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{L_{2}} \frac{\partial}{\partial L_{2}}) H (L_{1}, K; μ] H (L_{2}, K; μ] K d K \\ = - \int_{0}^{L_{2}} [H (L_{1}, K; μ] (\frac{1}{L_{2}} \frac{\partial}{\partial L_{2}} + \frac{1}{K} \frac{\partial}{\partial K}) H (L_{2}, K; μ] \\ + H (L_{2}, K; μ] (\frac{1}{L_{1}} \frac{\partial}{\partial L_{1}} + \frac{1}{K} \frac{\partial}{\partial K}) H (L_{1}, K; μ]] K d K . \end{array}

Since the leading coefficient in R of H(L₁, L₂; μ] satisfies , it follows that the same is true for the higher-order coefficients in R.

As a consequence of (62), $H (L_{1}, L_{2}; μ] = H (\sqrt{L_{1}^{2} - L_{2}^{2}}, 0; μ]$ and the claimed expression (63) follows from (64).□

C. Rewriting generating functions

Since the work of Mirzakhani¹⁵ it is known that the Weil–Petersson volumes V_g,n(L) are expressible in terms of intersection numbers as follows. The compactified moduli space

{\bar{M}}_{g, n}

of genus-g curves with n marked points comes naturally equipped with the Chern classes ψ₁, …, ψ_n associated with its n tautological line bundles, as well as the cohomology class κ₁ of the Weil–Petersson symplectic structure (up to a factor 2π²). The corresponding intersection numbers are given by the integrals

{⟨κ_{1}^{m} τ_{d_{1}} \dots τ_{d_{n}}⟩}_{g, n} = \int_{{\bar{M}}_{g, n}} κ_{1}^{m} ψ_{1}^{d_{1}} \dots ψ_{n}^{d_{n}},

where d₁, …, d_n ≥ 0 and n = d₁ + ⋯ d_n + m + 3 − 3g. For g ≥ 0 we denote the generating function of these intersection numbers by

G_{g} (s; x_{0}, x_{1}, \dots) = \sum_{n \geq 0} \frac{1}{n!} \sum_{\begin{array}{c} m, d_{1}, \dots, d_{n} \geq 0 \\ d_{1} + \dots + d_{n} + m = 3 g - 3 + n \end{array}} {⟨κ_{1}^{m} τ_{d_{1}} \dots τ_{d_{n}}⟩}_{g, n} \frac{s^{m}}{m!} x_{d_{1}} \dots x_{d_{n}} .

(65)

We may sum over all genera to arrive at the generating function

G (s; x_{0}, x_{1}, \dots) = \sum_{g = 0}^{\infty} λ^{2 g - 2} G_{g} (s; x_{0}, x_{1}, \dots) .

(66)

In order to lighten the notation we do not write the dependence on λ explicitly here, which only serves as a formal generating variable. Note that λ is actually redundant for organizing the series, since any monomial appears in at most one of the G_g as can be seen from . Then the generating function of Weil–Petersson volumes can be expressed as

\sum_{g = 0}^{\infty} λ^{2 g - 2} 2^{3 - 3 g} F_{g} [μ] = G (π^{2}; t_{0} [μ], t_{1} [μ], \dots),

(67)

where the times t_k[μ] are defined by

t_{k} [μ] = \int d μ (L) \frac{2 L^{2 k}}{4^{k} k!} .

(68)

See Ref. 21, Lemma 11 based on Ref. 15, where one should be careful that some conventions differ by some factors of two compared to the current work.

We will show that the (bivariate) generating function ${\tilde{F}}_{g} [ν, μ]$ of tight Weil–Petersson volumes, defined in (52), is also related to the intersection numbers, but with different times.

Proposition 12.

The generating function of the volumes T_g,n is related to the generating function of intersection numbers via

{\tilde{F}}_{g} [ν, μ] = 2^{3 g - 3} G_{g} (0; τ_{0} [ν, μ], τ_{1} [ν, μ], \dots),

(69)

where the shifted times τ_k[ν, μ] are defined by

τ_{k} [ν, μ] = t_{k} [ν] + δ_{k, 1} - 2^{1 - k} M_{k - 1} [μ] .

(70)

We remark that this result highlights the integrable properties of the tight Weil–Petersson volumes. It is well-known that exp G(0; t₀, t₁, …) is a τ-function of the KdV hierarchy.¹⁷ Since ${\tilde{F}}_{g} [ν, μ]$ and G_g are related by a ν-independent shift of the times, $\exp (\sum_{g = 0}^{\infty} λ^{2 g - 2} 2^{3 - 3 g} {\tilde{F}}_{g} [ν, μ])$ viewed as a formal power series in the times t_k = t_k[ν] is therefore a τ-function of the KdV hierarchy as well. The associated differential equations will play an important role in the Proof of Theorem 2 in Sec. III E.

This proposition will be proved in the remainder of this subsection, relying on an appropriate substitution of the weight ν. To this end, we informally introduce a linear mapping H_μ on measures on the half line [0, ∞) as follows. If ρ is a measure on [0, ∞) we let H_μρ be the measure given by

ρ + (\int H (L, K; μ] d ρ (L)) K d K .

(71)

The effect of H_μ on the times can be computed using the series expansion ,

\begin{align} t_{k} [H_{μ} ρ] & = t_{k} [ρ] + \frac{2}{4^{k} k!} \int_{0}^{\infty} (\int H (L, K; μ] d ρ (L)) K^{2 k + 1} d K \\ = t_{k} [ρ] + \sum_{p = 1}^{\infty} \frac{2 {(2 R [μ])}^{p}}{p! 4^{p + k} (p + k)!} \int d ρ (L) L^{2 p + 2 k} \\ = \sum_{p = 0}^{\infty} \frac{{(2 R [μ])}^{p}}{p!} t_{p + k} [ρ] . \end{align}

(72)

We observe that H_μ acts as an infinite upper-triangular matrix on the times. This matrix is easily inverted to give

t_{q} [ρ] = \sum_{p = 0}^{\infty} \frac{{(- 2 R [μ])}^{p}}{p!} t_{p + q} [H_{μ} ρ] .

(73)

This means that knowledge of the generating function

{\tilde{F}}_{g} [H_{μ} ρ, μ]

with substituted weight H_μρ is sufficient to recover the original generating function

{\tilde{F}}_{g} [ν, μ]

⁠. Luckily the former is within close reach.

Lemma 13.

The generating functions for tight Weil–Petersson volumes and regular Weil–Petersson volumes are related by

{\tilde{F}}_{g} [H_{μ} ρ, μ] = F_{g} [ρ + μ] - δ_{g, 0} F_{corr} [ρ, μ],

(74)

where the correction term

F_{corr} [ρ, μ] = \sum_{n = 0}^{2} \frac{1}{n!} \sum_{p = 0}^{\infty} \frac{1}{p!} \int d ρ (L_{1}) \dots d ρ (L_{n}) d μ (L_{n + 1}) \dots d μ (L_{n + p}) V_{0, n + p} (L)

(75)

is necessary to subtract the constant, linear and quadratic dependence on ρ in the genus-0 case.

Proof.

If g, n ≥ 0 (such that n ≥ 3 if g = 0) and L₁, …, L_n ∈ [0, ∞) ∪ i(0, π), then Proposition 7 allows us to compute

\begin{align} \sum_{p = 0}^{\infty} \frac{1}{p!} \int d μ (L_{n + 1}) \dots d μ (L_{n + p}) V_{g, n + p} (L) \\ = \sum_{p = 0}^{\infty} \frac{1}{p!} \int d μ (L_{n + 1}) \dots d μ (L_{n + p}) \sum_{I_{0} ⊔ \dots ⊔ I_{n} = {n + 1, \dots, n + p}} \int T_{g, n, | I_{0} |} (K, L_{I_{0}}) \prod_{\begin{array}{c} 1 \leq i \leq n \\ I_{i} \neq \emptyset \end{array}} H_{| I_{i} |} (L_{i}, K_{i}, L_{I_{i}}) K_{i} d K_{i} \\ = \sum_{p_{0}, \dots, p_{n} = 0}^{\infty} \frac{1}{p_{0}! \dots p_{n}!} \int d μ (L_{n + 1}) \dots d μ (L_{n + p_{0} + \dots + p_{n}}) T_{g, n, p_{0}} (K, L^{(0)}) \prod_{\begin{array}{c} 1 \leq i \leq n \\ p_{i} \neq 0 \end{array}} H_{p_{i}} (L_{i}, K_{i}, L^{(i)}) K_{i} d K_{i}, \end{align}

(76)

where we use the notation

L^{(j)} = (L_{n + p_{0} + \dots + p_{j - 1} + 1}, \dots, L_{n + p_{0} + \dots + p_{j}})

⁠. In terms of the tight Weil–Petersson volume generating function and the half-tight cylinder generating function this evaluates to

\sum_{p = 0}^{\infty} \frac{1}{p!} \int d μ (L_{n + 1}) \dots d μ (L_{n + p}) V_{g, n + p} (L) = \sum_{J \subset {1, \dots, n}} \int T_{g, n} (K; μ] \prod_{i \in J} K_{i} H (L_{i}, K_{i}; μ] d K_{i},

(77)

where it is understood that in the argument of T_g,n(K; μ] we take K_i = L_i for i∉J.

Expanding F_g[ρ + μ] from its definition we find

\begin{align} F_{g} [ρ + μ] - F_{corr} [ρ, μ] δ_{g, 0} & = \sum_{n, p = 0}^{\infty} 1_{g \geq 1 or n \geq 3} \frac{1}{n! p!} \int d ρ (L_{1}) \dots d ρ (L_{n}) d μ (L_{n + 1}) \dots d μ (L_{n + p}) V_{g, n + p} (L) \\ = \sum_{n = 0}^{\infty} \frac{1}{n!} 1_{g \geq 1 or n \geq 3} \int (\prod_{i = 1}^{n} d ρ (L_{i})) \sum_{p = 0}^{\infty} \frac{1}{p!} \int (\prod_{i = n + 1}^{n + p} d μ (L_{i})) V_{g, n + p} (L) . \end{align}

(78)

Plugging in and using that T_0,n(K; μ] = 0 for n < 3, yields

\begin{array}{l} F_{g} [ρ + μ] - F_{corr} [ρ, μ] δ_{g, 0} & = \sum_{n = 0}^{\infty} \frac{1}{n!} \int d ρ (L_{1}) \dots d ρ (L_{n}) \sum_{J \subset {1, \dots, n}} \int T_{g, n} (K; μ] \prod_{i \in J} K_{i} H (L_{i}, K_{i}; μ] d K_{i} \\ = \sum_{n = 0}^{\infty} \frac{1}{n!} \int d (H_{μ} ρ) (L_{1}) \dots d (H_{μ} ρ) (L_{n}) T_{g, n} (L; μ] \\ = {\tilde{F}}_{g} [H_{μ} ρ, μ] \end{array}

as claimed.□

Lemma 13 and together lead to the relation

\sum_{g = 0}^{\infty} λ^{2 g - 2} 2^{3 - 3 g} {\tilde{F}}_{g} [H_{μ} ρ, μ] = G (π^{2}; t_{0} [ρ + μ], t_{1} [ρ + μ], \dots) - 8 λ^{- 2} F_{corr} .

(79)

The right-hand side can be specialized, making use of a variety of identities between intersection numbers. Firstly, a relation between intersection numbers involving κ₁ and pure ψ-class intersection numbers^16,23 leads to the identity²⁴

G (s; x_{0}, x_{1}, \dots) = G (0; x_{0}, x_{1}, x_{2} + γ_{2} (s), \dots),

(80)

where the shifts are

γ_{k} (s) = \frac{{(- 1)}^{k}}{(k - 1)!} s^{k - 1} 1_{k \geq 2} .

(81)

For us this gives

\sum_{g = 0}^{\infty} λ^{2 g - 2} 2^{3 - 3 g} {\tilde{F}}_{g} [H_{μ} ρ, μ] = G (0; t [ρ + μ] + γ (π^{2})) - 8 λ^{- 2} F_{corr},

(82)

where we use the notation G(0; x) = G(0; x₀, x₁, x₂, …).

This can be further refined using Witten’s observation,¹⁶ proved by Kontsevich,¹⁷ that G(0; x) satisfies the string equation

(- \frac{\partial}{\partial x_{0}} + \sum_{p = 0}^{\infty} x_{p + 1} \frac{\partial}{\partial x_{p}} + \frac{x_{0}^{2}}{2 λ^{2}}) e^{G (0; x)} = 0 .

(83)

Following a computation of Itzykson and Zuber,²⁵ it implies the following identity.

Lemma 14.

The solution to the string equation (83) satisfies a formal power series identity in the parameter r,

G (0; x_{0}, x_{1}, \dots) = G (0; r + \sum_{k = 0}^{\infty} \frac{{(- r)}^{k}}{k!} x_{k}, \sum_{k = 0}^{\infty} \frac{{(- r)}^{k}}{k!} x_{k + 1}, \sum_{k = 0}^{\infty} \frac{{(- r)}^{k}}{k!} x_{k + 2}, \dots) - \frac{1}{2 λ^{2}} \int_{0}^{r} d s {(s + \sum_{k = 0}^{\infty} \frac{{(- s)}^{k}}{k!} x_{k})}^{2} .

(84)

Proof.

For x₀, x₁, … fixed, let us consider the sequence of functions

y (s) = (y_{0} (s), y_{1} (s), \dots), y_{i} (s) = δ_{i, 0} s + \sum_{k = 0}^{\infty} \frac{{(- s)}^{k}}{k!} x_{k + i},

(85)

such that

y_{p}^{'} (s) = δ_{p, 0} - y_{p + 1} (s)

⁠. The string equation then implies

\frac{d}{d s} G (0; y (s)) = \frac{\partial G}{\partial x_{0}} (0; y (s)) - \sum_{p = 0}^{\infty} y_{p + 1} (s) \frac{\partial G}{\partial x_{p}} (y (s)) = \frac{y_{0} {(s)}^{2}}{2 λ^{2}} .

(86)

Integrating from s = 0 to s = r gives the claimed identity.□

Before we can use this lemma, we establish a relation between τ_k[H_μρ, μ] and t_k[ρ + μ].

Lemma 15.

We can rewrite

t_{q} [ρ + μ] + γ_{q} (π^{2}) = δ_{q, 0} 2 R [μ] + \sum_{p = 0}^{\infty} \frac{{(- 2 R [μ])}^{p}}{p!} τ_{p + q} [H_{μ} ρ, μ],

(87)

where the shifted times τ_k[ν, μ] are defined in (70).

Proof.

We first relate the moments M_i[μ] defined in to the times t_i[μ]. Note that Z(u; μ] defined in can be expressed in the times as

Z (u; μ] = u - \sum_{k = 0}^{\infty} \frac{{(2 u)}^{k}}{2 k!} (t_{k} [μ] + γ_{k} (π^{2})) .

(88)

By taking p derivatives with respect to u, we get

\sum_{k = 0}^{\infty} \frac{{(2 R [μ])}^{k}}{k!} (t_{k + p} [μ] + γ_{k + p} (π^{2})) = \{\begin{cases} 2 R [μ] & if p = 0 \\ 1 - M_{0} [μ] & if p = 1 \\ - 2^{1 - p} M_{p - 1} [μ] & if p \geq 2 \end{cases} .

(89)

Just like before in obtaining , this can be inverted to

t_{q} [μ] + γ_{q} (π^{2}) = δ_{q, 1} - \sum_{p = 0}^{\infty} \frac{{(- 2 R [μ])}^{p}}{p!} 2^{1 - p - q} M_{p + q - 1} [μ] .

(90)

The right-hand side of can thus be expressed as

\begin{align} δ_{q, 0} 2 R [μ] + \sum_{p = 0}^{\infty} \frac{{(- 2 R [μ])}^{p}}{p!} τ_{p + q} [H_{μ} ρ, μ] & = δ_{q, 1} + \sum_{p = 0}^{\infty} \frac{{(- 2 R [μ])}^{p}}{p!} t_{p + q} [H_{μ} ρ] - \sum_{p = 0}^{\infty} \frac{{(- 2 R [μ])}^{p}}{p!} 2^{1 - p - q} M_{p + q - 1} [μ] \\ = t_{q} [μ] + γ_{q} (π^{2}) + \sum_{p = 0}^{\infty} \frac{{(- 2 R [μ])}^{p}}{p!} t_{p + q} [H_{μ} ρ] . \end{align}

(91)

From the last term is just t_q[ρ], so we have reproduced the left-hand side of , since t_q[ρ + μ] = t_q[ρ] + t_q[μ].□

The last two lemmas allow us to express as

\begin{align} \sum_{g = 0}^{\infty} λ^{2 g - 2} 2^{3 - 3 g} {\tilde{F}}_{g} [H_{μ} ρ, μ] & = G (0; τ [H_{μ} ρ, μ]) \\ + \frac{1}{2 λ^{2}} \int_{0}^{2 R [μ]} d s {(s + \sum_{k = 0}^{\infty} \frac{{(- s)}^{k}}{k!} τ_{k} [H_{μ} ρ, μ])}^{2} - \frac{8}{λ^{2}} F_{corr} [ρ, μ] . \end{align}

(92)

To finish the Proof of Proposition 12 we thus only need to check that the last two terms cancel.

Lemma 16.

We have

F_{corr} [ρ, μ] = \frac{1}{16} \int_{0}^{2 R [μ]} d s {(s + \sum_{k = 0}^{\infty} \frac{{(- s)}^{k}}{k!} τ_{k} [H_{μ} ρ, μ])}^{2} .

(93)

Proof.

Let us denote the right-hand side by G_corr. By the definition ,

G_{corr} = \frac{1}{16} \int_{0}^{2 R [μ]} d s {(\sum_{k = 0}^{\infty} \frac{{(- s)}^{k}}{k!} (t_{k} [H_{μ} ρ] - 2^{1 - k} M_{k - 1} [μ]))}^{2} .

(94)

Changing integration variables to r = R[μ] − s/2 gives

\begin{align} G_{corr} & = \frac{1}{8} \int_{0}^{R [μ]} d r {(\sum_{k = 0}^{\infty} \frac{{(2 r - 2 R [μ])}^{k}}{k!} (t_{k} [H_{μ} ρ] - 2^{1 - k} M_{k - 1} [μ]))}^{2} \\ = \frac{1}{8} \int_{0}^{R [μ]} d r {(\sum_{k = 0}^{\infty} \frac{{(2 r - 2 R [μ])}^{k}}{k!} t_{k} [H_{μ} ρ] - 2 Z (r; μ])}^{2} \\ = \frac{1}{8} \int_{0}^{R [μ]} d r {(\sum_{k = 0}^{\infty} \frac{{(2 r)}^{k}}{k!} t_{k} [ρ] - 2 Z (r))}^{2}, \end{align}

(95)

where in the second equality we use the series expansion of Z(r) = Z(r; μ] around r = R[μ] [recall from that

M_{k} [μ] = Z^{(k + 1)} (R [μ]; μ])

⁠, and in the third equality we expanded (2r − 2R[μ])^k as a polynomial in r and made use of .

In terms of the weight ρ this can be written as

G_{corr} = \frac{1}{2} \int_{0}^{R [μ]} d r {(Z (r) - \int d ρ (L) I_{0} (L \sqrt{2 r}))}^{2} .

(96)

In the constant, linear and quadratic term in ρ we then recognize exactly the expressions , , and ,

\begin{align} G_{corr} & = F_{0} [μ] + \int d ρ (L_{1}) \frac{δ F_{0} [μ]}{δ μ (L_{1})} + \frac{1}{2} \int d ρ (L_{1}) d ρ (L_{2}) \frac{δ F_{0} [μ]}{δ μ (L_{1}) δ μ (L_{2})} \\ = F_{corr} [ρ, μ] . \end{align}

(97)

□

D. Properties of the new kernel

Recall that the new kernel is given by

K (x, t, μ] = \int_{- \infty}^{\infty} K_{0} (x + z, t) d X (z), K_{0} (x, t) = \frac{1}{1 + \exp (\frac{x + t}{2})} + \frac{1}{1 + \exp (\frac{x - t}{2})},

(98)

where X(z) = X(z; μ] is determined by its two-sided Laplace transform

\hat{X} (u; μ]

⁠,

\hat{X} (u; μ] = \int_{- \infty}^{\infty} d X (z) e^{- u z} = \frac{\sin (2 π u)}{2 π u η (u; μ]}

(99)

and

η (u; μ] = \sum_{m = 0}^{\infty} \frac{M_{m} [μ]}{(2 m + 1) ‼} u^{2 m} .

(100)

To prove Theorem 2, we need to relate K(x, t; μ] to the moments M_k[μ], since they appear in the shifted times. We define the reverse moments β_m[μ] as the coefficients of the reciprocal series

\frac{1}{η (u; μ]} = \sum_{m = 0}^{\infty} β_{m} [μ] u^{2 m} .

(101)

Multiplying both series shows that the moments and reverse moments obey

\sum_{m = 0}^{p} \frac{M_{m} [μ]}{(2 m + 1) ‼} β_{p - m} [μ] = δ_{p, 0}

(102)

for each p ≥ 0. Note in particular that

β_{0} [μ] = \frac{1}{M_{0} [μ]}, β_{1} [μ] = - \frac{M_{1} [μ]}{3 M_{0} {[μ]}^{2}} .

(103)

Proposition 17.

For i, j ≥ 1, the new kernel satisfies

\int_{0}^{\infty} d x \frac{x^{2 i - 1}}{(2 i - 1)!} K (x, t; μ] = \sum_{m = 0}^{i} β_{m} [μ] \frac{t^{2 i - 2 m}}{(2 i - 2 m)!}

(104)

and

\int_{0}^{\infty} d x \int_{0}^{\infty} d y \frac{x^{2 i - 1} y^{2 j - 1}}{(2 i - 1)! (2 j - 1)!} K (x + y, t; μ] = \sum_{m = 0}^{i + j} β_{m} [μ] \frac{t^{2 i + 2 j - 2 m}}{(2 i + 2 j - 2 m)!} .

(105)

We need two lemmas to prove this proposition. First we examine the one-sided Laplace transforms

{\hat{K}}_{0} (u, t) : = \int_{0}^{\infty} d x e^{- u x} K_{0} (x, t),

(106)

\hat{K} (u, t; μ] : = \int_{0}^{\infty} d x e^{- u x} K (x, t; μ] .

(107)

Lemma 18.

{\hat{K}}_{0} (u, t) - {\hat{K}}_{0} (- u, t) = \frac{- 4 π \cosh (t u)}{\sin (2 π u)} + \frac{2}{u} .

(108)

Proof.

To compute the integral , we only need positive values for x, so we assume x > 0. Since K₀(x, −t) = K₀(x, t) we can also assume t ≥ 0. For x < t we may expand

\begin{align} K_{0} (x, t) & = \frac{\exp (\frac{- x - t}{2})}{1 + \exp (\frac{- x - t}{2})} + \frac{1}{1 + \exp (\frac{x - t}{2})} \\ = - \sum_{p = 1}^{\infty} {(- e^{\frac{- x - t}{2}})}^{p} + \sum_{p = 0}^{\infty} {(- e^{\frac{x - t}{2}})}^{p}, \end{align}

(109)

while for x > t we may use

\begin{align} K_{0} (x, t) & = \frac{\exp (\frac{- x - t}{2})}{1 + \exp (\frac{- x - t}{2})} + \frac{\exp (\frac{- x + t}{2})}{1 + \exp (\frac{- x + t}{2})} \\ = - \sum_{p = 1}^{\infty} {(- e^{\frac{- x - t}{2}})}^{p} - \sum_{p = 1}^{\infty} {(- e^{\frac{- x + t}{2}})}^{p} . \end{align}

(110)

This gives

\begin{align} {\hat{K}}_{0} (u, t) & = - \sum_{p = 1}^{\infty} \int_{0}^{\infty} d x e^{- u x} {(- e^{\frac{- x - t}{2}})}^{p} + \sum_{p = 0}^{\infty} \int_{0}^{t} d x e^{- u x} {(- e^{\frac{x - t}{2}})}^{p} - \sum_{p = 1}^{\infty} \int_{t}^{\infty} d x e^{- u x} {(- e^{\frac{- x + t}{2}})}^{p} \\ = - e^{- u t} \sum_{p = - \infty}^{\infty} \frac{{(- 1)}^{p}}{u - p / 2} - \sum_{p = 1}^{\infty} {(- 1)}^{p} \frac{\exp (\frac{- t p}{2})}{u + p / 2} + \sum_{p = 0}^{\infty} {(- 1)}^{p} \frac{\exp (\frac{- t p}{2})}{u - p / 2} \\ = \frac{- 2 π e^{- u t}}{\sin (2 π u)} + \frac{1}{u} + \sum_{p = 0}^{\infty} {(- 1)}^{p} e^{\frac{- t p}{2}} (\frac{1}{u - p / 2} - \frac{1}{u + p / 2}) . \end{align}

(111)

When subtracting

{\hat{K}}_{0} (- u, t)

it should be clear that the sum cancels and we easily obtain the claimed formula.□

Lemma 19.

\hat{K} (u, t; μ] - \hat{K} (- u, t; μ] = \hat{X} (u; μ] ({\hat{K}}_{0} (u, t) - {\hat{K}}_{0} (- u, t) - \frac{2}{u}) + \frac{2 \hat{X} (0; μ]}{u} .

(112)

Proof.

From the definition we obtain

\begin{align} \hat{K} (u, t; μ] - \hat{K} (- u, t; μ] & = \int_{- \infty}^{\infty} d X (z) \int_{0}^{\infty} d X (e^{- u x} - e^{u x}) K_{0} (x + z, t) \\ = \int_{- \infty}^{\infty} d X (z) \int_{z}^{\infty} d X (e^{(z - x) u} - e^{(x - z) u}) K_{0} (x, t) \\ = \int_{- \infty}^{\infty} d X (z) (e^{z u} {\hat{K}}_{0} (u, t) - e^{- z u} {\hat{K}}_{0} (- u, t)) \\ - \int_{- \infty}^{\infty} d X (z) \int_{0}^{z} d X (e^{(z - x) u} - e^{(x - z) u}) K_{0} (x, t) . \end{align}

(113)

The first integral evaluates to

\hat{X} (u; μ] ({\hat{K}}_{0} (u, t) - {\hat{K}}_{0} (- u, t))

⁠. By changing variables (x, z) → (−x, −z) and using the symmetry of dX(z), we observe that the second integral is unchanged when K₀(x, t) is replaced by K₀(−x, t). Since also K₀(x, t) + K₀(−x, t) = 2, the second integral can be calculated to give

\begin{align} \int_{- \infty}^{\infty} d X (z) \int_{0}^{z} d X (e^{(z - x) u} - e^{(x - z) u}) K_{0} (x, t) & = \int_{- \infty}^{\infty} d X (z) \int_{0}^{z} d X (e^{(z - x) u} - e^{(x - z) u}) \\ = - \frac{2}{u} \int_{- \infty}^{\infty} d X (z) (1 - e^{u z}) \\ = \frac{2}{u} \hat{X} (u; μ] - \frac{2 \hat{X} (0; μ]}{u} . \end{align}

(114)

Subtracting both integrals gives the desired result.□

Proof of Proposition 17.

We start by noting

\int_{0}^{\infty} d x \frac{x^{2 i - 1}}{(2 i - 1)!} K (x, t; μ] = - \frac{1}{2} [u^{2 i - 1}] (\hat{K} (u, t; μ] - \hat{K} (- u, t; μ]) .

(115)

Using Lemmas 19 and 18, we get for i ≥ 1

\begin{align} \int_{0}^{\infty} d x \frac{x^{2 i - 1}}{(2 i - 1)!} K (x, t; μ] & = - \frac{1}{2} [u^{2 i - 1}] (\hat{X} (u; μ] (\frac{- 4 π \cosh (t u)}{\sin (2 π u)}) + \frac{2 \hat{X} (0; μ]}{u}) \\ = [u^{2 i}] \frac{\cosh (t u)}{η (u; μ]} \\ = \sum_{m = 0}^{i} β_{m} [μ] \frac{t^{2 i - 2 m}}{(2 i - 2 m)!} . \end{align}

(116)

The second identity follows easily from the first by performing the integration at constant x + y, since

\int_{0}^{z} \frac{x^{2 i - 1} {(z - x)}^{2 j - 1}}{(2 i - 1)! (2 j - 1)!} d x = \frac{z^{2 i + 2 j - 1}}{(2 i + 2 j - 1)!} .

□

E. Proof of Theorem 2

We will prove the tight topological recursion by retracing Mirzakhani’s proof¹⁵ of Witten’s conjecture, which relies on the observation that her recursion formula , expressed as an identity on the coefficients of the volume polynomials V_g,n(L), is equivalent to certain differential equations for the generating function G(s; x₀, x₁, …) of intersection numbers (see also Ref. 18). These differential equations can be expressed as the Virasoro constraints^16,18,26

{\tilde{V}}_{p} e^{G (0; x_{0}, x_{1}, \dots)} = 0 .

(117)

Here the Virasoro operators

{\tilde{V}}_{- 1}, {\tilde{V}}_{0}, {\tilde{V}}_{1}, {\tilde{V}}_{2}, \dots

are the differential operators acting on the ring of formal power series in x₀, x₁, x₂, … via

\begin{align} {\tilde{V}}_{p} & = - \frac{(2 p + 3) ‼}{2^{p + 1}} \frac{\partial}{\partial x_{p + 1}} + \frac{1}{2^{p + 1}} \sum_{n = 0}^{\infty} \frac{(2 n + 2 p + 1) ‼}{(2 n - 1) ‼} x_{n} \frac{\partial}{\partial x_{n + p}} \\ + \frac{λ^{2}}{2^{p + 2}} \sum_{i + j = p - 1} (2 i + 1) ‼ (2 j + 1) ‼ \frac{\partial}{\partial x_{i}} \frac{\partial}{\partial x_{j}} \\ + δ_{p, - 1} (\frac{λ^{- 2} x_{0}^{2}}{2}) + \frac{δ_{p, 0}}{16} . \end{align}

(118)

They satisfy the Virasoro relations

[{\tilde{V}}_{m} {\tilde{V}}_{n}] = (m - n) {\tilde{V}}_{m + n} .

(119)

Proposition 12 suggests introducing the shift

x_{k} \to x_{k} + {\tilde{γ}}_{k}

in G with

{\tilde{γ}}_{k} = δ_{k, 1} - 2^{1 - k} M_{k - 1} [μ]

⁠, which satisfies

({\tilde{V}}_{p} + \frac{(2 p + 3) ‼}{2^{p + 1}} \frac{\partial}{\partial x_{p + 1}} - \frac{1}{2^{p + 1}} \sum_{n = 0}^{\infty} \frac{2^{- n} M_{n} [μ]}{(2 n + 1) ‼} (2 p + 2 n + 3) ‼ \frac{\partial}{\partial x_{p + n + 1}}) e^{G (0; x_{0}, x_{1} + {\tilde{γ}}_{1}, x_{2} + \tilde{γ_{2}}, \dots)} = 0 .

(120)

We use the reverse moments β_m[μ] of to introduce linear combinations

V_{p} = \sum_{m = 0}^{\infty} β_{m} [μ] 2^{p} ({\tilde{V}}_{p + m} + \frac{(2 p + 2 m + 3) ‼}{2^{p + m + 1}} \frac{\partial}{\partial x_{p + m + 1}} - \frac{1}{2^{p + m + 1}} \sum_{n = 0}^{\infty} \frac{2^{- n} M_{n} [μ]}{(2 n + 1) ‼} (2 p + 2 m + 2 n + 3) ‼ \frac{\partial}{\partial x_{p + m + n + 1}})

(121)

of these operators for all p ≥ −1, which therefore obey

V_{p} \exp (G (0; x_{0}, x_{1} + {\tilde{γ}}_{1}, x_{2} + \tilde{γ_{2}}, \dots)) = 0 .

(122)

Using the operators V_p can be expressed as

\begin{align} V_{p} & = \sum_{m = 0}^{\infty} β_{m} [μ] 2^{p} ({\tilde{V}}_{p + m} + \frac{(2 p + 2 m + 3) ‼}{2^{p + m + 1}} \frac{\partial}{\partial x_{p + m + 1}} - \frac{1}{2^{p + m + 1}} \sum_{n = 0}^{\infty} \frac{2^{- n} M_{n} [μ]}{(2 n + 1) ‼} (2 p + 2 m + 2 n + 3) ‼ \frac{\partial}{\partial x_{p + m + n + 1}}) \\ = - \frac{1}{2} (2 p + 3) ‼ \frac{\partial}{\partial x_{p + 1}} + \frac{λ^{2}}{4} \sum_{m = 0}^{\infty} \sum_{i + j = p + m - 1} 2^{- m} β_{m} [μ] (2 i + 1) ‼ (2 j + 1) ‼ \frac{\partial}{\partial x_{i}} \frac{\partial}{\partial x_{j}} \\ + \frac{1}{2} \sum_{n, m = 0}^{\infty} 2^{- m} β_{m} [μ] \frac{(2 n + 2 p + 2 m + 1) ‼}{(2 n - 1) ‼} x_{n} \frac{\partial}{\partial x_{n + p + m}} \\ + δ_{p, - 1} (\frac{λ^{- 2} x_{0}^{2} β_{0} [μ]}{4} + \frac{β_{1} [μ]}{32}) + δ_{p, 0} \frac{β_{0} [μ]}{16} . \end{align}

(123)

In particular, after some rearranging (and shifting p → p − 1) we observe the identity

\begin{align} \frac{1}{2} (2 p + 1) ‼ \frac{\partial G}{\partial x_{p}} = \frac{λ^{2}}{4} \sum_{m = 0}^{\infty} \sum_{i + j = p + m - 2} 2^{- m} β_{m} [μ] (2 i + 1) ‼ (2 j + 1) ‼ (\frac{\partial^{2} G}{\partial x_{i} \partial x_{j}} + \frac{\partial G}{\partial x_{i}} \frac{\partial G}{\partial x_{j}}) \\ + \frac{1}{2} \sum_{n, m = 0}^{\infty} 2^{- m} β_{m} [μ] \frac{(2 n + 2 p + 2 m - 1) ‼}{(2 n - 1) ‼} x_{n} \frac{\partial G}{\partial x_{n + p + m - 1}} \\ + δ_{p, 0} (\frac{λ^{- 2} x_{0}^{2} β_{0} [μ]}{4} + \frac{β_{1} [μ]}{32}) + δ_{p, 1} \frac{β_{0} [μ]}{16}, \end{align}

(124)

where G is understood to be evaluated at

G = G (0; x_{0}, x_{1} + {\tilde{γ}}_{1}, x_{2} + \tilde{γ_{2}}, \dots)

⁠.

Substituting x_k = t_k[ν] such that

x_{k} + {\tilde{γ}}_{k} = τ_{k} [ν, μ]

⁠, Proposition 12 links G to the generating function

\begin{align} G (0; τ_{0} [ν, μ], τ_{1} [ν, μ], \dots) & = \sum_{g = 0}^{\infty} λ^{2 g - 2} 2^{3 - 3 g} {\tilde{F}}_{g} [ν, μ] \\ = \sum_{g = 0}^{\infty} λ^{2 g - 2} 2^{3 - 3 g} \sum_{n = 0}^{\infty} \frac{1}{n!} \int d ν (L_{1}) \dots d ν (L_{n}) T_{g, n} (L; μ] \end{align}

(125)

of tight Weil–Petersson volumes. The differential equations can then be reformulated as the functional differential equation (here all partial derivates of G are evaluated at 0, τ₀[ν, μ], τ₁[ν, μ], …)

\begin{array}{l} \frac{δ}{δ ν (L_{1})} G (0; τ_{0} [ν, μ], τ_{1} [ν, μ], \dots) \\ = \sum_{p = 0}^{\infty} \frac{2 L_{1}^{2 p}}{4^{p} p!} \frac{\partial G}{\partial x_{p}} = \sum_{p = 0}^{\infty} \frac{1}{L_{1}} \int_{0}^{L_{1}} d t \frac{t^{2 p}}{(2 p)!} 2^{1 - p} (2 p + 1) ‼ \frac{\partial G}{\partial x_{p}} \\ = \sum_{p = 0}^{\infty} \frac{λ^{2}}{L_{1}} \int_{0}^{L_{1}} d t \frac{t^{2 p}}{(2 p)!} \sum_{m = 0}^{\infty} \sum_{i + j = p + m - 2} 2^{- i - j - 2} β_{m} [μ] (2 i + 1) ‼ (2 j + 1) ‼ (\frac{\partial^{2} G}{\partial x_{i} \partial x_{j}} + \frac{\partial G}{\partial x_{i}} \frac{\partial G}{\partial x_{j}}) \\ + \sum_{p = 0}^{\infty} \frac{1}{L_{1}} \int_{0}^{L_{1}} d t \frac{t^{2 p}}{(2 p)!} \int d ν (P) \sum_{n, m = 0}^{\infty} 2^{- m - p - n + 2} β_{m} [μ] \frac{(2 n + 2 p + 2 m - 1) ‼}{(2 n)!} P^{2 n} \frac{\partial G}{\partial x_{n + p + m - 1}} \\ + λ^{- 2} t_{0}^{2} [ν] β_{0} [μ] + \frac{β_{1} [μ]}{8} + \frac{β_{0} [μ] L_{1}^{2}}{48} . \end{array}

Inserting the integral identities of Proposition 17 this can also be expressed in terms of the kernel K(x, t; μ] as

\begin{array}{l} \frac{δ}{δ ν (L_{1})} G (0; τ_{0} [ν, μ], τ_{1} [ν, μ], \dots) \\ = \frac{λ^{2}}{4 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \int_{0}^{\infty} d y \sum_{i, j = 0}^{\infty} K (x + y, t; μ] \frac{x^{2 i + 1} y^{2 j + 1}}{4^{i} i! 4^{j} j!} (\frac{\partial^{2} G}{\partial x_{i} \partial x_{j}} + \frac{\partial G}{\partial x_{i}} \frac{\partial G}{\partial x_{j}}) \\ + \frac{1}{L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \int d ν (P) \sum_{q = 0}^{\infty} (K (x, t + P; μ] + K (x, t - P; μ]) \frac{x^{2 q + 1}}{4^{q} q!} \frac{\partial G}{\partial x_{q}} \\ + λ^{- 2} t_{0}^{2} [ν] β_{0} [μ] + \frac{β_{1} [μ]}{8} + \frac{β_{0} [μ] L_{1}^{2}}{48} \\ = \frac{λ^{2}}{16 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \int_{0}^{\infty} d y x y K (x + y, t; μ] (\frac{δ^{2} G}{δ ν (x) δ ν (y)} + \frac{δ G}{δ ν (x)} \frac{δ G}{δ ν (y)}) \\ + \frac{1}{2 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \int d ν (P) x (K (x, t + P; μ] + K (x, t - P; μ]) \frac{δ G}{δ ν (x)} \\ + \frac{δ}{δ ν (L_{1})} (\frac{λ^{- 2} t_{0}^{3} [ν]}{6 M_{0} [μ]} - \frac{M_{1} [μ] t_{0} [ν]}{48 M_{0} {[μ]}^{2}} + \frac{t_{1} [ν]}{24 M_{0} [μ]}), \end{array}

where in the last line we used . This equation at the level of the generating function is precisely equivalent to the recursion equation on its polynomial coefficients

\begin{align} T_{g, n} (L; μ] & = \frac{1}{2 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \int_{0}^{\infty} d y x y K (x + y, t; μ] [T_{g - 1, n + 1} (x, y, L_{\hat{{1}}}; μ] \\ + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} T_{g_{1}, 1 + | I |} (x, L_{I}; μ] T_{g_{2}, 1 + | J |} (y, L_{J}; μ]] \\ + \frac{1}{2 L_{1}} \int_{0}^{L_{1}} d t \int_{0}^{\infty} d x \sum_{j = 2}^{n} x (K (x, t + L_{j}; μ] + K (x, t - L_{j}; μ]) T_{g, n - 1} (x, L_{\hat{{1, j}}}; μ], \end{align}

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for (g, n) ∉ {(0, 3), (1, 1)} combined with the initial data

T_{0, 3} (L_{1}, L_{2}, L_{3}; μ] = \frac{1}{M_{0} [μ]}

(127)

T_{1, 1} (L; μ] = - \frac{M_{1} [μ]}{24 M_{0} {[μ]}^{2}} + \frac{L^{2}}{48 M_{0} [μ]} .

(128)

This completes the Proof of Theorem 2.

F. Proof of Theorem 4

We follow a strategy along the lines of the Proof of Theorem 2. Recall the relation between the intersection number generating function G and the tight Weil–Petersson volumes T_g,n. Let us denote by

G_{g, n} (x_{0}, x_{1}, \dots; M_{0}, M_{1}, \dots)

the homogeneous part of degree n in x₀, x₁, … of

G_{g} (0; x_{0}, x_{1} + 1 - M_{0}, x_{2} - \frac{1}{2} M_{1}, x_{3} - \frac{1}{4} M_{2}, \dots)

⁠. In other words, they are homogeneous polynomials of degree n in x₀, x₁, … with coefficients that are formal power series in

M_{0}, M_{1}, \dots

⁠, such that

G_{g} (0; x_{0}, x_{1} + 1 - M_{0}, x_{2} - \frac{1}{2} M_{1}, \dots) = \sum_{n = 0}^{\infty} G_{g, n} (x_{0}, x_{1}, \dots; M_{0}, M_{1}, \dots) .

(129)

We will prove that there exist polynomials

{\bar{P}}_{g, n} (x_{0}, x_{1}, \dots; m_{1}, m_{2} \dots)

such that

G_{g, n} (x_{0}, x_{1}, \dots; M_{0}, M_{1}, \dots) = \frac{1}{n!} \frac{1}{M_{0}^{2 g - 2 + n}} {\bar{P}}_{g, n} (x_{0}, x_{1}, \dots; \frac{M_{1}}{M_{0}}, \frac{M_{2}}{M_{0}}, \dots)

(130)

and deduce a recurrence in n.

For g ≥ 2 and n = 0, the existence of a polynomial

{\bar{P}}_{g, 0} (m_{1}, m_{2}, \dots)

follows from Ref. 21, Lemma 12, since

\begin{align} G_{g, 0} (x_{0}, x_{1}, \dots; M_{0}, M_{1}, \dots) & = G_{g} (0; 0, 1 - M_{0}, - \frac{1}{2} M_{1}, \dots) \\ = M_{0}^{2 - 2 g} G_{g} (0; 0, 0, - \frac{1}{2} \frac{M_{1}}{M_{0}}, - \frac{1}{4} \frac{M_{2}}{M_{0}}, \dots) \end{align}

(131)

and G_g(0; 0, 0, x₂, x₃, …) is polynomial by construction. Also

G_{0, 3} = \frac{x_{0}^{3}}{6} \frac{1}{M_{0}}, G_{1, 1} = \frac{x_{1}}{24} \frac{1}{M_{0}} - \frac{x_{0}}{48} \frac{M_{1}}{M_{0}^{2}} .

(132)

Let us now assume 2g − 2 + n ≥ 2 and aim to express G_g,n in tems of G_g,n−1. By construction the series G_g,n obeys for k ≥ 1,

\frac{\partial G_{g, n}}{\partial x_{k}} = - 2^{k - 1} \frac{\partial G_{g, n - 1}}{\partial M_{k - 1}} .

(133)

The string equation, i.e., at p = −1, written in terms of G_g,n reads

\sum_{k = 1}^{\infty} x_{k} \frac{\partial G_{g, n - 1}}{\partial x_{k - 1}} - \sum_{k = 0}^{\infty} 2^{- k} M_{k} \frac{\partial G_{g, n}}{\partial x_{k}} = 0,

(134)

which after rearranging gives the relation

\frac{\partial G_{g, n}}{\partial x_{0}} = \sum_{k = 1}^{\infty} (\frac{x_{k}}{M_{0}} \frac{\partial G_{g, n - 1}}{\partial x_{k - 1}} - 2^{- k} \frac{M_{k}}{M_{0}} \frac{\partial G_{g, n}}{\partial x_{k}}) .

(135)

Together with this is sufficient to identify the recursion relation

\begin{align} G_{g, n} (x_{0}, x_{1}, \dots; M_{0}, M_{1}, \dots) & = \frac{1}{n} \sum_{k = 0}^{\infty} x_{k} \frac{\partial G_{g, n}}{\partial x_{k}} \\ = \frac{1}{n} \sum_{k = 1}^{\infty} (\frac{x_{0} x_{k}}{M_{0}} \frac{\partial G_{g, n - 1}}{\partial x_{k - 1}} + \frac{x_{0} M_{k}}{2 M_{0}} \frac{\partial G_{g, n - 1}}{\partial M_{k - 1}} - 2^{k - 1} x_{k} \frac{\partial G_{g, n - 1}}{\partial M_{k - 1}}) . \end{align}

(136)

By induction, we now verify that G_g,n is of the form . If is granted for G_g,n−1, then

\begin{align} G_{g, n} (x_{0}, x_{1}, \dots; M_{0}, M_{1}, \dots) = \frac{1}{n!} \frac{1}{M_{0}^{2 g - 2 + n}} [\sum_{k = 1}^{\infty} x_{0} x_{k} \frac{\partial {\bar{P}}_{g, n - 1}}{\partial x_{k - 1}} - \sum_{k = 2}^{\infty} (- x_{0} \frac{M_{k}}{2 M_{0}} + 2^{k - 1} x_{k}) \frac{\partial {\bar{P}}_{g, n - 1}}{\partial m_{k - 1}} \\ + (- x_{0} \frac{M_{1}}{2 M_{0}} + x_{1}) ((2 g + n - 3) {\bar{P}}_{g, n - 1} - \sum_{k = 1}^{\infty} m_{k} \frac{\partial {\bar{P}}_{g, n - 1}}{\partial m_{k}})] \end{align}

(137)

is indeed of the form provided

{\bar{P}}_{g, n} (x_{0}, x_{1}, \dots; m_{1}, m_{2}, \dots) = \sum_{k = 1}^{\infty} x_{0} x_{k} \frac{\partial {\bar{P}}_{g, n - 1}}{\partial x_{k - 1}} - \sum_{p = 1}^{\infty} (- \frac{x_{0} m_{p + 1}}{2} + 2^{p} x_{p + 1} - \frac{x_{0} m_{1} m_{k}}{2} + x_{1} m_{k}) \frac{\partial {\bar{P}}_{g, n - 1}}{\partial m_{p}} + (- \frac{x_{0} m_{1}}{2} + x_{1}) (2 g + n - 3) {\bar{P}}_{g, n - 1} .

(138)

According to the series G_g,n and the tight Weil–Petersson volume T_g,n are related via

G_{g, n} (t_{0} [ν], t_{1} [ν], \dots; M_{0} [μ], M_{1} [μ], \dots) = \frac{2^{3 - 3 g}}{n!} \int d ν (L_{1}) \dots d ν (L_{n}) T_{g, n} (L; μ] .

(139)

This naturally leads to the existence of polynomials

P_{g, n} (L, m_{1}, m_{2}, \dots)

such that

T_{g, n} (L; μ] = \frac{1}{M_{0}^{2 g - 2 + n}} P_{g, n} (L, \frac{M_{1}}{M_{0}}, \dots, \frac{M_{3 g - 3 + n}}{M_{0}}),

(140)

to get

\begin{align} P_{g, n} (L, m) & = \sum_{p = 1}^{\infty} (m_{p + 1} - \frac{L_{1}^{2 p + 2}}{2^{p + 1} (p + 1)!} - m_{1} m_{p} + \frac{1}{2} L_{1}^{2} m_{p}) \frac{\partial P_{g, n - 1}}{\partial m_{p}} (L_{\hat{{1}}}, m) \\ + (2 g - 3 + n) (- m_{1} + \frac{1}{2} L_{1}^{2}) P_{g, n - 1} (L_{\hat{{1}}}, m) + 1_{{g = 0, n = 3}} + \sum_{i = 2}^{n} \int d L_{i} L_{i} P_{g, n - 1} (L_{\hat{{1}}}, m) . \end{align}

(141)

The claims about the degree of the polynomials

P_{g, n}

are easily checked to be valid for the initial conditions and to be preserved by the recursion formula . This proves theorem 4.

Proof of Corollary 5.

We note that

\begin{align} [L_{1}^{2 p}] P_{g, n} (L, m) & = δ_{p, 0} [\sum_{q = 1}^{\infty} (m_{q + 1} - m_{1} m_{q}) \frac{\partial P_{g, n - 1}}{\partial m_{q}} (L_{\hat{{1}}}, m) - (2 g - 3 + n) m_{1} P_{g, n - 1} (L_{\hat{{1}}}, m) \\ + 1_{{g = 0, n = 3}} + \sum_{i = 2}^{n} \int d L_{i} L_{i} P_{g, n - 1} (L_{\hat{{1}}}, m)] \\ + δ_{p, 1} [\sum_{q = 1}^{\infty} (\frac{1}{2} m_{q}) \frac{\partial P_{g, n - 1}}{\partial m_{q}} (L_{\hat{{1}}}, m) + \frac{1}{2} (2 g - 3 + n) P_{g, n - 1} (L_{\hat{{1}}}, m)] \\ + 1_{p > 1} [- \frac{1}{2^{p} p!} \frac{\partial P_{g, n - 1}}{\partial m_{p - 1}} (L_{\hat{{1}}}, m)] . \end{align}

(142)

Setting m₀ = 1 this gives

\sum_{p = 0}^{\infty} 2^{p} p! m_{p} [L_{1}^{2 p}] P_{g, n} (L, m) = 1_{{g = 0, n = 3}} + \sum_{i = 2}^{n} \int d L_{i} L_{i} P_{g, n - 1} (L_{\hat{{1}}}, m)

(143)

and

\sum_{p = 1}^{\infty} 2^{p} p! m_{p - 1} [L_{1}^{2 p}] P_{g, n} (L, m) = (2 g - 3 + n) P_{g, n - 1} (L_{\hat{{1}}}, m) .

(144)

Using Eq. , we get the desired result.□

IV. LAPLACE TRANSFORM, SPECTRAL CURVE AND DISK FUNCTION

A. Proof of Theorem 3

Let us consider the partial derivative operator

Δ (z) = 4 \sum_{p = 0}^{\infty} {(2 z^{2})}^{- 1 - p} (2 p + 1) ‼ \frac{\partial}{\partial x_{p}},

(145)

sometimes called the loop insertion operator in the literature, on the ring of formal power series in x₀, x₁, … and 1/z. For later purposes we record several identities for the power series coefficients around z = ∞, valid for a ≥ 0,

[u^{- 2 - 2 a}] Δ (u) = 2^{1 - a} (2 a + 1) ‼ \frac{\partial}{\partial x_{a}},

(146)

[u^{- 4 - 2 a}] Δ (u) Δ (- u) = 2^{2 - a} \sum_{i + j = a} (2 i + 1) ‼ (2 j + 1) ‼ \frac{\partial^{2}}{\partial x_{i} \partial x_{j}},

(147)

[u^{2 a - 1}] \frac{1}{u (z^{2} - u^{2}) η (u; μ]} = \sum_{m = 0}^{a} z^{2 m - 2 a - 2} β_{m} [μ],

(148)

where the reverse moments β[μ] were introduced in .

From the definition and the relation we deduce that for g ≥ 1 or n ≥ 3

\begin{align} ω_{g, n} (z) & = \int_{0}^{\infty} [\prod_{1 \leq i \leq n} L_{i} e^{- z_{i} L_{i}}] {\frac{δ^{n} {\tilde{F}}_{g} [ν, μ]}{δ ν (L_{1}) \dots δ ν (L_{n})}|}_{ν = 0} d L_{1} \dots d L_{n} \\ = 2^{3 g - 3} {Δ (z_{1}) \dots Δ (z_{n}) G_{g} (0; x_{0}, x_{1} + {\tilde{γ}}_{1}, x_{2} + {\tilde{γ}}_{2}, \dots)|}_{x_{0} = x_{1} = \dots = 0}, \end{align}

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where

{\tilde{γ}}_{k} = δ_{k, 1} - 2^{1 - k} M_{k - 1} [μ]

as before. Recall the differential equation satisfied by this (shifted) intersection number generating function

G (0; x_{0}, x_{1} + {\tilde{γ}}_{1}, x_{2} + {\tilde{γ}}_{2}, \dots)

⁠,

\begin{array}{l} \frac{1}{2} (2 p + 1) ‼ \frac{\partial G}{\partial x_{p}} = \frac{λ^{2}}{4} & \sum_{m = 0}^{\infty} \sum_{i + j = p + m - 2} 2^{- m} β_{m} [μ] (2 i + 1) ‼ (2 j + 1) ‼ (\frac{\partial^{2} G}{\partial x_{i} \partial x_{j}} + \frac{\partial G}{\partial x_{i}} \frac{\partial G}{\partial x_{j}}) \\ + \frac{1}{2} \sum_{n, m = 0}^{\infty} 2^{- m} β_{m} [μ] \frac{(2 n + 2 p + 2 m - 1) ‼}{(2 n - 1) ‼} x_{n} \frac{\partial G}{\partial x_{n + p + m - 1}} \\ + δ_{p, 0} (\frac{λ^{- 2} x_{0}^{2} β_{0} [μ]}{4} + \frac{β_{1} [μ]}{32}) + δ_{p, 1} \frac{β_{0} [μ]}{16} . \end{array}

With the help of the identities it can be recast in terms of the operator Δ(z) as

\begin{array}{l} Δ (z_{1}) G & = 2 λ^{2} \sum_{a = 0}^{\infty} \sum_{m = 0}^{a + 2} \sum_{i + j = a} {(2 z_{1}^{2})}^{m - a - 3} 2^{- m} β_{m} [μ] (2 i + 1) ‼ (2 j + 1) ‼ (\frac{\partial^{2} G}{\partial x_{i} \partial x_{j}} + \frac{\partial G}{\partial x_{i}} \frac{\partial G}{\partial x_{j}}) \\ + 4 \sum_{n, m, p = 0}^{\infty} {(2 z_{1}^{2})}^{- 1 - p} 2^{- m} β_{m} [μ] \frac{(2 n + 2 p + 2 m - 1) ‼}{(2 n - 1) ‼} x_{n} \frac{\partial G}{\partial x_{n + p + m - 1}} \\ + \frac{4}{z_{1}^{2}} (\frac{λ^{- 2} x_{0}^{2} β_{0} [μ]}{4} + \frac{β_{1} [μ]}{32}) + \frac{β_{0} [μ]}{8 z_{1}^{4}} \\ = \frac{λ^{2}}{16} \sum_{a = 0}^{\infty} ([u^{2 a + 3}] \frac{1}{u (z_{1}^{2} - u^{2}) η (u; μ]}) [u^{- 4 - 2 a}] (Δ (u) Δ (- u) G + (Δ (u) G) (Δ (- u) G)) \\ + \frac{1}{2} \sum_{q = 0}^{\infty} \sum_{n = 0}^{q + 1} \frac{2^{n} x_{n}}{(2 n - 1) ‼} ([u^{2 q - 2 n + 1}] \frac{1}{u (z_{1}^{2} - u^{2}) η (u; μ]}) [u^{- 2 q - 2}] Δ (u) G \\ + \frac{4}{z_{1}^{2}} (\frac{λ^{- 2} x_{0}^{2} β_{0} [μ]}{4} + \frac{β_{1} [μ]}{32}) + \frac{β_{0} [μ]}{8 z_{1}^{4}} \\ = \frac{λ^{2}}{16} \underset{u \to 0}{R e s} \frac{1}{u (z_{1}^{2} - u^{2}) η (u; μ]} (Δ (u) Δ (- u) G + (Δ (u) G) (Δ (- u) G)) \\ + \frac{1}{2} \underset{u \to 0}{R e s} \sum_{n = 0}^{\infty} \frac{{(2 u^{2})}^{n} x_{n}}{(2 n - 1) ‼} \frac{1}{u (z_{1}^{2} - u^{2}) η (u; μ]} Δ (u) G \\ + \frac{4}{z_{1}^{2}} (\frac{λ^{- 2} x_{0}^{2} β_{0} [μ]}{4} + \frac{β_{1} [μ]}{32}) + \frac{β_{0} [μ]}{8 z_{1}^{4}} . \end{array}

Extracting the genus-g contribution, which appears as the coefficient of λ^2g−2, the relation allows us to turn this into a recursion for ω_g,n,

\begin{array}{l} ω_{g, n} (z) & = \frac{1}{2} \underset{u \to 0}{R e s} \frac{1}{u (z_{1}^{2} - u^{2}) η (u; μ]} [ω_{g - 1, n + 1} (u, - u, z_{\hat{{1}}}) + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} ω_{g_{1}, | I |} (u, z_{I}) ω_{g_{2}, | J |} (- u, z_{J})] \\ + \overset{R e s}{u \to 0} \sum_{j = 2}^{n} \sum_{p = 0}^{\infty} u^{2 p} z_{j}^{- 2 - 2 p} (2 p + 1) \frac{1}{u (z_{1}^{2} - u^{2}) η (u; μ]} ω_{g, n - 1} (u, z_{\hat{{1, j}}}) \\ + \frac{δ_{g, 0} δ_{n, 3}}{M_{0} [μ] z_{1}^{2} z_{2}^{2} z_{3}^{2}} + δ_{g, 1} δ_{n, 1} (- \frac{M_{1} [μ]}{24 M_{0} {[μ]}^{2} z_{1}^{2}} + \frac{1}{8 M_{0} [μ] z_{1}^{4}}) \\ = \underset{u \to 0}{R e s} \frac{1}{2 u (z_{1}^{2} - u^{2}) η (u; μ]} [ω_{g - 1, n + 1} (u, - u, z_{\hat{{1}}}) + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} ω_{g_{1}, | I |} (u, z_{I}) ω_{g_{2}, | J |} (- u, z_{J}) \\ + \sum_{j = 2}^{n} (\frac{1}{{(z_{j} - u)}^{2}} + \frac{1}{{(z_{j} + u)}^{2}}) ω_{g, n - 1} (u, z_{\hat{{1, j}}})] \\ + \frac{δ_{g, 0} δ_{n, 3}}{M_{0} [μ] z_{1}^{2} z_{2}^{2} z_{3}^{2}} + δ_{g, 1} δ_{n, 1} (- \frac{M_{1} [μ]}{24 M_{0} {[μ]}^{2} z_{1}^{2}} + \frac{1}{8 M_{0} [μ] z_{1}^{4}}) . \end{array}

Finally, if we set

ω_{0, 2} (z) = {(z_{1} - z_{2})}^{- 2}

and ω_0,0(z) = ω_0,1(z) = 0, this reduces to

ω_{g, n} (z) = \underset{u \to 0}{R e s} \frac{1}{2 u (z_{1}^{2} - u^{2}) η (u; μ]} [ω_{g - 1, n + 1} (u, - u, z_{\hat{{1}}}) + \sum_{\begin{array}{c} g_{1} + g_{2} = g \\ I ⨿ J = {2, \dots, n} \end{array}} ω_{g_{1}, | I |} (u, z_{I}) ω_{g_{2}, | J |} (- u, z_{J})] .

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B. Disk function

Due to Proposition 7, there is a relation between the regular and tight Weil–Petersson volumes. In this subsection we will look at this relation in the Laplace transformed setting.

In particular, we are interested in the Laplace transformed generating functions of (regular) Weil–Petersson volumes

W_{g, n} (z) = \int_{0}^{\infty} d L_{1} L_{1} e^{- z_{1} L_{1}} \dots \int_{0}^{\infty} d L_{n} L_{n} e^{- z_{n} L_{n}} \frac{δ^{n} F_{g} [μ]}{δ μ (L_{1}) \dots δ μ (L_{n})},

where we recall that

\frac{δ^{n} F_{g} [μ]}{δ μ (L_{1}) \dots δ μ (L_{n})} = \sum_{p = 0}^{\infty} \frac{1}{p!} \int V_{g, n + p} (L, K) d μ (K_{1}) \dots d μ (K_{p}) .

Lemma 20.

We define

x_{i} = x_{i} (z_{i}; μ] = \sqrt{z_{i}^{2} - 2 R [μ]}

. For g ≥ 1 or n ≥ 3 we have

W_{g, n} (z) = ω_{g, n} (x) \prod_{i = 1}^{n} \frac{z_{i}}{x_{i}},

(151)

while for g = 0 and n = 1, 2,

W_{0, 1} (z) = - \int_{0}^{R} \frac{z_{1}}{{(z_{1}^{2} - 2 r)}^{3 / 2}} Z (r) d r,

(152)

W_{0, 2} (z) = \frac{z_{1}}{x_{1}} \frac{z_{2}}{x_{2}} ω_{0, 2} (x) - \frac{1}{{(z_{1} - z_{2})}^{2}} .

(153)

Proof.

For the first identity we wish to combine and . It requires an expression for the Laplace transform of the half-tight cylinder. Using

\int_{0}^{\infty} d y \frac{z}{\sqrt{4 π y^{3}}} e^{- \frac{z^{2}}{4 y} - y L^{2}} = e^{- z L}, \int_{0}^{\infty} d p e^{- y \frac{p^{2}}{2 R}} I_{1} (p) = e^{\frac{R}{2 y}} - 1,

(154)

allows us to compute

\begin{array}{l} \int_{K}^{\infty} d L L e^{- z L} (H (L, K) K + δ (L - K)) & = K e^{- z K} + \int_{K}^{\infty} d L L (\int_{0}^{\infty} d y \frac{z K}{\sqrt{4 π y^{3}}} e^{- \frac{z^{2}}{4 y} - y L^{2}}) \\ \sqrt{\frac{2 R}{L^{2} - K^{2}}} I_{1} (\sqrt{2 R (L^{2} - K^{2})}) \\ = K e^{- z K} + \int_{0}^{\infty} d y \frac{z K}{\sqrt{4 π y^{3}}} e^{- \frac{z^{2}}{4 y} - y K^{2}} \int_{0}^{\infty} d p e^{- y \frac{p^{2}}{2 R}} I_{1} (p) \\ = K e^{- z K} + \int_{0}^{\infty} d y \frac{z K}{\sqrt{4 π y^{3}}} e^{- \frac{z^{2}}{4 y} - y K^{2}} (e^{\frac{R}{2 y}} - 1) \\ = K \frac{z}{\sqrt{z^{2} - 2 R}} e^{- K \sqrt{z^{2} - 2 R}} . \end{array}

Therefore,

W_{g, n} (z) = \int_{0}^{\infty} T_{g, n} (K; μ] \prod_{i = 1}^{n} K_{i} \frac{z_{i}}{x_{i}} e^{- x_{i} K_{i}} d K_{i},

which by gives the first stated identity.

For the last two identities we use that the Laplace transform of the modified Bessel function I₀ is given by

\int_{0}^{\infty} I_{0} (L \sqrt{2 r}) L e^{- z L} d L = \frac{z}{{(z^{2} - 2 r)}^{3 / 2}} .

Then follows directly from , while for the cylinder case implies

\begin{array}{l} W_{0, 2} (z) & = \int_{0}^{R} \frac{z_{1}}{{(z_{1}^{2} - 2 r)}^{3 / 2}} \frac{z_{2}}{{(z_{2}^{2} - 2 r)}^{3 / 2}} d r \\ = \frac{z_{1}}{\sqrt{z_{1}^{2} - 2 R}} \frac{z_{2}}{\sqrt{z_{2}^{2} - 2 R}} \frac{1}{{(\sqrt{z_{1}^{2} - 2 R} - \sqrt{z_{2}^{2} - 2 R})}^{2}} - \frac{1}{{(z_{1} - z_{2})}^{2}} \\ = \frac{z_{1}}{x_{1}} \frac{z_{2}}{x_{2}} ω_{0, 2} (x) - \frac{1}{{(z_{1} - z_{2})}^{2}} . \end{array}

□

We finish this section by giving alternative expressions for the disk function and the series η(u; μ].

Lemma 21.

The disk function

W_{0, 1} (z)

is related to η via

W_{0, 1} (z) = - z_{1} \sqrt{z_{1}^{2} - 2 R} η (\sqrt{z_{1}^{2} - 2 R}) + \frac{z_{1}}{2 π} \sin 2 π z_{1} - \int d μ (L) \cosh (L z_{1}),

valid when 4|R| < |z₁|².

Note that μ = 0 gives $W_{0, 1} (z) = 0$ as expected.

Proof.

The starting point is the standard generating function [Ref. 27, Eq. (10.1.39)]

\frac{1}{u} \sin (\sqrt{u^{2} - 2 u t}) = \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} y_{n - 1} (u)

(155)

for the spherical Bessel functions y_k(u) valid when 2|t| < |u|. Restricting to 2|R| < |x|² and using the series expansion of the ordinary and spherical Bessel functions we find

\begin{align} \frac{x}{2 π} \sin (2 π λ \sqrt{x^{2} + 2 R}) & = \sum_{k = 0}^{\infty} \frac{{(- π)}^{k}}{k!} 2^{k} x^{2 - k} λ^{k + 1} y_{k - 1} (2 π λ x) R^{k} \\ = \sum_{k, m = 0}^{\infty} \frac{{(- 1)}^{m} π^{2 m + \frac{1}{2}}}{k! m!} \frac{2^{k - 1} x^{2 (m - k + 1)} λ^{2 m + 1} R^{k}}{Γ (m - k + 32)} \\ = \sum_{p = - \infty}^{\infty} x^{2 p} \sum_{m = \max (0, p - 1)}^{\infty} \frac{{(- 1)}^{m} π^{2 m + \frac{1}{2}}}{(m + 1 - p)! m!} \frac{2^{m - p} λ^{2 m + 1} R^{m + 1 - p}}{Γ (12 + p)} \\ = \sum_{p = - \infty}^{\infty} x^{2 p} λ^{p} \frac{Γ (1 / 2)}{Γ (p + 1 / 2) 2^{p}} {(\frac{- \sqrt{2} π}{\sqrt{R}})}^{p - 1} J_{p - 1} (2 π λ \sqrt{2 R}) . \end{align}

(156)

Setting λ = 1 now gives

\frac{x}{2 π} \sin (2 π \sqrt{x^{2} + 2 R}) = \sum_{p = - \infty}^{\infty} x^{2 p} \frac{Γ (1 / 2)}{Γ (p + 1 / 2) 2^{p}} {(\frac{- \sqrt{2} π}{\sqrt{R}})}^{p - 1} J_{p - 1} (2 π \sqrt{2 R}) .

(157)

On the other hand we can show that

\begin{array}{l} \frac{d}{d λ} [\frac{x}{\sqrt{x^{2} + 2 R}} \cos (2 π λ \sqrt{x^{2} + 2 R})] & = - 2 π x \sin (2 π λ \sqrt{x^{2} + 2 R}) \\ = - 4 π^{2} \sum_{p = - \infty}^{\infty} x^{2 p} λ^{p} \frac{Γ (1 / 2)}{Γ (p + 1 / 2) 2^{p}} {(\frac{- \sqrt{2} π}{\sqrt{R}})}^{p - 1} J_{p - 1} (2 π λ \sqrt{2 R}) \\ = \frac{d}{d λ} [\sum_{p = - \infty}^{\infty} x^{2 p} λ^{p} \frac{Γ (1 / 2)}{Γ (p + 1 / 2) 2^{p}} {(\frac{- 2 π}{\sqrt{2 R}})}^{p} J_{p} (2 π λ \sqrt{2 R})] . \end{array}

Integrating and setting λ = (iL)/(2π) gives

\frac{x}{\sqrt{x^{2} + 2 R}} \cosh (L \sqrt{x^{2} + 2 R}) = \sum_{p = - \infty}^{\infty} x^{2 p} \frac{Γ (1 / 2)}{Γ (p + 1 / 2) 2^{p}} {(\frac{L}{\sqrt{2 R}})}^{p} I_{p} (L \sqrt{2 R}),

(158)

valid for 2|R| < |x|².

Starting from and restricting to 2|R| < |x₁|² we find the series expansion

\begin{align} W_{0, 1} (\sqrt{x_{1}^{2} + 2 R}) \frac{x_{1}}{\sqrt{x_{1}^{2} + 2 R}} & = - \int_{0}^{R} \frac{x_{1}}{{(x_{1}^{2} + 2 R - 2 r)}^{3 / 2}} Z (r) d r \\ = - x_{1}^{- 2} \int_{0}^{R} \frac{1}{{(1 + \frac{2 R - 2 r}{x_{1}^{2}})}^{3 / 2}} Z (r) d r \\ = - \sum_{p = 1}^{\infty} \frac{{(- 2)}^{p} Γ (1 / 2)}{(p - 1)! Γ (1 / 2 - p)} x_{1}^{- 2 p} \int_{0}^{R} {(r - R)}^{p - 1} Z (r) d r . \end{align}

(159)

We can use Ref. 28, 6.567.1 and 6.567.18,

\begin{align} \int_{0}^{R} {(r - R)}^{p - 1} \frac{\sqrt{r}}{\sqrt{2} π} J_{1} (2 π \sqrt{2 r}) d r & = \frac{{(- 1)}^{p - 1} \sqrt{2} R^{p + 1 / 2}}{π} \int_{0}^{1} x^{2} {(1 - x^{2})}^{p - 1} J_{1} (2 π \sqrt{2 R} x) d x \\ = (p - 1)! {(\frac{- \sqrt{R}}{\sqrt{2} π})}^{p + 1} J_{p + 1} (2 π \sqrt{2 R}), \end{align}

(160)

\begin{align} \int_{0}^{R} {(r - R)}^{p - 1} I_{0} (L \sqrt{2 r}) d r & = {(- R)}^{p - 1} 2 R \int_{0}^{1} x {(1 - x^{2})}^{p - 1} I_{0} (L \sqrt{2 R} x) d x \\ = - (p - 1)! {(\frac{- \sqrt{2 R}}{L})}^{p} I_{p} (L \sqrt{2 R}) . \end{align}

(161)

This yields

\begin{align} W_{0, 1} (\sqrt{x_{1}^{2} + 2 R}) \frac{x_{1}}{\sqrt{x_{1}^{2} + 2 R}} & = - \sum_{p = 1}^{\infty} x_{1}^{- 2 p} \frac{{(- 2)}^{p} Γ (1 / 2)}{Γ (1 / 2 - p)} [{(\frac{- \sqrt{R}}{\sqrt{2} π})}^{p + 1} J_{p + 1} (2 π \sqrt{2 R}) \\ + \int d μ (L) {(\frac{- \sqrt{2 R}}{L})}^{p} I_{p} (L \sqrt{2 R})] \\ = \sum_{p = - \infty}^{- 1} x_{1}^{2 p} \frac{Γ (1 / 2)}{Γ (1 / 2 + p) 2^{p}} [{(\frac{- \sqrt{2} π}{\sqrt{R}})}^{p - 1} J_{p - 1} (2 π \sqrt{2 R}) \\ - \int d μ (L) {(\frac{L}{\sqrt{2 R}})}^{p} I_{p} (L \sqrt{2 R})] . \end{align}

(162)

On the other hand, and imply

x_{1}^{2} η (x_{1}) = \sum_{p = 1}^{\infty} x_{1}^{2 p} \frac{Γ (1 / 2)}{Γ (1 / 2 + p) 2^{p}} [{(\frac{- \sqrt{2} π}{\sqrt{R}})}^{p - 1} J_{p - 1} (2 π \sqrt{2 R}) - \int d μ (L) {(\frac{L}{\sqrt{2 R}})}^{p} I_{p} (L \sqrt{2 R})] .

(163)

Together with Z(R) = 0 we may now conclude that

\begin{align} W_{0, 1} (\sqrt{x_{1}^{2} + 2 R}) \frac{x_{1}}{\sqrt{x_{1}^{2} + 2 R}} + x_{1}^{2} η (x_{1}) \\ = \frac{x_{1}}{2 π} \sin (2 π \sqrt{x_{1}^{2} + 2 R}) - \int d μ (L) \frac{x_{1}}{\sqrt{x_{1}^{2} + 2 R}} \cosh (L \sqrt{x_{1}^{2} + 2 R}), \end{align}

(164)

valid for 2|R| < |x₁|². Substituting

x_{1} = \sqrt{z_{1}^{2} - 2 R}

gives the desired expression.□

For convenience, we record an explicit expression for η(u; μ] that follows from this proof. For a formal power series F(r, u) in r with coefficients that are Laurent polynomials in u, we denote by [u^≥0]F(r, u) the formal power series obtained by dropping the negative powers of u in the coefficients of F(r, u). Then we can write η(u; μ] as

η (u; μ] = [u^{\geq 0}] {(\frac{u}{2 π} \sin (2 π \sqrt{u^{2} + 2 r}) - \int d μ (L) \frac{u}{\sqrt{u^{2} + 2 r}} \cosh (L \sqrt{u^{2} + 2 r}))|}_{r = R [μ]} .

(165)

V. JT GRAVITY

The Weil–Petersson volumes play an important role in Jackiw–Teitelboim (JT) gravity,^29–31 a two-dimensional toy model of quantum gravity. JT gravity has received significant attention in recent years because of the holographic perspective on the double-scaled matrix model it is dual to Ref. 31. In this section we point to some opportunities to use our results in the context of JT gravity and its extensions in which hyperbolic surfaces with defects play a role.^32–34 But we start with a brief introduction to the JT gravity partition function in Euclidean signature.

JT gravity is governed by the (Euclidean) action

I_{JT, bulk} [g_{μ ν}, ϕ] = - \frac{1}{2} \int_{M} \sqrt{g} ϕ (R + 2),

(166)

where ϕ is the scalar dilaton field, g_μν is a two-dimensional Riemannian metric and R the corresponding Ricci scalar curvature, all living on a surface

M

⁠.

Since we want this action to make sense when the manifold has boundaries of lengths β = (β₁, …, β_n), the boundary term

I_{JT, boundary} [g_{μ ν}, ϕ] = - \int_{\partial M} \sqrt{h} ϕ (K - 1)

(167)

is included, where h_μν is the induced metric on the boundary and K is the extrinsic curvature at the boundary. Including the topological Einstein–Hilbert term, proportional to parameter S₀, gives the full (Euclidean) JT action

I_{JT} [g_{μ ν}, ϕ] = - S_{0} χ + I_{JT,bulk} [g_{μ ν}, ϕ] + I_{JT,boundary} [g_{μ ν}, ϕ],

(168)

where χ is the Euler characteristic of the manifold.

The JT gravity partition function can formally be written as

Z_{n} (β) = \int D g D ϕ \exp (I_{JT}) .

(169)

In the partition function, the dilation field ϕ acts as a Lagrange multiplier on (R + 2), therefore enforcing a constant negative curvature R = −2 in the bulk. This is why the relevant manifolds will be hyperbolic surfaces.

Due to the Einstein–Hilbert term, we can do a topological expansion by a formal power series expansion in

e^{- S_{0}}

⁠,

Z_{n} (β) = \sum_{g = 0}^{\infty} {(e^{- S_{0}})}^{2 g + n - 2} Z_{g, n} (β) .

(170)

It has been shown by Saad, Shenker, and Stanford³¹ that the JT partition functions Z_g,n for 2g + n − 2 > 0 can be further decomposed by splitting the surfaces into n trumpets and a hyperbolic surface of genus g and n geodesic boundaries with lengths b = (b₁, …, b_n), and that the partition function measure is closely related to the Weil–Petersson measure (Fig. 5). To be precise, it satisfies the identity

Z_{g, n} (β) = \int_{0}^{\infty} (\prod_{i = 1}^{n} b_{i} d b_{i} Z^{Trumpet} (β_{i}, b_{i})) V_{g, n} (b),

(171)

where V_g,n(b) are the Weil–Petersson volumes and the trumpet contributions are given by

Z^{Trumpet} (β, b) = \frac{1}{2 \sqrt{π β}} e^{- \frac{b^{2}}{4 β}} .

(172)

This formula is the link between JT gravity and Weil–Petersson volumes.

FIG. 5.

View large Download slide

Left: Visualization of decomposing the manifold into n JT trumpets and a hyperbolic surface with n geodesic boundaries. In this case g = 1 and n = 4. Right: A conical defect on a hyperbolic surface.

There are several natural extensions of the JT action. If we only allow up to two derivatives, the most general action can be transformed to³⁵

I_{bulk} [g_{μ ν}, ϕ] = - \frac{1}{2} \int_{M} \sqrt{g} [ϕ (R + 2) + U (ϕ)] .

(173)

In Subsection V A we will discuss a natural choice of the dilaton potential U(ϕ), which gives rise to defects in the hyperbolic surfaces.

A. Conical defects

One of the most natural dilaton potentials is

U (ϕ) = μ e^{- 2 π (1 - α) ϕ},

(174)

which adds a gas of conical defects of cone angle 2πα carrying weight μ each. It naturally arises³² from Kaluza–Klein instantons when performing dimensional reduction on three-dimensional black holes.

More generally, one can allow multiple types of defects by considering a measure μ on i[0, 2π) and setting

U (ϕ) = \int_{0}^{1} d μ (2 π i α) e^{- 2 π (1 - α) ϕ} .

(175)

For instance, the choice

μ = \sum_{j = 1}^{k} μ_{j} δ_{i α_{j}}

gives k types of defects with cone angles α₁, …, α_k ∈ [0, 2π],

U (ϕ) = \sum_{j} μ_{j} e^{- 2 π (1 - α_{j}) ϕ} .

(176)

The choice to consider the measure on the imaginary interval will be convenient later.

It can be shown³² that these potentials indeed lead to conical defects. For example, one can look at the term linear in μ in the integrand of the partition function for a single type of gas:

\begin{align} [μ^{1}] \exp (- I_{bulk}) = \frac{1}{2} \exp (- I_{JT, bulk}) \int_{M} d x_{1} \sqrt{g (x_{1})} \exp (- 2 π (1 - α) ϕ (x_{1})) \\ = \frac{1}{2} \int_{M} d x_{1} \sqrt{g (x_{1})} \exp (\frac{1}{2} \int_{M} d x \sqrt{g (x)} ϕ (x) (R (x) + 2 - 4 π (1 - α) δ^{2} (x - x_{1}))) . \end{align}

(177)

It follows that the surface has curvature R = −2 everywhere, except at point x₁, where we have a conical defect with cone angle 2πα. If one includes all orders of μ, any number of defects may appear and each defect carries a weight μ.³²

As already mentioned in the introduction, the Weil–Petersson volumes for surfaces with sharp cone points (cone angle 2πα < π) are obtained^1,2,13,36 from the usual Weil–Petersson volume polynomials by treating the defect angle 2πα as a geodesic boundary with imaginary boundary length 2πiα. This implies that the partition function Z_g,n(β) is closely related to the generating function F_g[μ] of Weil–Petersson volumes considered in this paper. To be precise, using and ,

\begin{align} Z_{g, n} (β) & = \int_{0}^{\infty} (\prod_{i = 1}^{n} b_{i} d b_{i} Z^{Trumpet} (β_{i}, b_{i})) \sum_{p = 0}^{\infty} \frac{1}{p!} \int d μ (b_{n + 1}) \dots d μ (b_{n + p}) V_{g, n + p} (b) \\ = \int_{0}^{\infty} (\prod_{i = 1}^{n} b_{i} d b_{i} Z^{Trumpet} (β_{i}, b_{i})) \frac{δ^{n} F_{g} [μ]}{δ μ (b_{1}) \dots δ μ (b_{n})} \\ = \int_{0}^{\infty} (\prod_{i = 1}^{n} b_{i} d b_{i} Z^{Trumpet} (β_{i}, b_{i})) (\prod_{i = 1}^{n} d K_{i} (K_{i} H (b_{i}, K_{i}; μ] + δ (K_{i} - b_{i}))) T_{g, n} (K; μ], \end{align}

(178)

which we can compute using the recursions described in this paper. In particular, its topological recursion can in principle be derived from that of T_g,n in Theorem 2.

We can simplify by considering the tight trumpet, which is a genus-0 hyperbolic surface with an asymptotic boundary of length β, a tight boundary of length K and an arbitrary number of extra geodesic boundaries, with the constraint that the tight boundary cannot be separated from the asymptotic one by a curve of length β. See Fig. 6. Since it can be obtained by gluing a trumpet to a half-tight cylinder, with the help of Lemma 11 we find that the partition function associated to a tight trumpet is given by

\begin{align} Z^{TT} (β, K) & = \int_{K}^{\infty} \frac{b}{K} d b Z^{Trumpet} (β, b) (K H (b, K; μ] + δ (K - b)) \\ = \frac{1}{2 \sqrt{π β}} e^{- \frac{K^{2}}{4 β}} + \int_{K}^{\infty} b d b \frac{1}{2 \sqrt{π β}} e^{- \frac{b^{2}}{4 β}} \sqrt{\frac{2 R [μ]}{b^{2} - K^{2}}} I_{1} (\sqrt{b^{2} - K^{2}} \sqrt{2 R [μ]}) \\ = \frac{1}{2 \sqrt{π β}} e^{- \frac{K^{2}}{4 β} + 2 R [μ] β} = e^{2 R [μ] β} Z^{Trumpet} (β, K) . \end{align}

(179)

Remarkably it differs from the JT trumpet only in a factor exponential in the boundary length β. We conclude that for g ≥ 1 or n ≥ 3,

Z_{g, n} (β) = \int_{0}^{\infty} (\prod_{i = 1}^{n} K_{i} d K_{i} Z^{TT} (β_{i}, K_{i})) T_{g, n} (K; μ],

(180)

which we understand as a gluing of tight trumpets to tight hyperbolic surfaces. In the case g = 0 and n = 2, we only need to glue two tight trumpets together to find the universal two-boundary correlator

\begin{align} Z_{0, 2} (β_{1}, β_{2}) & = \int_{0}^{\infty} Z^{TT} (β_{1}, K) Z^{TT} (β_{2}, K) K d K \\ = \frac{1}{2 π} \frac{\sqrt{β_{1} β_{2}}}{β_{1} + β_{2}} e^{2 (β_{1} + β_{2}) R [μ]} . \end{align}

(181)

In particular, a statistical interpretation of the two-boundary partition function leads to the observation that the shortest cycle separating the two boundaries has random length with density

\frac{1}{2} (\frac{1}{β_{1}} + \frac{1}{β_{2}}) K e^{- \frac{1}{4} (\frac{1}{β_{1}} + \frac{1}{β_{2}}) K} d K,

(182)

independent of the weight μ. This is the Rayleigh distribution with mean

\sqrt{π β_{1} β_{2} / (β_{1} + β_{2})}

⁠.

FIG. 6.

View large Download slide

Visualization of the tight trumpet.

We note that these expressions do not apply to the case of blunt cone points (cone angle 2πα ∈ [π, 2π]). The problem is that in the presence of such defects it is no longer true that every free homotopy class of closed curves necessarily contains a geodesic, because, informally, when shortening a closed curve it can be pulled across a blunt cone point, while that never happens for a short one. However, this is not an issue when considering tight cycles, because in that setting one is considering larger homotopy classes, namely of the manifold with its defects closed off. Such homotopy classes will always contain a shortest geodesic, which generically is unique. Whereas the JT trumpet cannot always be removed from a surface with blunt defects in a well-defined manner, the removal of a tight trumpet should pose no problem. It is natural to ask whether such reasoning can be used to connect to the recent works^37,38 in JT gravity dealing with blunt cone points.

B. FZZT-branes

Another well-studied extension of JT gravity, is the introduction of Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) branes. With this extension the hyperbolic surfaces can end on a FZZT brane. In the random matrix model description of JT gravity, this corresponds to fixing some eigenvalues of the random matrix.³³

In the partition function, this leads to the addition of an arbitrary number of geodesic boundaries as defects with a certain weight

M (L) = - e^{- z L}

⁠, where L is the length of the boundary,

Z_{g, n} {(β)}_{FZZT} = \sum_{p = 0}^{\infty} \frac{e^{- S_{0} p}}{p!} \int_{0}^{\infty} (\prod_{i = 1}^{n} d b_{i} b_{i} Z^{Trumpet} (β_{i}, b_{i})) (\prod_{i = n + 1}^{n + p} d b_{i} M (b_{i})) V_{g, n + p} (b) .

(183)

Such weights have been interpreted^34,39 as the action of a fermion with mass z.

Using our setup, we can rewrite this to:

Z_{g, n} {(β)}_{FZZT} = \int_{0}^{\infty} (\prod_{i = 1}^{n} d b_{i} b_{i} Z^{Trumpet} (β_{i}, b_{i})) (\prod_{i = 1}^{n} d K_{i} (K_{i} H (b_{i}, K_{i}; μ_{FZZT}] + δ (b_{i} - K_{i}))) T_{g, n} (K; μ_{FZZT}],

(184)

with

μ_{FZZT} = - e^{- S_{0} - z L} d L,

(185)

or again using the tight trumpet

Z_{g, n} {(β)}_{FZZT} = \int_{0}^{\infty} (\prod_{i = 1}^{n} K_{i} d K_{i} Z^{TT} (β_{i}, K_{i}; μ_{FZZT}]) T_{g, n} (K; μ_{FZZT}],

(186)

with

Z^{TT} (β, K; μ] = \frac{1}{2 \sqrt{π β}} e^{- \frac{K^{2}}{4 β} + 2 R [μ] β} .

(187)

The behavior of R[μ_FZZT] depends on z and S₀ and its critical points should give insight into critical phenomena of the partition function, see Ref. 40.

ACKNOWLEDGMENTS

We thank an anonymous referee for useful suggestions. This work is supported by the START-UP 2018 program with Project No. 740.018.017 and the VIDI program with Project No. VI.Vidi.193.048, which are financed by the Dutch Research Council (NWO).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Timothy Budd: Conceptualization (equal); Investigation (equal); Writing – original draft (equal). Bart Zonneveld: Conceptualization (equal); Investigation (equal); Writing – original draft (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Topological recursion of the Weil–Petersson volumes of hyperbolic surfaces with tight boundaries

I. INTRODUCTION

A. Topological recursion of Weil–Petersson volumes

B. Hyperbolic surfaces with tight boundaries

C. Intermezzo: Random (tight) hyperbolic surfaces

D. Generating functions

E. Main results

F. Idea of the proofs

G. Discussion

H. Outline

II. DECOMPOSITION OF TIGHT HYPERBOLIC SURFACES

A. Half-tight cylinder

B. Tight decomposition

C. Tight decomposition in the stable case

1. Shortest cycles

2. Integration on Moduli space

3. Shortest multicurves

D. Tight decomposition of the cylinder

III. GENERATING FUNCTIONS OF TIGHT WEIL–PETERSSON VOLUMES

A. Definitions

B. Volume of half-tight cylinder

C. Rewriting generating functions

D. Properties of the new kernel

E. Proof of Theorem 2

F. Proof of Theorem 4

IV. LAPLACE TRANSFORM, SPECTRAL CURVE AND DISK FUNCTION

A. Proof of Theorem 3

B. Disk function

V. JT GRAVITY

A. Conical defects

B. FZZT-branes

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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