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

^{1}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 $L=(L1,\u2026,Ln)\u2208R>0n$. Denoting [

*n*] = {1, 2, …,

*n*} and using the notation $LI=(Li)i\u2208I$,

*I*⊂ [

*n*], for a subsequence of

**L**and $LI\u0302=L[n]\I$, the recursion can be expressed for (

*g*,

*n*) ∉ {(0, 3), (1, 1)} as

*V*

_{0,3}(

**L**) = 1 and $V1,1(L)=L1248+\pi 212$ this completely determines

*V*

_{g,n}as a symmetric polynomial in $L12,\u2026,Ln2$ of degree 3

*g*− 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*, 2*π*). 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**).

^{3}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 3

*g*− 3 +

*n*≥ 0 the Laplace transformed

^{4}Weil–Petersson volumes

*g*− 4 + 2

*n*, while setting

*g*,

*n*≥ 1 and 3

*g*− 3 +

*n*≥ 0, which one may recognize as the recursion for the invariants $\omega g,n(0)(z)$ of the complex curve

### B. Hyperbolic surfaces with tight boundaries

*S*

_{g,n}be a fixed topological surface of genus

*g*with

*n*boundaries and $Tg,n(L)$ the Teichmüller space of hyperbolic structures on

*S*

_{g,n}with geodesic boundaries of lengths $L=(L1,\u2026,Ln)\u2208R>0n$. Denote the boundary cycles

^{5}by

*∂*

_{1}, …,

*∂*

_{n}and the free homotopy class of a cycle

*γ*in

*S*

_{g,n}by $[\gamma ]Sg,n$. For a hyperbolic surface $X\u2208Tg,n(L)$ and a cycle

*γ*, we denote by

*ℓ*

_{γ}(

*X*) the length of

*γ*, in particular $\u2113\u2202i(X)=Li$. The mapping class group of

*S*

_{g,n}is denoted Mod

_{g,n}and the quotient of $Tg,n(L)$ by its action leads to the moduli space

*S*

_{g,n+p}by capping off the last

*p*boundaries with disks (Fig. 1). Note that the free homotopy classes $[\u22c5]Sg,n+p$ of

*S*

_{g,n+p}are naturally partitioned into the free homotopy classes $[\u22c5]S\u2322g,n,p$ of $S\u2322g,n,p$. In particular, $[\u2202j]Sg,n+p$ for

*j*=

*n*+ 1, …,

*n*+

*p*are all contained in the null-homotopy class of $S\u2322g,n,p$. For

*i*= 1, …,

*n*the boundary

*∂*

_{i}of $X\u2208Mg,n+p(L)$ is said to be

*tight in*$S\u2322g,n,p$ if

*∂*

_{i}is the only simple cycle

*γ*in $[\u2202i]S\u2322g,n,p$ of length

*ℓ*

_{γ}(

*X*) ≤

*L*

_{i}. Remark that both $[\u2202i]Sg,n+p$ and $[\u2202i]S\u2322g,n,p$ 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*

*∂*

_{1}is null-homotopic and $M0,2,ptight(L)=\u2205$ because $[\u22021]S\u23220,2,p=[\u22022]S\u23220,2,p$ and therefore

*∂*

_{1}and

*∂*

_{2}can never both be the unique shortest cycle in their class. In general, it is an open subset of $Mg,n+p(L)$ and therefore it inherits the Weil–Petersson symplectic structure and Weil–Petersson measure d

*μ*

_{WP}from $Mg,n+p(L)$. The corresponding

*tight*Weil–Petersson volumes are denoted

*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.

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.

*For* *g*, *n*, *p* ≥ 0 *such that* 2*g* − 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* $L12,\u2026,Ln+p2$ *of degree* 3*g* − 3 + *n* + *p* *that is symmetric in* *L*_{1}, …, *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

*g*,

*n*and $L\u2208[0,\u221e\u222ai(0,\pi ))n$, then upon normalization by 1/

*V*

_{g,n}(

**L**) the Weil–Petersson measure d

*μ*

_{WP}provides a well-studied probability measure on $Mg,n(L)$ 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 $L\u2208[0,\u221e\u222ai(0,\pi ))n$ from the probability measure

*g*

*partition function*

^{11}

*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 work

^{12}for some of its statistical properties.

*g*Boltzmann hyperbolic surface with

*n*tight boundaries of length

**L**= (

*L*

_{1}, …,

*L*

_{n}), where the number

*p*of defects and their boundary lengths/cone angles

**K**= (

*K*

_{1}, …,

*K*

_{p}) are random. The corresponding partition function is

*p*with probability

*μ*

^{⊗p}(

*T*

_{g,n,p})(

**L**)/(

*p*!

*T*

_{g,n}(

**L**;

*μ*]) and then

**K**from the probability measure

*μ*

_{WP}/

*T*

_{g,n,p}(

**L**,

**K**) on $Mg,n,ptight(L,K)$. 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*^{2k}d*μ*(*L*), we can instead take these moments as the generating variables.

### D. Generating functions

*weight*

*μ*be a real linear function on the ring of even, real polynomials (i.e., $\mu \u2208R[K2]*$). For an even real polynomial

*f*we use the suggestive notation

*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*$(t0,t1,\u2026)\u2208RZ\u22650$ recording the evaluations of

*μ*on the even monomials, up to a conventional normalization,

*f*

_{1}, …,

*f*

_{p}and extending by linearity. More generally, we can view

*μ*

^{⊗p}as a linear map $R[L12,\u2026,Ln2,K12,\u2026,Kp2]\u2245R[L12,\u2026,Ln2][K12,\u2026,Kp2]\u2192R[L12,\u2026,Ln]$. We use the notation

*F*[

*μ*] of a collection of symmetric, even polynomials

*f*

_{1}(

*L*

_{1}),

*f*

_{2}(

*L*

_{1},

*L*

_{2}), … via $F[\mu ]=\u2211p\u22650\mu \u2297p(fp)p!$. Then the generating function of tight Weil–Petersson volumes is defined to be

*μ*and considering $Tg,n(L;x\mu ]\u2208R[[x]]$ 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 (

*t*

_{0},

*t*

_{1}, …) defined in (14).

*functional derivative*$\delta \delta \mu (L)$ on these types of series defined by

*f*(

**L**,

**K**), with

**L**= (

*L*

_{1}, …,

*L*

_{n}) and

**K**= (

*K*

_{1}, …,

*K*

_{p}), is an even polynomial that is symmetric in

*K*

_{1}, …,

*K*

_{p}then

### E. Main results

*T*

_{g,n}(

**L**;

*μ*], we need to introduce the generating function

*R*[

*μ*] as the unique formal power series solution satisfying

*R*[

*μ*] =

*∫*d

*μ*(

*L*) +

*O*(

*μ*

^{2}) to

*I*

_{0}and

*J*

_{1}are (modified) Bessel functions. Let also the

*moments*

*M*

_{k}[

*μ*] be the defined recursively via

*k*≥ 0 we may express

*M*

_{k}as

*Z*

^{(k+1)}(

*r*;

*μ*] denotes the (

*k*+ 1)th derivative of

*Z*(

*r*;

*μ*] with respect to

*r*.

*u*with coefficients that are formal power series in

*μ*. The reciprocal in the second definition is well-defined because

*η*(0) =

*M*

_{0}[

*μ*] = 1 +

*O*(

*μ*). We can now state our main result that generalizes Mirzakhani’s recursion formula.

*The tight Weil–Petersson volume generating functions*

*T*

_{g,n}(

**L**;

*μ*]

*satisfy*

*which is the same recursion formula (1) as for the Weil–Petersson volumes*

*V*

_{g,n}(

**L**)

*except that the kernel*

*K*

_{0}(

*x*,

*t*)

*is replaced by the “convolution”*

*where*

*X*(

*z*) =

*X*(

*z*;

*μ*]

*is a measure on*$R$

*determined by its two-sided Laplace transform*

*Furthermore, we have*

*X*(

*z*;

*μ*] that itself is a formal power series in

*μ*. Importantly its moments $\u222b\u2212\u221e\u221edX(z)zp=(\u22121)pp![up]X\u0302(u;\mu ]$ are formal power series in

*μ*, so for any $x,t\u2208R$

*μ*as well.

*μ*= 0, it is easily verified that

*X*(

*z*; 0] =

*δ*

_{0}(

*z*) and therefore one retrieves Mirzakhani’s kernel

*K*(

*x*,

*t*) =

*K*

_{0}(

*x*,

*t*). Given that the form of Mirzakhani’s recursion is unchanged except for the kernel, this strongly suggests that the Laplace transforms

*μ*= 0 this reduces to the Laplace-transformed Weil–Petersson volumes $\omega g,n(z;0]=\omega g,n(0)(z)$ defined in (3). The following theorem shows that this is the case in general.

*Setting*$\omega 0,2(z)=(z1\u2212z2)\u22122$

*and*

*ω*

_{0,0}(

**z**) =

*ω*

_{0,1}(

**z**) = 0

*, the Laplace transforms (33) satisfy for every*

*g*≥ 0

*,*

*n*≥ 1

*such that*3

*g*− 3 +

*n*≥ 0

*the recursion*

*These correspond precisely to the invariants of the curve*

Another consequence of Theorem 2 is that for all *g* ≥ 0, *n* ≥ 1, such that 3*g* − 3 + *n* ≥ 0, the tight Weil–Petersson volume *T*_{g,n}(**L**; *μ*] is expressible as a rational polynomial in $L12,\u2026,Ln2$ and $M0\u22121,M1,M2,\u2026$.

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

*For all*

*g*,

*n*≥ 0

*, such that*

*n*≥ 3

*for*

*g*= 0

*and*

*n*≥ 1

*for*

*g*= 1

*, we have that*

*where*$Pg,n(L,m)$

*is a rational polynomial in*$L12,\u2026,Ln2,m1,\u2026,m3g\u22123+n$

*. This polynomial is symmetric and of degree*3

*g*− 3 +

*n*

*in*$L12,\u2026,Ln2$

*, while*$Pg,n(\sigma L,\sigma m1,\sigma 2m2,\sigma 3m3,\u2026)$

*is homogeneous of degree*3

*g*− 3 +

*n*

*in*

*σ*

*. For all*

*g*≥ 0

*,*

*n*≥ 1

*such that*2

*g*− 3 +

*n*> 0

*the polynomial*$Pg,n(L,m)$

*can be obtained from*$Pg,n\u22121(L,m)$

*via the recursion relation*

*where we use the shorthand notation*$\u222bdLLf(L,\u2026)=\u222b0Ldxxf(x,\u2026)$

*. Furthermore, we have*

*and*$Pg,0$

*for*

*g*≥ 2

*is given by*

*where*$\u27e8\tau 2d2\tau 3d3\cdots \u27e9g$

*are the*

*ψ*

*-class intersection numbers on Deligne–Mumford compactification of the moduli space of curves*$M\u0304g,n$

*with*

*n*=

*∑*

_{k}

*d*

_{k}≤ 3

*g*− 3

*marked points.*

*V*

_{g,n}(

**L**) from the polynomial $Pg,0$, since

*string*and

*dilaton*equations generalizing those for the Weil–Petersson volumes derived by Do and Norbury in Ref. 13, Theorem 2.

*For all*

*g*≥ 0

*and*

*n*≥ 1

*, such that*2

*g*− 3 +

*n*> 0

*, we have the identities*

*where the notation*$[L12p]Tg,n(L;\mu ]$

*refers to the coefficient of*$L12p$

*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 work^{14} 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.

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 proof^{15} of Witten’s conjecture^{16} (proved first by Kontsevich^{17}). 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 Safnuk^{18} 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 2*L*_{1} the left-hand side 2*L*_{1}*V*_{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.

*c*

_{1},

*c*

_{2},

*c*

_{3}be unit-length horocycles around the three cusps and $\Delta (X)=dhyp(c1,c2)\u2212dhyp(c1,c3)$ the difference in hyperbolic distance between two pairs, then it is plausible that

*M*

_{0}[

*μ*]

*X*(

*z*;

*μ*] on $R$, which integrates to 1 due to (28), is the probability distribution of the random variable $2\Delta (X)$ in a genus-0 Boltzmann hyperbolic surface $X$ with weight

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

### 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

*∂*

_{i}of $X\u2208Mg,n+p(L)$ is said to be

*tight in*$S\u2322g,n,p$ if

*∂*

_{i}is the only simple cycle

*γ*in $[\u2202i]S\u2322g,n,p$ of length

*ℓ*

_{γ}(

*X*) ≤

*L*

_{i}and we defined the

*moduli space of tight hyperbolic surfaces*as

*g*= 0 and

*n*= 2 we have $M0,2,ptight(L)=\u2205$ because

*∂*

_{1}and

*∂*

_{2}belong to the same free homotopy class of $S\u23220,2,p$ 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*

*L*

_{1}>

*L*

_{2}> 0. We will also consider

*H*

_{p}(

**L**).

*When* *L*_{1} ≥ *L*_{2} > 0*,* $Hp(L)$ *is an open subset of* $M0,2+p(L)$*, and when it is non-empty* (*L*_{1} > *L*_{2}) *its closure is* $Hp\u0304(L)$*. In particular, both have the same finite Weil–Petersson volume* *H*_{p}(**L**) ≤ *V*_{0,2+p}(**L**) *when* *L*_{1} > *L*_{2}*, but* $Hp(L)$ *has 0 volume and* $Hp\u0304(L)$ *non-zero volume* *H*_{p}(**L**) *when* *L*_{1} = *L*_{2}.

For *L*_{1} > *L*_{2}, $Hp(L)$ is the intersection of the open sets {*ℓ*_{γ}(*X*) > *L*_{2}} indexed by the countable set of free homotopy classes *γ* in $[\u22022]S\u23220,2,p$. It is not hard to see that in a neighborhood of any $X\u2208Hp(L)$ 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*) ≥ *L*_{2}}, which is precisely $Hp\u0304(L)$.

In the case *L*_{1} = *L*_{2}, $Hp(L)$ is empty, because *∂*_{1} is a curve in $[\u22022]S\u23220,2,p$ of the same length as *∂*_{2}. On the other hand, $Hp\u0304(L)$ is non-empty and of full dimension. To see this, observe, for instance, that there exists an *ϵ* > 0, depending only on *L*_{1}, such that $Hp\u0304(L)$ includes all surfaces in $M0,2+p(L)$ 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 *L*_{1}. Hence, $Hp\u0304(L)$ has positive volume *H*_{p}(**L**) > 0.□

### B. Tight decomposition

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

*The Weil–Petersson volumes*

*T*

_{g,n,p}(

**L**)

*and*

*H*

_{p}(

**L**)

*satisfy*

*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.

*H*

_{1}(

**L**) =

*V*

_{0,3}(

**L**) = 1 for

*L*

_{1}≥

*L*

_{2}and

*T*

_{g,n,0}(

**L**) =

*V*

_{g,n}(

**L**). Rewriting the equations as

*V*

_{g,n}(

**L**). Moreover, by induction we easily verify that

*H*

_{p}(

**L**) in the region

*L*

_{1}≥

*L*

_{2}is a polynomial in $L12,\u2026,L2+p2$ of degree

*p*− 1 that is symmetric in

*L*

_{3}, …,

*L*

_{2+p}, and

*T*

_{g,n,p}is a polynomial in $L12,\u2026,Ln+p2$ of degree 3

*g*− 3 +

*n*+

*p*that is symmetric in

*L*

_{1}, …,

*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.

*Given a hyperbolic surface* $X\u2208Mg,n+p$ *for* *g* ≥ 1 *or* *n* ≥ 2*, then for each* *i* = 1, …, *n* *there exists a unique innermost shortest cycle* $\sigma S\u2322g,n,pi(X)$ *on* *X**, meaning that it has minimal length in* $[\u2202i]S\u2322g,n,p$ *and such that all other cycles of minimal length (if they exist) are contained in the region of* *X* *delimited by* *∂*_{i} *and* $\sigma S\u2322g,n,pi(X)$*. Moreover, if* *g* ≥ 1 *or* *n* ≥ 3*, the curves* $\sigma S\u2322g,n,p1(X),\u2026,\sigma S\u2322g,n,pn(X)$ *are disjoint.*

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 $\u2264Li$ in $[\u2202i]S\u2322g,n,p$. Since $\u2202i\u2208[\u2202i]S\u2322g,n,p$ has length *L*_{i}, this proves the existence of at least one cycle in $[\u2202i]S\u2322g,n,p$ with minimal length.

Regarding the existence and uniqueness of a well-defined innermost shortest cycle, suppose $\alpha ,\beta \u2208[\u2202i]S\u2322g,n,p$ are two distinct simple closed geodesics with minimal length *ℓ* (see left side of Fig. 3). Since $\alpha \u2208[\u2202i]S\u2322g,n,p$, 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 *γ*_{1} and *γ*_{2} by switching between *α* and *β* at the chosen intersections, such that *γ*_{1} and *γ*_{2} are still in $[\u2202i]S\u2322g,n,p$. Since the total length is still 2*ℓ*, at least one of the new cycles has length $\u2264\u2113$. This cycle is not geodesic, so there will be a closed cycle in $[\u2202i]S\u2322g,n,p$ with length $<\u2113$, which contradicts that *α* and *β* have minimal length. We conclude that *α* and *β* are disjoint. Since all cycles in $[\u2202i]S\u2322g,n,p$ with minimal length are disjoint and separating, the notion of being innermost is well-defined.

Consider $\alpha i=\sigma S\u2322g,n,pi(X)$ and $\alpha j=\sigma S\u2322g,n,pj(X)$ 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 $[\u2202i]S\u2322g,n,p$ and $[\u2202j]S\u2322g,n,p$ 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 $[\u2202a]S\u2322g,n,p$ 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 $\alpha \u2208[\u2202i]S\u2322g,n,p$ is the innermost shortest cycle $\sigma S\u2322g,n,pi(X)$;

For a simple closed geodesic $\alpha \u2208[\u2202i]S\u2322g,n,p$ we have

*ℓ*(*α*) ≤*L*_{i}and each simple closed geodesic $\beta \u2208\sigma S\u2322g,n,pi(X)$ 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}.

#### 2. Integration on Moduli space

*multicurve*Γ = (

*γ*

_{1}, …,

*γ*

_{k}) is a collection of disjoint simple closed curves Γ = (

*γ*

_{1}, …,

*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

*∂*

_{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

*γ*in

*S*

_{g,n}and $X\u2208Mg,n$, let

*ℓ*

_{γ}(

*X*) be the length of the geodesic representative in the free homotopy class of

*γ*. For $K=(K1,\u2026,Kk)\u2282R>0k$ we can restrict the lengths of the geodesic representatives of curves in Γ by setting

*K*

_{i}=

*L*

_{j}. Denote by $\pi \Gamma :Mg,n(L)\Gamma \u2192Mg,n(L)$ 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 $(S1)p$ obtained by twisting along each of these

*p*cycles proportional to their length. The quotient space is denoted

*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 $M(Sg,n(\Gamma ),L,K)$ its moduli space, then according to Ref. 1, Lemma 8.3, the canonical mapping

#### 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 = (*γ*_{1}, …, *γ*_{n}) on *S*_{g,n+p} for *n* ≥ 1, *p* ≥ 0. Namely, we require that $\gamma i\u2208[\u2202i]S\u2322g,n,p$ is freely homotopic to the boundary *∂*_{i} in the capped-off surface $S\u2322g,n,p$ for *i* = 1, …, *n*. Then there exists a partition *I*_{0} ⊔⋯ ⊔ *I*_{n} = {*n* + 1, …, *n* + *p*} such that *S*_{g,n+p}(Γ) has *n* + 1 connected components *s*_{0}, …, *s*_{n}, where *s*_{0} is of genus *g* and is adjacent to all curves Γ as well as the boundaries $(\u2202j)j\u2208I0$ while for each *i* = 1, …, *n*, *s*_{i} is of genus 0 and contains the *i*th boundary *∂*_{i} as well as $(\u2202j)j\u2208Ii$ and is adjacent to *γ*_{i}. Note that *I*_{i} = ∅ if and only if $\gamma i\u2208[\u2202i]Sg,n+p$. Finally, we observe that mapping class group orbits ${Modg,n+p\u22c5[\Gamma ]Sg,n+p}$ of these multicurves Γ are in bijection with the set of partitions {*I*_{0} ⊔ ⋯ ⊔ *I*_{n} = {*n* + 1, …, *n* + *p*}}.

*γ*

_{i}to be (freely homotopic to) the innermost shortest cycle in $[\u2202i]S\u2322g,n,p$,

*The natural projection*

*where the disjoint union is over (representatives of) the mapping class group orbits of multicurves*Γ

*, is a bijection.*

If $X,X\u2032\u2208Tg,n+p(L)$ are representatives of hyperbolic surfaces in $M\u0302g,n,p(L)\Gamma $ and $M\u0302g,n,p(L)\Gamma \u2032$ respectively, then by definition $[\gamma i]=[\sigma S\u2322g,n,pi(X)]$ and $[\gamma i\u2032]=[\sigma S\u2322g,n,pi(X\u2032)]$. If *X* and *X*′ represent the same surface in $Mg,n+p(L)$, they are related by an element *h* of the mapping class group, *X*′ = *h* · *X*, and therefore also $\sigma S\u2322g,n,pi(X)=h\u22c5\sigma S\u2322g,n,pi(X\u2032)$ and $[\gamma i]=h\u22c5[\gamma i\u2032]$. 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 $X\u2208Tg,n+p(L)$ is a representative of $M\u0302g,n,p(L)\Gamma $ if we take $\Gamma =(\sigma S\u2322g,n,p1(X),\u2026,\sigma S\u2322g,n,pn(X))$, which is a valid multicurve due to Lemma 8.□

We can introduce the length-restricted version $M\u0302g,n,p(L)\Gamma (K)\u2282Mg,n+p(L)\Gamma (K)$ as before.

*The subset*$M\u0302g,n,p(L)\Gamma (K)\u2282Mg,n+p(L)\Gamma (K)$

*is invariant under twisting [the torus-action on*$Mg,n+p(L)\Gamma (K)$

*described above]. The image of the quotient*$M\u0302g,n,p(L)\Gamma *(K)$

*under the symplectomorphism (47) is precisely*

Let $X\u2208Mg,n+p(L)\Gamma (K)$ 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 $\sigma S\u2322g,n,pi(X)$ is thus also preserved under twisting, showing that the subset $M\u0302g,n,p(L)\Gamma (K)$ is invariant.

Let $X0\u2208Mg,n,|I0|(K,LI0)$ and $Xi\u2208M0,2+|Ii|(Li,Ki,LIi)$ for those *i* = 1, …, *n* for which *I*_{i} ≠ ∅ be the hyperbolic structures on the connected components *s*_{0}, …, *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 $\sigma S\u2322g,n,pi(X)$ is equivalent to the following two conditions holding:

the

*i*th boundary of*X*_{0}is tight in the capped-off surface associated to*s*_{0};*I*_{i}= ∅ (meaning*γ*_{i}=*∂*_{i}) or $Xi\u2208H|Ii|\u0304(Li,Ki,LIi)$ [recall the definition in (46)].

Hence, we have $X\u2208M\u0302g,n,p(L)\Gamma (K)$ precisely when $X0\u2208Mg,n,|I0|tight(K,LI0)$ and $Xi\u2208H|Ii|\u0304(Li,Ki,LIi)$ when *I*_{i} ≠ ∅. This proves the second statement of the lemma.□

*K*

_{i}=

*L*

_{i}whenever

*I*

_{i}= ∅. This proves the first relation of Proposition 7.

### D. Tight decomposition of the cylinder

*g*= 0 and

*n*= 2, because

*∂*

_{1}and

*∂*

_{2}are in the same free homotopy class of the capped surface $S\u23220,2,p$. Instead, we should consider a multicurve Γ = (

*γ*

_{1}) consisting of a single curve

*γ*

_{1}on

*S*

_{0,2+p}in the free homotopy class $[\u22021]S\u23220,2,p=[\u22022]S\u23220,2,p$, see Fig. 4. In this case there exists a partition

*I*

_{1}⊔

*I*

_{2}= {3, …,

*p*+ 2} such that

*S*

_{0,2+p}(Γ) has two connected components

*s*

_{1}and

*s*

_{2}, with

*s*

_{i}a genus-0 surface with 2 + |

*I*

_{i}| boundaries corresponding to

*∂*

_{i},

*γ*

_{1}and $(\u2202j)j\u2208Ii$. We consider the restricted moduli space

*∂*

_{1}and

*∂*

_{2}asymmetrically, by requiring that

*γ*

_{1}is the shortest curve farthest from

*∂*

_{1}. Lemma 9 goes through unchanged: the projection

*γ*

_{1}), is a bijection. Assuming

*L*

_{1}≥

*L*

_{2}, we cannot have $\gamma 1\u2208[\u22021]S0,2+p$ so

*I*

_{1}≠ ∅. There are two cases to consider:

$\gamma 1\u2208[\u22022]S0,2+p$ and therefore

*I*_{2}= ∅: this means that*∂*_{2}has minimal length in $[\u22022]S\u23220,2,p$, so $M\u03020,2,p(L)\Gamma =Hp\u0304(L)$.*I*_{2}≠ ∅: by reasoning analogous to that of Lemma 10 we have that $M\u03020,2,p(L)\Gamma *(K)$ is symplectomorphic toHence, when$H|I1|\u0304(L1,K,LI1)\xd7H|I2|(L2,K,LI2).$*L*_{1}≥*L*_{2}we haveThis proves the second relation of Proposition 7.$V0,2+p(L)=Hp(L)+\u2211I1\u2294I2={3,\u2026,2+p}I1,I2\u2260\u2205\u222b0L2H|I1|(L1,K,LI1)H|I2|(L2,K,LI2)KdK.$

## III. GENERATING FUNCTIONS OF TIGHT WEIL–PETERSSON VOLUMES

### A. Definitions

*g*≥ 2 we recall the polynomial

*ψ*-class intersection numbers on Deligne–Mumford compactification of the moduli space of curves $M\u0304g,n$ with

*n*=

*∑*

_{k}

*d*

_{k}≤ 3

*g*− 3 marked points. Then according to Ref. 21, Theorem 3

^{22}

*M*

_{k}[

*μ*] are defined in Eq. (22).

### B. Volume of half-tight cylinder

*H*(

*L*

_{1},

*L*

_{2};

*μ*] uniquely. The left-hand side depends on

*μ*only through the quantity

*R*[

*μ*], and the dependence on

*R*is analytic,

*H*(

*L*

_{1},

*L*

_{2};

*μ*] and one may easily calculate order by order in

*R*that

*H*(

*L*

_{1},

*L*

_{2};

*μ*] depends on

*L*

_{1}and

*L*

_{2}only through the combination $L12\u2212L22$. Let’s prove this.

*The half-tight cylinder generating function satisfies*

*and is therefore given by*

*L*

_{2}= 0 evaluates to

*R*of

*H*(

*L*

_{1},

*L*

_{2};

*μ*] satisfies (62), it follows that the same is true for the higher-order coefficients in

*R*.

As a consequence of (62), $H(L1,L2;\mu ]=HL12\u2212L22,0;\mu $ and the claimed expression (63) follows from (64).□

### C. Rewriting generating functions

^{15}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 $M\u0304g,n$ of genus-

*g*curves with

*n*marked points comes naturally equipped with the Chern classes

*ψ*

_{1}, …,

*ψ*

_{n}associated with its

*n*tautological line bundles, as well as the cohomology class

*κ*

_{1}of the Weil–Petersson symplectic structure (up to a factor 2

*π*

^{2}). The corresponding intersection numbers are given by the integrals

*d*

_{1}, …,

*d*

_{n}≥ 0 and

*n*=

*d*

_{1}+ ⋯

*d*

_{n}+

*m*+ 3 − 3

*g*. For

*g*≥ 0 we denote the generating function of these intersection numbers by

*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 (65). Then the generating function of Weil–Petersson volumes can be expressed as

*times*

*t*

_{k}[

*μ*] are defined by

We will show that the (bivariate) generating function $F\u0303g[\nu ,\mu ]$ of *tight* Weil–Petersson volumes, defined in (52), is also related to the intersection numbers, but with different times.

*The generating function of the volumes*

*T*

_{g,n}

*is related to the generating function of intersection numbers via*

*where the shifted times*

*τ*

_{k}[

*ν*,

*μ*]

*are defined by*

We remark that this result highlights the integrable properties of the tight Weil–Petersson volumes. It is well-known that exp *G*(0; *t*_{0}, *t*_{1}, …) is a *τ*-function of the KdV hierarchy.^{17} Since $F\u0303g[\nu ,\mu ]$ and *G*_{g} are related by a *ν*-independent shift of the times, $exp\u2211g=0\u221e\lambda 2g\u2212223\u22123gF\u0303g[\nu ,\mu ]$ 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.

*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*

_{μ}

*ρ*be the measure given by

*H*

_{μ}on the times can be computed using the series expansion (63),

*H*

_{μ}acts as an infinite upper-triangular matrix on the times. This matrix is easily inverted to give

*H*

_{μ}

*ρ*is sufficient to recover the original generating function $F\u0303g[\nu ,\mu ]$. Luckily the former is within close reach.

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

*where the correction term*

*is necessary to subtract the constant, linear and quadratic dependence on*

*ρ*

*in the genus-0 case.*

*g*,

*n*≥ 0 (such that

*n*≥ 3 if

*g*= 0) and

*L*

_{1}, …,

*L*

_{n}∈ [0,

*∞*) ∪

*i*(0,

*π*), then Proposition 7 allows us to compute

*T*

_{g,n}(

**K**;

*μ*] we take

*K*

_{i}=

*L*

_{i}for

*i*∉

*J*.

*F*

_{g}[

*ρ*+

*μ*] from its definition (10) we find

*T*

_{0,n}(

**K**;

*μ*] = 0 for

*n*< 3, yields

*κ*

_{1}and pure

*ψ*-class intersection numbers

^{16,23}leads to the identity

^{24}

*G*(0;

**x**) =

*G*(0;

*x*

_{0},

*x*

_{1},

*x*

_{2}, …).

^{16}proved by Kontsevich,

^{17}that

*G*(0;

**x**) satisfies the

*string equation*

^{25}it implies the following identity.

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

*r*

*,*

*x*

_{0},

*x*

_{1}, … fixed, let us consider the sequence of functions

*s*= 0 to

*s*=

*r*gives the claimed identity.□

Before we can use this lemma, we establish a relation between *τ*_{k}[*H*_{μ}*ρ*, *μ*] and *t*_{k}[*ρ* + *μ*].

*We can rewrite*

*where the shifted times*

*τ*

_{k}[

*ν*,

*μ*]

*are defined in (70).*

*M*

_{i}[

*μ*] defined in (22) to the times

*t*

_{i}[

*μ*]. Note that

*Z*(

*u*;

*μ*] defined in (21) can be expressed in the times as

*p*derivatives with respect to

*u*, we get

*t*

_{q}[

*ρ*], so we have reproduced the left-hand side of (87), since

*t*

_{q}[

*ρ*+

*μ*] =

*t*

_{q}[

*ρ*] +

*t*

_{q}[

*μ*].□

*We have*

*G*

_{corr}. By the definition (70),

*r*=

*R*[

*μ*] −

*s*/2 gives

*Z*(

*r*) =

*Z*(

*r*;

*μ*] around

*r*=

*R*[

*μ*] [recall from (23) that $Mk[\mu ]=Z(k+1)R[\mu ];\mu ]$, and in the third equality we expanded (2

*r*− 2

*R*[

*μ*])

^{k}as a polynomial in

*r*and made use of (73).

*ρ*this can be written as

*ρ*we then recognize exactly the expressions (54), (57), and (58),

### D. Properties of the new kernel

*X*(

*z*) =

*X*(

*z*;

*μ*] is determined by its two-sided Laplace transform $X\u0302(u;\mu ]$,

*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

*p*≥ 0. Note in particular that

*For*

*i*,

*j*≥ 1

*, the new kernel satisfies*

*and*

*x*, so we assume

*x*> 0. Since

*K*

_{0}(

*x*, −

*t*) =

*K*

_{0}(

*x*,

*t*) we can also assume

*t*≥ 0. For

*x*<

*t*we may expand

*x*>

*t*we may use

*x*,

*z*) → (−

*x*, −

*z*) and using the symmetry of d

*X*(

*z*), we observe that the second integral is unchanged when

*K*

_{0}(

*x*,

*t*) is replaced by

*K*

_{0}(−

*x*,

*t*). Since also

*K*

_{0}(

*x*,

*t*) +

*K*

_{0}(−

*x*,

*t*) = 2, the second integral can be calculated to give

*i*≥ 1

*x*+

*y*, since

### E. Proof of Theorem 2

^{15}of Witten’s conjecture, which relies on the observation that her recursion formula (1), 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*;

*x*

_{0},

*x*

_{1}, …) of intersection numbers (see also Ref. 18). These differential equations can be expressed as the Virasoro constraints

^{16,18,26}

*x*

_{0},

*x*

_{1},

*x*

_{2}, … via

*G*with $\gamma \u0303k=\delta k,1\u221221\u2212kMk\u22121[\mu ]$, which satisfies

*β*

_{m}[

*μ*] of (101) to introduce linear combinations

*p*≥ −1, which therefore obey

*V*

_{p}can be expressed as

*p*→

*p*− 1) we observe the identity

*G*is understood to be evaluated at $G=G(0;x0,x1+\gamma \u03031,x2+\gamma 2\u0303,\u2026)$.

*x*

_{k}=

*t*

_{k}[

*ν*] such that $xk+\gamma \u0303k=\tau k[\nu ,\mu ]$, Proposition 12 links

*G*to the generating function

*G*are evaluated at 0,

*τ*

_{0}[

*ν*,

*μ*],

*τ*

_{1}[

*ν*,

*μ*], …)

*K*(

*x*,

*t*;

*μ*] as

*g*,

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

### F. Proof of Theorem 4

*G*and the tight Weil–Petersson volumes

*T*

_{g,n}. Let us denote by $Gg,n(x0,x1,\u2026;M0,M1,\u2026)$ the homogeneous part of degree

*n*in

*x*

_{0},

*x*

_{1}, … of $Gg(0;x0,x1+1\u2212M0,x2\u221212M1,x3\u221214M2,\u2026)$. In other words, they are homogeneous polynomials of degree

*n*in

*x*

_{0},

*x*

_{1}, … with coefficients that are formal power series in $M0,M1,\u2026$, such that

*n*.

*g*≥ 2 and

*n*= 0, the existence of a polynomial $P\u0304g,0(m1,m2,\u2026)$ follows from Ref. 21, Lemma 12, since

*G*

_{g}(0; 0, 0,

*x*

_{2},

*x*

_{3}, …) is polynomial by construction. Also

*g*− 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,

*p*= −1, written in terms of

*G*

_{g,n}reads

*G*

_{g,n}is of the form (130). If (130) is granted for

*G*

_{g,n−1}, then

*G*

_{g,n}and the tight Weil–Petersson volume

*T*

_{g,n}are related via

*m*

_{0}= 1 this gives

## IV. LAPLACE TRANSFORM, SPECTRAL CURVE AND DISK FUNCTION

### A. Proof of Theorem 3

*loop insertion operator*in the literature, on the ring of formal power series in

*x*

_{0},

*x*

_{1}, … and 1/

*z*. For later purposes we record several identities for the power series coefficients around

*z*=

*∞*, valid for

*a*≥ 0,

*β*[

*μ*] were introduced in (101).

*g*≥ 1 or

*n*≥ 3

*z*) as

*g*contribution, which appears as the coefficient of

*λ*

^{2g−2}, the relation (149) allows us to turn this into a recursion for

*ω*

_{g,n},

*ω*

_{0,0}(

**z**) =

*ω*

_{0,1}(

**z**) = 0, this reduces to

### 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.

*Laplace transformed generating functions of (regular) Weil–Petersson volumes*

*We define*$xi=xi(zi;\mu ]=zi2\u22122R[\mu ]$

*. For*

*g*≥ 1

*or*

*n*≥ 3

*we have*

*while for*

*g*= 0

*and*

*n*= 1, 2

*,*

*I*

_{0}is given by

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

*The disk function*$W0,1(z)$

*is related to*

*η*

*via*

*valid when*4|

*R*| < |

*z*

_{1}|

^{2}.

Note that *μ* = 0 gives $W0,1(z)=0$ as expected.

*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

*λ*= 1 now gives

*λ*= (

*iL*)/(2

*π*) gives

*R*| < |

*x*|

^{2}.

*R*| < |

*x*

_{1}|

^{2}we find the series expansion

*Z*(

*R*) = 0 we may now conclude that

*R*| < |

*x*

_{1}|

^{2}. Substituting $x1=z12\u22122R$ gives the desired expression.□

*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

## 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.

*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 $M$.

**= (**

*β**β*

_{1}, …,

*β*

_{n}), the boundary term

*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

*S*

_{0}, gives the full (Euclidean) JT action

*χ*is the Euler characteristic of the manifold.

*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.

^{31}that the JT partition functions

*Z*

_{g,n}for 2

*g*+

*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**= (

*b*

_{1}, …,

*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

*V*

_{g,n}(

**b**) are the Weil–Petersson volumes and the trumpet contributions are given by

^{35}

*U*(

*ϕ*), which gives rise to defects in the hyperbolic surfaces.

### A. Conical defects

*πα*carrying weight

*μ*each. It naturally arises

^{32}from Kaluza–Klein instantons when performing dimensional reduction on three-dimensional black holes.

*μ*on

*i*[0, 2

*π*) and setting

*k*types of defects with cone angles

*α*

_{1}, …,

*α*

_{k}∈ [0, 2

*π*],

^{32}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:

*R*= −2 everywhere, except at point

*x*

_{1}, where we have a conical defect with cone angle 2

*πα*. If one includes all orders of

*μ*, any number of defects may appear and each defect carries a weight

*μ*.

^{32}

*πα*<

*π*) are obtained

^{1,2,13,36}from the usual Weil–Petersson volume polynomials by treating the defect angle 2

*πα*as a geodesic boundary with imaginary boundary length 2

*πiα*. This implies that the partition function

*Z*

_{g,n}(

**) is closely related to the generating function**

*β**F*

_{g}[

*μ*] of Weil–Petersson volumes considered in this paper. To be precise, using (10) and (60),

*T*

_{g,n}in Theorem 2.

*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

*β*. We conclude that for

*g*≥ 1 or

*n*≥ 3,

*g*= 0 and

*n*= 2, we only need to glue two tight trumpets together to find the universal two-boundary correlator

*μ*. This is the Rayleigh distribution with mean $\pi \beta 1\beta 2/(\beta 1+\beta 2)$.

We note that these expressions do not apply to the case of blunt cone points (cone angle 2*πα* ∈ [*π*, 2*π*]). 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}

*L*is the length of the boundary,

^{34,39}as the action of a fermion with mass

*z*.

*R*[

*μ*

_{FZZT}] depends on

*z*and

*S*

_{0}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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