Topological recursion

In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory.

## Introduction

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The topological recursion is a construction in algebraic geometry.[1] It takes as initial data a spectral curve: the data of  ( Σ , Σ 0 , x , ω 0 , 1 , ω 0 , 2 ) {\displaystyle \left(\Sigma ,\Sigma _{0},x,\omega _{0,1},\omega _{0,2}\right)} , where:  x : Σ → Σ 0 {\displaystyle x:\Sigma \to \Sigma _{0}} is a covering of Riemann surfaces with ramification points;  ω 0 , 1 {\displaystyle \omega _{0,1}} is a meromorphic differential 1-form on  Σ {\displaystyle \Sigma } , regular at the ramification points;  ω 0 , 2 {\displaystyle \omega _{0,2}} is a symmetric meromorphic bilinear differential form on  Σ 2 {\displaystyle \Sigma ^{2}} having a double pole on the diagonal and no residue.

The topological recursion is then a recursive definition of infinite sequences of symmetric meromorphic n-forms  ω g , n {\displaystyle \omega _{g,n}} on  Σ n {\displaystyle \Sigma ^{n}} , with poles at ramification points only, for integers g≥0 such that 2g-2+n>0\. The definition is a recursion on the integer 2g-2+n.

In many applications, the n-form  ω g , n {\displaystyle \omega _{g,n}} is interpreted as a generating function that measures a set of surfaces of genus g and with n boundaries. The recursion is on 2g-2+n the Euler characteristics, whence the name "topological recursion".

Schematic illustration of the topological recursion: recursively adding pairs of pants to build a surface of genus g with n boundaries

## Origin

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The topological recursion was first discovered in random matrices. One main goal of random matrix theory, is to find the large size asymptotic expansion of n-point correlation functions, and in some suitable cases, the asymptotic expansion takes the form of a power series. The n-form  ω g , n {\displaystyle \omega _{g,n}} is then the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found[2][3][4] that the coefficients  ω g , n {\displaystyle \omega _{g,n}} always obey a same recursion on 2g-2+n. The idea to consider this universal recursion relation beyond random matrix theory, and to promote it as a definition of algebraic curves invariants, occurred in Eynard-Orantin 2007[1] who studied the main properties of those invariants.

An important application of topological recursion was to Gromov–Witten invariants. Marino and BKMP[5] conjectured that Gromov–Witten invariants of a toric Calabi–Yau 3-fold  X {\displaystyle {\mathfrak {X}}} are the TR invariants of a spectral curve that is the mirror of  X {\displaystyle {\mathfrak {X}}} .

Since then, topological recursion has generated a lot of activity in particular in enumerative geometry. The link to Givental formalism and Frobenius manifolds has been established.[6]

## Definition

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(Case of simple branch points. For higher order branchpoints, see the section Higher order ramifications below)

* For  n ≥ 1 {\displaystyle n\geq 1} and  2 g − 2 + n > 0 {\displaystyle 2g-2+n>0} :

ω g , n ( z 1 , z 2 , … , z n ) = ∑ a = branchpoints Res z → a ⁡ K ( z 1 , z , σ a ( z ) ) ( ω g − 1 , n + 1 ( z , σ a ( z ) , z 2 , … , z n ) + ∑ ′ I 1 ⊎ I 2 = { z 2 , … , z n } g 1 + g 2 = g ⁡ ω g 1 , 1 + # I 1 ( z , I 1 ) ω g 2 , 1 + # I 2 ( σ a ( z ) , I 2 ) ) {\displaystyle {\begin{aligned}\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})&=\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}K(z_{1},z,\sigma _{a}(z)){\Big (}\omega _{g-1,n+1}(z,\sigma _{a}(z),z_{2},\dots ,z_{n})\\\&\qquad \qquad \qquad +\mathop {{\sum }'} _{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\\{z_{2},\dots ,z_{n}\\}}}\omega _{g_{1},1+\\#I_{1}}(z,I_{1})\omega _{g_{2},1+\\#I_{2}}(\sigma _{a}(z),I_{2}){\Big )}\end{aligned}}} where  K ( z 1 , z 2 , z 3 ) {\displaystyle K(z_{1},z_{2},z_{3})} is called the recursion kernel:  K ( z 1 , z 2 , z 3 ) = 1 2 ∫ z ′ = z 3 z 2 ω 0 , 2 ( z 1 , z ′ ) ω 0 , 1 ( z 2 ) − ω 0 , 1 ( z 3 ) {\displaystyle K(z_{1},z_{2},z_{3})={\frac {{\frac {1}{2}}\int _{z'=z_{3}}^{z_{2}}\omega _{0,2}(z_{1},z')}{\omega _{0,1}(z_{2})-\omega _{0,1}(z_{3})}}}
and  σ a {\displaystyle \sigma _{a}} is the local Galois involution near a branch point  a {\displaystyle a} , it is such that  x ( σ a ( z ) ) = x ( z ) {\displaystyle x(\sigma _{a}(z))=x(z)} . The primed sum  ∑ ′ {\displaystyle {\sum }'} means excluding the two terms  ( g 1 , I 1 ) = ( 0 , ∅ ) {\displaystyle (g_{1},I_{1})=(0,\emptyset )} and  ( g 2 , I 2 ) = ( 0 , ∅ ) {\displaystyle (g_{2},I_{2})=(0,\emptyset )} .

* For  n = 0 {\displaystyle n=0} and  2 g − 2 > 0 {\displaystyle 2g-2>0} :

F g = ω g , 0 = 1 2 − 2 g   ∑ a = branchpoints Res z → a ⁡ F 0 , 1 ( z ) ω g , 1 ( z ) {\displaystyle F_{g}=\omega _{g,0}={\frac {1}{2-2g}}\ \sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\omega _{g,1}(z)}
with  d F 0 , 1 = ω 0 , 1 {\displaystyle dF_{0,1}=\omega _{0,1}} any antiderivative of  ω 0 , 1 {\displaystyle \omega _{0,1}} .

* The definition of  F 0 = ω 0 , 0 {\displaystyle F_{0}=\omega _{0,0}} and  F 1 = ω 1 , 0 {\displaystyle F_{1}=\omega _{1,0}} is more involved and can be found in the original article of Eynard-Orantin.[1]

## Main properties

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* Symmetry: each  ω g , n {\displaystyle \omega _{g,n}} is a symmetric  n {\displaystyle n} -form on  Σ n {\displaystyle \Sigma ^{n}} .
* poles: each  ω g , n {\displaystyle \omega _{g,n}} is meromorphic, it has poles only at branchpoints, with vanishing residues.
* Homogeneity: ω g , n {\displaystyle \omega _{g,n}} is homogeneous of degree  2 − 2 g − n {\displaystyle 2-2g-n} . Under the change  ω 0 , 1 → λ ω 0 , 1 {\displaystyle \omega _{0,1}\to \lambda \omega _{0,1}} , we have  ω g , n → λ 2 − 2 g − n ω g , n {\displaystyle \omega _{g,n}\to \lambda ^{2-2g-n}\omega _{g,n}} .
* Dilaton equation:

∑ a = branchpoints Res z → a ⁡ F 0 , 1 ( z )   ω g , n + 1 ( z 1 , … , z n , z ) = ( 2 g − 2 + n ) ω g , n ( z 1 , … , z n ) {\displaystyle \sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\ \omega _{g,n+1}(z_{1},\dots ,z_{n},z)=(2g-2+n)\omega _{g,n}(z_{1},\dots ,z_{n})}
where  d F 0 , 1 = ω 0 , 1 {\displaystyle dF_{0,1}=\omega _{0,1}} .

* Loop equations: The following forms have no poles at branchpoints

∑ z ∈ x − 1 ( x ) ω g , n + 1 ( z , z 1 , … , z n ) {\displaystyle \sum _{z\in x^{-1}(x)}\omega _{g,n+1}(z,z_{1},\dots ,z_{n})}
∑ { z ≠ z ′ } ⊂ x − 1 ( x ) ( ω g , n + 1 ( z , z ′ , z 2 , … , z n ) + ∑ I 1 ⊎ I 2 = { z 2 , … , z n } g 1 + g 2 = g ω g 1 , 1 + # I 1 ( z , I 1 ) ω g 2 , 1 + # I 2 ( z ′ , I 2 ) ) {\displaystyle \sum _{\\{z\neq z'\\}\subset x^{-1}(x)}{\Big (}\omega _{g,n+1}(z,z',z_{2},\dots ,z_{n})+\sum _{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\\{z_{2},\dots ,z_{n}\\}}}\omega _{g_{1},1+\\#I_{1}}(z,I_{1})\omega _{g_{2},1+\\#I_{2}}(z',I_{2}){\Big )}}
where the sum has no prime, i.e. no term excluded.

* Deformations: The  ω g , n {\displaystyle \omega _{g,n}} satisfy deformation equations
* Limits: given a family of spectral curves  S t {\displaystyle {\mathcal {S}}_{t}} , whose limit as  t → 0 {\displaystyle t\to 0} is a singular curve, resolved by rescaling by a power of  t μ {\displaystyle t^{\mu }} , then  lim t → 0 t ( 2 − 2 g − n ) μ ω g , n ( S t ) = ω g , n ( lim t → 0 t μ S t ) {\displaystyle \lim _{t\to 0}t^{(2-2g-n)\mu }\omega _{g,n}({\mathcal {S}}_{t})=\omega _{g,n}(\lim _{t\to 0}t^{\mu }{\mathcal {S}}_{t})} .
* Symplectic invariance: In the case where  Σ {\displaystyle \Sigma } is a compact algebraic curve with a marking of a symplectic basis of cycles,  x {\displaystyle x} is meromorphic and  ω 0 , 1 = y d x {\displaystyle \omega _{0,1}=ydx} is meromorphic and  ω 0 , 2 = B {\displaystyle \omega _{0,2}=B} is the fundamental second kind differential normalized on the marking, then the spectral curve  S = ( Σ , C , x , y d x , B ) {\displaystyle {\mathcal {S}}=(\Sigma ,\mathbb {C} ,x,ydx,B)} and  S ~ = ( Σ , C , y , − x d y , B ) {\displaystyle {\tilde {\mathcal {S}}}=(\Sigma ,\mathbb {C} ,y,-xdy,B)} , have the same  F g {\displaystyle F_{g}} shifted by some terms.
* Modular properties: In the case where  Σ {\displaystyle \Sigma } is a compact algebraic curve with a marking of a symplectic basis of cycles, and  ω 0 , 2 = B {\displaystyle \omega _{0,2}=B} is the fundamental second kind differential normalized on the marking, then the invariants  ω g , n {\displaystyle \omega _{g,n}} are quasi-modular forms under the modular group of marking changes. The invariants  ω g , n {\displaystyle \omega _{g,n}} satisfy BCOV equations.[clarification needed]

## Generalizations

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### Higher order ramifications

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In case the branchpoints are not simple, the definition is amended as follows[7] (simple branchpoints correspond to k=2):

ω g , n ( z 1 , z 2 , … , z n ) = ∑ a = branchpoints Res z → a ⁡ ∑ k = 2 o r d e r x ( a ) ∑ J ⊂ x − 1 ( x ( z ) ) ∖ { z } , # J = k − 1 K k ( z 1 , z , J ) ⋅ ∑ J 1 , … , J ℓ ⊢ J ∪ { z } ∑ I 1 ⊎ … I ℓ = { z 2 , … , z n } g 1 + ⋯ + g ℓ = g + ℓ − k ′ ∏ i = 1 l ω g i , # J i + # I i ( J i , I i ) {\displaystyle {\begin{aligned}\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})=&\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}\sum _{k=2}^{{\rm {order}}_{x}(a)}\sum _{J\subset x^{-1}(x(z))\setminus \\{z\\},\,\\#J=k-1}K_{k}(z_{1},z,J)\\\&\qquad \cdot \sum _{J_{1},\dots ,J_{\ell }\vdash J\cup \\{z\\}}\sum '_{\overset {g_{1}+\dots +g_{\ell }=g+\ell -k}{I_{1}\uplus \dots I_{\ell }=\\{z_{2},\dots ,z_{n}\\}}}\prod _{i=1}^{l}\omega _{g_{i},\\#J_{i}+\\#I_{i}}(J_{i},I_{i})\end{aligned}}}

The first sum is over partitions  J 1 , … , J ℓ {\displaystyle J_{1},\dots ,J_{\ell }} of  J ∪ { z } {\displaystyle J\cup \\{z\\}} with non empty parts  J i ≠ ∅ {\displaystyle J_{i}\neq \emptyset } , and in the second sum, the prime means excluding all terms such that  ( g i , # J i + # I i ) = ( 0 , 1 ) {\displaystyle (g_{i},\\#J_{i}+\\#I_{i})=(0,1)} .

K k {\displaystyle K_{k}} is called the recursion kernel:
K k ( z 0 , z 1 , … , z k ) = ∫ z ′ = ∗ z 1 ω 0 , 2 ( z 0 , z ′ ) ∏ i = 2 k ( ω 0 , 1 ( z 1 ) − ω 0 , 1 ( z i ) ) {\displaystyle K_{k}(z_{0},z_{1},\dots ,z_{k})={\frac {\int _{z'=*}^{z_{1}}\omega _{0,2}(z_{0},z')}{\prod _{i=2}^{k}(\omega _{0,1}(z_{1})-\omega _{0,1}(z_{i}))}}}
The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants  ω g , n {\displaystyle \omega _{g,n}} will not depend on it.

## Topological recursion invariants and intersection numbers

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The invariants  ω g , n {\displaystyle \omega _{g,n}} can be written in terms of intersection numbers of tautological classes:[8]
(*)  ω g , n ( z 1 , … , z n ) = 2 3 g − 3 + n ∑ G = Graphs 1 # Aut ( G ) ∫ ( ∏ v = vertices M ¯ g v , n v ) ∏ v = vertices e ∑ k t ^ σ ( v ) , k κ k ∏ ( p , p ′ ) = nodal points ( ∑ d , d ′ B σ ( p ) , 2 d ; σ ( p ′ ) , 2 d ′ ψ p d ψ p ′ d ′ ) ∏ p i = marked points i = 1 , … , n ( ∑ d i ψ p i d i d ξ σ ( p i ) , d i ( z i ) ) {\displaystyle {\begin{aligned}\omega _{g,n}(z_{1},\dots ,z_{n})=2^{3g-3+n}&\sum _{G={\text{Graphs}}}{\frac {1}{\\#{\text{Aut}}(G)}}\int _{\left(\prod _{v={\text{vertices}}}{\overline {\mathcal {M}}}_{g_{v},n_{v}}\right)}\,\,\prod _{v={\text{vertices}}}e^{\sum _{k}{\hat {t}}_{\sigma (v),k}\kappa _{k}}\\\&\prod _{(p,p')={\text{nodal points}}}\left(\sum _{d,d'}B_{\sigma (p),2d;\sigma (p'),2d'}\psi _{p}^{d}\psi _{p'}^{d'}\right)\prod _{p_{i}={\text{marked points}}\,i=1,\dots ,n}\left(\sum _{d_{i}}\psi _{p_{i}}^{d_{i}}d\xi _{\sigma (p_{i}),d_{i}}(z_{i})\right)\end{aligned}}}
where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus  g {\displaystyle g} , and  n {\displaystyle n} smooth labeled marked points  p 1 , … , p n {\displaystyle p_{1},\dots ,p_{n}} , and equipped with a map  σ : { vertices } → { branchpoints } {\displaystyle \sigma :\\{{\text{vertices}}\\}\to \\{{\text{branchpoints}}\\}} .  ψ p = c 1 ( L p ) {\displaystyle \psi _{p}=c_{1}({\mathcal {L}}_{p})} is the Chern class of the cotangent line bundle L p {\displaystyle {\mathcal {L}}_{p}} whose fiber is the cotangent plane at  p {\displaystyle p} .  κ k {\displaystyle \kappa _{k}} is the  k {\displaystyle k} th Mumford's kappa class. The coefficients  t ^ a , k {\displaystyle {\hat {t}}_{a,k}} ,  B a , k ; a ′ , k ′ {\displaystyle B_{a,k;a',k'}} ,  d ξ a , k ( z ) {\displaystyle d\xi _{a,k}(z)} , are the Taylor expansion coefficients of  ω 0 , 1 {\displaystyle \omega _{0,1}} and  ω 0 , 2 {\displaystyle \omega _{0,2}} in the vicinity of branchpoints as follows: in the vicinity of a branchpoint  a {\displaystyle a} (assumed simple), a local coordinate is  ζ a ( z ) = x ( z ) − a {\displaystyle \zeta _{a}(z)={\sqrt {x(z)-a}}} . The Taylor expansion of  ω 0 , 2 ( z , z ′ ) {\displaystyle \omega _{0,2}(z,z')} near branchpoints  z → a {\displaystyle z\to a} ,  z ′ → a ′ {\displaystyle z'\to a'} defines the coefficients  B a , d ; a ′ , d ′ {\displaystyle B_{a,d;a',d'}}
ω 0 , 2 ( z , z ′ ) ∼ z → a ,   z ′ → a ′ ⁡ ( δ a , a ′ ( ζ a ( z ) − ζ a ′ ( z ′ ) ) 2 + 2 π ∑ d , d ′ = 0 ∞ B a , d ; a ′ , d ′ Γ ( d + 1 2 ) Γ ( d ′ + 1 2 ) ζ a ( z ) d ζ a ′ ( z ′ ) d ′ ) d ζ a ( z ) d ζ a ′ ( z ′ ) {\displaystyle \omega _{0,2}(z,z')\mathop {\sim } _{z\to a,\ z'\to a'}\left({\frac {\delta _{a,a'}}{(\zeta _{a}(z)-\zeta _{a'}(z'))^{2}}}+2\pi \sum _{d,d'=0}^{\infty }{\frac {B_{a,d;a',d'}}{\Gamma ({\frac {d+1}{2}})\Gamma ({\frac {d'+1}{2}})}}\,\zeta _{a}(z)^{d}\zeta _{a'}(z')^{d'}\right)d\zeta _{a}(z)d\zeta _{a'}(z')} .
The Taylor expansion at  z ′ → a {\displaystyle z'\to a} , defines the 1-forms coefficients  d ξ a , d ( z ) {\displaystyle d\xi _{a,d}(z)}
d ξ a , d ( z ) = − Γ ( d + 1 2 ) Γ ( 1 2 ) Res z ′ → a ⁡ ( x ( z ′ ) − a ) − d − 1 2 ω 0 , 2 ( z , z ′ ) {\displaystyle d\xi _{a,d}(z)={\frac {-\Gamma (d+{\frac {1}{2}})}{\Gamma ({\frac {1}{2}})}}\operatorname {Res} _{z'\to a}(x(z')-a)^{-d-{\frac {1}{2}}}\omega _{0,2}(z,z')} whose Taylor expansion near a branchpoint  a ′ {\displaystyle a'} is
d ξ a , d ( z ) ∼ z → a ′ ⁡ − δ a , a ′ ( 2 d + 1 ) ! ! d ζ a ( z ) 2 d ζ a ( z ) 2 d + 2 + ∑ k = 0 ∞ B a , 2 d ; a ′ , 2 k 2 k + 1 ( 2 k − 1 ) ! ! ζ a ′ ( z ) 2 k d ζ a ′ ( z ) {\displaystyle d\xi _{a,d}(z)\mathop {\sim } _{z\to a'}{\frac {-\delta _{a,a'}(2d+1)!!d\zeta _{a}(z)}{2^{d}\zeta _{a}(z)^{2d+2}}}+\sum _{k=0}^{\infty }{\frac {B_{a,2d;a',2k}2^{k+1}}{(2k-1)!!}}\zeta _{a'}(z)^{2k}d\zeta _{a'}(z)} .
Write also the Taylor expansion of  ω 0 , 1 {\displaystyle \omega _{0,1}}
ω 0 , 1 ( z ) ∼ z → a ⁡ ∑ k = 0 ∞ t a , k   Γ ( 1 2 ) ( k + 1 ) Γ ( k + 1 2 )   ζ a ( z ) k d ζ a ( z ) {\displaystyle \omega _{0,1}(z)\mathop {\sim } _{z\to a}\sum _{k=0}^{\infty }t_{a,k}\ {\frac {\Gamma ({\frac {1}{2}})}{(k+1)\Gamma ({\frac {k+1}{2}})}}\ \zeta _{a}(z)^{k}d\zeta _{a}(z)} .
Equivalently, the coefficients  t a , k {\displaystyle t_{a,k}} can be found from expansion coefficients of the Laplace transform, and the coefficients  t ^ a , k {\displaystyle {\hat {t}}_{a,k}} are the expansion coefficients of the log of the Laplace transform
∫ x ( z ) − x ( a ) ∈ R + ω 0 , 1 ( z ) e − u x ( z ) = e − u x ( a ) π 2 u 3 / 2 ∑ k = 0 ∞ t a , k u − k = e − u x ( a ) π 2 u 3 / 2 e − ∑ k = 0 ∞ t ^ a , k u − k {\displaystyle \int _{x(z)-x(a)\in \mathbb {R} _{+}}\omega _{0,1}(z)e^{-ux(z)}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}\sum _{k=0}^{\infty }t_{a,k}u^{-k}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}e^{-\sum _{k=0}^{\infty }{\hat {t}}_{a,k}u^{-k}}} .

For example, we have
ω 0 , 3 ( z 1 , z 2 , z 3 ) = ∑ a e t ^ a , 0 d ξ a , 0 ( z 1 ) d ξ a , 0 ( z 2 ) d ξ a , 0 ( z 3 ) . {\displaystyle \omega _{0,3}(z_{1},z_{2},z_{3})=\sum _{a}e^{{\hat {t}}_{a,0}}d\xi _{a,0}(z_{1})d\xi _{a,0}(z_{2})d\xi _{a,0}(z_{3}).}

ω 1 , 1 ( z ) = 2 ∑ a e t ^ a , 0 ( 1 24 d ξ a , 1 ( z ) + t ^ a , 1 24 d ξ a , 0 ( z ) + 1 2 B a , 0 ; a , 0 d ξ a , 0 ( z ) ) . {\displaystyle \omega _{1,1}(z)=2\sum _{a}e^{{\hat {t}}_{a,0}}\left({\frac {1}{24}}d\xi _{a,1}(z)+{\frac {{\hat {t}}_{a,1}}{24}}d\xi _{a,0}(z)+{\frac {1}{2}}B_{a,0;a,0}d\xi _{a,0}(z)\right).}

The formula (*) generalizes ELSV formula as well as Mumford's formula and Mariño-Vafa formula.

## Some applications in enumerative geometry

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### Mirzakhani's recursion

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M. Mirzakhani's recursion for hyperbolic volumes of moduli spaces is an instance of topological recursion. For the choice of spectral curve  ( C ;   C ;   x : z ↦ z 2 ;   ω 0 , 1 ( z ) = 4 π z sin ⁡ ( π z ) d z ; ω 0 , 2 ( z 1 , z 2 ) = d z 1 d z 2 ( z 1 − z 2 ) 2 ) {\displaystyle \left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)={\frac {4}{\pi }}z\sin {(\pi z)}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)}
the n-form  ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is the Laplace transform of the Weil-Petersson volume
F g , n ( z 1 , … , z n ) = ∫ 0 ∞ e − z 1 L 1 d L 1 … ∫ 0 ∞ e − z n L n d L n ∫ M g , n ( L 1 , … , L n ) w {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=\int _{0}^{\infty }e^{-z_{1}L_{1}}dL_{1}\dots \int _{0}^{\infty }e^{-z_{n}L_{n}}dL_{n}\quad \int _{{\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})}w}
where  M g , n ( L 1 , … , L n ) {\displaystyle {\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})} is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths  L 1 , … , L n {\displaystyle L_{1},\dots ,L_{n}} , and  w {\displaystyle w} is the Weil-Petersson volume form.
The topological recursion for the n-forms  ω g , n ( z 1 , … , z n ) {\displaystyle \omega _{g,n}(z_{1},\dots ,z_{n})} , is then equivalent to Mirzakhani's recursion.

### Witten–Kontsevich intersection numbers

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For the choice of spectral curve  ( C ;   C ;   x : z ↦ z 2 ;   ω 0 , 1 ( z ) = 2 z 2 d z ; ω 0 , 2 ( z 1 , z 2 ) = d z 1 d z 2 ( z 1 − z 2 ) 2 ) {\displaystyle \left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)=2z^{2}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)}
the n-form  ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is
F g , n ( z 1 , … , z n ) = 2 2 − 2 g − n ∑ d 1 + ⋯ + d n = 3 g − 3 + n ∏ i = 1 n ( 2 d i − 1 ) ! ! z i 2 d i + 1 ⟨ τ d 1 … τ d n ⟩ g {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=2^{2-2g-n}\sum _{d_{1}+\dots +d_{n}=3g-3+n}\prod _{i=1}^{n}{\frac {(2d_{i}-1)!!}{z_{i}^{2d_{i}+1}}}\quad \left\langle \tau _{d_{1}}\dots \tau _{d_{n}}\right\rangle _{g}}
where  ⟨ τ d 1 … τ d n ⟩ g {\displaystyle \left\langle \tau _{d_{1}}\dots \tau _{d_{n}}\right\rangle _{g}} is the Witten-Kontsevich intersection number of Chern classes of cotangent line bundles in the compactified moduli space of Riemann surfaces of genus g with n smooth marked points.

### Hurwitz numbers

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For the choice of spectral curve  ( C ;   C ;   x : z ↦ − z + ln ⁡ z ;   ω 0 , 1 ( z ) = ( 1 − z ) d z ; ω 0 , 2 ( z 1 , z 2 ) = d z 1 d z 2 ( z 1 − z 2 ) 2 ) {\displaystyle \left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto -z+\ln {z};\ \omega _{0,1}(z)=(1-z)dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)}
the n-form  ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is
F g , n ( z 1 , … , z n ) = ∑ ℓ ( μ ) ≤ n m μ ( e x ( z 1 ) , … , e x ( z n ) ) h g , μ 1 , … , μ n {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\ell (\mu )\leq n}m_{\mu }(e^{x(z_{1})},\dots ,e^{x(z_{n})})\quad h_{g,\mu _{1},\dots ,\mu _{n}}}
where  h g , μ {\displaystyle h_{g,\mu }} is the connected simple Hurwitz number of genus g with ramification  μ = ( μ 1 , … , μ n ) {\displaystyle \mu =(\mu _{1},\dots ,\mu _{n})} : the number of branch covers of the Riemann sphere by a genus g connected surface, with 2g-2+n simple ramification points, and one point with ramification profile given by the partition  μ {\displaystyle \mu } .

### Gromov–Witten numbers and the BKMP conjecture

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Let  X {\displaystyle {\mathfrak {X}}} a toric Calabi–Yau 3-fold, with Kähler moduli  t 1 , … , t b 2 ( X ) {\displaystyle t_{1},\dots ,t_{b_{2}({\mathfrak {X}})}} . Its mirror manifold is singular over a complex plane curve  Σ {\displaystyle \Sigma } given by a polynomial equation  P ( e x , e y ) = 0 {\displaystyle P(e^{x},e^{y})=0} , whose coefficients are functions of the Kähler moduli. For the choice of spectral curve  ( Σ ;   C ∗ ;   x ;   ω 0 , 1 = y d x ; ω 0 , 2 ) {\displaystyle \left(\Sigma ;\ \mathbb {C} ^{*};\ x;\ \omega _{0,1}=ydx;\,\omega _{0,2}\right)} with  ω 0 , 2 {\displaystyle \omega _{0,2}} the fundamental second kind differential on  Σ {\displaystyle \Sigma } ,
According to the BKMP[5] conjecture, the n-form  ω g , n = d 1 … d n F g , n {\displaystyle \omega _{g,n}=d_{1}\dots d_{n}F_{g,n}} is
F g , n ( z 1 , … , z n ) = ∑ d ∈ H 2 ( X , Z ) ∑ μ 1 , … , μ n ∈ H 1 ( L , Z ) t d ∏ i = 1 n e x ( z i ) N g ( X , L ; d , μ 1 , … , μ n ) {\displaystyle F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\mathbf {d} \in H_{2}({\mathfrak {X}},\mathbb {Z} )}\sum _{\mu _{1},\dots ,\mu _{n}\in H_{1}({\mathcal {L}},\mathbb {Z} )}t^{d}\prod _{i=1}^{n}e^{x(z_{i})}{\mathcal {N}}_{g}({\mathfrak {X}},{\mathcal {L}};\mathbf {d} ,\mu _{1},\dots ,\mu _{n})}
where  N g ( X , L ; d , μ 1 , … , μ n ) = ∫ [ M ¯ g , n ( X , L , d , μ 1 , … , μ n ) ] v i r 1 {\displaystyle {\mathcal {N}}_{g}({\mathfrak {X}},{\mathcal {L}};\mathbf {d} ,\mu _{1},\dots ,\mu _{n})=\int _{[{\overline {\mathcal {M}}}_{g,n}({\mathfrak {X}},{\mathcal {L}},\mathbf {d} ,\mu _{1},\dots ,\mu _{n})]^{\rm {vir}}}1}
is the genus g Gromov–Witten number, representing the number of holomorphic maps of a surface of genus g into  X {\displaystyle {\mathfrak {X}}} , with n boundaries mapped to a special Lagrangian submanifold  L {\displaystyle {\mathcal {L}}} .  d = ( d 1 , … , d b 2 ( X ) ) {\displaystyle \mathbf {d} =(d_{1},\dots ,d_{b_{2}({\mathfrak {X}})})} is the 2nd relative homology class of the surface's image, and  μ i ∈ H 1 ( L , Z ) {\displaystyle \mu _{i}\in H_{1}({\mathcal {L}},\mathbb {Z} )} are homology classes (winding number) of the boundary images.
The BKMP[5] conjecture has since then been proven.

## Notes

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1. ^ a b c Invariants of algebraic curves and topological expansion, B. Eynard, N. Orantin, math-ph/0702045, ccsd-hal-00130963, Communications in Number Theory and Physics, Vol 1, Number 2, p347-452.
2. ^ B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, JHEP/024A/0904, hep-th/0407261 A short overview of the ”Topological recursion”, math-ph/arXiv:1412.3286
3. ^ A. Alexandrov, A. Mironov, A. Morozov, Solving Virasoro Constraints in Matrix Models, Fortsch.Phys.53:512-521,2005, arXiv:hep-th/0412205
4. ^ L. Chekhov, B. Eynard, N. Orantin, Free energy topological expansion for the 2-matrix model, JHEP 0612 (2006) 053, math-ph/0603003
5. ^ a b c Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178,2009
6. ^ P. Dunin-Barkowski, N. Orantin, S. Shadrin, L. Spitz, "Identification of the Givental formula with the spectral curve topological recursion procedure", Commun.Math.Phys. 328 (2014) 669-700.
7. ^ V. Bouchard, B. Eynard, "Think globally, compute locally", JHEP02(2013)143.
8. ^ B. Eynard, Invariants of spectral curves and intersection theory of moduli spaces of complex curves, math-ph: arxiv.1110.2949, Journal Communications in Number Theory and Physics, Volume 8, Number 3.

## References

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[1]

1. ^ O. Dumitrescu and M. Mulase, Lectures on the topological recursion for Higgs bindles and quantum curves, https://www.math.ucdavis.edu/~mulase/texfiles/OMLectures.pdf