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.

## Contents

* 1 Introduction
* 2 Origin
* 3 Definition
* 4 Main properties
* 5 Generalizations
* 5.1 Higher order ramifications
* 6 Topological recursion invariants and intersection numbers
* 7 Some applications in enumerative geometry
* 7.1 Mirzakhani's recursion
* 7.2 Witten–Kontsevich intersection numbers
* 7.3 Hurwitz numbers
* 7.4 Gromov–Witten numbers and the BKMP conjecture
* 8 Notes
* 9 References

## Introduction

The topological recursion is a construction in algebraic geometry.[1] It takes as initial data a spectral curve: the data of [math]\displaystyle{ \left(\Sigma,\Sigma_0,x,\omega_{0,1},\omega_{0,2}\right) }[/math], where: [math]\displaystyle{ x:\Sigma\to\Sigma_0 }[/math] is a covering of Riemann surfaces with ramification points; [math]\displaystyle{ \omega_{0,1} }[/math] is a meromorphic differential 1-form on [math]\displaystyle{ \Sigma }[/math], regular at the ramification points; [math]\displaystyle{ \omega_{0,2} }[/math] is a symmetric meromorphic bilinear differential form on [math]\displaystyle{ \Sigma^2 }[/math] 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 [math]\displaystyle{ \omega_{g,n} }[/math] on [math]\displaystyle{ \Sigma^n }[/math], 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 [math]\displaystyle{ \omega_{g,n} }[/math] is interpreted as a generating function that measures a set of surfaces of genus g and with n boundaries. The recursion is on 2-2g+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

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 [math]\displaystyle{ \omega_{g,n} }[/math] is then the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found[2][3][4] that the coefficients [math]\displaystyle{ \omega_{g,n} }[/math] 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 [math]\displaystyle{ \mathfrak X }[/math] are the TR invariants of a spectral curve that is the mirror of [math]\displaystyle{ \mathfrak X }[/math].

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

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

* For [math]\displaystyle{ n\geq 1 }[/math] and [math]\displaystyle{ 2g-2+n\gt 0 }[/math]:

[math]\displaystyle{ \begin{align}\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{align} }[/math] where [math]\displaystyle{ K(z_1,z_2,z_3) }[/math] is called the recursion kernel: [math]\displaystyle{ K(z_1,z_2,z_3) = \frac{\frac12 \int_{z'=z_3}^{z_2} \omega_{0,2}(z_1,z')}{\omega_{0,1}(z_2)-\omega_{0,1}(z_3)} }[/math]
and [math]\displaystyle{ \sigma_a }[/math] is the local Galois involution near a branch point [math]\displaystyle{ a }[/math], it is such that [math]\displaystyle{ x(\sigma_a(z))=x(z) }[/math]. The primed sum [math]\displaystyle{ {\sum}' }[/math] means excluding the two terms [math]\displaystyle{ (g_1,I_1)=(0,\emptyset) }[/math] and [math]\displaystyle{ (g_2,I_2)=(0,\emptyset) }[/math].

* For [math]\displaystyle{ n=0 }[/math] and [math]\displaystyle{ 2g-2\gt 0 }[/math]:

[math]\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) }[/math]
with [math]\displaystyle{ dF_{0,1}=\omega_{0,1} }[/math] any antiderivative of [math]\displaystyle{ \omega_{0,1} }[/math].

* The definition of [math]\displaystyle{ F_0=\omega_{0,0} }[/math] and [math]\displaystyle{ F_1=\omega_{1,0} }[/math] is more involved and can be found in the original article of Eynard-Orantin.[1]

## Main properties

* Symmetry: each [math]\displaystyle{ \omega_{g,n} }[/math] is a symmetric [math]\displaystyle{ n }[/math]-form on [math]\displaystyle{ \Sigma^n }[/math].
* poles: each [math]\displaystyle{ \omega_{g,n} }[/math] is meromorphic, it has poles only at branchpoints, with vanishing residues.
* Homogeneity: [math]\displaystyle{ \omega_{g,n} }[/math] is homogeneous of degree [math]\displaystyle{ 2-2g-n }[/math]. Under the change [math]\displaystyle{ \omega_{0,1}\to \lambda \omega_{0,1} }[/math], we have [math]\displaystyle{ \omega_{g,n}\to \lambda^{2-2g-n}\omega_{g,n} }[/math].
* Dilaton equation:

[math]\displaystyle{ \sum_{a=\text{branchpoints}} \operatorname{Res}_{z\to a} F_{0,1}(z)\ \omega_{g,n+1}(z_1,\dots,z_n,z) = (2-2g-n) \omega_{g,n}(z_1,\dots,z_n) }[/math]
where [math]\displaystyle{ dF_{0,1}=\omega_{0,1} }[/math].

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

[math]\displaystyle{ \sum_{z\in x^{-1}(x)} \omega_{g,n+1}(z,z_1,\dots,z_n) }[/math]
[math]\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) }[/math] where the sum has no prime, i.e. no term excluded.

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

## Generalizations

### Higher order ramifications

In case the branchpoints are not simple, the definition is amended as follows[7] (simple branchpoints correspond to k=2):
[math]\displaystyle{ \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) \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) }[/math]
The first sum is over partitions [math]\displaystyle{ J_1,\dots,J_\ell }[/math] of [math]\displaystyle{ J\cup\\{z\\} }[/math] with non empty parts [math]\displaystyle{ J_i\neq \emptyset }[/math], and in the second sum, the prime means excluding all terms such that [math]\displaystyle{ (g_i,\\# J_i+\\# I_i)=(0,1) }[/math].

[math]\displaystyle{ K_k }[/math] is called the recursion kernel:
[math]\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))} }[/math]
The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants [math]\displaystyle{ \omega_{g,n} }[/math] will not depend on it.

## Topological recursion invariants and intersection numbers

The invariants [math]\displaystyle{ \omega_{g,n} }[/math] can be written in terms of intersection numbers of tautological classes

[8]

(*) [math]\displaystyle{ \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) }[/math]
where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus [math]\displaystyle{ g }[/math], and [math]\displaystyle{ n }[/math] smooth labeled marked points [math]\displaystyle{ p_1,\dots,p_n }[/math], and equipped with a map [math]\displaystyle{ \sigma:\\{\text{vertices}\\}\to \\{\text{branchpoints}\\} }[/math]. [math]\displaystyle{ \psi_p=c_1(\mathcal L_p) }[/math] is the Chern class of the cotangent line bundle [math]\displaystyle{ \mathcal L_p }[/math] whose fiber is the cotangent plane at [math]\displaystyle{ p }[/math]. [math]\displaystyle{ \kappa_k }[/math] is the [math]\displaystyle{ k }[/math]th Mumford's kappa class. The coefficients [math]\displaystyle{ \hat t_{a,k} }[/math], [math]\displaystyle{ B_{a,k;a',k'} }[/math], [math]\displaystyle{ d\xi_{a,k}(z) }[/math], are the Taylor expansion coefficients of [math]\displaystyle{ \omega_{0,1} }[/math] and [math]\displaystyle{ \omega_{0,2} }[/math] in the vicinity of branchpoints as follows: in the vicinity of a branchpoint [math]\displaystyle{ a }[/math] (assumed simple), a local coordinate is [math]\displaystyle{ \zeta_a(z)=\sqrt{x(z)-a} }[/math]. The Taylor expansion of [math]\displaystyle{ \omega_{0,2}(z,z') }[/math] near branchpoints [math]\displaystyle{ z\to a }[/math], [math]\displaystyle{ z'\to a' }[/math] defines the coefficients [math]\displaystyle{ B_{a,d;a',d'} }[/math]
[math]\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') }[/math].
The Taylor expansion at [math]\displaystyle{ z'\to a }[/math], defines the 1-forms coefficients [math]\displaystyle{ d\xi_{a,d}(z) }[/math]
[math]\displaystyle{ d\xi_{a,d}(z) = \frac{-\Gamma(d+\frac12)}{\Gamma(\frac12)} \operatorname{Res}_{z'\to a} (x(z')-a)^{-d-\frac12}\omega_{0,2}(z,z') }[/math] whose Taylor expansion near a branchpoint [math]\displaystyle{ a' }[/math] is
[math]\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) }[/math].
Write also the Taylor expansion of [math]\displaystyle{ \omega_{0,1} }[/math]
[math]\displaystyle{ \omega_{0,1}(z) \mathop{{\sim}}_{z\to a} \sum_{k=0}^\infty t_{a,k}\ \frac{\Gamma(\frac12)}{(k+1)\Gamma(\frac{k+1}{2})}\ \zeta_{a}(z)^{k} d\zeta_{a}(z) }[/math].
Equivalently, the coefficients [math]\displaystyle{ t_{a,k} }[/math] can be found from expansion coefficients of the Laplace transform, and the coefficients [math]\displaystyle{ \hat t_{a,k} }[/math] are the expansion coefficients of the log of the Laplace transform
[math]\displaystyle{ \int_{x(z)-x(a)\in \mathbb R_+} \omega_{0,1}(z) e^{-u x(z)} = \frac{e^{-ux(a)}\sqrt\pi}{2 u^{3/2}} \sum_{k=0}^\infty t_{a,k} u^{-k} = \frac{e^{-ux(a)}\sqrt\pi}{2 u^{3/2}} e^{-\sum_{k=0}^\infty \hat t_{a,k} u^{-k}} }[/math] .

For example, we have
[math]\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). }[/math]

[math]\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) +\frac12 B_{a,0;a,0} d\xi_{a,0}(z)\right). }[/math]

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

## Some applications in enumerative geometry

### Mirzakhani's recursion

M. Mirzakhani's recursion for hyperbolic volumes of moduli spaces is an instance of topological recursion. For the choice of spectral curve [math]\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) }[/math]
the n-form [math]\displaystyle{ \omega_{g,n} = d_1 \dots d_n F_{g,n} }[/math] is the Laplace transform of the Weil-Petersson volume
[math]\displaystyle{ F_{g,n}(z_1,\dots,z_n) = \int_0^\infty e^{-z_1L_1} dL_1 \dots \int_0^\infty e^{-z_nL_n} dL_n \quad \int_{\mathcal M_{g,n}(L_1,\dots,L_n)} w }[/math]
where [math]\displaystyle{ \mathcal M_{g,n}(L_1,\dots,L_n) }[/math] is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths [math]\displaystyle{ L_1,\dots,L_n }[/math], and [math]\displaystyle{ w }[/math] is the Weil-Petersson volume form.
The topological recursion for the n-forms [math]\displaystyle{ \omega_{g,n}(z_1,\dots,z_n) }[/math], is then equivalent to Mirzakhani's recursion.

### Witten–Kontsevich intersection numbers

For the choice of spectral curve [math]\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) }[/math]
the n-form [math]\displaystyle{ \omega_{g,n}= d_1 \dots d_n F_{g,n} }[/math] is
[math]\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 }[/math]
where [math]\displaystyle{ \left\langle\tau_{d_1}\dots\tau_{d_n}\right\rangle_g }[/math] 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

For the choice of spectral curve [math]\displaystyle{ \left(\mathbb C; \ \mathbb C;\ x: -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) }[/math]
the n-form [math]\displaystyle{ \omega_{g,n}= d_1 \dots d_n F_{g,n} }[/math] is
[math]\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} }[/math]
where [math]\displaystyle{ h_{g,\mu} }[/math] is the connected simple Hurwitz number of genus g with ramification [math]\displaystyle{ \mu=(\mu_1,\dots,\mu_n) }[/math]: 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 [math]\displaystyle{ \mu }[/math].

### Gromov–Witten numbers and the BKMP conjecture

Let [math]\displaystyle{ \mathfrak X }[/math] a toric Calabi–Yau 3-fold, with Kähler moduli [math]\displaystyle{ t_1,\dots,t_{b_2(\mathfrak X)} }[/math]. Its mirror manifold is singular over a complex plane curve [math]\displaystyle{ \Sigma }[/math] given by a polynomial equation [math]\displaystyle{ P(e^x,e^y)=0 }[/math], whose coefficients are functions of the Kähler moduli. For the choice of spectral curve [math]\displaystyle{ \left(\Sigma; \ \mathbb C^*;\ x ;\ \omega_{0,1}=y dx;\,\omega_{0,2} \right) }[/math] with [math]\displaystyle{ \omega_{0,2} }[/math] the fundamental second kind differential on [math]\displaystyle{ \Sigma }[/math],
According to the BKMP[5] conjecture, the n-form [math]\displaystyle{ \omega_{g,n}= d_1 \dots d_n F_{g,n} }[/math] is
[math]\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) }[/math]
where [math]\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 }[/math]
is the genus g Gromov–Witten number, representing the number of holomorphic maps of a surface of genus g into [math]\displaystyle{ \mathfrak X }[/math], with n boundaries mapped to a special Lagrangian submanifold [math]\displaystyle{ \mathcal L }[/math]. [math]\displaystyle{ \mathbf d=(d_1,\dots,d_{b_2(\mathfrak X)}) }[/math] is the 2nd relative homology class of the surface's image, and [math]\displaystyle{ \mu_i\in H_1(\mathcal L,\mathbb Z) }[/math] are homology classes (winding number) of the boundary images.
The BKMP[5] conjecture has since then been proven.

## Notes

1. ↑ 1.0 1.1 1.2 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. ↑ 5.0 5.1 5.2 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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Original source: https://en.wikipedia.org/wiki/Topological recursion. Read more

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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

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