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



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 $$\omega_{g,n}$$ is then the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found  that the coefficients $$\omega_{g,n}$$ 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 who studied the main properties of those invariants.

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

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.

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

$$\begin{align}\omega_{g,n}(z_1,z_2,\dots,z_{n}) &=\sum_{a=\text{branchpoints}} \operatorname{Res}_{z\to a} K(z_1,z,\sigma_a(z)) \Big( \omega_{g-1,n+1}(z,\sigma_a(z),z_2,\dots,z_n) \\ &\qquad\qquad\qquad + \mathop{{\sum}'}_{\overset{g_1+g_2=g}{I_1\uplus I_2=\{z_2,\dots,z_n\} }} \omega_{g_1,1+\# I_1}(z,I_1)\omega_{g_2,1+\# I_2}(\sigma_a(z),I_2) \Big) \end{align}$$ where $$K(z_1,z_2,z_3)$$ is called the recursion kernel: $$ K(z_1,z_2,z_3) = \frac{\frac12 \int_{z'=z_3}^{z_2} \omega_{0,2}(z_1,z')}{\omega_{0,1}(z_2)-\omega_{0,1}(z_3)} $$
 * For $$n\geq 1$$ and $$2g-2+n>0$$:

and $$\sigma_a$$ is the local Galois involution near a branch point $$a$$, it is such that $$x(\sigma_a(z))=x(z)$$. The primed sum $${\sum}'$$ means excluding the two terms $$(g_1,I_1)=(0,\emptyset)$$ and $$(g_2,I_2)=(0,\emptyset)$$.


 * For $$n=0$$ and $$2g-2>0$$:

$$ F_g = \omega_{g,0} = \frac{1}{2-2g}\ \sum_{a=\text{branchpoints}} \operatorname{Res}_{z\to a} F_{0,1}(z) \omega_{g,1}(z) $$

with $$dF_{0,1}=\omega_{0,1}$$ any antiderivative of $$\omega_{0,1}$$.


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

Main properties
$$\sum_{a=\text{branchpoints}} \operatorname{Res}_{z\to a} F_{0,1}(z)\ \omega_{g,n+1}(z_1,\dots,z_n,z) = (2g-2+n) \omega_{g,n}(z_1,\dots,z_n) $$ where $$dF_{0,1}=\omega_{0,1}$$.
 * Symmetry: each $$\omega_{g,n}$$ is a symmetric $$n$$-form on $$\Sigma^n$$.
 * poles: each $$\omega_{g,n}$$ is meromorphic, it has poles only at branchpoints, with vanishing residues.
 * Homogeneity: $$\omega_{g,n}$$ is homogeneous of degree $$2-2g-n$$. Under the change $$\omega_{0,1}\to \lambda \omega_{0,1}$$, we have $$\omega_{g,n}\to \lambda^{2-2g-n}\omega_{g,n}$$.
 * Dilaton equation:

$$\sum_{z\in x^{-1}(x)} \omega_{g,n+1}(z,z_1,\dots,z_n) $$
 * Loop equations: The following forms have no poles at branchpoints

$$\sum_{\{z\neq z'\} \subset x^{-1}(x)} \Big(\omega_{g,n+1}(z,z',z_2,\dots,z_n) + \sum_{\overset{g_1+g_2=g}{I_1\uplus I_2=\{z_2,\dots,z_n\} }} \omega_{g_1,1+\# I_1}(z,I_1)\omega_{g_2,1+\# I_2}(z',I_2) \Big)$$ where the sum has no prime, i.e. no term excluded.


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

Higher order ramifications
In case the branchpoints are not simple, the definition is amended as follows (simple branchpoints correspond to k=2):

$$\omega_{g,n}(z_1,z_2,\dots,z_{n})= \sum_{a=\text{branchpoints}} \operatorname{Res}_{z\to a} \sum_{k=2}^{{\rm order}_x(a)} \sum_{J \subset x^{-1}(x(z))\setminus\{z\},\,\# J=k-1} K_k(z_1,z,J) \sum_{J_1,\dots, J_\ell \vdash J\cup\{z\}} \sum'_{\overset{g_1+\dots+g_\ell=g+\ell-k}{I_1\uplus \dots I_\ell=\{z_2,\dots,z_n\} }} \prod_{i=1}^l \omega_{g_i,\# J_i+\# I_i}(J_i,I_i) $$

The first sum is over partitions $$J_1,\dots,J_\ell$$ of $$J\cup\{z\}$$ with non empty parts $$J_i\neq \emptyset$$, and in the second sum, the prime means excluding all terms such that $$(g_i,\# J_i+\# I_i)=(0,1)$$.

$$K_k$$ is called the recursion kernel:

$$ K_k(z_0,z_1,\dots,z_k) = \frac{\int_{z'=*}^{z_1} \omega_{0,2}(z_0,z')}{\prod_{i=2}^k (\omega_{0,1}(z_1)-\omega_{0,1}(z_i))} $$

The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants $$\omega_{g,n}$$ will not depend on it.

Topological recursion invariants and intersection numbers
The invariants $$\omega_{g,n}$$ can be written in terms of intersection numbers of tautological classes

(*) $$ \omega_{g,n}(z_1,\dots,z_n) = 2^{3g-3+n}\sum_{G=\text{Graphs}} \frac{1}{\#\text{Aut}(G)} \int_{\left(\prod_{v=\text{vertices}} {\overline{\mathcal M}}_{g_v,n_v} \right)}\,\, \prod_{v=\text{vertices}} e^{\sum_k \hat t_{\sigma(v),k} \kappa_k} \prod_{(p,p')=\text{nodal points}} \left(\sum_{d,d'} B_{\sigma(p),2d;\sigma(p'),2d'} \psi_p^d \psi_{p'}^{d'}\right) \prod_{p_i=\text{marked points}\, i=1,\dots,n} \left(\sum_{d_i} \psi_{p_i}^{d_i} d\xi_{\sigma(p_i),d_i}(z_i) \right) $$

where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus $$g$$, and $$n$$ smooth labeled marked points $$p_1,\dots,p_n$$, and equipped with a map $$\sigma:\{\text{vertices}\}\to \{\text{branchpoints}\} $$. $$\psi_p=c_1(\mathcal L_p)$$ is the Chern class of the cotangent line bundle $$\mathcal L_p$$ whose fiber is the cotangent plane at $$p$$. $$\kappa_k$$ is the $$k$$th Mumford's kappa class. The coefficients $$\hat t_{a,k}$$, $$B_{a,k;a',k'}$$, $$d\xi_{a,k}(z)$$, are the Taylor expansion coefficients of $$\omega_{0,1}$$ and $$\omega_{0,2}$$ in the vicinity of branchpoints as follows: in the vicinity of a branchpoint $$a$$ (assumed simple), a local coordinate is $$\zeta_a(z)=\sqrt{x(z)-a}$$. The Taylor expansion of $$\omega_{0,2}(z,z')$$ near branchpoints $$z\to a$$, $$z'\to a'$$ defines the coefficients $$B_{a,d;a',d'}$$

$$\omega_{0,2}(z,z') \mathop_{z\to a,\ z'\to a'} \left( \frac{\delta_{a,a'} }{(\zeta_a(z)-\zeta_{a'}(z'))^2}+ 2\pi \sum_{d,d'=0}^\infty \frac{B_{a,d;a',d'}}{\Gamma(\frac{d+1}{2})\Gamma(\frac{d'+1}{2})}\, \zeta_a(z)^d \zeta_{a'}(z')^{d'} \right) d\zeta_a(z)d\zeta_{a'}(z')$$.

The Taylor expansion at $$z'\to a$$, defines the 1-forms coefficients $$d\xi_{a,d}(z)$$

$$d\xi_{a,d}(z) = \frac{-\Gamma(d+\frac12)}{\Gamma(\frac12)} \operatorname{Res}_{z'\to a} (x(z')-a)^{-d-\frac12}\omega_{0,2}(z,z')$$ whose Taylor expansion near a branchpoint $$a'$$ is

$$ d\xi_{a,d}(z) \mathop_{z\to a'} \frac{-\delta_{a,a'} (2d+1)!! d\zeta_a(z)}{2^d \zeta_a(z)^{2d+2}}+ \sum_{k=0}^\infty \frac{B_{a,2d;a',2k} 2^{k+1}}{(2k-1)!!}\zeta_{a'}(z)^{2k} d\zeta_{a'}(z) $$.

Write also the Taylor expansion of $$\omega_{0,1}$$

$$ \omega_{0,1}(z) \mathop_{z\to a} \sum_{k=0}^\infty t_{a,k}\ \frac{\Gamma(\frac12)}{(k+1)\Gamma(\frac{k+1}{2})}\ \zeta_{a}(z)^{k} d\zeta_{a}(z) $$.

Equivalently, the coefficients $$t_{a,k}$$ can be found from expansion coefficients of the Laplace transform, and the coefficients $$\hat t_{a,k}$$ are the expansion coefficients of the log of the Laplace transform

$$ \int_{x(z)-x(a)\in \mathbb R_+} \omega_{0,1}(z) e^{-u x(z)} = \frac{e^{-ux(a)}\sqrt\pi}{2 u^{3/2}} \sum_{k=0}^\infty t_{a,k} u^{-k} = \frac{e^{-ux(a)}\sqrt\pi}{2 u^{3/2}} e^{-\sum_{k=0}^\infty \hat t_{a,k} u^{-k}} $$.

For example, we have

$$ \omega_{0,3}(z_1,z_2,z_3) = \sum_a e^{\hat t_{a,0}}  d\xi_{a,0}(z_1)d\xi_{a,0}(z_2)d\xi_{a,0}(z_3). $$

$$ \omega_{1,1}(z) = 2 \sum_a e^{\hat t_{a,0}} \left( \frac{1}{24} d\xi_{a,1}(z) + \frac{\hat t_{a,1}}{24} d\xi_{a,0}(z) +\frac12 B_{a,0;a,0} d\xi_{a,0}(z)\right). $$

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

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 $$\left(\mathbb C; \ \mathbb C;\ x: z\mapsto z^2 ;\ \omega_{0,1}(z)=\frac{4}{\pi} z \sin{(\pi z)} dz;\,\omega_{0,2}(z_1,z_2) = \frac{dz_1 dz_2}{(z_1-z_2)^2}\right) $$

the n-form $$\omega_{g,n} = d_1 \dots d_n F_{g,n}$$ is the Laplace transform of the Weil-Petersson volume

$$ F_{g,n}(z_1,\dots,z_n) = \int_0^\infty e^{-z_1L_1} dL_1 \dots \int_0^\infty e^{-z_nL_n} dL_n    \quad \int_{\mathcal M_{g,n}(L_1,\dots,L_n)} w $$

where $$\mathcal M_{g,n}(L_1,\dots,L_n)$$ is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths $$L_1,\dots,L_n$$, and $$w$$ is the Weil-Petersson volume form.

The topological recursion for the n-forms $$\omega_{g,n}(z_1,\dots,z_n)$$, is then equivalent to Mirzakhani's recursion.

Witten–Kontsevich intersection numbers
For the choice of spectral curve $$\left(\mathbb C; \ \mathbb C;\ x: z\mapsto z^2 ;\ \omega_{0,1}(z)=2z^2 dz;\,\omega_{0,2}(z_1,z_2) = \frac{dz_1 dz_2}{(z_1-z_2)^2}\right) $$

the n-form $$\omega_{g,n}= d_1 \dots d_n F_{g,n}$$ is

$$ F_{g,n}(z_1,\dots,z_n) = 2^{2-2g-n}\sum_{d_1+\dots+d_n=3g-3+n} \prod_{i=1}^n \frac{(2d_i-1)!! }{z_i^{2d_i+1}} \quad \left\langle\tau_{d_1}\dots\tau_{d_n}\right\rangle_g $$

where $$\left\langle\tau_{d_1}\dots\tau_{d_n}\right\rangle_g$$ 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
For the choice of spectral curve $$\left(\mathbb C; \ \mathbb C;\ x: -z+\ln{z} ;\ \omega_{0,1}(z)=(1-z) dz;\,\omega_{0,2}(z_1,z_2) = \frac{dz_1 dz_2}{(z_1-z_2)^2}\right) $$

the n-form $$\omega_{g,n}= d_1 \dots d_n F_{g,n}$$ is

$$ F_{g,n}(z_1,\dots,z_n) = \sum_{\ell(\mu)\leq n} m_\mu(e^{x(z_1)},\dots,e^{x(z_n)}) \quad h_{g,\mu_1,\dots,\mu_n} $$

where $$h_{g,\mu}$$ is the connected simple Hurwitz number of genus g with ramification $$\mu=(\mu_1,\dots,\mu_n)$$: 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 $$\mu$$.

Gromov–Witten numbers and the BKMP conjecture
Let $$\mathfrak X$$ a toric Calabi–Yau 3-fold, with Kähler moduli $$t_1,\dots,t_{b_2(\mathfrak X)}$$. Its mirror manifold is singular over a complex plane curve $$\Sigma$$ given by a polynomial equation $$P(e^x,e^y)=0$$, whose coefficients are functions of the Kähler moduli. For the choice of spectral curve $$\left(\Sigma; \ \mathbb C^*;\ x ;\ \omega_{0,1}=y dx;\,\omega_{0,2} \right) $$ with $$\omega_{0,2}$$ the fundamental second kind differential on $$\Sigma$$,

According to the BKMP conjecture, the n-form $$\omega_{g,n}= d_1 \dots d_n F_{g,n}$$ is

$$ F_{g,n}(z_1,\dots,z_n) = \sum_{\mathbf d\in H_2(\mathfrak X,\mathbb Z)} \sum_{\mu_1,\dots,\mu_n\in H_1(\mathcal L,\mathbb Z)} t^d \prod_{i=1}^n e^{x(z_i)} \mathcal N_{g}(\mathfrak X,\mathcal L;\mathbf d,\mu_1,\dots,\mu_n) $$

where $$ \mathcal N_{g}(\mathfrak X,\mathcal L;\mathbf d,\mu_1,\dots,\mu_n) = \int_{[{\overline{\mathcal M}}_{g,n}(\mathfrak X,\mathcal L, \mathbf d,\mu_1,\dots,\mu_n)]^{\rm vir}} 1 $$

is the genus g Gromov–Witten number, representing the number of holomorphic maps of a surface of genus g into $$\mathfrak X$$, with n boundaries mapped to a special Lagrangian submanifold $$\mathcal L$$. $$\mathbf d=(d_1,\dots,d_{b_2(\mathfrak X)})$$ is the 2nd relative homology class of the surface's image, and $$\mu_i\in H_1(\mathcal L,\mathbb Z)$$ are homology classes (winding number) of the boundary images.

The BKMP conjecture has since then been proven.