Topological recursion

In mathematics, Topological Recursion is a recursive definition of invariants of spectral curves. It has applications to enumerative geometry, random matrix theory, string theory, knot theory.

Introduction

The topological recursion is a construction in algebraic geometry, initially developed by Eynard and Orantin. 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 $\Sigma ^{2}$, which has a double pole on the diagonal and no residue.

The topological recursion is then a recursive definition of inifinite 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. 2-2g-n is its Euler characteristics, and the recursion is thus on the Euler characteristics, whence the name "topological recursion".

It is often illustrated by the following picture:

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 g^th coefficient in the asymptotic expansion of the n-point correlation function. It was first noticed for 1-hermitian random matrix, then for 2-hermitian random matrices, then many other cases, that the coefficients $\omega _{g,n}$ can be found by recursion on 2g-2+n. The idea to consider this universal recursion relation beyond random matrix theory, and to promote is as a definition of algebraic curves invariants, occurred in Eynard-Orantin 2007. They 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.

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 $\left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)={\frac {4}{\pi }}z\sin {(\pi z)}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)$

the n-form $\omega _{g,n}$ is

$F_{g,n}(z_{1},\dots ,z_{n})=\int _{0}^{\infty }e^{-z_{1}L_{1}}dL_{1}\dots \int _{0}^{\infty }e^{-z_{n}L_{n}}dL_{n}\quad \int _{{\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})}w$ $\omega _{g,n}(z_{1},\dots ,z_{n})=d_{1}\dots d_{n}F_{g,n}(z_{1},\dots ,z_{n})$ 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 $\omega _{g,n}$ is

$F_{g,n}(z_{1},\dots ,z_{n})=2^{2-2g-n}\sum _{d_{1}+\dots +d_{n}=3g-3+n}\prod _{i=1}^{n}{\frac {(2d_{i}-1)!!}{z_{i}^{2d_{i}+1}}}\quad \left\langle \tau _{d_{1}}\dots \tau _{d_{n}}\right\rangle _{g}$ $\omega _{g,n}(z_{1},\dots ,z_{n})=d_{1}\dots d_{n}F_{g,n}(z_{1},\dots ,z_{n})$

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 $\omega _{g,n}$ is

$F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\ell (\mu )\leq n}m_{\mu }(e^{x(z_{1})},\dots ,e^{x(z_{n})})\prod _{i=1}^{n}{\frac {\mu _{i}^{\mu _{i}}}{\mu _{i}!}}\quad H_{g}^{*}(\mu )$ $\omega _{g,n}(z_{1},\dots ,z_{n})=d_{1}\dots d_{n}F_{g,n}(z_{1},\dots ,z_{n})$

where $\mathbb {H} _{g}^{*}(\mu )$ is the connected simple Hurwitz number of genus g with ramification $\mu$: 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 $\left(\Sigma ;\ \mathbb {C} ^{*};\ x;\ \omega _{0,1}=ydx;\,\omega _{0,2}\right)$ with $\omega _{0,2}$ the fundamental second kind differential on $\Sigma$,

According to the BKMP conjecture, the n-form $\omega _{g,n}$ is

$F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\mathbf {d} \in H_{2}({\mathfrak {X}},\mathbb {Z} )}\sum _{\mu _{1},\dots ,\mu _{n}\in H_{1}({\mathcal {L}},\mathbb {Z} )}t^{d}\prod _{i=1}^{n}e^{x(z_{i})}{\mathcal {N}}_{g}({\mathfrak {X}},{\mathcal {L}};\mathbf {d} ,\mu _{1},\dots ,\mu _{n})$ $\omega _{g,n}(z_{1},\dots ,z_{n})=d_{1}\dots d_{n}F_{g,n}(z_{1},\dots ,z_{n})$ 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 proved.

Definition

Invariants: (case of simple branch points) for $2g-2+n>0$ and $n\geq 1$: $\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})=\sum _{a,\,dx(a)=0}\operatorname {Res} _{z\to a}K(z_{1},z,\sigma _{a}(z)){\Big (}\omega _{g,n+1}(z,\sigma _{a}(z),z_{2},\dots ,z_{n})+\sum '_{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},|I_{1}|+1}(z,I_{1})\omega _{g_{2},|I_{2}|+1}(\sigma _{a}(z),I_{2}){\Big )}$ where $K(z_{1},z_{2},z_{3})$ is called the recursion kernel: $K(z_{1},z_{2},z_{3})={\frac {{\frac {1}{2}}\int _{z'=z_{3}}^{z_{2}}\omega _{0,2}(z_{1},z')}{\omega _{0,1}(z_{2})-\omega _{0,1}(z_{3})}}$ and $\sigma _{a}$ 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 $2g-2>0$: $F_{g}=\omega _{g,0}=\sum _{a,\,dx(a)=0}\operatorname {Res} _{z\to a}F_{0,1}(z)\omega _{g,1}(z)$ 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

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 ramification points, 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 $\omega _{g,n}\to \lambda ^{2-2g-n}\omega _{g,n}$.

Generalizations

Higher order ramifications

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

$\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})=\sum _{a,\,dx(a)=0}\operatorname {Res} _{z\to a}\sum _{k=2}^{{\rm {order}}_{x}(a)}\sum _{J\subset x^{-1}(x(z))\setminus \{z\},\,|J|=k-1}K_{k}(z_{1},z,J)\sum _{J_{1},\dots ,J_{\ell }\vdash J\cup \{z\}}\sum '_{\overset {g_{1}+\dots +g_{\ell }=g+\ell -k}{I_{1}\uplus \dots I_{\ell }=\{z_{2},\dots ,z_{n}\}}}\prod _{i=1}^{k}\omega _{g_{i},|J_{i}|+|I_{i}|}(J_{i},I_{i})$ The first sum is over partitions $J_{1},\dots ,J_{\ell }$ of $J\cup \{z\}$ with non empty parts Failed to parse (unknown function "\nea"): {\displaystyle J_i\nea \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.

Blobbed topological recursion

To be written