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 $\Sigma ^{2}$ having 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. The recursion is on 2-2g-n the Euler characteristics, whence the name "topological recursion".

It is often illustrated by the following picture:

Schematic representation of the topological recursion: recursively removing a short pair of pants

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 it as a definition of algebraic curves invariants, occurred in Eynard-Orantin 2007 in which were 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

Invariants: (case of simple branch points) for $2g-2+n>0$ and $n\geq 1$:
$\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})=\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}K(z_{1},z,\sigma _{a}(z)){\Big (}\omega _{g,n+1}(z,\sigma _{a}(z),z_{2},\dots ,z_{n})+\sum '_{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},1+\#I_{1}}(z,I_{1})\omega _{g_{2},1+\#I_{2}}(\sigma _{a}(z),I_{2}){\Big )}$
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 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 $\omega _{g,n}\to \lambda ^{2-2g-n}\omega _{g,n}$.

Generalizations

Higher order ramifications

In case the branchpoints are not simple, the definition is amended as follows (simple branchpoints correspond to k=2):
$\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})=\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}\sum _{k=2}^{{\rm {order}}_{x}(a)}\sum _{J\subset x^{-1}(x(z))\setminus \{z\},\,\#J=k-1}K_{k}(z_{1},z,J)\sum _{J_{1},\dots ,J_{\ell }\vdash J\cup \{z\}}\sum '_{\overset {g_{1}+\dots +g_{\ell }=g+\ell -k}{I_{1}\uplus \dots I_{\ell }=\{z_{2},\dots ,z_{n}\}}}\prod _{i=1}^{k}\omega _{g_{i},\#J_{i}+\#I_{i}}(J_{i},I_{i})$
The first sum is over partitions $J_{1},\dots ,J_{\ell }$ of $J\cup \{z\}$ with non empty parts $J_{i}\neq \emptyset$, 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.

Blobbed topological recursion

To be written

Topological recursion invariants and intersection numbers

The invariants $\omega _{g,n}$ can be written in terms of intersection numbers of tautological classes :
(*) $\omega _{g,n}(z_{1},\dots ,z_{n})=2^{3g-3+n}\sum _{G={\text{Graphs}}}{\frac {1}{\#{\text{Aut}}(G)}}\int _{\left(\prod _{v={\text{vertices}}}{\overline {\mathcal {M}}}_{g_{v},n_{v}}\right)}\,\,\prod _{v={\text{vertices}}}e^{\sum _{k}{\hat {t}}_{\sigma (v),k}\kappa _{k}}\prod _{(p,p')={\text{nodal points}}}e^{{\frac {1}{2}}\sum _{d,d'}B_{\sigma (p),2d;\sigma (p'),2d'}\psi _{p}^{d}\psi _{p'}^{d'}}\prod _{p_{i}={\text{marked points}}\,i=1,\dots ,n}\left(\sum _{d_{i}}\psi _{p_{i}}^{d_{i}}d\xi _{\sigma (p_{i}),d_{i}}(z_{i})\right)$
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 in some sense the Taylor expansion coefficients of $\omega _{0,1}$ and $\omega _{0,2}$ in the vicinity of branchpoints:

In the vicinity of a branchpoint $a$ (assumed simple), a local coordinate is $\zeta _{a}(z)={\sqrt {x(z)-a}}$,
write the Taylor expansion of $\omega _{0,2}(z,z')$ near branchpoints $z\to a$, $z'\to a'$ as
$\omega _{0,2}(z,z')\mathop {\sim } _{z\to a,\ z'\to a'}\left({\frac {\delta _{a,a'}}{(\zeta _{a}(z)-\zeta _{a'}(z'))^{2}}}+2\pi \sum _{d,d'=0}^{\infty }{\frac {B_{a,d;a',d'}}{\Gamma ({\frac {d+1}{2}})\Gamma ({\frac {d'+1}{2}})}}\,\zeta _{a}(z)^{d}\zeta _{a'}(z')^{d'}\right)d\zeta _{a}(z)d\zeta _{a'}(z')$.
and then let the family of 1-forms
$d\xi _{a,d}(z)={\frac {-\Gamma (d+{\frac {1}{2}})}{\Gamma ({\frac {1}{2}})}}\operatorname {Res} _{z'\to a}(x(z')-a)^{-d-{\frac {1}{2}}}\omega _{0,2}(z,z')$ whose Taylor expansion near a branchpoint $a'$ is
$d\xi _{a,d}(z)\mathop {\sim } _{z\to a'}{\frac {-\delta _{a,a'}(2d+1)!!d\zeta _{a}(z)}{2^{d}\zeta _{a}(z)^{2d+2}}}+\sum _{k=0}^{\infty }{\frac {B_{a,2d;a',2k}2^{k+1}}{(2k-1)!!}}\zeta _{a'}(z)^{2k}d\zeta _{a'}(z)$.
Write also the Taylor expansion of $\omega _{0,1}$
$\omega _{0,1}(z)\mathop {\sim } _{z\to a}\sum _{k=0}^{\infty }t_{a,k}\ {\frac {\Gamma ({\frac {1}{2}})}{(k+1)\Gamma ({\frac {k+1}{2}})}}\ \zeta _{a}(z)^{k}d\zeta _{a}(z)$.
Equivalently, the coefficients $t_{a,k}$ can be found from the Laplace transform, and recasted as $\int _{x(z)-x(a)\in \mathbb {R} _{+}}\omega _{0,1}(z)e^{-ux(z)}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}e^{-\sum _{k=0}^{\infty }{\hat {t}}_{a,k}u^{-k}}$ .

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

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

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

Some applications in enumerative geometry

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
$F_{g,n}(z_{1},\dots ,z_{n})=\int _{0}^{\infty }e^{-z_{1}L_{1}}dL_{1}\dots \int _{0}^{\infty }e^{-z_{n}L_{n}}dL_{n}\quad \int _{{\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})}w$
where ${\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})$ 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 $\mathbb {H} _{g}^{*}(\mu )$ 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 $\omega _{g,n}=d_{1}\dots d_{n}F_{g,n}$ 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 proved.