Topological recursion

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

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

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]

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

• For ${\displaystyle n\geq 1}$ and ${\displaystyle 2g-2+n>0}$:

$${\displaystyle {\begin{aligned}\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})&=\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}K(z_{1},z,\sigma _{a}(z)){\Big (}\omega _{g-1,n+1}(z,\sigma _{a}(z),z_{2},\dots ,z_{n})\\&\qquad \qquad \qquad +\mathop {{\sum }'} _{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},1+\#I_{1}}(z,I_{1})\omega _{g_{2},1+\#I_{2}}(\sigma _{a}(z),I_{2}){\Big )}\end{aligned}}}$$

where ${\displaystyle K(z_{1},z_{2},z_{3})}$ is called the recursion kernel: ${\displaystyle K(z_{1},z_{2},z_{3})={\frac {{\frac {1}{2}}\int _{z'=z_{3}}^{z_{2}}\omega _{0,2}(z_{1},z')}{\omega _{0,1}(z_{2})-\omega _{0,1}(z_{3})}}}$
and ${\displaystyle \sigma _{a}}$ is the local Galois involution near a branch point ${\displaystyle a}$, it is such that ${\displaystyle x(\sigma _{a}(z))=x(z)}$. The primed sum ${\displaystyle {\sum }'}$ means excluding the two terms ${\displaystyle (g_{1},I_{1})=(0,\emptyset )}$ and ${\displaystyle (g_{2},I_{2})=(0,\emptyset )}$.

• For ${\displaystyle n=0}$ and ${\displaystyle 2g-2>0}$:

${\displaystyle F_{g}=\omega _{g,0}={\frac {1}{2-2g}}\ \sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\omega _{g,1}(z)}$
with ${\displaystyle dF_{0,1}=\omega _{0,1}}$ any antiderivative of ${\displaystyle \omega _{0,1}}$.

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

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

${\displaystyle \sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\ \omega _{g,n+1}(z_{1},\dots ,z_{n},z)=(2-2g-n)\omega _{g,n}(z_{1},\dots ,z_{n})}$
where ${\displaystyle dF_{0,1}=\omega _{0,1}}$.

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

${\displaystyle \sum _{z\in x^{-1}(x)}\omega _{g,n+1}(z,z_{1},\dots ,z_{n})}$
${\displaystyle \sum _{\{z\neq z'\}\subset x^{-1}(x)}{\Big (}\omega _{g,n+1}(z,z',z_{2},\dots ,z_{n})+\sum _{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},1+\#I_{1}}(z,I_{1})\omega _{g_{2},1+\#I_{2}}(z',I_{2}){\Big )}}$ where the sum has no prime, i.e. no term excluded.

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

The invariants ${\displaystyle \omega _{g,n}}$ can be written in terms of intersection numbers of tautological classes

[8]

(*) ${\displaystyle \omega _{g,n}(z_{1},\dots ,z_{n})=2^{3g-3+n}\sum _{G={\text{Graphs}}}{\frac {1}{\#{\text{Aut}}(G)}}\int _{\left(\prod _{v={\text{vertices}}}{\overline {\mathcal {M}}}_{g_{v},n_{v}}\right)}\,\,\prod _{v={\text{vertices}}}e^{\sum _{k}{\hat {t}}_{\sigma (v),k}\kappa _{k}}\prod _{(p,p')={\text{nodal points}}}\left(\sum _{d,d'}B_{\sigma (p),2d;\sigma (p'),2d'}\psi _{p}^{d}\psi _{p'}^{d'}\right)\prod _{p_{i}={\text{marked points}}\,i=1,\dots ,n}\left(\sum _{d_{i}}\psi _{p_{i}}^{d_{i}}d\xi _{\sigma (p_{i}),d_{i}}(z_{i})\right)}$
where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus ${\displaystyle g}$, and ${\displaystyle n}$ smooth labeled marked points ${\displaystyle p_{1},\dots ,p_{n}}$, and equipped with a map ${\displaystyle \sigma$ :\{{\text{vertices}}\}\to \{{\text{branchpoints}}\}} . ${\displaystyle \psi _{p}=c_{1}({\mathcal {L}}_{p})}$ is the Chern class of the cotangent line bundle ${\displaystyle {\mathcal {L}}_{p}}$ whose fiber is the cotangent plane at ${\displaystyle p}$. ${\displaystyle \kappa _{k}}$ is the ${\displaystyle k}$^th Mumford's kappa class. The coefficients ${\displaystyle {\hat {t}}_{a,k}}$, ${\displaystyle B_{a,k;a',k'}}$, ${\displaystyle d\xi _{a,k}(z)}$, are the Taylor expansion coefficients of ${\displaystyle \omega _{0,1}}$ and ${\displaystyle \omega _{0,2}}$ in the vicinity of branchpoints as follows: in the vicinity of a branchpoint ${\displaystyle a}$ (assumed simple), a local coordinate is ${\displaystyle \zeta _{a}(z)={\sqrt {x(z)-a}}}$. The Taylor expansion of ${\displaystyle \omega _{0,2}(z,z')}$ near branchpoints ${\displaystyle z\to a}$, ${\displaystyle z'\to a'}$ defines the coefficients ${\displaystyle B_{a,d;a',d'}}$
${\displaystyle \omega _{0,2}(z,z')\mathop {\sim } _{z\to a,\ z'\to a'}\left({\frac {\delta _{a,a'}}{(\zeta _{a}(z)-\zeta _{a'}(z'))^{2}}}+2\pi \sum _{d,d'=0}^{\infty }{\frac {B_{a,d;a',d'}}{\Gamma ({\frac {d+1}{2}})\Gamma ({\frac {d'+1}{2}})}}\,\zeta _{a}(z)^{d}\zeta _{a'}(z')^{d'}\right)d\zeta _{a}(z)d\zeta _{a'}(z')}$.
The Taylor expansion at ${\displaystyle z'\to a}$, defines the 1-forms coefficients ${\displaystyle d\xi _{a,d}(z)}$
${\displaystyle d\xi _{a,d}(z)={\frac {-\Gamma (d+{\frac {1}{2}})}{\Gamma ({\frac {1}{2}})}}\operatorname {Res} _{z'\to a}(x(z')-a)^{-d-{\frac {1}{2}}}\omega _{0,2}(z,z')}$ whose Taylor expansion near a branchpoint ${\displaystyle a'}$ is
${\displaystyle d\xi _{a,d}(z)\mathop {\sim } _{z\to a'}{\frac {-\delta _{a,a'}(2d+1)!!d\zeta _{a}(z)}{2^{d}\zeta _{a}(z)^{2d+2}}}+\sum _{k=0}^{\infty }{\frac {B_{a,2d;a',2k}2^{k+1}}{(2k-1)!!}}\zeta _{a'}(z)^{2k}d\zeta _{a'}(z)}$.
Write also the Taylor expansion of ${\displaystyle \omega _{0,1}}$
${\displaystyle \omega _{0,1}(z)\mathop {\sim } _{z\to a}\sum _{k=0}^{\infty }t_{a,k}\ {\frac {\Gamma ({\frac {1}{2}})}{(k+1)\Gamma ({\frac {k+1}{2}})}}\ \zeta _{a}(z)^{k}d\zeta _{a}(z)}$.
Equivalently, the coefficients ${\displaystyle t_{a,k}}$ can be found from expansion coefficients of the Laplace transform, and the coefficients ${\displaystyle {\hat {t}}_{a,k}}$ are the expansion coefficients of the log of the Laplace transform
${\displaystyle \int _{x(z)-x(a)\in \mathbb {R} _{+}}\omega _{0,1}(z)e^{-ux(z)}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}\sum _{k=0}^{\infty }t_{a,k}u^{-k}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}e^{-\sum _{k=0}^{\infty }{\hat {t}}_{a,k}u^{-k}}}$ .

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

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

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

[1]