Topological recursion

The topological recursion is a construction in algebraic geometry.

is a covering of Riemann surfaces with ramification points;

is a symmetric meromorphic bilinear differential form on

is interpreted as a generating function that measures a set of surfaces of genus g and 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.

is then the gth coefficient in the asymptotic expansion of the n-point correlation function.

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

are the TR invariants of a spectral curve that is the mirror of

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.

is the local Galois involution near a branch point

, and in the second sum, the prime means excluding all terms such that

The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants

can be written in terms of intersection numbers of tautological classes:[8]

{\displaystyle {\begin{aligned}\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)\end{aligned}}}

where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus

is the Chern class of the cotangent line bundle

(assumed simple), a local coordinate is

are the expansion coefficients of the log of the Laplace transform

is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths

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.

is the connected simple Hurwitz number of genus g with ramification

: 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

Its mirror manifold is singular over a complex plane curve

, with n boundaries mapped to a special Lagrangian submanifold

is the 2nd relative homology class of the surface's image, and

are homology classes (winding number) of the boundary images.