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.

## Contents

* 1 Introduction
* 2 Origin
* 3 Definition
* 4 Main properties
* 5 Generalizations
* 5.1 Higher order ramifications
* 6 Topological recursion invariants and intersection numbers
* 7 Some applications in enumerative geometry
* 7.1 Mirzakhani's recursion
* 7.2 Witten–Kontsevich intersection numbers
* 7.3 Hurwitz numbers
* 7.4 Gromov–Witten numbers and the BKMP conjecture
* 8 Notes
* 9 References

## Introduction

The topological recursion is a construction in algebraic geometry.[1] It takes as initial data a spectral curve: the data of (Σ,Σ0,x,ω0,1,ω0,2), where: x:Σ->Σ0 is a covering of Riemann surfaces with ramification points; ω0,1 is a meromorphic differential 1-form on Σ, regular at the ramification points; ω0,2 is a symmetric meromorphic bilinear differential form on Σ2 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 ωg,n on Σn, 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 ωg,n is interpreted as a generating function that measures a set of surfaces of genus g and with n boundaries. The recursion is on 2g-2+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

## 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 ωg,n is then the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found[2][3][4] that the coefficients ωg,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 𝔛 are the TR invariants of a spectral curve that is the mirror of 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]

## Definition

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

* For n≥1 and 2g−2+n>0:

ωg,n(z1,z2,…,zn)=∑a=branchpointsResz->aK(z1,z,σa(z))(ωg−1,n+1(z,σa(z),z2,…,zn)+∑′I1⊎I2={z2,…,zn}g1+g2=gωg1,1+#I1(z,I1)ωg2,1+#I2(σa(z),I2)) where K(z1,z2,z3) is called the recursion kernel: K(z1,z2,z3)=12∫z′=z3z2ω0,2(z1,z′)ω0,1(z2)−ω0,1(z3)
and σa is the local Galois involution near a branch point a, it is such that x(σa(z))=x(z). The primed sum ∑' means excluding the two terms (g1,I1)=(0,∅) and (g2,I2)=(0,∅).

* For n=0 and 2g−2>0:

Fg=ωg,0=12−2g ∑a=branchpointsResz->aF0,1(z)ωg,1(z)
with dF0,1=ω0,1 any antiderivative of ω0,1.

* The definition of F0=ω0,0 and F1=ω1,0 is more involved and can be found in the original article of Eynard-Orantin.[1]

## Main properties

* Symmetry: each ωg,n is a symmetric n-form on Σn.
* poles: each ωg,n is meromorphic, it has poles only at branchpoints, with vanishing residues.
* Homogeneity: ωg,n is homogeneous of degree 2−2g−n. Under the change ω0,1->λω0,1, we have ωg,n->λ2−2g−nωg,n.
* Dilaton equation:

∑a=branchpointsResz->aF0,1(z) ωg,n+1(z1,…,zn,z)=(2g−2+n)ωg,n(z1,…,zn)
where dF0,1=ω0,1.

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

∑z∈x−1(x)ωg,n+1(z,z1,…,zn)
∑{z≠z′}⊂x−1(x)(ωg,n+1(z,z′,z2,…,zn)+∑I1⊎I2={z2,…,zn}g1+g2=gωg1,1+#I1(z,I1)ωg2,1+#I2(z′,I2))
where the sum has no prime, i.e. no term excluded.

* Deformations: The ωg,n satisfy deformation equations
* Limits: given a family of spectral curves 𝒮t, whose limit as t->0 is a singular curve, resolved by rescaling by a power of tμ, then limt->0t(2−2g−n)μωg,n(𝒮t)=ωg,n(limt->0tμ𝒮t).
* Symplectic invariance: In the case where Σ is a compact algebraic curve with a marking of a symplectic basis of cycles, x is meromorphic and ω0,1=ydx is meromorphic and ω0,2=B is the fundamental second kind differential normalized on the marking, then the spectral curve 𝒮=(Σ,ℂ,x,ydx,B) and 𝒮~=(Σ,ℂ,y,−xdy,B), have the same Fg shifted by some terms.
* Modular properties: In the case where Σ is a compact algebraic curve with a marking of a symplectic basis of cycles, and ω0,2=B is the fundamental second kind differential normalized on the marking, then the invariants ωg,n are quasi-modular forms under the modular group of marking changes. The invariants ωg,n satisfy BCOV equations.[clarification needed]

## Generalizations

### Higher order ramifications

In case the branchpoints are not simple, the definition is amended as follows[7] (simple branchpoints correspond to k=2):

ωg,n(z1,z2,…,zn)=∑a=branchpointsResz->a∑k=2orderx(a)∑J⊂x−1(x(z))∖{z},#J=k−1Kk(z1,z,J)⋅∑J1,…,Jℓ⊢J∪{z}∑'I1⊎…Iℓ={z2,…,zn}g1+…+gℓ=g+ℓ−k∏i=1lωgi,#Ji+#Ii(Ji,Ii)

The first sum is over partitions J1,…,Jℓ of J∪{z} with non empty parts Ji≠∅, and in the second sum, the prime means excluding all terms such that (gi,#Ji+#Ii)=(0,1).

Kk is called the recursion kernel:
Kk(z0,z1,…,zk)=∫z′=*z1ω0,2(z0,z′)∏i=2k(ω0,1(z1)−ω0,1(zi))
The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants ωg,n will not depend on it.

## Topological recursion invariants and intersection numbers

The invariants ωg,n can be written in terms of intersection numbers of tautological classes:[8]
(*) ωg,n(z1,…,zn)=23g−3+n∑G=Graphs1#Aut(G)∫(∏v=verticesℳ‾gv,nv)∏v=verticese∑kt^σ(v),kκk∏(p,p′)=nodal points(∑d,d′Bσ(p),2d;σ(p′),2d′ψpdψp′d′)∏pi=marked pointsi=1,…,n(∑diψpididξσ(pi),di(zi))
where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus g, and n smooth labeled marked points p1,…,pn, and equipped with a map σ:{vertices}->{branchpoints}. ψp=c1(ℒp) is the Chern class of the cotangent line bundle ℒp whose fiber is the cotangent plane at p. κk is the kth Mumford's kappa class. The coefficients t^a,k, Ba,k;a′,k′, dξa,k(z), are the Taylor expansion coefficients of ω0,1 and ω0,2 in the vicinity of branchpoints as follows: in the vicinity of a branchpoint a (assumed simple), a local coordinate is ζa(z)=x(z)−a. The Taylor expansion of ω0,2(z,z′) near branchpoints z->a, z′->a′ defines the coefficients Ba,d;a′,d′
ω0,2(z,z′)∼z->a, z′->a′(δa,a′(ζa(z)−ζa′(z′))2+2π∑d,d′=0∞Ba,d;a′,d′Γ(d+12)Γ(d′+12)ζa(z)dζa′(z′)d′)dζa(z)dζa′(z′).
The Taylor expansion at z′->a, defines the 1-forms coefficients dξa,d(z)
dξa,d(z)=−Γ(d+12)Γ(12)Resz′->a(x(z′)−a)−d−12ω0,2(z,z′) whose Taylor expansion near a branchpoint a′ is
dξa,d(z)∼z->a′−δa,a′(2d+1)!!dζa(z)2dζa(z)2d+2+∑k=0∞Ba,2d;a′,2k2k+1(2k−1)!!ζa′(z)2kdζa′(z).
Write also the Taylor expansion of ω0,1
ω0,1(z)∼z->a∑k=0∞ta,k Γ(12)(k+1)Γ(k+12) ζa(z)kdζa(z).
Equivalently, the coefficients ta,k can be found from expansion coefficients of the Laplace transform, and the coefficients t^a,k are the expansion coefficients of the log of the Laplace transform
∫x(z)−x(a)∈ℝ+ω0,1(z)e−ux(z)=e−ux(a)π2u3/2∑k=0∞ta,ku−k=e−ux(a)π2u3/2e−∑k=0∞t^a,ku−k .

For example, we have
ω0,3(z1,z2,z3)=∑aet^a,0dξa,0(z1)dξa,0(z2)dξa,0(z3).

ω1,1(z)=2∑aet^a,0(124dξa,1(z)+t^a,124dξa,0(z)+12Ba,0;a,0dξa,0(z)).

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 (ℂ; ℂ; x:z↦z2; ω0,1(z)=4πzsin⁡(πz)dz;ω0,2(z1,z2)=dz1dz2(z1−z2)2)
the n-form ωg,n=d1…dnFg,n is the Laplace transform of the Weil-Petersson volume
Fg,n(z1,…,zn)=∫0∞e−z1L1dL1…∫0∞e−znLndLn∫ℳg,n(L1,…,Ln)w
where ℳg,n(L1,…,Ln) is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths L1,…,Ln, and w is the Weil-Petersson volume form.
The topological recursion for the n-forms ωg,n(z1,…,zn), is then equivalent to Mirzakhani's recursion.

### Witten–Kontsevich intersection numbers

For the choice of spectral curve (ℂ; ℂ; x:z↦z2; ω0,1(z)=2z2dz;ω0,2(z1,z2)=dz1dz2(z1−z2)2)
the n-form ωg,n=d1…dnFg,n is
Fg,n(z1,…,zn)=22−2g−n∑d1+…+dn=3g−3+n∏i=1n(2di−1)!!zi2di+1⟨τd1…τdn⟩g
where ⟨τd1…τdn⟩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 (ℂ; ℂ; x:z↦−z+ln⁡z; ω0,1(z)=(1−z)dz;ω0,2(z1,z2)=dz1dz2(z1−z2)2)
the n-form ωg,n=d1…dnFg,n is
Fg,n(z1,…,zn)=∑ℓ(μ)≤nmμ(ex(z1),…,ex(zn))hg,μ1,…,μn
where hg,μ is the connected simple Hurwitz number of genus g with ramification μ=(μ1,…,μn): 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 μ.

### Gromov–Witten numbers and the BKMP conjecture

Let 𝔛 a toric Calabi–Yau 3-fold, with Kähler moduli t1,…,tb2(𝔛). Its mirror manifold is singular over a complex plane curve Σ given by a polynomial equation P(ex,ey)=0, whose coefficients are functions of the Kähler moduli. For the choice of spectral curve (Σ; ℂ*; x; ω0,1=ydx;ω0,2) with ω0,2 the fundamental second kind differential on Σ,
According to the BKMP[5] conjecture, the n-form ωg,n=d1…dnFg,n is
Fg,n(z1,…,zn)=∑𝐝∈H2(𝔛,ℤ)∑μ1,…,μn∈H1(ℒ,ℤ)td∏i=1nex(zi)𝒩g(𝔛,ℒ;𝐝,μ1,…,μn)
where 𝒩g(𝔛,ℒ;𝐝,μ1,…,μn)=∫[ℳ‾g,n(𝔛,ℒ,𝐝,μ1,…,μn)]vir1
is the genus g Gromov–Witten number, representing the number of holomorphic maps of a surface of genus g into 𝔛, with n boundaries mapped to a special Lagrangian submanifold ℒ. 𝐝=(d1,…,db2(𝔛)) is the 2nd relative homology class of the surface's image, and μi∈H1(ℒ,ℤ) are homology classes (winding number) of the boundary images.
The BKMP[5] conjecture has since then been proven.

## Notes

1. ↑ 1.0 1.1 1.2 Invariants of algebraic curves and topological expansion, B. Eynard, N. Orantin, math-ph/0702045, ccsd-hal-00130963, Communications in Number Theory and Physics, Vol 1, Number 2, p347-452.
2. ↑ B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, JHEP/024A/0904, hep-th/0407261 A short overview of the ”Topological recursion”, math-ph/arXiv:1412.3286
3. ↑ A. Alexandrov, A. Mironov, A. Morozov, Solving Virasoro Constraints in Matrix Models, Fortsch.Phys.53:512-521,2005, arXiv:hep-th/0412205
4. ↑ L. Chekhov, B. Eynard, N. Orantin, Free energy topological expansion for the 2-matrix model, JHEP 0612 (2006) 053, math-ph/0603003
5. ↑ 5.0 5.1 5.2 Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178,2009
6. ↑ P. Dunin-Barkowski, N. Orantin, S. Shadrin, L. Spitz, "Identification of the Givental formula with the spectral curve topological recursion procedure", Commun.Math.Phys. 328 (2014) 669-700.
7. ↑ V. Bouchard, B. Eynard, "Think globally, compute locally", JHEP02(2013)143.
8. ↑ B. Eynard, Invariants of spectral curves and intersection theory of moduli spaces of complex curves, math-ph: arxiv.1110.2949, Journal Communications in Number Theory and Physics, Volume 8, Number 3.

## References

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1. ↑ O. Dumitrescu and M. Mulase, Lectures on the topological recursion for Higgs bindles and quantum curves, https://www.math.ucdavis.edu/~mulase/texfiles/OMLectures.pdf

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