Eynard-Orantin Topological Recursion: A Brief Surveynjhlai/research/papers/toprecursion.pdf · qÑh...

Eynard-Orantin Topological Recursion: A Brief Survey

Nicholas Jian Hao Lai

University of WaterlooWaterloo, Ontario Canada

December 22, 2015

Abstract: In this article, we aim to give a brief survey of the theory of Eynard-Orantin topologicalrecursion, from its original formulation in random matrix theory to its current state of the art.

Contents

1 Prerequiste: Geometry of Riemann Surface 2

2 Random Matrix Theory: A Brief Introduction 3

2.1 Random Matrix in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Measure-Theoretic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Gaussian Ensembles and Its Probabilty Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Spectral Distribution and Equilibrium Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5 Formal Matrix Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Random Matrix Formulation of Eynard-Orantin Topological Recursion 8

3.1 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Connected n-Point Functions and Its Loop Equation . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Large N Limit of Connected n-Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Topological Expansion of Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.6 Spectral Curve of Matrix Model and Associated Differential Forms . . . . . . . . . . . . . . . . 13

3.7 Topological Recursion of Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Eynard-Orantin Topological Recursion 15

4.1 Spectral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Bergmann Kernel and Recursion Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Definition of Topological Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Applications of Topological Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 A Combinatorial View of Topological Recursion 18

5.1 Diagramatic Interpretation For Random Matrix Topological Recursion . . . . . . . . . . . . . . 18

5.2 Diagramatic Interpretation of Eynard-Orantin Topological Recursion . . . . . . . . . . . . . . . 21

6 Open Problems and Future Work 22

1

2 Nicholas Jian Hao Lai

1 Prerequiste: Geometry of Riemann Surface

We begin by recalling the basic definitions of Riemann surfaces.

Definition 1.1. Let M be a connected Hausdorff topological space. A complex holomorphic structure,on the manifold M is the system tpUv, hvqu of pairs pUv, hvq, known as charts, such that tUvu is an opencovering of M and hv is a homeomorphism from Uv to open hvpUvq Ď C satisfying: if Uv X Uµ ‰ H, thenhv ˝ h

´1µ : hµpUv X Uµq Ñ hvpUv X Uµq is holomorphic.

Definition 1.2. A Riemann surface is the pair R “ pM, tpUv, hvquq, where M is a connected Hausdorfftopological space and tpUv, hvqu is a complex holomorphic system on M .

One can equipped a compact Riemann surface Σ of genus g ě 1 with a symplectic basis of 2g non-contractiblecycles with

Ai X B〉 “ δi,j , Ai XAj “ 0, B〉 X B| “ 0

We want to define a differential form as differential forms are the main object of study in the Eynard-Orantintopological recursion theory. Hence, we will quickly recall the definition of a differential form.

Let M be a smooth manifold, and let p P M . Recall that the cotangent space is the dual spaceT˚pM “ pTpMq

˚ of the tangent space TpM . The cotangent bundle of M is defined as

T˚M “ğ

pPM

T˚pM

The smooth map

α : M Ñ T˚M

such that for π : T˚M Ñ M , we have that π ˝ α “ idM is called a 1-form on M . Note that a 1-form has inlocal coordinates the form

α “nÿ

k“1

αk dxk

Let 0 ď k ď n. Then, we denote ΛkpT˚pMq to be the space of k-forms at p, i.e. the space of alternatingmaps

αp : pTpMqk Ñ R

As before, we define the bundle of k-forms as the space

ΛkpT˚Mq “ğ

pPM

ΛkpT˚pMq

A k-form is then a smooth map α : M Ñ ΛkpT˚Mq such that for π : ΛkpT˚Mq ÑM , we have that π ˝α “ idM .Finally, we define a special bilinear meromorphic form.

Definition 1.3. Let Σ be a Riemann surface. The Bergmann kernel Bpp, qq is a bilinear meromorphic formon Σ with one double pole at p “ q and no other pole, and normalised such that

Bpp, qq „pÑq

dzppqdzpqq

pzppq ´ zpqqq2` fpp, qqdzppqdzpqq,

ż

AiB “ 0

where fpp, qq is regular and z is a local coordinate at q.

Note that the Bergmann kernel depends only on the complex structure of Σ. Further, Bergmann kernel isclearly unique, as the difference of two Bergmann kernel has no pole and vanishes at Ai cycle integrals, and soit is 0.

Eynard-Orantin Topological Recursion: A Brief Survey 3

2 Random Matrix Theory: A Brief Introduction

Random matrix theory plays an essential part in the study of the Eynard-Orantin topological recursion.Indeed, it is through the study of random matrices that Eynard and Orantin discovered a mysterious yetbeautiful universal recursion formula which dictate the behaviour of correlation functions of random matrixlaw. Motivated by its universality, Eynard and Orantin developed the so-called Eynard-Orantin topologicalrecursion theory independent of the underlying random matrix formulation. In this section, we will provide abrief introduction of random matrix theory. Our treatment will mainly be measure-theoretic to make thingsmore precise and formal, and we largely follow the treatment by Tao in [Tao12].

2.1 Random Matrix in Statistics

Random matrix theory is the study of matrices whose entries are all random variables, and as such was firstmotivated primarily in the statistics and probability. It first arose in the work of Wishart [Wis28] in his attemptto generalise the theory of univariate random variable to multivariate random variable. Here, we will give abrief overview of its statistical formulation.

Let N, p P N be such that N ě p, and let X be a N ˆ p matrix, i.e.

X ““

xij‰

1ďiďN1ďjďp

where each xij are random variables sampled from the univariate normal distribution with mean 0 and varianceσ2j , i.e. xij „ N p0, σ2

j q. In particular, note that for each 1 ď i ď N , we do not assume any independencebetween the random variables xi1, ¨ ¨ ¨ , xip, so by a basic theory in statistics this implies that each row

Xi ““

xi1 ¨ ¨ ¨ xip‰

„ Npp0, V q

where Npp0, V q is the p-dimensional multivariate distribution with mean 0 and covariance matrix V of sizepˆ p.

We defined the following, which is a basic object in multivariate statistic.

Definition 2.1. Let X be a random matrix as defined above. The correlation matrix of X is the matrix

M “ XTX

Note that the correlation matrix M is a real symmetric matrix of size pˆ p, which follows via some routinelinear algebraic calculation. Another way to view this is to realise that M´1 can be proved to be a scaledcovariance matrix of X, which is usually proven in a course in statistics. Hence intuitively we will like M to bea real symmsetric matrix, and our intuition is supported by the linear algebraic calculation.

The probability law for the correlation matrix M is given by the Wishart distribution

P pMqdM “1

2Np2 pdetV q

N2 Γp

`

N2

˘

pdetMqN´p´1

2 e12 TrV ´1M dM

Ginibre then generalised the result to include other matrix ensemble in 1960, to include complex not necessarilyHermitian matrices.

2.2 Measure-Theoretic Formulation

For our purpose, we require a more formal formulation of random matrix models. As such, we will utilise themeasure-theoretic formulation of random matrix models here, as well as specifying the notation we will adoptfor the rest of this paper.

We start by recalling the following basic definition in probability theory.

Definition 2.2. A probability space is a measure space pΩ,Fq with a measure

P : F Ñ r0, 1s

known as the probability measure, where Ω is known as the sample space and F is a σ-algebra of subsetsof Ω whose elements are known as events. is a measure known as the probability measure.


We will usually denote a probabilty space by pΩ,F , P q.A basic object in statistic is the notion of a random variable. We can define a random variable in this

measure-theoretic framework as follows.

Definition 2.3. Let pΩ,F , P q be a probability space, and let pR,Rq be a measurable space. A pppR,Rqqq-valuedrandom variable is a FL

R -measurable function

X : pΩ,F , P q Ñ pR,Rq

i.e., for each S P R, X´1pSq P F .

We will usually denote R “ pR,Rq when the underlying σ-algebra is clear, and speak of a R-valued randomvariable.

To make the connection to the usual notation found in probability and statistics, we will adopt the followingconvention. Namely, we will adopt the convention that for a R-valued random variable X and for S P R,

P pX P Sq :“ P pt ω P Ω; Xpωq P S uq “ P pX´1pSqq

Given a R-valued random variable X, in probability and statistic we are interested in the distributionassociated to X. As such, we will need a definition of this concept in the measure-theoretic framework.

Definition 2.4. A distribution of X is the probabilty measure µX on R such that for all S P R

µXpSq “ P pX P Sq “ P pX´1pSqq

Again, if the underlying random variable X is clear, we will write µ “ µX . Now, let dm be a measure on pR,R.If the random variable X is absolutely continuous, i.e. for all null sets S P R,

P pX P Sq “ 0

then, by Radon-Nikodym Theorem, there exists a non-negative, absolutely integrable function f P L1pR, dmqwith

ż

R

f dm “ 1

such that for all S P R,

µXpSq “

ż

S

f dm ùñ dµX “ f dm

Finally, we will define an important statistical value associated to a random variable.

Definition 2.5. Let pΩ,F , P q be a probabilty space, and let R “ pR,Rq be a measurable space. Let X be aR-valued random variable. The expectation of X is the quantity

ErXs “

ż

R

r dµXprq

A fundamental property of the expectation of X is that it is linear, which easily follows by the linearity of theintegration operator. More non-trivially, one can extend the definition of the expection to incorporate somewell-behaved transformation of random variables. In particular, for F P L1pR,dµXq, we define

ErF pXqs “

ż

R

F prqdµXprq

Some expectations of transformations of random variable of particular interest in probabilty and statisticinclude the moments

ErXks “

ż

R

rk dµXprq

where k ě 1 and the exponential moment

EretX s “

ż

R

etr dµXprq

In this paper, we are specifically interested in matrix-valued random variables, which we define as follows:


Definition 2.6. Let pΩ,F , P q be a probabilty space. Let pY, t Y XB; B P B uq be a subspace of the measurablespace pMNˆppFq,Bq, where B is the usual Borel σ-algebra of MNˆppFq. A Y -valued random matrix is aY -valued random variable M .

In this note, we shall only consider the space of square matrices, MNˆN pFq. Finally, we can define the randommatrix model as follows:

Definition 2.7. A random matrix model, or a matrix ensemble, is a pair E “ pM,dµq, where M is arandom matrix and dµ is a distribution.

In the literature, the random matrix model is usually known as the matrix ensemble, and so in this paper wewill use both terminology interchangeably.

2.3 Gaussian Ensembles and Its Probabilty Measures

There are many matrix ensembles in the literature. Here, we list four matrix ensembles which are studiedextensively.

1. The iid matrix ensembles, in which each Mij are iid random variables with a common distribution.

2. The symmetric Wigner matrix ensembles, in which M is a SN -valued random matrix, where SN isthe set of symmetric real N ˆN matrices. Each Mij for j ě i are jointly independent and real.

3. The complex Hermitian Wigner matrix ensembles, in which M is a HN -valued random matrix,where HN is the set of Hermitian N ˆN matrices. Each Mij for j ě i are joinly independent, with Mii

being real.

4. The quaternionic Hermitian Wigner matrix ensembles, in which M is a QN -valued random matrix,where QN is the set of quaternionic Hermitian N ˆN matrices. Each Mij for j ě i are joinly independent,with Mii being real.

In this note, we will be focusing on the symmetric and Hermitian Wigner matrix ensembles. In particular,we will be focusing on the Gaussian ensembles, i.e. the Gaussian Othorgonal Ensemble (GOE), the GaussianUnitary Ensemble (GUE), and the Gaussian Symplectic Ensemble (GSE), which are special cases of thesymmetric, Hermitian and quaternionic Hermitian Wigner matrix ensembles respectively. Indeed, we can unifythe treatment of both GOE and GUE. It turns out for our purpose, this is sufficient to study the symmetricand Hermitian random matrices, regardless of the probabilty measure endowed on it. We will discuss this inthe next section.

As the name suggested, GOE, GUE and GSE are matrix ensembles which has as its probability measurethe simple Gaussian measure. That is, if µ is the simple Gaussian measure of the matrix ensemble, then

dµpMq “1

Ze´

n2 TrpM2

q dM

where dM is the usual Lesbegue measure Z is the normalisation constant associated to the class of matrices inconsideration.

But we can simplify the probability measure further. Now, note that each of the three classes of matricescan be decomposed as

M “ UΛ´1U´1

where Λ “ diagpλ1, ¨ ¨ ¨ , λN q P RN , and

1. U is orthgonal if M is symmetric

2. U is unitary if M is Hermitian

3. U is sympletic if M is quaternionic Hermitian


Let us write

MβN “

$

’

&

’

%

SN , if β “ 1

HN , if β “ 2

QN , if β “ 4

, UβN “

$

’

&

’

%

ON , if β “ 1

UN , if β “ 2

SpN , if β “ 4

Let EβN be the subset of EβNThen, with a little basic linear algebra we see the decomposition

EβN –

´

UβNM

T pUβN q ˆ RN¯

N

WeylpUβN q

where T pUβN q is the maximal torus of UβN and WeylpUβN q is the Weyl group of UβN .Now, with the decomposition above, the Lebesgue measure can be written as

dM “ dλ1 ¨ ¨ ¨ dλNdνpUβN q

where dνpUβN q is the probabilty measure on UβN . But note that UβN is a compact Lie group, so it has a uniqueHaar measure. Thus, we can write it more explicitly as

dM “

ˇ

ˇ

ˇ

ˇ

ˇ

ź

1ďiăjďN

pλi ´ λjq

ˇ

ˇ

ˇ

ˇ

ˇ

β

dλ1 ¨ ¨ ¨ dλNdUβN

where dUβN is the Haar measure on UβN . Hence

dµpMq “1

Ze´

n2 TrpM2

q dM “1

Z

ˇ

ˇ

ˇ

ˇ

ˇ

ź

1ďiăjďN

pλi ´ λjq

ˇ

ˇ

ˇ

ˇ

ˇ

β

e´n2

řNi“1 λ

2i dλ1 ¨ ¨ ¨ dλNdUβN

As a final remark, it is easy to see how we can extend the above analysis to include any probability measureof the form

dµpMq “ e´TrpV pMqq dM “1

Z

ˇ

ˇ

ˇ

ˇ

ˇ

ź

1ďiăjďN

pλi ´ λjq

ˇ

ˇ

ˇ

ˇ

ˇ

β

e´TrpV pdiagpλ1,¨¨¨ ,λN qqq dλ1 ¨ ¨ ¨ dλNdUβN

where V is any function on M , called the potential.

2.4 Spectral Distribution and Equilibrium Measure

With a matrix ensemble E, there are many different statistics which are of interest, such as the density ofeigenvalues and the operator norm of a matrix. For our purposes, we are interested in the spectral distributionof the matrix ensemble.

Let MβN be as defined in the previous subsection. We define the following.

Definition 2.8. Let MN P MβN , and let λ1 ď λ2 ď ¨ ¨ ¨ ď λN be its complete set of eigenvalues counting

multiplicities. The normalised empirical spectral distribution (ESD) is the probability measure

µ 1?NMN

“1

N

Nÿ

i“1

δ λi?N

where δx is a Dirac measure.

When EN “ pMN , µq is a matrix ensemble, then the map

MN ÞÑ µ 1?NMN


becomes a random measure. That is, the map is now a PpRq-valued random variable, where PpRq is the spaceof probability measures on the real line. By abuse of notation we will call this map the ESD. Note that thismeans that a distribution of the ESD is then a probability measure on probability measures!

Now, let pEβN “ pMβN , µqqNě1 be a sequence of matrix ensemble, where β P t1, 2, 4u. It turns out that if

for every ϕ P CpRq, we have thatż

Rϕdµ 1?

NMβN

converge in probability toż

Rϕdµ

where µ P PpRq, then the sequence of ESD´

µ 1?NMβN

¯

Ně1converges in probability to µ. The same holds

true for almost surely convergence. It is well known that almost surely convergence implies convergence inprobability. We thus have the following definition.

Definition 2.9. The almost convergence limit µ “ limNÑ8 µ 1?NMβN

is called the equilibrium measure for

the sequence of matrix ensembles pEβN “ pMβN , µqqNě1.

In particular, if ρpxq is the probability density function of the equilibrium measure, succintly known as theequilibrium density, then

ρpxqdx “ dµ “ limNÑ8

dµ 1?NMβNùñ ρpxq “

C

1

N

Nÿ

i“1

δ λi?N

pxq

G

where x¨y is the expectation operator. That is

x¨y “

ż

MβN

p¨qe´TrpV pMqq dM

ż

MβN

e´TrpV pMqq dM

We thus define the partition function to be

Z “

ż

MβN

e´TrpV pMqq dM

A surprising fact is that the equilibrium measure is actually not random, that it is actually a deterministicmeasure.

A very important result in the study of Wigner matrix ensembles is the Wigner semicircular law, which wequote here.

Wigner Semicircular Law. Let pMN , µq be a sequence of Wigner matrix ensembles. Then, the ESD µ 1?NMN

converges almost surely to the Wigner semicircular distribution

1

2π

a

4´ |x|2 dx

For proof, c.f. for example [TAO].

2.5 Formal Matrix Integrals

The last ingredient we need from random matrix theory is the notion of a formal matrix theory, to endow somecombinatorial structure to random matrix models. We view the potential as

V pMq “1

2M2 ´

ÿ

kě3

tkkMk


and so for our purpose, we view non-Gaussian measures as pertubations of the Gaussian measure.

Let us define the following.

Definition 2.10. A formal matrix integral is the formal power series

ż

formal

e´NTrpV pMqq dM “ Zÿ

ně0

ÿ

k1,¨¨¨ ,kn

1

pk1 ` ¨ ¨ ¨ ` knq!

C

nź

m“1

NTrpMkmq

kmtkm

G

Note that the formal matrix integral is usually not the same as the usual matrix integral. Formal matrixintegral has a nice intepretation as the generating function of ribbon graphs, which we will discuss in Section 5.For more details in formal matrix integrals, see for example [Eyn06].

3 Random Matrix Formulation of Eynard-OrantinTopological Recursion

One of the major focus of random matrix theory is following: What is the asymptotic expansion of thecorrelation function of a given matrix ensemble? This is the question that is addressed by Eynard-Orantintheory of topological recursion. Briefly, Eynard and Orantin observed that the knowledge of the equilibriummeasure is sufficient to recover the asymptotic expansion of the correlation functions. Moreover, this expansionis universal, in the sense we will define later. The remarkable observation made by Eynard and Orantin is thatin many enumerative geometric question, such a phenomenon is observed as well, leading them to formulatetheir theory of topological recursion.

In this section, we will detail the random matrix formulation of the Eynard-Orantin topological recursion.We are mainly following the arguments outlined by in Eynard, Kimura and Ribault’s [??], as well Eynard’s[Eyn14].

3.1 Universality

We define the following.

Definition 3.1. Let E “ pM,µq be a matrix ensemble. A property of a matrix ensemble is universal if itdoes not depend on µ.

Universality is extremely powerful in random matrix theory, because a universal property of a matrixensemble is invariant under change of probability measure. This implies that for us to study a universal propertyof a matrix ensemble, we need only study the property under a simple probability measure. For example, wecan study the property under Gaussian measure, since they are easier to understand and to work with. Aprominent example is the Lindebery replacement trick, which exploits the universality in proving results likethe central limit theorem.

For our purpose, it turns out that large N limits exhibits universality under appropriate assumptions onthe large N behaviour of the potential V . For example, consider the partition function, defined as

Z “

ż

MN

dµpMq “

ż

M

e´TrpV pMqq dM

where MN is a N ˆN random matrix. Then, if V pxq has a finite large N limit, Z have a non-trivial large Nlimit.

Eynard, Kimura and Ribault observed in [??] that matrix models tend to be related to integrable systems.This suggest an important role of integrable systems in random matrix theory, and indeed since topologicalrecursion is closely related to universality, Eynard remarked [??] that integrable system has some currentlyunknown relation with the topological recursion theory.


3.2 Correlation Functions

From here on, we will be performing all computations formally. That is, our matrix integral is formal, and inparticular our partition function is

Z “

ż

formal

eŃTrpV pMqq dM

We will also abuse more notation and denote

xÿ “

ż

MβN ,formal

p¨qe´TrpV pMqq dM

ż

MβN ,formal

e´TrpV pMqq dM

We now define the correlation functions of order k ě 1.

Definition 3.2. Let fkpx1, ¨ ¨ ¨ , xkq be a probability density function. A k-point correlation function is thefunction

Rkpx1, ¨ ¨ ¨ , xkq “N !

pN ´ kq!

ż

Rkfkpx1, ¨ ¨ ¨ , xkqdx1 ¨ ¨ ¨ dxk

In our case, the definition of correlation functions specialises to the spectral distribution UβN , we get thatthe k-point correlation is of form

Nk´2 log

C

kź

i“1

˜

Nÿ

n“1

δ λn?N

pxiq

¸G

but since Trppx´Mq´1q is just the Stieltjes transform of Nρpxq, we get that it has the asymptotic expansiongiven by

Tr

ˆ

1

x´M

˙

“ÿ

kě0

mk

xk`1` o

ˆ

1

xn`1

˙

where mk is the kth moment of Nρpxq. Thus, wxTrppx´Mq´1qy contains information about the spectrum ofeigenvalues. Hence, we will be focusing on this quantity.

3.3 Connected n-Point Functions and Its Loop Equation

We are now interested in computing correlation functions and deduce its universal recursive relation. For moredetailed computation, see for example [??], [EO07] and [EO08].

First, let β “ 2. Consider the moments,@

Tr`

Mk˘D

We require that the potential V is chosen such that

Tr`

Mk˘

eŃTrV pMq

vanishes at the boundary of MβN . Then, we get that

ÿ

1ďi,jďN

ż

UβN

B

BMij

´

pMkqijeŃTrpV pMqq

¯

dM “ 0

By computing the derivate, we get that the 0th loop equation

0 “ÿ

1ďi,jďN

ż

UβN

B

BMij

´

pMkqijeŃTrpV pMqq

¯

dM

“ÿ

1ďi,jďN

ż

UβN

˜

k´1ÿ

l“0

pM lqiipMk´1´lqjj `NpM

kqijV1pMqji

¸

eŃTrpV pMqq dM

“

k´1ÿ

l“0

ÿ

1ďi,jďN

ż

UβN

pM lqiipMk´1´lqjje

ŃTrpV pMqq dM `ÿ

1ďi,jďN

ż

UβN

NpMkqijV1pMqjie

ŃTrpV pMqq dM


ùñ N@

TrpV 1pMqMkqD

“

k´1ÿ

l“0

@

TrpM lqTrpMk´1´lqD

In general, one can derive the pppn ´ 2qqq-th loop equation

N

C

TrpV 1pMqMµ1q

nź

i“2

TrMµi

G

“

µ1´1ÿ

l“0

C

TrpM lqTrpMµ1´l´1q

nź

i“2

TrpMµiq

G

`

nÿ

j“2

µj

C

TrpMµ1`µj´1qź

i“2i‰j

TrpMµiq

G

In Section 5, we will demonstrate a combinatorial and geometrical interpretation of the loop equation.To simplify the presentation of these loop equations, one can encode all these correlation functions into

generating functions. Recall that

Tr

ˆ

1

x´M

˙

“ÿ

µě0

x´µ´1TrpMµq

We can therefore define the following.

Definition 3.3. The connected n-point function is

Wnpx1, ¨ ¨ ¨ , xnq “ log

B

Tr

ˆ

1

x1 ´M

˙

¨ ¨ ¨Tr

ˆ

1

xn ´M

˙F

The disconnected n-point function is

yWnpx1, ¨ ¨ ¨ , xnq “

B

Tr

ˆ

1

x1 ´M

˙

¨ ¨ ¨Tr

ˆ

1

xn ´M

˙F

Denote

Pnpx;x1, ¨ ¨ ¨ , xnq “ log

C

Tr

ˆ

V 1pxq ´ V 1pMq

x´M

˙ nź

i“1

Tr

ˆ

1

xi ´M

˙

G

xPnpx;x1, ¨ ¨ ¨ , xnq “

C

Tr

ˆ

V 1pxq ´ V 1pMq

x´M

˙ nź

i“1

Tr

ˆ

1

xi ´M

˙

G

which are polynomials in x.

With this notation, we can sumarise the 0th loop equation as

B

Tr

ˆ

1

x´M

˙

Tr

ˆ

1

x´M

˙F

“ÿ

kě1

1

xk`1

k´1ÿ

l“0

@

TrpM lqTrpMk´1´lqD

“ÿ

kě1

N

xk`1

@

TrpV 1pMqMkqD

“ N

B

Tr

ˆ

V 1pMq

x´M

˙F

By noting that

W1pxq “ xW1pxq, P0pxq “ xP0pxq, xW2px1, x2q “W1px1qW2px2q `W2px1, x2q

we get the more compact form of the 0th loop equation

W2px, xq `W1pxq2 “ xW2px, xq “

B

Tr

ˆ

1

x´M

˙

Tr

ˆ

1

x´M

˙F

“ N

B

Tr

ˆ

V 1pMq

x´M

˙F

“ N

ˆB

Tr

ˆ

V 1pxq

x´M

˙F

´

B

Tr

ˆ

V 1pxq ´ V 1pMq

x´M

˙F˙

“ N

ˆ

V 1pxq

B

Tr

ˆ

1

x´M

˙F

´

B

Tr

ˆ

V 1pxq ´ V 1pMq

x´M

˙F˙

“ N`

V 1pxqW1pxq ´ P0pxq˘


In general, higher loop equations under similar calculations can be sumarised into

N´

V 1pxqWn`1px, x1, ¨ ¨ ¨ , xnq ´ xPnpx;x1, ¨ ¨ ¨ , xnq¯

“ Wn`2px, x, x1, ¨ ¨ ¨ , xnq `nÿ

i“1

B

Bxi

yWnpx, x1, ¨ ¨ ¨ , pxi, ¨ ¨ ¨ , xnq ´yWnpx1, ¨ ¨ ¨ , xnq

x´ xi

Let I “ t x1, ¨ ¨ ¨ , xn u. By rewriting the above in terms of the connected correlation functions, we get

N`

V 1pxqWn`1px, Iq ´ Pnpx; Iq˘

“Wn`2px, x, Iq`ÿ

JĎI

W1`|J|px, JqW1`|IzJ|px, IzJq`nÿ

i“1

B

Bxi

Wnpx, Iztxiuq ´WnpIq

x´ xi

This expression can now be easily seen to be universal, as its expression is independent of the probabilitymeasure. In particular, it is hidden away by the connected n-point functions.

For β P t1, 24u, we can factor out β2 from the potential function for convenience. In particular, the

probability measure is written as

e´N2 βTrpV pMqq

and so the nth loop equations become

N

C

TrpV 1pMqMµ1q

nź

i“2

TrMµi

G

“

µ1´1ÿ

l“0

C


nź

i“2

TrpMµiq

G

`2

β

nÿ

j“2

µj

C

TrpMµ1`µj´1qź

i“2i‰j

TrpMµiq

G

`

ˆ

2

β´ 1

˙

µ1

C


nź

i“2

TrpMµiq

G

For connected n-point function, for convenience we will scale it by´

β2

¯n2

, i.e.

Wnpx1, ¨ ¨ ¨ , xnq “

ˆ

β

2

˙n2

C

nź

i“1

Tr

ˆ

1

xi ´M

˙

G

c

Then, the recursion satisfied by the connected n-point function is

N

c

β

2

`


“Wn`2px, x, Iq `ÿ

JĎI

W1`|J|px, JqW1`|IzJ|px, IzJq `nÿ

i“1

B

Bxi


x´ xi

`

˜

c

β

2´

c

2

β

¸

B

BxWn`1px, Iq

Again, this expression is universal.

3.4 Large N Limit of Connected n-Point Functions

Each connected n-point functions is related to the connected correlation function of the ESD µ 1?NMβN

via

Stieltjes transform. A natural question to ask is what is the large N limit of the connected n-point functions?It turns out that it is also related to the connected correlation function of the equilibrium measure due toStieltjes transform.

We thus define the following:

Definition 3.4. The large N limit of the connected n-point functions is

ĄWnpx1, ¨ ¨ ¨ , xnq “ limNÑ8

Nn´2Wnpx1, ¨ ¨ ¨ , xnq

and similarly, the large N limit of Pn is

ĂPnpx;x1, ¨ ¨ ¨ , xnq “ limNÑ8

Nn´1Pnpx;x1, ¨ ¨ ¨ , xnq


Utilising the result in Section 3.3, the nth loop equation is

c

β

2

´

V 1pxqČWn`1px, Iq ´ ĂPnpx; Iq¯

“ limNÑ8

Nn´2N

c

β

2

`


“ limNÑ8

Nn´2

˜

Wn`2px, x, Iq `ÿ

JĎI

W1`|J|px, JqW1`|IzJ|px, IzJq

`

nÿ

i“1

B

Bxi


x´ xi`

˜

c

β

2´

c

2

β

¸

B

BxWn`1px, Iq

¸

“ limNÑ8

˜

1

N2ČWn`2px, x, Iq `

ÿ

JĎI

ČW1`|J|px, Jq ČW1`|IzJ|px, IzJq

`

nÿ

i“1

B

Bxi

ĄWnpx, Iztxiuq ´ĄWnpIq

x´ xi`

1

N

˜

c

β

2´

c

2

β

¸

B

BxČWn`1px, Iq

¸

“ÿ

JĎI

ČW1`|J|px, Jq ČW1`|IzJ|px, IzJq `nÿ

i“1

B

Bxi

ĄWnpx, Iztxiuq ´ĄWnpIq

x´ xi

which again, is universal.

3.5 Topological Expansion of Correlation Functions

Consider the connected n-point functions Wn. Since the partition function is a formal matrix integral, Wn

will be a formal power series in N . Indeed, the coefficients are polynomial moments of Gaussian integrals. Wewrite the topological expansion of Wn as

Wnpx1, ¨ ¨ ¨ , xnq “ÿ

ghě0

ˆ

N

t

˙2´2g´n

Wg,npx1, ¨ ¨ ¨ , xnq

Our curious method of indexing is due to the geometrical interpretation, which we will discuss in Section 5.This is also why the expression above is the topological expansion of Wn, as it is the expansion whereby thecoefficients denote the graphical, and hence topological, breakdown of Wn. See Section 5 for more details.

We may also define similarly the topological expansion of Pn,

Pnpx;x1, ¨ ¨ ¨ , xnq “ÿ

gě0

ˆ

N

t

˙1´2g´n

Pg,npx;x1, ¨ ¨ ¨ , xnq

By comparing coefficients, it is immediate that Wg,n and Pg,n satisfy the universal recursion

Wg´1,n`2px, x, x1, ¨ ¨ ¨ , xnq `ÿ

h`h1“gIYJ“tx1,¨¨¨ ,xnu

Wh,1`|I|px, IqWh1,1`|J|px, Jq

“

c

β

2

`

V 1pxqWg,n`1px, x1, ¨ ¨ ¨ , xnq ´ Pg,npx, x1, ¨ ¨ ¨ , xnqdxdx1 ¨ ¨ ¨ dxn˘

dx

Finally, we can extend the definition of Wn to n “ 0, by defining Wg,0 to be the coefficient in the topologicalexpansion of

W0 “ logZ “ÿ

gě0

N2´2gWg,0

Now, the loop equations in Section 3.3 is not effective, and in particular the loop equation for Wn is noteffective. That is, in order to compute W0, one needs W1, which in turn needs W2, and so on. However, byconsidering a certain transformation of the coefficients Wg,n of the topological expansion of Wn, we do indeed


obtain a universal recursion expression which is in fact effective. The computations required forms the randommatrix formulation of the topological recursion, which is the subject of Section 3.7.

Here, we will instead demonstrate the computation for a few small values of g and n. In particular, we willdemonstrate the computation for pg, nq “ p0, 1q, p0, 2q and p1, 1q.

Computation of W0,1

Consider the 0th loop equationÿ

h`h1“g

Wh,1pxqWh1,1pxq `Wg´1,2px, xq “ V 1pxqWg,1pxq ´ Pg,0pxq

Now, note that for g “ 0, the loop equation simplies further, and we get

W0,1pxq2 “ V 1pxqW0,1pxq ´ P0,0pxq

This leads to the well-known result

W0,1pxq “1

2

´

V 1pxq ´Mpxqa

σpxq¯

where Mpxq and σpxq are such thatpV 1q2 ´ 4P0,0 “M2σ

with σ having only simple roots.

Computation of W0,2

Consider the 1st loop equation

2W0,2px, x1qW0,1pxq `

B

Bx1W0,1pxq ´W0,1px

1q

x´ x1“ V 1pxqW0,2px, x

1q ´ P0,1px;x1q

Since we know W0,1, we can compute W0,2, and we get

W0,2 “

BBx1

W0,1pxq´W0,1px1q

x´x1 ` P0,1px;x1q

V 1pxq ´ 2W0,1pxq“

1

2px´ x1q2`

BBx1

Mpx1q?σpx1q

2px´x1q ` BBx1

V 1pxq´V 1px1q2px´x1q ` P0,1px;x1q

V 1pxq ´ 2W0,1pxq

Computation of W1,1

Consider the 0th loop equation

2W0,1pxqW1,1pxq `W0,2px, xq “ V 1pxqW1,1pxq ´ P1,0pxq

and so the solution becomes

W1,1pxq “W0,2px, xq ` P1,0pxq

Mpxq?σx

3.6 Spectral Curve of Matrix Model and Associated Differential Forms

The final ingredient to the topological recursion is the notion of spectral curve and some associated differentialforms on the spectral curve. Specifically, we will like to attach a geometric object to the equilibrium densityand employ geometric methods to encode the correlation functions and to model its recursion relation as theresult of a cut-and-join type operation on the geometric object.

We first need to attach a geometric object to the equilibrium measure. Now, be noting that in the case ofn “ 1, the 0th loop equation in Section 3.5 specialises to

W0,12pxq “ V 1pxqW0,1pxq ´ P0,0pxq

We will define the geometric object such that it has the 0th loop equation built into it.


Definition 3.5. Let ρpxq be the equilibrium density of a sequence of matrix ensemble. The spectral curveassociated to ρpxq is the quadruple S “ pΣ, x, yq, where Σ is a Riemann surface and x, y : Σ Ñ C, such that forall z P Σ,

ypzq2 “ V 1pxpzqqypzq ` P0pxpzqq

It turns out if the potential V pxq is a rational function, then the equilibrium density ρpxq has compact support,and so its spectral curve is a compact Riemann surface. Also note that by definition

ypzq “W0,1pxq

An interesting point to note, which feature more prominently in Section 4, is the local behaviour of xpzqnear a branch point. Now, if dx has simple zero at a branch point, then x is of form

xpzq “ xpaq `O`

pz ´ aq2˘

Thus, x is a double-sheeted cover over a, and that there exists a nontrivial local map σapzq such thatxpσapzqq “ xpzq. This involution is known as the local Galois involution near a, and it exchanges two sheetsof Σ.

Now, we will encode the correlation functions into the spectral curve, by building the differential forms

ωg,npz1, ¨ ¨ ¨ , znq “ ωg,npxpz1q, ¨ ¨ ¨ , xpznqq “

ˆ

Wg,npxpz1q, ¨ ¨ ¨ , xpznqq `δn,2δg,0

pxpz1q ´ xpz2qq2

˙

dxpz1q ¨ ¨ ¨dxpznq

Note that in the case of W0,2, we get that

ω0,2pz1, z2q “ Bpz1, z2q

where B is the Bergmann kernel of Σ. In particular, this implies that ω0,2 depends only on the Riemann surfaceΣ, and hence is universal.

3.7 Topological Recursion of Correlation Functions

We will now study the recursion formulae defined in Section 3.5. Specificially, we want to find an effective formof the loop equation. For this, we need to perform a transformation on Wg,n into a form which does have aneffective recursion formulae.

Let S “ pΣ, x, yq be the associated spectral curve. By considering the differential forms ωg,n, the loopequations in Section 3.5 becomes

ωg´1,n`2pz, σapzq, z1, ¨ ¨ ¨ , znq `ÿ

h`h1“gIYJ“tx1,¨¨¨ ,xnu

ωh,1`|I|pz, Iqωh1,1`|J|pσapzq, Jq

“

c

β

2

`

V 1pxpzqqωg,n`1pz, z1, ¨ ¨ ¨ , znq ´ Pg,npxpzq, xpz1q, ¨ ¨ ¨ , xpznqqdxpzqdxpz1q ¨ ¨ ¨dxpznq˘

dxpzq

´

nÿ

j“1

d

dxpzjq

ωg,npz1, ¨ ¨ ¨ , znq

pxpzq ´ xpzjqqdxpzjqdxpzq2

One can then show that

ωg,npz1, ¨ ¨ ¨ , znq “ÿ

a branch point

ReszÑaKapz1, zq

¨

˚

˝

ωg´1,n`1pz, σapzq, z2, ¨ ¨ ¨ , znq

“ `

stableÿ

h`h1“gIYI1“t z2,¨¨¨ ,zn u

ωh,1`|I|pz, Iqωh1,1`|I1|pσapzq, I1q

˛

‹

‹

‚


whereřstable

indicates a sum ignoring ph, Iq “ p0,Hq and pg, t z1, ¨ ¨ ¨ , zn uq, and

Kapz1, zq “1

2

ż z

σapzq

ω0,2pz1, ¨q

ω0,1pzq ´ ω0,1pσapzqq

is the recursion kernel. We will discuss this in detail later in Section 4.The recursion relation defined above is the topological recursion. This result can be extended to any

spectral curve, and is the starting point of the Eynard-Orantin topological recursion.

4 Eynard-Orantin Topological Recursion

Eynard and Orantin’s work in random matrix, which culminated to the result in Section 3.7 in 2007 [EO07]and 2008 [EO08], is a very beautiful result. As physicists their goal was to figure out what their result canyield. It was not immediately obvious: while they have used it solve some combinatorial problem in physics,it was usually applied to problems for which solutions are known. In most cases, topological recursion mayunearth some interesting insights, but the essence of what topological recursion reveals are usually known.

The work of Eynard and Orantin begin to draw attention from physicts, especially string theorists. Indeed,Bouchard, Klemm, Marino and Pasquetti conjectured in [BKMP09] as part of the BKMP Conjecture thattopological recursion allows computation of various geometric quantities, such as the Hurwitz numbers, theGromov-Witten invariants and knot invariants. As such, topological recursion began to gain attention inenumerative geometry and knot theory, as a potentially powerful tool to discover new insights.

In this section, we will finally present Eynard-Orantin theory of topological recursion. This is essentially ageneralisation, or rather an abstraction, of the results in Section 3.7 to a general spectral curve and certainfamilies of its differential forms, which under suitable condition has a surprising amount of structure.

4.1 Spectral Curves

We will now generalise the definition of spectral curve that was introduced in Section 3.6.

Definition 4.1. A spectral curve is a pair S “ pΣ, πq, where Σ is a Riemann surface and

π : z P Σ ÞÑ πpzq “ pxpzq, ypzqq P P

is such that x and y are analytic on some open domain in Σ.

Eynard and Orantin remarked in [EO08] that this is not the usual spectral curve one encounters in the studyof integrable system. In fact, our spectral curve is the classical limit of the spectral curve in integrable system.While topological recursion is closely tied to the study of integrable system, we will not address that in thispaper, and so there will be no confusion in terminology.

The geometry of the spectral curve will be the one inherited from the riemannian geometry on Σ. Inparticular, we inherit the notion of genus, its symplectic basis of 2g non-contractible cycles Ai, its differentialforms and the cotangent sheaf KpΣq, which for our purpose is the canonical sheaf on Σ

There are a few special classes of spectral curve, which exhibit specific relationship between x and y.

Definition 4.2. A spectral curve pΣ, πq is algebraic if Σ is a compact Riemann surface, and π is meromorphic.

An algebraic spectral curve is particularly nice, as there is then a polynomial P such that

P px, yq “ 0

Definition 4.3. An algebraic spectral curve pΣ, πq is rational if Σ “ P is the Riemann sphere.

In the case of a rational spectral curve, the relationship can be parameterised by two rational functions of z.The class of spectral curve we will consider for the remainder of this paper will be the regular spectral

curve, which we define as follows.


Definition 4.4. A spectral curve pΣ, πq is regular if the differential form dx has a finite number of zeroes ai,where each zeroes are simple and are not zeroes of the differential form dy. The zeroes of dx is the branchpoints of the spectral curve.

The branch points of a regular spectral curve are the simplied ramified points of π. As such, locally around thebranch points ai π is a double covering, ramified only at ai. For each z ‰ ai near ai, there is a z ‰ z, calledthe local Galois conjugate of z, near ai such that

xpzq “ xpzq

Further, locally at each branch point ai the curve ypzq behaves like a square root, i.e.

ypzq „ ypaiq ` Cia

xpzq ´ xpaiq

4.2 Bergmann Kernel and Recursion Kernel

Recall the definition of the Bergmann kernel B of Σ from Section 1: It is the unique bilinear differential withdouble pole at z1 “ z2, and has the local form

Bpz1, z2q „z1Ñz2

dz1dz2

pz1 ´ z2q2` fpz1, z2qdz1dz2

where fpz1, z2q is regular.Finally, we define the recursion kernel.

Definition 4.5. Let ai be a branch point. The recursion kernel associated to ai is defined by

Kipz, z1q “1

2

şz

zBp¨, z1q

pypzq ´ ypzqq dxpzq

where z is close to ai and z1 P Σ.

The recursion kernel Ki is defined globally and is meromorphic in z1, but is only defined locally around ai.

4.3 Definition of Topological Recursion

We will finally define the Eynard-Orantin topological recursion for a regular spectral curve.

Definition 4.6. Let pΣ, πq be a regular spectral curve with branch points R “ t a1, ¨ ¨ ¨ , ar u. A familyωg,npz1, ¨ ¨ ¨ , xnq of meromorphic differential forms on Σ, where g ě 0 and n ě 0, is said to satisfy theEynard-Orantin topological recursion if

1. ω0,1pzq P H0pΣ,KpΣqq

2. ω0,2pz1, z2q “ Bpz1, z2q

3. For 2g ´ 2` n ą 0, ωg,n P H0pΣn,Kp˚Rq‘nq is a symmetric meromorphic differential forms with poles

only at R, and satisfy the recursion

ωn`1,gpz0, z1, ¨ ¨ ¨ , znq

“1

2πı

rÿ

j“1

ReszÑajKpz, z0q

¨

˚

˚

˝

ωg´1,n`2pz, z, z1, ¨ ¨ ¨ , znq `stableÿ

h`h1“gIYJ“tz1,¨¨¨ ,znu

ωh,1`|I|pz, Iqωh1,1`|J|pz, Jq

˛

‹

‹

‚

whereřstable

indicates a sum ignoring ph, Iq “ p0,Hq and pg, t z1, ¨ ¨ ¨ , zn uq.


Note that there are some other equivalent definitions of the Eynard-Orantin topological recursion. We willdiscuss one other in Section 5, but c.f. [Eyn14] for more information.

The correlation functions defined in Section 3.5 satisfies the Eynard-Orantin topological recursion, for it isindeed the model for which the general topological recursion is formulated. Many other counting problem inenumerative geometry also satisfy this recursion, as well as Gromov-Witten invariants and knot invariants. Wequote a few examples here.

1. Eynard-Orantin topological recursion is used in the context of the counting problem of Grothendieck’sdessins d’enfants. In this case, the spectral curve is given by

x “ z `1

z

2. The Laplace transform of the single Hurwitz numbers are also known to be satisfy the Eynard-Orantintopological recursion. In this case, the spectral curve is the Lamber curve, given by

x “ ze´1z, y “z

x“ ez´1

As a closing remark to this section, note that Eynard-Orantin topological recursion can be defined for alarger class of spectral curves. In order to deal with spectral curve with higher order branched points, careneeds to be taken with dealing with these branched points, but it turn outs that the result will be a limit ofthe case of the regular spectral curve. For more information, see [BHL`14].

4.4 Applications of Topological Recursion

Now that we have a definition of Eynard-Orantin topological recursion, there are a few natural questions to ask.One of the first questions, and the one which will be addressed here, is how would one use the Eynard-Orantintopological recursion.

Suppose you have a family of coefficients Wg,npµ1, ¨ ¨ ¨ , µnq which are symmetric in µ1, ¨ ¨ ¨ , µn. By viewingeach of Wg,npz1, ¨ ¨ ¨ , znq as a function, theoretically one can build a system

Wg,npz1, ¨ ¨ ¨ , znqdz1 ¨ ¨ ¨ dzn

of differential forms, and test if it satisfy the Eynard-Orantin topological recursion. If the system does satisfythe Eynard-Orantin topological recursion, then we can deduce each higher genera and higher n differentialforms recursively, once we figure out the pg, nq “ p0, 1q and p0, 2q case. However, in practice the determinationof the coefficients will usually be easier than the verification of Eynard-Orantin topological recursion, as it isperceivably hard to deduce a recursive structure without solving the system beforehand. This is usually thecase in all enumerative application of the Eynard-Orantin topological recursion currently known. In all of theknown cases, the problem is usually solved, and the Eynard-Orantin topological recursion is used more as aconfirmation in these cases.

The Eynard-Orantin topological recursion is not without its merits however. At the very least, it reveals adeeper structure of the problem that is possibly not visible before hand. The universality of the Eynard-Orantintopological recursion also implies that to the class which do satisfy the Eynard-Orantin topological recursion,these problems may share a deeper connection, and perhaps possess a deeper invariant which manifest as thetopological recursion.

A form of the Eynard-Orantin topological recursion has also been utilised to simplify the calculation ofthe hyperbolic volume of moduli spaces by Mirzakhani. See [Mir07] for more information. Also, going backto the random matrix formulation, if we have the spectral curve associated to an equilibrium measure, theEynard-Orantin recursion allows one to calculate the correlation functions up to every order, and so one canrecover the random matrix law of the underlying matrix ensemble. Going further, Eynard remarked thatin [Eyn14] if the formulation in reverse, i.e. starting with a spectral curve along with a family of differentialforms which satisfies the Eynard-Orantin topological, under suitable assumptions on the spectral curve one canderive what is known as the pseudo-random matrix law.


Another possible application of the Eynard-Orantin topological recursion is the following. Suppose if bydeducing the solution for the pg, nq “ p0, 1q and p0, 2q case, one can construct the spectral curve. Then, we haveessentially bypass the problem mentioned earlier, and by utilising the Eynard-Orantin topological recursion,one can effectively solve for any pg, nq. Unfortunately, the theory of Eynard-Orantin topological recursion isnot constructive at the current state of the art, as it does not specify how one would construct the spectralcurve or the differential forms that satisfy the Eynard-Orantin topological recursion. In the case of randommatrix theory, the spectral curve is determined by the equilibrium distributiona and the correlation functionsare constructed by performing the Stieltjes transform.

There is, fortunately, a breakthrough. In 2012, Dumitrescu, Mulase, Safnuk, and Sorkin conjecturedin [DMSS12] that if the counting problem for pg, nq “ p0, 1q and p0, 2q makes sense, then the Laplace transformdetermines the spectral curve and the recursion kernel of the Eynard-Orantin formalism. The conjecture is stillvague as it still unclear how to precisely formulate this conjecture, but this conjecture as some hope as Laplacetransform are known to be the way to construct the spectral curve and the recursion kernel in many knowncases. For full details, please see [DMSS12].

5 A Combinatorial View of Topological Recursion

In this section, we will provide some combinatorial interpretation to the Eynard-Orantin topological recursion.For this section, some basic knowledge of ribbon graph is assumed.

5.1 Diagramatic Interpretation For Random Matrix Topological Recursion

Recall the formal matrix integral in Section 2.5

ż

formal

e´NTrpV pMqq dM “ Zÿ

ně0

ÿ

k1,¨¨¨ ,kn

1

pk1 ` ¨ ¨ ¨ ` knq!

C

nź

m“1

NTrpMkmq

kmtkm

G

By a theorem of ’t Hooft, we have that

log

C

nź

m“1

NTrpMkmq

kmtkm

G

“ÿ

GPGpk1,¨¨¨ ,knq

NχpGq

|AutpGq|

nź

i“1

t|k-vertices|ki

where Gpk1, ¨ ¨ ¨ , knq is the collection of connected ribbon graphs with n vertices and vertex degree partitionpk1, ¨ ¨ ¨ , knq, and χpGq is the Euler characteristic of G. One can also view the ribbon graphs dually, whichgives us a punctured Riemann surface, i.e. a Riemann surface with n punctures. Recall that in this case, wecan separate the regions of the Riemann surface into faces centered around the punctures by embedding theribbon graph onto the surface. In this case, the above equation can be rephrased as

log

C

nź

m“1

NTrpMkmq

kmtkm

G

“ÿ

ΣPΣpk1,¨¨¨ ,knq

NχpΣq

|AutpΣq|

nź

i“1

t|k-vertices|ki

where Σpk1, ¨ ¨ ¨ , knq is the collection of n-punctured connected Riemann surface with face degree partitionpk1, ¨ ¨ ¨ , knq, and χpΣq “ 2´ 2g, where g is the genus of Σ. In general,

C

nź

m“1

NTrpMkmq

kmtkm

G

“ÿ


NχpΣq

|AutpΣq|

nź

i“1

t|k-vertices|ki

where Σpk1, ¨ ¨ ¨ , knq is the collection of n-punctured not necessarily connected Riemann surface with facedegree partition pk1, ¨ ¨ ¨ , knq


Figure 5.1: An example of a ribbon graph. Its dual is the separation of regions of the torus by the ribbon graphitself.

ConsiderC

nź

m“1

TrpMkmqtkm

G

“ Nń

˜

nź

i“1

ki

¸C

nź

m“1

NTrpMkmq

kmtkm

G

“ÿ


NχpΣqń

|AutpΣq|

nź

i“1

kit|k-vertices|ki

Combinatorially, the factor pśni“1 kiq shifts the counting to marked surface, where the edge of each face is

marked according to the orientation. The factor Nń on the other hand changes the Euler characteristic of Σto the Euler characteristic of the n-punctured Σ. Similarly, in the case of connected Riemann surface we get

log

C

nź

m“1

TrpMkmqtkm

G

“ÿ


NχpΣqń

|AutpΣq|

nź

i“1

kit|k-vertices|ki

Now, by considering the loop equation

N

C

TrpV 1pMqMµ1q

nź

i“2

TrMµi

G

“

µ1´1ÿ

l“0

C


nź

i“2

TrpMµiq

G

`

nÿ

j“2

µj

C

TrpMµ1`µj´1qź

i“2i‰j

TrpMµiq

G

and by noting that in Section 2.5 the potential is of the form

V pxq “1

2x2 ´

ÿ

kě3

tkkxk ùñ V 1pxq “ x´

ÿ

kě3

tkxk´1

and so the loop equation becomes

N

C

TrpMµ1`1q

nź

i“2

TrMµi

G

“

µ1´1ÿ

l“0

C


nź

i“2

TrpMµiq

G

`

nÿ

j“2

µj

C

TrpMµ1`µj´1qź

i“2i‰j

TrpMµiq

G

`Nÿ

kě3

tk

C

TrpMµ1`k´1q

nź

i“2

TrMµi

G


The left hand side of this equation counts the number of surfaces with n-marked faces of face partitionpµ1 ` 1, µ2, ¨ ¨ ¨ , µnq. On the right hand side, each of the summand counts the number of surface where one ofthe marked edge is delete, and is as follows:

1.@

TrpM lqTrpMµ1´l´1qśni“2 TrpMµiq

D

counts the number of surfaces with n-marked faces of face partitionpl, µ1 ´ l´ 1, µ2, ¨ ¨ ¨ , µnq. This is the case where a marked face got split into to marked faces, and so themarked edge deleted is within a single face.

2. µj

B

TrpMµ1`µj´1qś

i“2i‰j

TrpMµiq

F

counts the number of surfaces with n´1-marked faces of face partition

pl, µ1 ` µj ´ 1, µ2, ¨ ¨ ¨ , µj´1, µj`1, ¨ ¨ ¨µnq with µj multiplicities. This is the case where the marked edgedeleted lies in two marked faces, hence deleting merge to the two marked faces.

3. tk@

TrpMµ1`k´1qśni“2 TrMµi

D

counts the number of surface with n marked faces of face partitionpµ1 ` k ´ 1, µ2, ¨ ¨ ¨ , µnq. this the case where the marked edge deleted is adjacent to a marked face andan unmarked face.


This gives us a way of recursively count the number of surfaces. Indeed, this counting technique is well-known,and is known as the Tutte’s recursion.

5.2 Diagramatic Interpretation of Eynard-Orantin Topological Recursion

The diagramatic interpretation in Section 5.1 can be extended to the Eynard-Orantin topological recursion.Let n ě 0 and g ě 0, with 2g ´ 2` n ą 0. Define Gg,n`1pp, p1, ¨ ¨ ¨ , pnq be the set of connected trivalent

graph G, such that there are

1. 2g ´ 1` n trivalent vertices

2. one special univalent vertex p called the root

3. n univalent vertices labelled p1, ¨ ¨ ¨ , pn called the leaves

4. 3g ´ 1` 2 edges, of which n` g are not directed and 2g ` n´ 1 are directed

5. the edge adjacent to p is directed away from p

6. the n edges adjacent to pi are not directed

7. the directed edges form a spanning planar binary tree T with root p of G with pi removed.

8. if two vertices, i.e. not the root or the leaves, are joined by one of the g remaining non-directed edges,then one is the parent of the other along the T

9. If a directed and non-directed edge is adjacent to the same vertex, then the directed edge is the left childif the non-directed edge links this vertex to its children in T

Now, define the weight on Gg,n`1pp, p1, ¨ ¨ ¨ , pnq to be such that for all G P Gg,n`1pp, p1, ¨ ¨ ¨ , pnq,

wpGq “ź

v vertex

ResqvÑavź

pp,qq undirected

Bpp, qqź

pp,qq directed

Kappp, qq

where the order of taking residues is by following along T from the leaves to the root. Then, a remarkable factis that

ωg,npp1, ¨ ¨ ¨ , pnq “ÿ

GPGg,n`1pp,p1,¨¨¨ ,pnq

wpGq

The graphs are defined to encode the recursion into a more combinatorial structure. This interpretation isparticularly useful for intuition.


As an example, consider

ω0,3pp1, p2, p3q “ÿ

a

ResqÑaKapp1, qq pBpq, p2qBpσapqq, p2q `Bpq, p1qBpσapqq, p1qq

which has the corresponding graph:

6 Open Problems and Future Work

In this article, we have presented a brief overview of an exciting area of mathematics. The theory of Eynard-Orantin topological recursion, with is roots in random matrix theory, has progressed to occupy a special placein the intersection of mathematical physics, enumerative geometry and knot theory. It has evolved to become atopic rich with interesting problems to physicts and mathematicians alike. It will be exciting to see how thistheory will influence the progress of mathematics and physics in the future.

There are many open problems in this theory, most of them we have yet to mention and all of them veryinteresting problem to study at their own right. Firstly, we have already mentioned the Laplace TransformConjecture in Section 4.4. A positive solution to this conjecture will shed light to some very peculiar andpowerful strutural nature of enumerative geometric problems in general. Indeed, any progress towards aconstruction of a spectral curve and the appropriate differential forms given an enumerative problem will be agiant step forward in our understanding of classes of problems which do exhibit the Eynard-Orantin topologicalrecursion.

We did not address the knot theory for which the methods of Eynard-Orantin has some substantial successin recent years. Specifically, Mulase and his collabrators have been working feverishly in the question of


quantum curves and how the theory of Eynard-Orantin topological recursion fits into that framework. It willcertainly be interesting to see how far the method of Eynard-Orantin topological research can push the envelopeon that fron.

Lastly, we will mention a problem which goes back to the root of the Eynard-Orantin theory. Namely, is itpossible to formulate a topological recursion over non-Hermitian model? In our brief review, the importance ofthe properties of the Hermitian matrices made it possible to write the probability measure in a nice compactway, which we then studied to obtain the topological recursion. Non-Hermitian matrices generally do nothave as much structure as the Hermitian matrices, but it will be interesting to see if there are other classes ofmatrices for which there is a notion of topological recursion.

Acknowledgement

This paper represents the result of a reading course on Eynard-Orantin topological recursion. The authorof this paper wishes to express gratitude to Professor David Jackson, of the University of Waterloo, for thewonderful opportunity to work in this interesting problem. The author will also like to thank Raymond Cheng,of the University of Waterloo, and Professor Bertrand Eynard, of the Institut de Physique Theorique, for theirinsights into the subject.

References

[BHL`14] Vincent Bouchard, Joel Hutchinson, Prachi Loliencar, Michael Meiers, and Matthew Rupert,A generalized topological recursion for arbitrary ramification, Annales Henri Poincare,vol. 15, Springer, 2014, pp. 143–169.

[BKMP09] Vincent Bouchard, Albrecht Klemm, Marcos Marino, and Sara Pasquetti, Remodeling theb-model, Communications in Mathematical Physics 287 (2009), no. 1, 117–178.

[DMSS12] Olivia Dumitrescu, Motohico Mulase, Brad Safnuk, and Adam Sorkin, The spectral curve of theeynard-orantin recursion via the laplace transform, arXiv preprint arXiv:1202.1159 (2012).

[EO07] Bertrand Eynard and Nicolas Orantin, Invariants of algebraic curves and topological ex-pansion, arXiv preprint math-ph/0702045 (2007).

[EO08] , Algebraic methods in random matrices and enumerative geometry, arXivpreprint arXiv:0811.3531 (2008).

[Eyn06] Bertrand Eynard, Formal matrix integrals and combinatorics of maps, arXiv preprintmath-ph/0611087 (2006).

[Eyn14] , A short overview of the” topological recursion”, arXiv preprint arXiv:1412.3286(2014).

[Mir07] Maryam Mirzakhani, Simple geodesics and weil-petersson volumes of moduli spaces ofbordered riemann surfaces, Inventiones mathematicae 167 (2007), no. 1, 179–222.

[Tao12] Terence Tao, Topics in random matrix theory, vol. 132, American Mathematical Soc., 2012.

[Wis28] John Wishart, The generalised product moment distribution in samples from a normalmultivariate population, Biometrika (1928), 32–52.

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