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KP INTEGRABILITY OF TRIPLE HODGE INTEGRALS. III.

CUT-AND-JOIN DESCRIPTION, KDV REDUCTION, AND TOPOLOGICAL

RECURSIONS

ALEXANDER ALEXANDROV

Abstract. In this paper, we continue our investigation of the triple Hodge integrals satisfyingthe Calabi–Yau condition. For the tau-functions, which generate these integrals, we derive thecomplete families of the Heisenberg–Virasoro constraints. We also construct several equivalentversions of the cut-and-join operators. These operators describe the algebraic version of topolog-ical recursion. For the specific values of parameters associated with the KdV reduction, we provethat these tau-functions are equal to the generating functions of intersection numbers of ψ andκ classes. We interpret this relation as a symplectic invariance of the Chekhov–Eynard–Orantintopological recursion and prove this recursion for the general Θ-case.

Keywords: enumerative geometry, Hodge integrals, tau-functions, KP hierarchy, Virasoroconstraints, cut-and-join operator, topological recursion

Contents

Introduction 20.1. Algebraic topological recursion 20.2. Intersection theory and KP integrability 20.3. KdV reduction 31. Symmetries of KP hierarchy and Heisenberg–Virasoro algebra 42. Algebraic topological recursion and cut-and-join operators 52.1. General cut-and-join description 52.2. Algebraic topological recursion 63. Cut-and-join operators for KW and BGW tau-functions 83.1. Intersection numbers 83.2. Heisenberg–Virasoro constraints 103.3. Heisenberg–Virasoro group action and cut-and-join operators 124. Triple Hodge integrals and KP hierarchy 144.1. KP integrability 144.2. Heisenberg–Virasoro and dimension constraints 174.3. Factorizations of the Virasoro group elements 194.4. Cut-and-join description of the triple Hodge integrals 225. KdV reduction 245.1. Translations and κ-classes 245.2. KdV reduction for triple Hodge integrals 256. Tau-functions identification and topological recursion 276.1. Chekhov–Eynard–Orantin topological recursion 276.2. Topological recursion for triple Hodge integrals 306.3. Topological recursion for triple Θ-Hodge integrals 32

Appendix A. Topological expansion of log τ(α)q,p 33

References 35

2020 Mathematics Subject Classification. 37K10, 14N35, 81R10, 14H70, 14N10.

1

http://arxiv.org/abs/2108.10023v1

2 ALEXANDER ALEXANDROV

Introduction

0.1. Algebraic topological recursion. The Chekhov–Eynard–Orantin topological recursionis an effective and universal tool in mathematical physics, enumerative geometry, and combina-torics [EO07, EO09]. For the basic models, this recursion provides the solution to the Virasoroconstraints. However, there are other ways to solve them, and sometimes they suit better for theparticular needs. For instance, the Kontsevich–Witten and Brezin–Gross–Witten tau-functions,which are the basic elements of the Chekhov–Eynard–Orantin topological recursion, can bedescribed by explicit formulas in terms of the second order differential operators [Ale11, Ale18].

This description results from a solution of an equation given by the combination of the Vira-soro constraints. It was recently noted [MMMR21] that the partition functions of some interest-ing matrix models are completely specified by a unique equation of this type. In this paper, weprove a vast generalization of this observation. Namely, we prove that the solution of a rathergeneral equation, which includes the Euler operator as the leading term, exists and is unique.This solution can be constructed recursively.

For the generating functions with the natural topological expansion this recursive proceduremay have a transparent topological meaning. In this case the recursion can be interpreted asthe algebraic topological recursion. As this recursion is given by the differential operators (alsoknown as cut-and-join operators), it is rather simple computationally. It is not known in generalhow to construct the equation leading to the cut-and-join solution. In some cases this equationcan be constructed from the Virasoro and W-constrains, satisfied by the generating functions.One of the main goals of this paper is to provide the cut-and-join description for an interestingfamily of the enumerative geometry invariants, namely for the triple Hodge integrals satisfyingthe Calabi–Yau condition.

0.2. Intersection theory and KP integrability. Let Mg,n be the Deligne–Mumford com-pactification of the moduli space of all compact Riemann surfaces of genus g with n distinctmarked points. It is empty unless the stability condition

2g − 2 + n > 0(0.1)

is obeyed.On Mg,n consider four families of cohomological classes: ψk, κk, λk, and Θg,n. For each

marking index i we consider the cotangent line bundle Li → Mg,n, whose fiber over a point

[Σ, z1, . . . , zn] ∈ Mg,n is the complex cotangent space T ∗ziΣ of Σ at zi. Then ψi ∈ H2(Mg,n,Q)

is the first Chern class of Li. With the forgetful map for the marked point n+1, π : Mg,n+1 →Mg,n, we define the Miller–Morita–Mumford tautological classes [Mum83], κk := π∗ψ

k+1n+1 ∈

H2k(Mg,n,Q). For the Hodge bundle E, a rank g vector bundle over Mg,n, we consider the ithChern classes λi = ci(E) and their generating function Λg(u) =

∑gi=0 u

iλi. Finally, following

Norbury [Nor17], we introduce Θ-classes, Θg,n ∈ H4g−4+2n(Mg,n), which are also related to thesuper Riemann surfaces [Nor20]. We refer the reader to [Nor17, Nor18, Nor20] for a detaileddescription of Θ-classes.

By the Kontsevich–Witten theorem [Kon92, Wit91] the generating function of the intersectionnumbers

(0.2)

∫

Mg,n

ψa11 ψ

a22 · · ·ψan

n

is a tau-function of the KdV hierarchy. For their Θ-version,

(0.3)

∫

Mg,n

Θg,nψa11 ψ

a22 · · ·ψan

n ,

KdV integrability of the generating function was established by Norbury [Nor17]. Relationbetween the intersection theory over the moduli spaces and solitonic integrable hierarchies israther mysterious, and only a few instances of this relation are known so far.

KP INTEGRABILITY OF TRIPLE HODGE INTEGRALS. III. 3

The Kontsevich–Witten (KW) and Brezin–Gross–Witten (BGW) tau-functions, correspond-ing to the intersection numbers of ψ and Θ classes, play very important role in the mod-ern mathematical physics and enumerative geometry. In particular, they are the main ingre-dients of such universal constructions as the Chekhov–Eynard–Orantin topological recursion[EO07, EO09] and the Givental decomposition [Giv01, Tel12]. These tau-functions share alot of common properties, in particular they satisfy the similar looking Virasoro constraints[FKN91, DVV91, GN92], and it is convenient to consider them simultaneously. The Virasoroconstraints allow us [Ale11, Ale18] to derive the cut-and-join description of these tau-functionsand to prove the algebraic version of topological recursion, associated with the expansion withrespect to the negative Euler characteristics of the punctured Riemann surface. We claim thatit is natural to consider all types of intersection numbers and their Θ-versions simultaneously.In many respects Θ-versions are a bit simpler, perhaps because of the supersymmetry behindthem.

In this paper, we investigate triple Hodge integrals satisfying the Calabi–Yau condition,

(0.4)

∫

Mg,n

Θ1−αg,n Λg(−q)Λg(−p)Λg

Åpq

p+ q

ãψa11 ψ

a22 · · ·ψan

n

for α = 0 and α = 1. For α = 1 these integrals appear at the localization techniques for theGromov–Witten invariants on the three-dimensional target manifolds [GP99]. They constitutethe topological vertex, which provides a diagrammatic method to compute topological stringamplitudes on non-compact toric Calabi–Yau three-folds [AKMnV05, LLLZ09]. String theoryinterpretation of these integrals for α = 0 is not known yet.

It appears that the generating functions of these intersection numbers have a surprisinglysimple description in terms of integrable hierarchies. Namely, these generating functions aretau-functions of the KP hierarchy [Ale20]. These tau-functions are certain deformations of theKW and BGW tau-functions. This generalizes the result of Kazarian for the linear Hodgeintegrals [Kaz09].

In this paper, we continue our investigation of the tau-functions for the triple Hodge integralssatisfying the Calabi–Yau condition. In particular, for these tau-functions using their relationto the KW and BGW tau-functions, we derive a complete set of the Heisenberg–Virasoro con-straints, see Theorem 1. These linear constrains completely specify a formal series solution.However, they are not sufficient to construct the cut-and-join description. In Theorem 2 weconstruct this description using the KP hierarchy group of symmetries. Resulting algebraictopological recursion, given by the third order differential operators, is very convenient com-putationally. We use it to compute the first few terms of the topological expansion of thetau-functions for the triple Hodge integrals satisfying the Calabi–Yau condition, see AppendixA.

0.3. KdV reduction. Let us also consider the intersection numbers of ψ, κ and Θ classes

(0.5)

∫

Mg,n

Θ1−αg,n e

∑∞

j=1 sjκjψa11 ψ

a22 · · ·ψan

n .

According to [MZ00, Nor20] the generating functions of these intersection numbers can be iden-tified with the KW (for α = 1) and BGW (for α = 0) tau-functions with shifted arguments.

For p = −2q the generating functions of the triple Hodge integrals satisfying the Calabi–Yaucondition (0.4) are the tau-functions of the KdV hierarchy [Ale21]. In Theorem 4 we prove thatin this case the tau-functions coincide with the generating functions of intersection numbers (0.5)for the specific values of sk variables. The proof is based on the Virasoro constraints and theexplicit form of the group element of the KP hierarchy symmetry. A geometric interpretationof this relation is not clear yet.

This theorem allows us to identify the intersection numbers (0.4) with (0.5) for arbitrarygenus g and number of marked points n and specific values of parameters. We interpret thisidentification in terms of the symplectic invariance of the Chekhov–Eynard–Orantin topologicalrecursion. For α = 1, where the topological recursion for (0.4) is known [BMn08, Che18, Zho09],we use the symplectic invariance to find an independent proof of the relations between the


intersection numbers. For α = 0 we prove the topological recursion for the general triple Θ-Hodge integrals satisfying the Calabi–Yau condition.

Organization of the paper. In Section 1 we remind the reader some basic properties of theHeisenberg–Virasoro symmetry algebra of the KP hierarchy. Section 2 is devoted to the origin ofthe cut-and-join description. In particular, we derive it as a solution of the differential equationgiven by a rather general deformation of the Euler operator and its relation to the topologicalexpansion. In Section 3 we remind the main properties of the KW and BGW tau-functions, andinvestigate how these properties transform under the action of the Virasoro group. In Section4 we construct the Heisenberg–Virasoro constraints and the cut-and-join operators for the tau-functions, which describe the triple Hodge integrals satisfying the Calabi–Yau condition. Section5 is devoted to the reduction to the KdV hierarchy for the specific values of the parameters.We prove that in this case the tau-functions coincide with the tau-functions for the intersectionnumbers containing the Miller–Morita–Mumford classes. In Section 6 we describe this relationusing symplectic invariance of the Chekhov–Eynard–Orantin topological recursion and provethis recursion for the general triple Θ-Hodge integrals satisfying the Calabi–Yau condition.

Acknowledgments. This work was supported by the Institute for Basic Science (IBS-R003-D1).

1. Symmetries of KP hierarchy and Heisenberg–Virasoro algebra

It is well known that a certain central extension of the algebra gl(∞) and correspondinggroup GL(∞) act on the space of KP tau-functions [FKN92, MJD00]. From the boson-fermioncorrespondence we also know how to identify the operators, acting on the tau-functions with theoperators, acting on the Sato Grassmannian. Often the Sato Grassmannian parametrization ismore convenient, and we will use it for computations. In this section, we remind the reader theproperties of the important Heisenberg–Virasoro subalgebra of gl(∞); for more details see, e.g.,[Ale15].

Consider the space C[[t]] of formal power series of infinitely many variables t = {t1, t2, . . . }.We will denote by . . . differential operators acting on this space. The Heisenberg–Virasorosubalgebra L of gl(∞) is generated by the operators

(1.1) Jk =

∂

∂tkfor k > 0,

0 for k = 0,

−kt−k for k < 0,

the unit, and the Virasoro operators

(1.2) Lm =1

2

∑

a+b=−m

abtatb +∞∑

k=1

ktk∂

∂tk+m+

1

2

∑

a+b=m

∂2

∂ta∂tb.

These operators satisfy the commutation relationsîJk, Jm

ó= kδk,−m,î

Lk, Jmó= −mJk+m,(1.3)

îLk, Lm

ó= (k −m)Lk+m +

1

12δk,−m(k3 − k)

and generate a Lie algebra. We decompose L as L = L+⊕L0⊕L−, where L+ (L−) are generatedby the operators Lk and Jk with positive (negative) k, and L0 is generated by L0 and a unit.

The Heisenberg–Virasoro group V is generated by the operators Jk, Lk and a unit,

(1.4) V := {C e∑

akJk+bkLk

∣∣∣ ak, bk, C ∈ C}.

For this group we also have a decomposition, V = V+ ⊕ V0 ⊕ V−. This group acts on the spaceof the tau-functions of the KP hierarchy.


The Heisenberg–Virasoro algebra (1.3) is a central extension of the algebra of operators, actingon the Sato Grassmannian and generated by

(1.5) jm = zm, lm = −zmÅz∂

∂z+m+ 1

2

ã.

Operators lm generate the Witt algebra. These operators obey the commutation relations (1.3)with omitted central term. Thanks to this relation between the commutation relations it isconvenient to use the operators jm and lm for computations in the group V.

The action of the V0 and V− groups on the space of the functions of t is rather simple –it is equivalent to the linear change of the variables supplemented by the multiplication by afunction. Below we will mostly focus on the group V+.

2. Algebraic topological recursion and cut-and-join operators

In this section, we describe how a single equation of a certain type satisfied by the generatingfunctions can lead to the recursion relations and the cut-and-join description which specifya unique solution. Moreover, for the generating functions of the enumerative geometry andmathematical physics this recursion can have a natural topological interpretation.

2.1. General cut-and-join description. Let us consider (in general, infinitely many) for-mal variables q = {q1, q2, q3, . . . }. We associate with them positive integer degrees ak ∈ Z>0.Consider corresponding Euler operator

(2.1) E =∑

k

akqk∂

∂qk.

We assume that there are only finitely many variables with any finite degree, that is, #qk(ak <n) < ∞ for any 0 < n < ∞. We associate the degree −ak to the operators ∂

∂qkand consider a

differential equation

(2.2)

(E −

∞∑

k=1

kVk

)· Z(q) = 0,

where Vk are some differential operators of degree k with coefficients given by formal series invariables q. It appears that such an equation always has a unique solution.

Proposition 2.1. Equation (2.2) has a unique solution in C[[q]] up to a normalization.

Proof. Let us find this solution explicitly. Consider the expansion

(2.3) Z(q) =

∞∑

k=0

Zk(q),

where Z0 ∈ C is a constant, and the degree of Zk is k, that is, E · Zk = kZk. Then the term ofdegree k on the left hand side of (2.2) should vanish,

(2.4) E · Zk −k−1∑

j=0

(k − j)Vk−j · Zj = 0.

Since E · Zk = kZk, the unique solution of (2.2) is given by the recursion

(2.5) Zk =1

k

k−1∑

j=0

(k − j)Vk−jZj .

Because of the finiteness of the variables with a given finite degree, all Zj are polynomials, andthe action on them of all finite degree operators is well defined. Arbitrary constant Z0 fixes thetotal normalization of the solution, which completes the proof. �

Equation (2.5) provides a recursion which completely specifies the solution for a given initialcondition Z0 = Z(0) which can be fixed by Z(0) = 1. This solution is described by the operatorsVk. There are several possibilities depending on the relations between the operators Vk.


• If Vk = δk,rVr for some r > 0, we have an explicit solution

(2.6) Z(q) = exp (Vr) · 1.

• In general, if the operators Vk constitute a commutative family, [Vk, Vm] = 0 for allk,m > 0, then

(2.7) Z(q) = exp

( ∞∑

k=1

Vk

)· 1.

If Vk = 0 for all k > r for some r, then the sum in the right hand side of (2.5) containsat most r terms.

• If the operators Vk do not commute, then the solution is given by the ordered exponential,which we denote by “ exp (. . . ) ”,

(2.8) Z(q) = “ exp

( ∞∑

k=1

Vk

)” · 1.

Remark 2.1. Equation of the form (2.2) is never unique. Therefore, in general it is not clearhow to choose the best representative of equations of this type and the corresponding solution(2.8). At least for some interesting models arising in enumerative geometry and matrix modelstheory the generating functions can be represented by (2.6).

In particular, one can consider the functions of the variables tk. We introduce a Z grading onfunctions and differential operators on this space by assigning

(2.9) deg tk = k, deg∂

∂tk= −k.

Then deg Lk = deg Jk = −k and degree of the central element of the Heisenberg–Virasoro

algebra is zero. The Euler operator E =∑∞

k=1 ktk∂∂tk

coincides with L0. Let us consider a

differential equation

(2.10)

(L0 −

∞∑

k=1

k“Vk)

· Z(t) = 0,

where “Vk are some differential operators (not necessarily from the gl(∞) algebra associated with

the KP hierarchy) with deg“Vk = k. From Proposition 2.1 it follows that such an equation alwayshave a unique solution. The following simple statement is a generalization of the observation of[MMMR21]:

Corollary 2.2. Equation (2.10) has a unique solution in C[[t]] up to a normalization.

Remark 2.2. For the KP hierarchy one can construct the equation of the form (2.10) using the

product of the canonical Kac–Schwarz operators [Ale21]. In this case the constructed operators “Vkbelong to the gl(∞) algebra of KP symmetries. However, already for the Kontsevich–Witten andBrezin–Gross–Witten tau-functions the answer, provided by this construction, does not coincidewith the simplest description obtained from the solution of the Virasoro constraints, see Section3.2.

2.2. Algebraic topological recursion. Generating functions in enumerative geometry andmathematical physics often depend on a parameter ~ associated to some topological expansion.Often it is related to the Euler characteristics χ of certain Riemann surface with punctures, inthis case the topological expansion is given by Z = exp

∑χ ~

−χFχ, or

(2.11) Z =

∞∑

k=0

~kZ(k).


Assume that the generating function Z is described by the equation (2.10), the more generalequation (2.2) can be considered similarly. Let the generating function Z also satisfy the dimen-sion constraint

(2.12) a~∂

∂~Z = L0 · Z

for some a ∈ Z>0.

Remark 2.3. The constraint (2.12) makes the parameter ~ redundant, however, it is convenientto use it.

Assume that the operators in (2.10) are of the form “Vk = ~ka Wk, where Wk do not depend on

~. Then they are nontrivial only if k/a ∈ Z, and the equation (2.10) leads to

(2.13) a~∂

∂~Z =

∞∑

k=1

ka ~kWka · Z.

Proposition 2.3. For any initial condition Z(0) if a solution to the equation (2.13) of the form(2.11) exists, it is unique.

Proof. Indeed, if we substitute (2.11) into (2.13), we get a recursion

(2.14) Z(k) =1

k

k−1∑

j=0

(k − j)Wa(k−j) · Z(j),

which allows us to find all Z(k) recursively starting from the initial condition Z(0). �

Note that we do not use Proposition 2.1, therefore, Proposition 2.3 is valid even if the gen-erating function does not satisfy any equation (2.2). However, if this equation is satisfied, then

Z(0) is a constant and the solution always exists.Equation of the form (2.14) describes the algebraic topological recursion. The solution is given

by ordered exponential

(2.15) Z = “exp

( ∞∑

k=1

~kWka

)” · Z(0),

which simplifies as in (2.6) or (2.7) for the particular cases. Operators Wk describe the change oftopology, that’s why below slightly abusing notation we will call them the cut-and-join operators.

In the simplest case Wka = δk,1Wa, and the generating function is given by

(2.16) Z = expÄ~Wa

ä· Z(0).

In this case the algebraic topological recursion have a particularly simple form

(2.17) Z(k) =1

kWa · Z(k−1).

Let us list some enumerative geometry problems (and associated matrix models), which canbe described by the cut-and-join operators

• Fat graphs or maps (Hermitian matrix model), [MS09];• Intersection theory of ψ classes on the moduli spaces (Kontsevich model), [Ale11];• Intersection theory of ψ and Norbury’s classes on the moduli spaces (Brezin–Gross–Witten model), [Ale18];

• Weighted Hurwitz numbers (certain multi-matrix models), [GJ97] for ordinary Hurwitznumbers, [ACEH18] and references therein for the weighted Hurwitz numbers;

• Open intersection theory (Kontsevich–Penner model), [Ale17];• Intersection theory of ψ and Witten’s classes (Generalized Kontsevich model), [Zho13,MMM21].


The last two examples are described by the non-commuting families of the cut-and-join operators.The Chekhov–Eynard–Orantin topological recursion [EO07, EO09] also describes a recursive

procedure in the negative Euler characteristics. However, many properties of this recursion,briefly outlined in Section 6, are essentially different from those of the algebraic topologicalrecursion (2.14).

The algebraic recursion (2.14) is given by differential operators with polynomial coefficients,while the Chekhov–Eynard–Orantin recursion is given in terms of the residues on the spectralcurve. The topological recursion (2.14) is formulated in terms of the coefficients of the gener-ating function Z, or non-connected correlation functions, while the Chekhov–Eynard–Orantintopological recursion is naturally formulated in terms of connected correlation functions, or thecoefficients of expansion of logZ. The recursion (2.14) is linear in the non-connected contri-butions, while the Chekhov–Eynard–Orantin recursion in quadratic in connected correlationfunctions.

The topological recursion (2.14) is often very convenient for computations. It would beinteresting to find a relation between the algebraic topological recursion, described in this paper,and geometrical topological recursion of Chekhov–Eynard–Orantin. In particular, there shouldbe a way to construct the cut-and-join operators from the topological recursion data. We believethat it is given by some version of the inverse Laplace transform [EMS11].

Conjecture 2.1. The Chekhov–Eynard–Orantin topological recursion with simple ramification

points can always be described by the cut-and-join formula (2.16) with operator Wa cubic in Jk.Higher ramification points correspond to the higher cut-and-join operators.

It is also unclear how to directely relate the cut-and-join description to the matrix integrals.If two generating functions Z and Z∗ are related to each other by

(2.18) Z∗ = “U · Z,where the operator “U does not depend on ~, then their cut-and-join operators are also relatedby conjugation. Namely, if Z satisfies (2.13), then

(2.19) Z∗ = “exp

( ∞∑

k=1

~kW ∗ka

)” · Z(0)∗,

where W ∗k = “U Wk

“U−1 and Z(0)∗ = “U · Z(0) are independent of ~. For the case (2.16) we have

(2.20) Z∗ = expÄ~W ∗

a

ä· Z(0)∗.

For a more general case with “U dependent on ~ sometimes it is also possible to relate thecut-and-join descriptions.

In Sections 3 and 4 we consider the generating functions of the intersection numbers on themoduli spaces with a natural topological expansion. Namely, the KW and BGW tau-functionssatisfy (2.12) for a = 3 and a = 1 respectively. From the Virasoro constraints it follows thatthey are given by a cut-and-join description of the form (2.16), see (3.21) below. The generatingfunctions of the triple Hodge integrals satisfying the Calabi–Yau condition satisfy the dimensionconstraints given by Proposition 4.1. For the corresponding Euler operator we do not have adescription of the form (2.2). However, these tau-functions are related to the KW and BGW

tau-functions by (2.18), where “U are the elements of the Virasoro group, see (4.50). Using theconjugation, we derive the cut-and-join description (2.20) for these generating functions.

3. Cut-and-join operators for KW and BGW tau-functions

3.1. Intersection numbers. Denote by Mg,n the Deligne–Mumford compactification of themoduli space of all compact Riemann surfaces of genus g with n distinct marked points. It is anon-singular complex orbifold of dimension 3g− 3+n. It is empty unless the stability condition

2g − 2 + n > 0(3.1)

is satisfied.


For each marking index i consider the cotangent line bundle Li → Mg,n, whose fiber over

a point [Σ, z1, . . . , zn] ∈ Mg,n is the complex cotangent space T ∗ziΣ of Σ at zi. Let ψi ∈

H2(Mg,n,Q) denote the first Chern class of Li. We consider the intersection numbers

〈τa1τa2 · · · τan〉g =∫

Mg,n

ψa11 ψ

a22 · · ·ψan

n .(3.2)

The integral on the right hand side of (3.2) vanishes unless the stability condition (3.1) is obeyed,all ai are non-negative integers, and the dimension constraint

(3.3) 3g − 3 + n =

n∑

i=1

ai

holds. Let Ti, i ≥ 0, be formal variables and consider

(3.4) τKW = exp

Ñ∞∑

g=0

∞∑

n=0

~2g−2+nFg,n

é,

where

(3.5) Fg,n =∑

a1,...,an≥0

〈τa1τa2 · · · τan〉g∏Tain!

.

Witten’s conjecture [Wit91], proved by Kontsevich [Kon92], states that the partition func-tion τKW becomes a tau-function of the KdV hierarchy after the change of variables Tk =(2k + 1)!!t2k+1. Integrability of the Kontsevich–Witten tau-function immediately follows fromKontsevich’s matrix integral representation, for more details see e.g. [Ale21] and referencestherein.

The Brezin–Gross–Witten tau-function is another solution of the KdV hierarchy, which isclosely related to the moduli spaces. Namely, this tau-function governs the intersection theorywith the insertions of the fascinating Norbury’s Θ-classes. Norbury’s Θ-classes are the coho-mology classes, Θg,n ∈ H4g−4+2n(Mg,n), which are also related to the super Riemann surfaces[Nor20]. We refer the reader to [Nor17, Nor18] for a detailed description. Let us consider theintersection numbers

(3.6) 〈τa1τa2 · · · τan〉Θg =

∫

Mg,n

Θg,nψa11 ψ

a22 · · ·ψan

n .

Remark 3.1. For the Θ-classes, higher degrees vanish because of the dimension constrains.Indeed, dimMg,n = 3g − 3 + n and Θg,n ∈ H4g−4+2n(Mg,n), therefore Θ2

g,n = 0.

Again, the integral on the right hand side vanishes unless the stability condition (3.1) issatisfied, all ai are non-negative integers, and the dimension constraint

(3.7) g − 1 =

n∑

i=1

ai

holds. Consider the generating function of the intersection numbers of Θ and ψ classes

(3.8) FΘg,n =

∑

a1,...,an≥0

〈τa1τa2 · · · τan〉Θg∏Tain!

then, we have a direct analog of the Kontsevich–Witten tau-function:

Theorem ([Nor17]). The generating function

(3.9) τΘ = exp

Ñ∞∑

g=0

∞∑

n=0

~2g−2+nFΘg,n

é

becomes a tau-function of the KdV hierarchy after the change of variables Tk = (2k + 1)!!t2k+1.


Norbury also proved, that τΘ is nothing but a tau-function of the BGW model,

(3.10) τΘ = τBGW .

KdV integrability of the BGW model follows from the relation to the generalized Kontsevichmodel and was established by Mironov, Morozov and Semenoff [MMS96].

3.2. Heisenberg–Virasoro constraints. It appears that the KW and BGW tau-functionsshare a lot of common properties and it is natural to consider them simultaneously. Let usdenote

(3.11) τ1 = τKW , τ0 = τBGW .

The dimension constrains (3.3), (3.7) can be represented as

(1 + 2α)~∂

∂~τα = L0 · τα, α = 0, 1.(3.12)

Below we always assume that the index α runs from 0 to 1,

(3.13) α ∈ {0, 1}.

Remark 3.2. Tau-functions τα are KdV tau-functions; however, it is convenient to considerthem in the context of the KP hierarchy. So, below we use the Heisenberg–Virasoro algebra ofKP symmetries, introduced in Section 1.

Both τα are KdV tau-functions, hence satisfy the Heisenberg constraints (or the KdV reductionconstraints)

(3.14)∂

∂t2kτα = 0, k > 0.

Moreover, they satisfy the Virasoro constraints [FKN91, DVV91, GN92]

Lαk · τα = 0, k ≥ −α,(3.15)

where the Virasoro operators are given by

Lαk =

1

2L2k −

1

2~

∂

∂t2k+1+2α+δk,016

∈ L.(3.16)

These operators satisfy the commutation relations

(3.17)îLαk , L

αk

ó= (k −m)Lα

k+m, k,m ≥ −α.

Combining (3.14) with (3.15) one gets

(3.18) L0 · τα = (1 + 2α)~Wα · τα,

where

(3.19) Wα =1

2α+ 1

∞∑

k=0

(2k + 1)t2k+1

ÅL2k−2α +

δk,α8

ã.

Therefore, from (3.12) we have

(3.20) ~∂

∂~τα = ~Wα · τα.

This is the relation of the form (2.13), it leads to the cut-and-join description for the KWand BGW tau-functions [Ale11, Ale18]

(3.21) τα = expÄ~Wα

ä· 1.


Remark 3.3. There is a certain arbitrariness in the construction of the cut-and-join operators.

Since the tau-functions τα depend only on odd times, it is possible to consider operators Wα withomitted even times t2k and corresponding derivatives,

W ∗0 =

∞∑

k,m∈Z+

odd

Åkmtktm

∂

∂tk+m−1+

1

2(k +m+ 1)tk+m+1

∂2

∂tk∂tm

ã+t18,

W ∗1 =

1

3

∞∑

k,m∈Z+

odd

Åkmtktm

∂

∂tk+m−3+

1

2(k +m+ 3)tk+m+3

∂2

∂tk∂tm

ã+t313!

+t38.

(3.22)

Then an equivalent cut-and-join description is given by the these operators,

(3.23) τα = expÄ~W ∗

α

ä· 1.

If we consider the topological expansion of the tau-functions τα =∑∞

k=0 τ(k)α ~k, then we have

the algebraic topological recursion

τ (k)α =1

kWα · τ (k−1)

α

=1

kW ∗

α · τ (k−1)α

(3.24)

with the initial condition τ(0)α = 1. Moreover, we can separate the contributions with different

numbers of marked points. Namely, let τ(k)α =

∑(2α+1)(k−1)n=1 τ

(k)α,n, where τ

(k)α,n is a homogeneous

polynomial in t of degree n. Let also

W ∗0,1 =

∞∑

k,m∈Z+

odd

kmtktm∂

∂tk+m−1+t18,

W ∗0,−1 =

1

2

∞∑

k,m∈Z+

odd

(k +m+ 1)tk+m+1∂2

∂tk∂tm,

W ∗1,1 =

1

3

∞∑

k,m∈Z+

odd

kmtktm∂

∂tk+m−3+t38,

W ∗1,−1 =

1

6

∞∑

k,m∈Z+

odd

(k +m+ 3)tk+m+3∂2

∂tk∂tm,

(3.25)

and W ∗0,3 = 0, W ∗

1,3 =t313! . Then we have the recursion

(3.26) τ (k)α,n =1

k

ÄW ∗

α,3 · τ(k−1)α,n−3 + W ∗

α,1 · τ(k−1)α,n−1 + W ∗

α,−1 · τ(k−1)α,n+1

ä

with the initial condition τ(0)α,n = δn,0.

The cut-and-join description allows us to find the topological expansion of the tau-functions

(3.27)

log τ0 =t18~+

t12

16~2 +

Åt1

3

24+

9 t3128

ã~3 +

Åt1

4

32+

27 t3t1128

ã~4 +

Åt1

5

40+

27 t3t12

64+

225 t51024

ã~5

+

Åt1

6

48+

45 t3t13

64+

1125 t5t11024

+567 t3

2

1024

ã~6 +O(~7),

(3.28) log τ1 =

Åt1

3

6+t38

ã~+

Åt3t1

3

2+

5 t5t18

+3 t3

2

16

ã~2

+

Å15 t1t3t5

4+

3 t32t1

3

2+

5 t5t14

8+

35 t7t12

16+

3 t33

8+

105 t9128

ã~3 +O(~4).


3.3. Heisenberg–Virasoro group action and cut-and-join operators. In this section, weinvestigate how the subgroup V+ of the Heisenberg–Virasoro group, see Section 1, acts on theKdV tau-functions τα, α ∈ {0, 1}. In general, the action of this subgroup is not well-defined andcan lead to divergences. In this section, we do not address this potential issue, assuming thatall considered functions are well defined.

Let us consider the subgroups of the Heisenberg–Virasoro group, generated by the operatorsof the form

(3.29) V+ ∋ “Gα := exp

( ∞∑

k=1

akLk

)exp

(~−1

∞∑

k=2+2α

vkJk

).

Remark 3.4. One can easily relax the constraint on the range of summation and consider moregeneral operators. However, this makes the formulas less compact, so we prefer to deal with aless general case, which is sufficient for our needs.

We consider the functions “Gα ·τα. The tau-functions τα satisfy the KdV reduction constraints,

J2k ·τα = 0 for k > 0, therefore we may assume that v2k = 0. Using the Virasoro constrains (3.15)we can rewrite action of the operator (3.29) on the tau-function τα in terms of the Virasoro grouponly. For this purpose we introduce a bijection between the space of the Virasoro operators

(3.30) “Va = exp

(∑

k>0

akLk

)

and the space of all formal series of the form f(z) ∈ z+ zC[[z]]. Recall that the operators lk aregiven by (1.5).

Definition 1. For the Virasoro group element “V consider

(3.31) Ξ(“V ) = e∑

k>0aklk z e−

∑k>0

aklk ∈ z + zC[[z]].

For a given set of parameters ak ∈ C, k ∈ Z>0 we denote such a series associated to the operator“Va by f(z;a) = Ξ(“Va). We also introduce the inverse formal series h(z;a),

(3.32) f(h(z;a);a) = z.

For any series f(z) ∈ z + zC[[z]] we construct a corresponding element of the Virasoro group

(3.33) “V = Ξ−1(f(z)),

where ak are defined implicitly by

(3.34) f(z) = e∑

k>0aklk z e−

∑k>0

aklk .

For the inverse group element we have

(3.35) “V −1 = Ξ−1(h(z)).

Multiplication of the Virasoro group elements corresponds to the composition of the associatedfunctions

(3.36) Ξ(“V1“V2) = f2(f1(z)).

To any sequence of coefficients ak ∈ C, k ∈ Z>0, we also associate the following formal series

(3.37) vα(z) =1

(2α + 1)(z2α+1 − f(z;a)2α+1) ∈ z2+2αC[[z]]

with the coefficients

(3.38) vαk (a) = [zk]vα(z),

where for any formal Laurent series F (z) =∑bkz

k we introduce [zk]F (z) := bk.

Lemma 3.1. We have

(3.39) exp

(~−1

∑

k=α+1

vαk (a)∂

∂t2k+1

)· τα = exp

(∑

k>0

a2kL2k

)· τα.


Proof. The statement of the lemma follows from Lemma 1.1 of [Ale15]. �

From this lemma it follows that

“Gα · τα = exp

( ∞∑

k=1

akLk

)exp

(~−1

∞∑

k=α+1

v2k+1J2k+1

)· τα

= exp

( ∞∑

k=1

akLk

)exp

( ∞∑

k=1

a2kL2k

)· τα,

(3.40)

where the coefficients ak are given implicitly by the relation (3.37) with v2k+1 = vα2k+1(a).Therefore, we always can reduce the action of the operator (3.29) to the action of equivalentoperator (3.30) from the Virasoro group.

Let us consider arbitrary operator “Va of the form (3.30) and its action on the tau-function

(3.41) τaα = “Va · τα.for some coefficients ak. τ

a

α are tau-functions of the KP hierarchy. Let us derive the Heisenberg–Virasoro constraints and the cut-and-join description for the tau-functions τaα .

We introduce

Ja

k = “VaJk“V −1a, La

k = “VaLk“V −1a.(3.42)

From the commutation relations of the Heisenberg–Virasoro algebra L it follows that these

operators are given by the linear combinations of Jm and Lm operators. Let us also introducethe coefficients

(3.43) ρ[k,m] = [zm]f(z;a)k, σ[k,m] = [zm+1]f(z;a)k+1

f ′(z;a),

where f(z,a) = Ξ(“Va). They are polynomials in a.

From the relation between the operators Jk, Lk, lk, jk and Definition 1 we have

Lemma 3.2.

(3.44)

Ja

k =

∞∑

m=k

ρ[k,m]Jm, k ∈ Z,

La

k =

∞∑

m=k

σ[k,m]Lm +a22δk,−2, k ≥ −2.

Consider the operators

Ja

k =1

2Ja

2k,

Lα,ak =

1

2La

2k −1

2~Ja

2k+1+2α +δk,016

.

(3.45)

Then

Lemma 3.3. Tau-functions τaα satisfy the Heisenberg constraints

(3.46) Ja

2k · τaα = 0, k > 0,

and the Virasoro constraints

Lα,ak · τaα = 0, k ≥ −α.(3.47)

These operators satisfy the commutation relations of the Heisenberg–Virasoro algebra

(3.48)

îJa

k , Ja

m

ó= 0, k,m > 0,

îLα,ak , Ja

m

ó= kJa

k+m, k ≥ −α, m > 0,îLα,ak , Lα,a

m

ó= (k −m)Lα,a

k+m, k,m ≥ −α.


Proof. Operators Ja

k and Lα,ak are given by the conjugation of the constraints (3.14) and (3.15),

in particular

(3.49) Lα,ak = “VaLα

k“V −1a.

�

If a do not depend on ~, then the relation (3.41) is of the form (2.18), therefore we can derivethe cut-and-join description of τaα from the expression (3.19). Let

(3.50) W a

α =1

2α+ 1

∞∑

k=0

Ja

−2k−1

ÅLa

2k−2α +δk,α8

ã.

This is a cut-and-join operator for τaα .

Lemma 3.4.

(3.51) τaα = expÄ~W a

α

ä· 1.

Proof. We use the cut-and-join representation (3.21). Since Lk · 1 = 0 for k ≥ 0, we have

“Va · τα = “Va expÄ~Wα

ä“V −1a

· 1

= expÄ~W a

α

ä· 1,

(3.52)

where W a

α := “VaWα“V −1a

coincides with (3.50). �

Because of the certain arbitrariness of the cut-and-join description of the tau-functions τα, seeRemark 3.3, we also construct another version of the cut-and-join operators for the tau-functionsτaα . Let

W ∗a0 =

∞∑

k,m∈Z+

odd

ÅJa

−kJa

−mJa

k+m−1 +1

2Ja

−k−m−1Ja

k Ja

m

ã+Ja

−1

8,

W ∗a1 =

1

3

∞∑

k,m∈Z+

odd

ÅJa

−kJa

−mJa

k+m−3 +1

2Ja

−k−m−3Ja

k Ja

m

ã+

(Ja

−1)3

3!+Ja

−3

24.

(3.53)

Lemma 3.5.

(3.54) τaα = expÄ~ W ∗a

α

ä· 1.

Note that the operators W a

α and W ∗aα only contain the cubic and linear terms in Jk.

4. Triple Hodge integrals and KP hierarchy

In [Ale20, Ale21] the author has proven that the generating functions of the triple Hodgeintegrals satisfying the Calabi–Yau condition are tau-functions of the KP hierarchy. These tau-functions can be related to the tau-functions τα considered in Section 3 by certain elementsof the subgroup V+ of the Heisenberg–Virasoro group. The Virasoro constraints allow us toreduce these group elements to the elements of the Virasoro group. In this section, using thearguments of Section 3.3 we will derive the cut-and-join description and the Heisenberg–Virasoroconstraints for the generating functions of the triple Hodge integrals satisfying the Calabi–Yaucondition.

4.1. KP integrability. Let us remind the description of the generating functions of the tripleHodge integrals satisfying the Calabi–Yau condition by KP tau-functions [Ale20, Ale21]. Con-sider the linear change of variables

T q,p0 (t) = t1,

T q,pk (t) =

∞∑

m=1

mtm

Åq∂

∂tm+

2q + p√p+ q

∂

∂tm−1+

∂

∂tm−2

ã· T q,p

k−1(t), k > 1,(4.1)


where q and p are two parameters with q + p 6= 0 if q 6= 0.We also introduce the coefficients aq,pk . They are given implicitly through the Definition 1,

(4.2) f(z;aq,p) = f(z),

where the series on the right hand side is given by

(4.3) f(z)df(z) =zdz

(1 +√p+ qz)(1 + qz/

√p+ q)

.

In particular,

(4.4) aq,p1 =1

3

p+ 2 q√p+ q

, aq,p2 = − 1

12

p2 + pq + q2

p+ q, aq,p3 =

31 p3 + 69 p2q + 21 pq2 + 14 q3

1080 (p+ q)3/2.

Let x(z) = f2(z)2 = z2/2 + . . . . If p 6= 0 and q 6= 0 we have

(4.5) x(z) =p+ q

pqlog

Å1 +

qz√p+ q

ã− 1

plog(1 +

√p+ qz)

and

(4.6) f(z) =

2p+ q

pqlog

Å1 +

qz√p+ q

ã− 2

plog(1 +

√p+ qz).

For q = 0 it degenerates to

(4.7) x(z) =z√p− 1

plog(1 +

√pz),

and for p = 0 it degenerates to

(4.8) x(z) =1

qlog(1 +

√qz)− 1√

q

z

1 +√qz.

Let us note that the change of variables (4.1) is generated by the operator

(4.9)∂

∂x=

(1 +√p+ qz)(1 + qz/

√p+ q)

z

∂

∂z.

Namely, the functions

(4.10) Φk(z) :=

Å− ∂

∂x

ãk· 1z

define this change of variables if we associate ktk with z−k and T q,pk with Φk(z).

Let

(4.11) y0(z) =1

z

and

(4.12) y1(z) =

∫ z

0y0(η)dx(η) =

∫ z

0

dx(η)

η,

where dx(η) = f(η)df(η). For p 6= 0 and q 6= 0 we have

(4.13) y1(z) =

√p+ q

p

Ålog(1 +

√p+ qz

)− log

Å1 +

qz√p+ q

ãã.

For q = 0 it reduces to

(4.14) y1(z) =1√plog (1 +

√pz)

and for p = 0 it degenerates to

(4.15) y1(z) =z

1 +√qz.


The series x and y1 satisfy the equation

(4.16) p exp (−qx(z)) = (p+ q) exp

Åq√p+ q

y1(z)

ã− q exp

(√p+ qy1(z)

).

Let us consider the Hodge bundle E, a rank g vector bundle over Mg,n. For λi = ci(E) we set

(4.17) Λg(u) =

g∑

i=0

uiλi.

Below we focus on the case of cubic Hodge integrals

(4.18)

∫

Mg,n

Θ1−αg,n Λg(u1)Λg(u2)Λg(u3)ψ

a11 ψ

a22 · · ·ψan

n

satisfying an additional Calabi–Yau condition

(4.19)1

u1+

1

u2+

1

u3= 0.

For this case it is convenient to use the parametrization

(4.20) u1 = −p, u2 = −q, u3 =pq

p+ q.

We introduce the generating functions of the triple Hodge integrals satisfying the Calabi–Yaucondition, with (for α = 0) and without (for α = 1) Norbury’s Θ-classes, namely

(4.21) Z(α)q,p (T) = exp

Ñ∞∑

g=0

∞∑

n=0

~2g−2+nFαg,n

é,

where

(4.22) Fαg,n =

∑

a1,...,an≥0

∏Tain!

∫

Mg,n

Θ1−αg,n Λg(−q)Λg(−p)Λg

Åpq

p+ q

ãψa11 ψ

a22 · · ·ψan

n .

Define the coefficients v(α)j by

∞∑

j=2+2α

v(α)j zj =

∫ z

0

(f(η)2α−1 − yα(η)

)dx(η).(4.23)

In particular, for α = 0 one has

v(0)k = [zk] (f(z)− y1(z)) .(4.24)

Let us introduce an element of the Virasoro group

(4.25) “Vq,p = Ξ−1(f(z)) = exp

Ñ∑

k∈Z>0

aq,pk Lk

é.

Then we have

Theorem ([Ale20]). The generating functions of triple Hodge integrals and Θ-Hodge integralssatisfying the Calabi–Yau condition in the variables Tq,p(t) are tau-functions of the KP hierar-chy,

(4.26) τ (α)q,p (t) = Z(α)q,p (T

q,p(t)).

They are related to the KW and BGW tau-functions by the elements of the Heisenberg–Virasorogroup

(4.27) τ (α)q,p (t) = exp

Ñ~−1

∞∑

j=2+2α

v(α)j

∂

∂tj

é“Vq,p · τα(t).


The proof of the theorem is based on the comparison between the Givental operators andthe elements of the Heisenberg–Virasoro group V of KP symmetries. For the p = 0 and α = 1case, corresponding to the linear Hodge integrals, the KP integrability was proved by Kazarian[Kaz09]. Relation between tau-functions given by the elements of the Heisenberg–Virasorogroup elements for this case was derived in [Ale15]. Identification of this group element and theGivental group element has been found in [LW16]. Alternative proof of the integrability part ofthis theorem for α = 1, based on the Marino–Vafa formula, is given by Kramer [Kra21].

Remark 4.1. In this paper we focus on the tau-functions τ(α)q,p and do not investigate the proper-

ties of the generating functions Z(α)q,p in the original variables. Virasoro constraints and cut-and-

join description for them can be found with the help of the Givental formalism [Giv01, Tel12]and will be considered elsewhere.

4.2. Heisenberg–Virasoro and dimension constraints. We start with the permutation ofthe translation operator with the element of the Virasoro group in (4.27). Namely, Lemma 3.2leads to the relation

(4.28) τ (α)q,p (t) =“Vq,p exp

Ñ~−1

∑

j=2+2α

v(α)j

∂

∂tj

é· τα(t),

where from (4.23) for the coefficients v(α)j we have

∞∑

j=2+2α

v(α)j zj =

∫ h(z)

0

(f(η)2α−1 − yα(η)

)dx(η).(4.29)

Here h(z) is a series, inverse to f(z) of (4.5), that is h(f(z)) = f(h(z)) = z, hence

(4.30)∞∑

j=2+2α

v(α)j zj =

∫ z

0

(η2α − ηyα(h(η))

)dη.

Since τα(t) are tau-function of the KdV hierarchy, we can neglect the translations of the eventimes t2k and

(4.31) τ (α)q,p (t) =“Vq,p exp

Ñ~−1

∑

j=1+α

v(α)2j+1

∂

∂t2j+1

é· τα(t).

To describe the Heisenberg–Virasoro constraints for the tau-functions τ(α)q,p let us consider

(4.32)

J(q,p)k =

1

2

∞∑

m=2k

ρ[2k,m]Jm,

Lα,(q,p)k =

1

2

∞∑

m=2k

σ[2k,m]Lm − 1

2~

∞∑

m=2k+1+2α

χα[2k,m]Jm +δk,016

− δk,−1

48

p2 + pq + q2

p+ q.

Here the coefficients σ[k,m], ρ[k,m], and χα[k,m] are given by

(4.33) ρ[k,m] = [zm]f(z)k, σ[k,m] = [zm+1]f(z)k+1

f ′(z), χα[k,m] = [zm]f(z)k+2yα(z),

where the function f(z) is given by (4.6).

Theorem 1. The tau-functions of triple Hodge integrals satisfy the Heisenberg–Virasoro con-straints

(4.34)J(q,p)k · τ (α)q,p = 0, k > 0,

Lα,(q,p)k · τ (α)q,p = 0, k ≥ −α.


These operators satisfy the commutation relations of the Heisenberg–Virasoro algebra

(4.35)

îJ(q,p)k , J (q,p)

m

ó= 0, k,m > 0,

îLα,(q,p)k , J (q,p)

m

ó= kJ

(q,p)k+m, k ≥ −α, m > 0,

îLα,(q,p)k , Lα,(q,p)

m

ó= (k −m)L

α,(q,p)k+m , k,m ≥ −α.

Proof. Consider the conjugation of the constraints (3.14) and (3.15) by the translation operator

(4.36) exp

Ñ~−1

∑

j=1+α

v(α)2j+1

∂

∂t2j+1

é.

The Heisenberg constraints do not change, and the conjugated Virasoro constraints are givenby the operators

(4.37)1

2L2k −

1

2~

∞∑

j=2k+1+2α

[zj ]z2k+2yα(h(z))∂

∂tj+δk,016

.

Then the conjugation by the Virasoro group element “Vq,p is described by Lemma 3.2. Note thatthe value of a2 in (3.44) is provided by (4.4). �

For the case of linear Hodge integrals (p = 0, α = 1) the Heisenberg–Virasoro constraints werederived in [Ale15]. A more convenient basis of constraints was obtained in [GW17]. Constraintsin Theorem 1 provide a generalization of this basis for arbitrary p and α ∈ {0, 1}.Remark 4.2. The Heisenberg–Virasoro constraints can also be derived from the Kac–Schwarzdescription, constructed in [Ale21].

The sting equation for α = 1 has a particularly simple form. We have

(4.38)

σ[−2,m] = [zm+1]1

f ′(z)f(z)

= [zm+2](1 +√p+ qz)(1 + qz/

√p+ q)

= δm,−2 +2q + p√p+ q

δm,−1 + qδm,0

and

(4.39) χ1[−2,m] = [zm]y1 = [zm]

√p+ q

p

Ålog(1 +

√p+ qz

)− log

Å1 +

qz√p+ q

ãã.

Then

(4.40) L1,(q,p)−1 =

1

2L−2 +

2q + p

2√p+ q

L−1 +q

2L0 −

1

2~

∞∑

m=1

χ1[−2,m]Jm − 1

48

p2 + pq + q2

p+ q

annihilates the tau-function

(4.41) L1,(q,p)−1 · τ (1)q,p = 0.

From the dimension constraints we have(q∂

∂q+ p

∂

∂p+

∞∑

a=0

Åa+

1

2

ãTa

∂

∂Ta− 2α+ 1

2~∂

∂~

)· Z(α)

q,p = 0.(4.42)

Let

(4.43) E = 2q∂

∂q+ 2p

∂

∂p+ L0

be the Euler operator. Then from relations between Z(α)’s and τα’s we have

Proposition 4.1.

E · τ (α)q,p = (2α+ 1)~∂

∂~τ (α)q,p .(4.44)


Remark 4.3. From this proposition it immediately follows that one of the parameters p, q and

~ in the tau-functions τ(α)q,p is redundant.

For the Euler operator (4.43) we do not know the relation of the form (2.2). However, tau-

functions τ(α)q,p have a simple cut-and-join description, which leads to the algebraic topological

recursion. To derive it we will use (2.18), where “U is a Virasoro group element described in thenext section.

4.3. Factorizations of the Virasoro group elements. In this section, we will describe the

relation between the tau-functions τα and τ(α)q,p given by the Virasoro group element. Let us use

Lemma 3.1 to substitute the translation operators in (4.31) with the equivalent elements of theVirasoro group,

(4.45) exp

Ñ~−1

∞∑

j=1+α

v(α)2j+1

∂

∂t2j+1

é· τα(t) = “V (α)

∆ · τα(t).

Here the Virasoro group operators “V (α)∆ contain only even generators L2k. By Definition 1 these

operators can be described by the formal series

(4.46) f(α)∆ (z) = Ξ[“V (α)

∆ ].

Here from (4.30) we have

f(0)∆ (z) = Y (z),

f(1)∆ (z)3

3=

∫ z

0Y (η)ηdη,(4.47)

where

(4.48) Y (z) =1

2(y1(h(z)) − y1(h(−z)))

is the odd part of y1(h(z)). To derive the first equation we use

(4.49) y1(h(z)) =

∫ h(z)

0

f(z)df(z)

η=

∫ z

0

ηdη

h(η).

Combining (4.45) with (4.28) one gets

(4.50) τ (α)q,p (t) =“V (α)q,p · τα(t),

where

(4.51) “V (α)q,p = “Vq,p“V (α)

∆ ∈ V+

is an element of the Virasoro group.

Let us find a different, more convenient factorization of this Virasoro group element “V (α)q,p . For

p 6= 0 and q 6= 0 we introduce

(4.52) h(z) =p

q√p+ q

expÄ2 q√

p+qzä− 1

1− expÄ−2 p√

p+qzä − 1√

p+ q.

For p = 0 it degenerates to

(4.53) h(z) =exp

(2√qz)− 1

2qz− 1√

q

and for q = 0 it degenerates to

(4.54) h(z) =2z

1− exp(−2

√pz) − 1√

p.


Inverse function is given by the series

f(z) = z − 1

3

p+ 2 q√p+ q

z2 +2 p2 + 5 pq + 5 q2

9(p + q)z3 − 2

135

11 p3 + 39 p2q + 51 pq2 + 34 q3

(p + q)3/2z4

+52 p4 + 236 qp3 + 429 p2q2 + 386 pq3 + 193 q4

405 (p+ q)2z5 +O(z6).

(4.55)

It is easy to see that both h(z) and f(z) are formal series in z of the form z+∑∞

k=2ck

(p+q)k−12

zk,

where ck ∈ Q[q, p] is a polynomial of degree k − 1. Set

(4.56) “V = Ξ−1(f(z)).

Consider an element of the Virasoro group “V −1“Vq,p. Then(4.57) Ξ[“V −1“Vq,p] = f(h(z))

is a composition of (4.52) and (4.6). Let the coefficients b(α) be given by

(4.58) exp

Ñ∑

k∈Z>0

b(α)k Lk

é= “V −1“Vq,p“V (α)

∆ ,

then the Virasoro group element “V (α)q,p can be factorized as

(4.59) “V (α)q,p = “V exp

Ñ∑

k∈Z>0

b(α)k Lk

é.

We have

(4.60) f(z,b(α)) = Ξ(“V −1“Vq,p“V (α)∆ ) = f

(α)∆ (f(h(z))).

The reason to introduce the Virasoro group element “V is the following

Lemma 4.2. The function Ξ[“V −1“Vq,p] given by (4.57) is inverse to Y (z), given by (4.48), or

(4.61) f(h(Y (z))) = z.

Proof. Assume that p 6= 0 and q 6= 0. Recall that x and y1 satisfy the equation (4.16),

(4.62) p exp (−qx(z)) = (p+ q) exp

Åq√p+ q

y1(z)

ã− q exp

(√p+ qy1(z)

).

Let us change the variable z 7→ h(z). We get

(4.63) p exp(−qz2/2

)= (p + q) exp

Åq√p+ q

y1(h(z))

ã− q exp

(√p+ qy1(h(z))

),

or, if we change the sign of z,

(4.64) p exp(−qz2/2

)= (p+ q) exp

Åq√p+ q

y1(h(−z))ã− q exp

(√p+ qy1(h(−z))

).

Making the inverse transformation z 7→ f(z), we have

(4.65) p exp (−qx(z)) = (p + q) exp

Åq√p+ q

χ

ã− q exp

(√p+ qχ

),

where we denote

(4.66) χ = y1(h(−f(z))).If we substitute the expression for x(z) here, we get

(4.67) p(1 +

√p+ qz)

q

p

(1 + qz/√p+ q)

p+q

p

= (p+ q) exp

Åq√p+ q

χ

ã− q exp

(√p+ qχ

),


which, after a straightforward computation leads to the identity

(4.68)p

q√p+ q

Ä1+

√p+qz

1+qz/√p+q

ä q

p expÄ− q√

p+qχä− 1

1−Ä1+qz/

√p+q

1+√p+qz

äexpÄ

p√p+q

χä − 1√

p+ q= z.

The left hand side of this equation is equal to h(12 (y1(z)− χ)

); therefore, we have

(4.69) h

Å1

2(y1(z)− y1(h(−f(z))))

ã= z.

If we apply f(z) to the both sides of this equation, we have

(4.70) f

Åh

Å1

2(y1(z)− y1(h(−f(z))))

ãã= f(z).

After the change of variables z 7→ h(z) we have

(4.71) f

Åh

Å1

2(y1(h(z)) − y1(h(−z)))

ãã= z.

Then the statement of the lemma for q 6= 0 and p 6= 0 follows from the definition of Y (z), (4.48).Special cases with p = 0 or q = 0 can be considered similarly as the certain limits. �

From this lemma and (4.47) it follows that the Virasoro group elements (4.58) are describedby the functions

(4.72)

f(z;b(0)) = Y (f(h(z))) = z,

f(z;b(1))3

3=

∫ z

0η dx(h(η)).

In particular, b(0) = 0 and “V −1“Vq,p“V (0)∆ = 1, therefore, “V (0)

q,p = “V .The last integral in (4.72) can be computed explicitly as a series,

(4.73)f(z;b(1))3

3=

∞∑

k=1

4kB2k

(2k + 1)!

(p + q)2k+1 − q2k+1 − p2k+1

pq(p+ q)kz2k+1.

Here B2k are the Bernoulli numbers defined by

(4.74)zez

ez − 1= 1 +

z

2+

∞∑

k=1

B2kz2k

(2k)!.

Remark 4.4. For p = 0 we have

(4.75)f(z;b(1))3

3

∣∣∣∣∣p=0

=z2√qcoth(

√qz)− z

q5

2

which, up to inversion and rescaling of the variable, coincides with the function describing one ofthe components of the Virasoro group element in [Ale15, Section 2.2]. This observation indicatesthat the factorization (4.59) is a generalization for p 6= 0 of the factorization of the group elementconsidered in [Ale15].

Proposition 4.3. The Virasoro group element “V (α)q,p by Definition 1 corresponds to the series

(4.76) fα(z) = Ξ(“V (α)q,p ) = f(f(z);b(α)),

where f(z,b(α)) and f are given by (4.72) and the function inverse to (4.52). We have

(4.77) f0(z) = f(z),f1(z)

3

3=

∫ z

0f(η)dx(η).


4.4. Cut-and-join description of the triple Hodge integrals. The cut-and-join descriptionof the tau-function for the triple Hodge integrals satisfying the Calabi–Yau condition follows fromthe cut-and-join description for the KW and BGW tau-functions, see Section 3.2. Let

(4.78) W q,pα =

1

2α + 1

∞∑

k=0

Jα−2k−1

ÅLα2k−2α +

δk,α8

ã.

Here

(4.79)Jαk = “V (α)

q,p JkÄ“V (α)

q,p

ä−1,

Lαk = “V (α)

q,p Lk

Ä“V (α)q,p

ä−1,

are the conjugated operators of the Heisenberg–Virasoro algebra. By Lemma 3.2 they are givenby the linear combinations

(4.80)

Jαk =

∞∑

m=k

ρα[k,m]Jm, k ∈ Z,

Lαk =

∞∑

m=k

σα[k,m]Lm + cαδk,−2, k ≥ −2,

where

(4.81) ρα[k,m] = [zm]fα(z)k, σα[k,m] = [zm+1]

fα(z)k+1

f ′α(z)

with fα given by Proposition 4.3 and

(4.82) cα =

Åδα,190

− 1

18

ãp2 + pq + q2

p+ q.

Because of a certain arbitrariness of the cut-and-join description of the tau-functions τα, seeRemark 3.3, we can also construct different versions of the cut-and-join operators for the tau-

functions τ(α)q,p . For example, consider the operators W ∗q,p

α given by (3.53) with Jak substituted

by Jαk .

Theorem 2.

τ (α)q,p = expÄ~ W q,p

α

ä· 1

= expÄ~ W ∗q,p

α

ä· 1.

(4.83)

Proof. The statement immediately follows from Lemma 3.4, Lemma 3.5, and Proposition 4.3. �

Operators W q,pα and W ∗q,p

α do not belong to the gl(∞) algebra of KP hierarchy symmetries.Expressions for the coefficients ρα and σα can be simplified. Namely, for α = 0, since f0(z) =

f(z), we have

ρ0[k,m] = [zm]f(z)k =1

2πi

∮dh(z)

h(z)m+1zk,

σ0[k,m] = [zm+1]f(z)k+1

f ′(z)=

1

2πi

∮h′(z)dh(z)

h(z)m+2zk+1.

(4.84)

Both (p+ q)(m−k)/2ρ0[k,m] ∈ Q[q, p] and (p+ q)(m−k)/2σ0[k,m] ∈ Q[q, p] are polynomials of thetotal degree m− k.

Let us expand the cut-and-join operator

(4.85) W q,p0 =

∞∑

k=0

W q,p0,1−k,


where deg W q,p0,k = k. The leading term coincides with the cut-and-join operator for the BGW

model (3.19), W q,p0,1 = W0. For the sub-leading term

(4.86) W q,p0,0 =

∞∑

k=0

Äρ0[−2k − 1,−2k]J−2kL2k + σ0[2k, 2k + 1]J−2k−1L2k+1

ä

it is possible to find corresponding coefficients ρ0 and σ0 explicitly. From (4.84) and (4.52) forthe non-negative k we immediately get

(4.87) ρ0[−2k − 1,−2k] =2k + 1

3

p+ 2q√p+ q

, σ0[2k, 2k + 1] = −2k − 1

3

p+ 2q√p+ q

.

Therefore, we have

W q,p0,0 =

1

3

p+ 2q√p+ q

∞∑

k=0

Ä(2k + 1)J−2kL2k − (2k − 1)J−2k−1L2k+1

ä.(4.88)

In the same way for the next term we have

W q,p0,−1 =

∞∑

k=0

Åρ0[−2k − 1,−2k + 1]J−2k+1

ÅL2k +

δk,08

ã

+ρ0[−2k − 1,−2k]σ0[2k, 2k + 1]J−2kL2k+1 + σ0[2k, 2k + 2]J−2k−1L2k+2

ä,

(4.89)

where

ρ0[−2k − 1,−2k + 1] = −2k + 1

9

Å(k − 1)

p2

p + q+ (4k − 1)q

ã,

σ0[2k, 2k + 2] =1

9

Å(2 k2 + k − 2

) p2

p+ q+(8k2 − 2k − 2

)q

ã.

(4.90)

For α = 1 the coefficients ρ1 and σ1 can be expressed in terms of the residues of the combi-nations of functions h(z) and f(z;b(1)), given by (4.52) and (4.73), namely

ρ1[k,m] = [zm]f1(z)k =

1

2πi

∮dh(z)

h(z)m+1f(z;b(1))k,

σ1[k,m] = [zm+1]f1(z)

k+1

f ′1(z)=

1

2πi

∮h′(z)dh(z)

h(z)m+2

f(z;b(1))k+1

f ′(z;b(1)).

(4.91)

By construction, the cut-and-join operators W q,pα and W ∗q,p

α are cubic in Jk. Moreover, theydo not depend on ~. Hence, the cut-and-join formulas (4.83) describe the algebraic topologicalrecursion. Namely, if we consider the topological expansion of the tau-functions

(4.92) τ (α)q,p =∞∑

k=0

~kτ (α,k)q,p ,

then the coefficients of this expansion satisfy

τ (α,k)q,p =1

kW q,p

α · τ (α,k−1)q,p

=1

kW ∗q,p

α · τ (α,k−1)q,p

(4.93)

with the initial condition τ(α,0)q,p = 1. The operators W q,p

α and W ∗q,pα contain infinitely many

terms, however, only a finite number of them contribute to each step of recursion. Moreover, fora contributions with the fixed numbers of marked points we have a recursion similar to (3.26).

In Appendix A we provide the first few terms Fαk (t) of the expansion of log τ

(α)q,p ,

(4.94) τ (α)q,p (t) = exp

( ∞∑

k=1

~kFαk (t)

)

obtained by this recursion with the cut-and-join operators W q,pα using Maple.


For q = 0 or p = 0 we get the cut-and-join description for the tau-function of the linear Hodgeintegrals.

Remark 4.5. Using this approach one can also restore other types of cut-and-join-like repre-sentations of the generating functions of the triple Hodge integrals. For example, it is possibleto find a description similar to the one constructed in [GHW19] for the KW tau-function.

Remark 4.6. We believe that there is a universal distinguished choice of the cut-and-join op-erators associated to some natural representation theory interpretation. It may be related to thequantum Airy structures [BBC+18, BKS20].

5. KdV reduction

For p = −2q the tau-functions τ(α)q,p do not depend on even times, hence, they are solutions

of the KdV hierarchy, a 2-reduction of the KP hierarchy [Ale20]. These tau-functions haveparticularly nice properties, and can be identified with the KW and BGW tau-functions withshifted arguments. We will investigate them in this and the next sections.

Remark 5.1. In [DLYZ20] it was shown that in this case the tau-function τ(1)q,−2q is related to

the discrete KdV hierarchy (aka the Volterra lattice hierarchy). We expect that τ(0)q,−2q is also

related to the Volterra hierarchy in the similar way.

5.1. Translations and κ-classes. With the forgetful map π : Mg,n+1 → Mg,n we define the

Miller–Morita–Mumford tautological classes [Mum83], κk := π∗ψk+1n+1 ∈ H2k(Mg,n,Q). Accord-

ing to Manin and Zograf [MZ00], insertion of these classes into the intersection numbers (3.2)can be described by the translation of the variables tk responsible for the insertion of the ψ-classes. Let us consider the generating functions of the intersection numbers of ψ and κ classeswith (for α = 0) and without (for α = 1) Norbury’s Θ-classes

(5.1) Fαg,n(T, s) =

∑

a1,...,an≥0

∫

Mg,n

Θ1−αg,n e

∑∞

j=1 sjκjψa11 ψ

a22 · · ·ψan

n

∏Tain!

and

(5.2) τα(t, s) = exp

Ñ∞∑

g=0

∞∑

n=0

~2g−2+nFαg,n(T, s)

é∣∣∣∣∣∣Tk=(2k+1)!!t2k+1

.

We introduce the polynomials qj(s), defined by the generating function

(5.3) 1− exp

Ñ−

∞∑

j=1

sjzj

é=

∞∑

j=1

qj(s)zj ,

and we put qk(s) = 0 for k < 1. For k ≥ 1 they are nothing but the negative of the elementarySchur functions

(5.4) qj(s) = −pj(−s),

where

(5.5) exp

Ñ∞∑

j=1

sjzj

é=

∞∑

j=0

pj(s)zj .

Then the generating functions τα(t, s) can be obtained from the generating functions τα(t)without κ-classes, see Section 3.1, by translation

Theorem 3 ([MZ00, Nor20]). For α ∈ {0, 1}

(5.6) τα(t, s) = τα

Åßt2k+1 +

1

~

qk−α(s)

(2k + 1)!!

™ã.


5.2. KdV reduction for triple Hodge integrals. From the change of the variables (4.1) it

is clear that for p = −2q the tau-functions τ(α)q,p (t) do not depend on even variables t2k, therefore

they are tau-function of the KdV hierarchy. Let us put q = −u2 with new parameter u, so thatp = 2u2. This parametrization corresponds to the insertion of the triple Hodge class of the formΛ(−2u2)Λ(−2u2)Λ(u2).

Remark 5.2. From Proposition 4.1 it follows that the parameter u is redundant. However, wewill leave it as a free parameter to make the limit u = 0 more transparent.

In this case the series f(z), see (4.55), have a simple explicit form

f(z) =1

2ulog

Å1 + uz

1− uz

ã

=1

uarctanh(uz)

=∞∑

k=0

u2kz2k+1

2k + 1.

(5.7)

It is an odd function. The Virasoro group operators “V (α)q,p in (4.50), relating τα to the tau-

functions for the triple Hodge integrals, contain only even positive components of the Virasorooperators. Therefore, using Lemma 3.1 one equivalently describes them by translation operators

(5.8) τ(α)−u2,2u2(t) = exp

Ñ~−1

∑

j=1+α

v(α)2j+1(u)

∂

∂t2j+1

é· τα(t).

Let us find the coefficients v(α)2j+1(u) using Lemma 3.1 and Proposition 4.3.

For α = 0 these coefficients are given by the generating function

(5.9)

∞∑

k=1

v(0)2k+1(u)z

2k+1 = z − f(z),

therefore,

(5.10) v(0)2k+1(u) = − u2k

2k + 1.

By Theorem 3 this translation of the variables corresponds to the insertion of the κ-classeswith parameters s0k given by qk(s

0) = −(2k − 1)!!u2k . From (5.3) we have

(5.11)

∞∑

k=1

s0kzk = − log

(1 +

∞∑

k=1

(2k − 1)!!(zu2)k).

For instance, s01 = −u2, s02 = −52u

4, s03 = −373 u

6. In general, s0k = −akk u

2k, where the sequenceak is described in [SI20, A004208].

To find the coefficients for α = 1 we need the specialization of (4.3),

(5.12) dx =zdz

1− u2z2.

From Proposition 4.3 we have

1

3f1(z)

3 =

∫ z

0f(η)dx(η)

=1

u

∫ z

0arctanh(uη)

ηdη

1 − u2η2

=1

2u2

∫ z

0η

Å∂

∂ηarctanh(uη)2

ãdη.

(5.13)


From Lemma 3.1 we know that the coefficients satisfy

(5.14)

∞∑

k=1

v(1)2k+1(u)z

2k+1 =1

3

(z3 − f1(z)

3),

therefore,

(5.15) v(1)2k+3(u) = − u2k

2k + 3

k∑

j=0

1

2j + 1.

Here we use the series expansion of arctanh(z)2, see e.g. [SI20, A004041].Using Theorem 3 one can find the values of the parameters sk, associated to these translations,

exp

Ñ−

∞∑

j=1

s1jzj

é= 1−

∞∑

k=1

(2k + 3)!!v(1)2j+3(u)z

k

=

∞∑

k=0

(2k + 1)!!

Ñk∑

j=0

1

2j + 1

é(zu2)k.

(5.16)

In particular, s11 = −4u2, s22 = −15u4, s13 = −3163 u

6.Therefore, for sα given by (5.11) and (5.16) we have the following relation between the gen-

erating functions of different types of intersection numbers:

Theorem 4. The generating functions of the triple Hodge integrals satisfying Calabi–Yau con-dition for p = −2q = 2u2 coincide with the generation functions of the intersection numberscontaining κ classes,

(5.17) τ(α)−u2,2u2(t) = τα(t, s

α).

Remark 5.3. Geometric interpretation of this identity is not clear yet. It would be interestingto find if it follows from the relations between the cohomology classes on Mg,n.

This theorem implies that for investigation of the tau-functions τ(α)−u2,2u2(t) we can use known

properties of the KW and BGW tau-functions. In particular, the tau-functions τ(α)−u2,2u2(t)

obey the Virasoro constraints, obtained from the constraints (3.15) by a translation of variables

tk 7→ tk + ~−1v(α)2j+1(u). This basis in the space of constraints can be more convenient when one,

constructed in Section 4.2.Let us consider the identification of the intersection numbers for the given genus g and number

of the marked points n. Using the change of variables (4.1) we can reformulate Theorem 4 interms of these intersection numbers. Namely, for p = −2q = 2u2 the change of the variables(4.1) is given by

(5.18) T−u2,2u2

k (t) =

∞∑

m=1

mtm

Å∂

∂tm−2− u2

∂

∂tm

ãT−u2,2u2

k−1 (t)

with T−u2,2u2

0 = t1. It is generated by

(5.19)∂

∂x=

1− u2z2

z

∂

∂z.

Namely, the functions (4.10),

(5.20) Φk(z) :=

Å− ∂

∂x

ãk· 1z,

define a change of variables, if we associate Tk = (2k + 1)!!t2k+1 with (2k − 1)!!z−2k−1 and

T−u2,2u2

k with Φk(z).


For g, n ≥ 0 let us introduce the correlation functions

(5.21) Wαg,n(z1, . . . , zn) =

∑

a1,...,an≥0

∫

Mg,n

Θ1−αg,n Λ(−2u2)Λ(−2u2)Λ(u2)

n∏

j=1

ψajj Φaj (zj)

and

(5.22) Wαg,n(z1, . . . , zn) =

∑

a1,...,an≥0

∫

Mg,n

Θ1−αg,n e

∑∞

j=1 sαj κj

n∏

j=1

ψajj

Å− 1

zj

∂

∂zj

ãajz−1j .

Then the following statement is equivalent to Theorem 4:

Corollary 5.1. For g, n ≥ 0

(5.23) Wαg,n(z1, . . . , zn) = Wα

g,n(z1, . . . , zn).

For u = 0 this relation is trivial. Note that for the non-stable cases with 2g − 2 + n ≤ 0 bothWα

g,n and Wαg,n vanish identically. For n = 0 we have

(5.24)

∫

Mg

Θ1−αg,0 Λg(−2)Λg(−2)Λg(1) =

∫

Mg

Θ1−αg,0 e

∑∞

j=1 sαj κj .

For α = 1 according to Faber and Pandharipande [FP00] one has

(5.25)

∫

Mg

Λg(−q)Λg(−p)Λg(pq

p+ q) =

1

2

Åq2p2

q + p

ãg−1B2g

2g

B2g−2

2g − 2

1

(2g − 2)!

for g ≥ 2. For p = −2q = 2u2 it reduces to

(5.26)

∫

Mg

Λg(−2u2)Λg(−2u2)Λg(u2) = 22g−3u6g−6B2g

2g

B2g−2

2g − 2

1

(2g − 2)!.

It would be interesting to derive this expression using the right hand side of (5.24).

6. Tau-functions identification and topological recursion

In this section, we will describe the Chekhov–Eynard–Orantin topological recursion for thetriple Hodge integrals satisfying the Calabi–Yau condition and consider Theorem 4 from thepoint of view of this recursion.

6.1. Chekhov–Eynard–Orantin topological recursion. The Chekhov–Eynard–Orantintopological recursion is a universal construction, which allows to recover an infinite tower ofcorrelation function starting from a (non necessarily globally defined) spectral curve Σ, twomeromorphic differentials on it, dx and dy, and a canonical bilinear differential B(z1, z2) forz1, z2 ∈ Σ. We call the set

(6.1) S = (Σ, B, x, y)

the topological recursion data. Let us briefly review the formalism of topological recursion, see[EO07, EO09, CN19] for more details.

In this paper we restrict ourselves to the spectral curves of genus zero, Σ = CP1. ConsiderCP1 with a fixed global coordinate z, then we can specify the spectral curve by the functionsx(z) and y(z). Consider the differential dx with an assumption that dx is a rational differentialwith simple critical points p1, . . . , pN . On CP1 the canonical bilinear differential is the Cauchydifferentiation kernel

(6.2) BC(z1, z2) =dz1dz2

(z1 − z2)2.

We assume that the function y is meromorphic in the neighbourhoods of the zeros of dx. Weallow the spectral curve to be irregular [CN19], that is, the poles of y may coincide with thezeroes of dx.


With this input, on the Cartesian product Σn we construct a system of symmetric differentials(or correlators) ωg,n, g ≥ 0, n ≥ 1, given by

ω0,1(z1) = y(z1)dx(z1);(6.3)

ω0,2(z1, z2) = BC(z1, z2);

and for 2g − 2 + n > 0 we use the recursion

ωg,n(z1, . . . , zn) :=1

2

N∑

i=1

Resz→pi

∫ zσi(z)

BC(z1, ·)ω0,1(z1)− ω0,1(σi(z1))

(ωg−1,n+1(z, σi(z), zJnK\{1})(6.4)

+∑

g1+g2=g,I1⊔I2=JnK\{1}(g1,|I1|),(g2,|I2|)6=(0,0)

ωg1,1+|I1|(z, zI1)ωg2,1+|I2|(σi(z), zI2)

),

where σi is the deck transformation of x near pi, i = 1, . . . , N . For the stable cases 2g−2+n > 0these differentials are meromorphic. We wouldn’t go into the discussion of this version of thistopological recursion, for more detail e.g. [EO07]. As the spectral curve Σ = CP1 and thecanonical bilinear differential B = BC are the same for all considered cases, we will denote thetopological recursion data by S = (x(z), y(z)), and also call it the spectral curve.

The gth symplectic invariant for g ≥ 2 associated to the spectral curve S = (x, y) is definedby

(6.5) F (g) =1

2− 2g

N∑

i=1

Resz→pi Φ(z)ωg,1(z),

where Φ(z) =∫ zydx is a primitive of y(z)dx(z). Definitions of F (0) and F (1) can be found in

[EO07]. We will identify ωg,0 = F (g). All correlators ωg,n with 2− 2g − n < 0 are called stable,and the others are called unstable.

The symplectic invariants are expected to be invariant under the symplectic transformationswhich preserve the symplectic form dx∧ dy. However, the invariance under the exchange trans-formation x ↔ y is still challengeable. It is known that in general the correlators ωg,n arenot invariant under the symplectic transformations. Nevertheless, there are some symplectictransformations which leave them invariant. Let us consider an interesting class of such trans-formations.

Namely, consider the transformation [EO09] of the form

(6.6) x = g(x), y =1

g′(x)y,

where g(x) is an analytic function near the zeros of dx. This transformation preserves thesymplectic form dx ∧ dy = dx ∧ dy, therefore, it is symplectic. We assume that both spectralcurves S = (x, y) and S = (x, y) are of genus zero.

Proposition 6.1. If the zeroes of dx are simple and the set of these zeroes coincides with theset of the zeroes of dx, then for the two spectral curves S = (x, y) and S = (x, y) we have

(6.7) ωg,n(z1, . . . , zn) = ωg,n(z1, . . . , zn)

for all stable cases.

Proof. For n = 0 the statement was established in [EO09]. The canonical bilinear differentialsfor two spectral curves are the same and coincide with the Cauchy differentiation kernel (6.2).Since x(σi(z)) = x(z) in the neighbourhood of pi, we have x(σi(z)) = g(x(σi(z))) = x(z).Therefore, σi(z) = σi(z) and

(6.8) ω0,1(z1)− ω0,1(σi(z1)) = ω0,1(z1)− ω0,1(σi(z1)).

We see that the initial conditions and the kernel of the topological recursion for S and S arethe same, which completes the proof. �


With a topological recursion data we associate a generating function made of only stablecorrelators with neglected unstable terms,

(6.9) exp

Ñ∑

2g−2+n>0

~2g−2+n

n!ωg,n(z1, . . . , zn)|Ψk(zi)=Tk

é,

where Ψk(z) = d(− ∂

∂x

)kz−1.

From now on we consider only the case N = 1, that is, the differential dx has a unique zero.The KW tau-function τ1(t) is described by the topological recursion on the Airy curve x = 1

2y2

[EO09]

(6.10) S =

Å1

2z2, z

ã.

It was conjectured in [Ale18] and proven in [DN18] that the BGW tau-function τ0(t) is describedby the topological recursion on the Bessel curve xy2 = 1

2 with the topological recursion data

(6.11) S =

Å1

2z2,

1

z

ã.

For these cases the stable correlators of topological recursion are related to the intersectionnumbers

(6.12) ωα,0g,n(z1, . . . , zn) = d1 . . . dn

∑

a1,...,an≥0

∫

Mg,n

Θ1−αg,n

n∏

j=1

ψajj

Å− 1

zj

∂

∂zj

ãajz−1j .

for α = 1 and α = 0 respectively.For these two tau-functions translation of variables t leads to the change of y described by

Proposition 4.8 of Chekhov–Norbury:

Proposition 6.2 ([CN19]). The generating function of the topological recursion data

(6.13) S =

(1

2z2, z2α−1 +

∞∑

k=2α

ykzk

)

is obtained via translation of the appropriate tau-function

(6.14) τα

Åßt2k+1 −

1

~

y2k−1

2k + 1

™ã.

Note that only t2k+1 with k ≥ α+ 1 are translated.

Remark 6.1. Actually, the proposition of Chekhov–Norbury is more general and describes y =∑∞k=2α−1 ykz

k with y2α−1 6= 1. We do not need this more general version here.

Corollary 6.3. The functions τα(t, sα) in Theorem 4 are described by the topological recursion

on the genus zero spectral curves Sα with

(6.15) S0 =

(1

2z2, z−1 +

∞∑

k=1

u2kz2k−1

)=

Å1

2z2,

1

z(1 − u2z2)

ã

and

(6.16) S1 =

Ñ1

2z2, z +

∞∑

k=1

u2kz2k+1k∑

j=0

1

2j + 1

é=

Å1

2z2,

1

2u(1− u2z2)log

1 + uz

1− uz

ã.

The stable correlators of topological recursion for these curves are related to the functions(5.22) by

(6.17) ωα,ug,n = d1 . . . dnW

αg,n(z1, . . . , zn).

Spectral curves in Corollary 6.3 are described by the equations

(6.18) x(1− 2u2x)2y2 =1

2


and

(6.19) u−2Ä1− e2u(1−2u2x)y

ä2= 2x

Ä1 + e2u(1−2u2x)y

ä2

respectively. Locally at x = 0 their behavior is described by the Bessel and Airy curves corre-spondingly. For both curves the only zero of dx = zdz is simple and is located at z = 0. Thecurve for α = 1 is not algebraic, similar to the case of the Weil–Petersson volumes [EO09].

6.2. Topological recursion for triple Hodge integrals. In this section, we will focus onthe case α = 1, the case α = 0 is considered in Section 6.3. Topological recursion for thetriple Hodge integrals of the form (0.4) for α = 1 is described by Bouchard–Marino conjecture[BMn08], which was proven by Chen and Zhou [Che18, Zho09]. In this section we remind thereader these results basically following [Zho09].

To formulate the results of Chen and Zhou let us describe the relation between the combinationof the Hodge classes used in these papers and Λg(p),

(6.20) Λ∨g (a) = agΛg

Å−1

a

ã.

Let

(6.21) φb(z) =

ÅX

∂

∂X

ãb 1z − (a+ 1)

(a+ 1)2,

where

(6.22) X =aa

(a+ 1)a+1(1− (a+ 1)z)

Å1 +

(a+ 1)z

a

ãa.

Then X and

(6.23) Y =a

a+ 1+ z

satisfy the spectral curve equation

(6.24) X = Y a − Y a+1.

Consider the topological recursion for the genus zero spectral curve

(6.25) S = (logX, log Y )

and

ω0,1(z1) = log Y (z1) d logX(z1);(6.26)

ω0,2(z1, z2) = BC(z1, z2).

Then the stable correlators are related to the triple Hodge integrals satisfying the Calabi–Yaucondition by [Che18, Zho09]

(6.27) ωg,n(z1, . . . , zn)

= (−1)n(a(a+ 1))g+n−1∑

a1,...,an≥0

∫

Mg,n

Λg(−1)Λg

Å−1

a

ãΛg

Å1

a+ 1

ã n∏

j=1

ψajj dφai(zi).

Under the rescaling y 7→ ǫy the correlators of the topological recursion change as

(6.28) ωg,n 7→ ǫ2−2g−nωg,n.

Let us consider this transformation with ǫ = −√a(a+ 1) for the spectral curves (6.25), then

for the new spectral curve S =ÄlogX, log Y

äthe stable correlators (6.27) acquire the form

(6.29) ωg,n(z1, . . . , zn) =∑

a1,...,an≥0

∫

Mg,n

Λg(−1)Λg

Å−1

a

ãΛg

Å1

a+ 1

ã n∏

j=1

ψajj dφai(zi).


Here Y = (Y )−√

a(a+1) and

φb(z) =»a(a+ 1)φb(z)

=

ÅX

∂

∂X

ãb √a

(a+ 1)3

2

Å1

z− (a+ 1)

ã.

(6.30)

To identify the functions φk(z) with the coefficients Φk(z) in (4.10) let us make the change ofthe variable

(6.31)

√a

(a+ 1)3

2

Å1

z− (a+ 1)

ã7→ 1

z.

Then X(z) and Y (z) get the form

(6.32) X(z) =aa

(a+ 1)a+1

(1 +»

a+1a z)a

Ä1 +»

aa+1z

äa+1 , Y =

Ña

a+ 1

1 +»

a+1a z

1 +»

aa+1z

é−√

a(a+1)

.

To identify the triple Hodge class of (6.29) with the one in (4.22) one should consider q, p ∈¶1, 1a ,− 1

a+1

©. Put q = 1, p = − 1

a+1 or p = 1a . Then (4.5) and (4.13) are given by

x(z) = (a+ 1) log

Å1 +

…a

a+ 1z

ã− a log

Ç1 +

…a+ 1

az

å,

y1(z) =»a(a+ 1)

Çlog

Ç1 +

…a+ 1

az

å− log

Å1 +

…a

a+ 1z

ãå.

(6.33)

This pair is symplectically equivalent to the pair logX(z) and log Y (z),

(6.34) − logX(z) = x(z)− log

Åaa

(a+ 1)a+1

ã, − log Y (z) = y1(z)−

»a(a+ 1) log

a+ 1

a,

that is d logX ∧d log Y = dx∧dy1. It is easy to see that forÄlogX, log Y

äand (x, y1) all stable

correlators of topological recursion coincide. In particular,

(6.35)

ÅX

∂

∂X

ãb 1z=

Å− ∂

∂x

ãk· 1z.

and the functions Φk(z) and φk(z), given (4.10) and (6.21) coincide with each other.For arbitrary p and q let us consider

(6.36) ω(α),p,qg,n (z1, . . . , zn) = d1 . . . dn

∑

a1,...,an≥0

∫

Mg,n

Λg(−p)Λg (−q)Λg

Åpq

p+ q

ã n∏

j=1

ψajj Φai(zi),

where Φa’s are given by (4.10). If we transform p 7→ ǫ2p, q 7→ ǫ2q and z 7→ ǫ−1z, then x 7→ ǫ−2x

and yα 7→ ǫ1−2αyα, therefore ω(α),p,qg,n (z1, . . . , zn) 7→ ǫ(2α+1)(2g−2+n)ω

(α),p,qg,n (z1, . . . , zn). Using this

transformation for α = 1 we can substitute the values q0 = 1 and p0 = − 1a+1 or p0 =

1a in (6.29)

by arbitrary q and p. Therefore, the tau-function τ(1)q,p and correlators ω

(1),p,qg,n are governed by

topological recursion with S = (x, y1). This spectral curve is given by the equation (4.16).Let us put p = −2q = 2u2. Then for α = 1 the triple Hodge classes in (6.36) coincide with

the triple Hodge classes in (5.21), and for the stable cases with n > 0 we have

(6.37) ω(1),2u2,−u2

g,n (z1, . . . , zn) = d1 . . . dnW1g,n(z1, . . . , zn).

We have

x = − 1

2u2log(1− u2z2

),

y1 =1

2ulog

1 + uz

1− uz.

(6.38)


If we compare the curve S = (x, y1) to the curve S1 in (6.16) with x = 12z

2 and y =1

2u(1−u2z2)log 1+uz

1−uz , we have

(6.39) x = g(x), y1 =1

g′(x)y,

where x = z2

2 and

(6.40) g(x) = − 1

2u2log(1− 2u2x).

We see that the curve (x, y1) is related to (x, y) by a symplectic transformation of the form (6.6).Now from Proposition 6.1 it follows that stable correlators for the topological recursion data

S = (x, y1) and (6.16) coincide. This provides an independent proof of Corollary 5.1 for α = 1.

6.3. Topological recursion for triple Θ-Hodge integrals. In this section, we will considerthe Chekhov–Eynard–Orantin topological recursion for the triple Θ-Hodge integrals satisfyingthe Calabi–Yau condition (α = 0). Namely, for arbitrary q and p we prove an analog of theBouchard–Marino topological recursion for this case.

Let us consider the triple Θ-Hodge integrals satisfying the Calabi–Yau condition for arbitrary

q and p. For ω(0),p,qg,n (z1, . . . , zn), given by (6.36), we have

Proposition 6.4. Stable correlators ω(0),p,qg,n are given by topological recursion for

S = (x, y0)

=

Åp+ q

pqlog

Å1 +

qz√p+ q

ã− 1

plog(1 +

√p+ qz),

1

z

ã.

(6.41)

Proof. The statement follows from the results of Chekhov and Norbury [CN19] and almostliterally repeats the proof of Theorem 12 in [Nor18]. Let us give only a brief outline of the proof.

The generating function of the triple Hodge integrals Z(1)p,q given by (4.21) can be described by

Givental’s formalism. For Z(0)p,q , according to [Nor18, Proposition 3.9], we have a similar looking

Givental description with τ1 substituted by τ0 and a different translation operator. The detailsof both Givental’s descriptions are given in [Ale20, Section 3].

By [DBOSS14, CN19] Givental’s description is related to the topological recursion. The

topological recursion for τ(1)p,q is described by S = (x, y1), see Section 6.2. Then, according to

[Nor18], the Θ-version of the same tau-function is described by substitution of y1 with∂y1∂x . From

(4.12) we have

(6.42)∂y1∂x

= y0

which completes the proof.�

For for p = −2q = 2u2 the stable correlators are related to (5.21) with α = 0,

ω(0),2u2,−u2

g,n (z1, . . . , zn) = d1 . . . dnW0g,n(z1, . . . , zn)(6.43)

Then these correlators are given by topological recursion:

Corollary 6.5. The stable correlators ω(0),2u2,−u2

g,n are given by the topological recursion with

(6.44) S =

Å− 1

2u2log(1− u2z2),

1

z

ã.

Proof. This corollary has an independent proof, based on the symplectic invariance. From

Corollary 5.1 it follows that the stable correlators ω(0),2u2,−u2

g,n are equal to d1 . . . dnW0g,n. From

Corollary 6.3 these correlators are given by the topological recursion with the data

(6.45) S =

Å1

2z2,

1

z(1− u2z2)

ã.


These x and y are symplectically equivalent to (6.44) for the symplectic transformation (6.6)with g(x) given by (6.40), therefore, the statement follows from Proposition 6.1. �

The curve (6.44) is described by the equation

(6.46) u−2y2(1− e−2u2x) = 1,

which for u = 0 reduces to the Bessel curve 2xy2 = 1.

Appendix A. Topological expansion of log τ(α)q,p

F 01 =

1

8t1,(A.1)

F 02 =

1

16t1

2 − 1

128

p2 + pq + q2

p+ q,(A.2)

F 03 =

9

128t3 +

1

24t1

3 +3

64

(p+ 2 q) t2√p+ q

− 1

128

(2 p2 − pq − q2

)t1

p+ q(A.3)

F 04 =

27

128t3t1 +

1

32t1

4 +9

64

(p+ 2 q) t1t2√p+ q

− 3

128

(p2 − 2 pq − 2 q2

)t1

2

p+ q

+1

512

p4 + 2 p3q + 3 p2q2 + 2 pq3 + q4

(p+ q)2

(A.4)

F 05 =

225

1024t5 +

27

64t3t1

2 +1

40t1

5 +75

256

(p+ 2 q) t4√p+ q

+9

32

(p+ 2 q) t2t12

√p+ q

+3

512

t3(4 p2 + 79 pq + 79 q2

)

p+ q− 1

64

(2 p2 − 7 pq − 7 q2

)t1

3

p+ q

− 1

512

(22 p3 + 21 p2q − 69 pq2 − 46 q3

)t2

(p+ q)3/2

+1

1024

(8 p4 − 6 p3q − 5 p2q2 + 2 pq3 + q4

)t1

(p+ q)2

(A.5)

F 06 =

1125

1024t5t1 +

567

1024t3

2 +45

64t3t1

3 +1

48t1

6

+375

256

(p+ 2 q) t1t4√p+ q

+189

256

(p+ 2 q) t2t3√p+ q

+15

32

(p+ 2 q) t2t13

√p+ q

+3

256

(10 p2 + 229 pq + 229 q2

)t3t1

p+ q+

63

256

(p2 + 4 pq + 4 q2

)t2

2

p+ q

− 5

128

(p2 − 5 pq − 5 q2

)t1

4

p+ q

− 1

512

(110 p3 − 21 p2q − 723 pq2 − 482 q3

)t2t1

(p+ q)3/2

+1

512

(10 p4 − 35 p3q − 11 p2q2 + 48 pq3 + 24 q4

)t1

2

(p+ q)2

− 1

12288

17 p6 + 51 p5q + 105 p4q2 + 125 p3q3 + 105 p2q4 + 51 pq5 + 17 q6

(p+ q)3

(A.6)


F 07 =

55125 t732768

+3375

1024t5t1

2 +1701

512t3

2t1 +135

128t3t1

4 +1

56t1

7 +55125

16384

(p+ 2 q) t6√p+ q

+1125

256

(p+ 2 q) t4t12

√p+ q

+567

128

(p+ 2 q) t3t2t1√p+ q

+45

64

(p+ 2 q) t2t14

√p+ q

+1875

32768

(26 p2 + 173 pq + 173 q2

)t5

p+ q+

9

512

(20 p2 + 521 pq + 521 q2

)t3t1

2

p+ q

+189

128

(p2 + 4 pq + 4 q2

)t2

2t1

p+ q− 3

128

(2 p2 − 13 pq − 13 q2

)t1

5

p+ q

− 25

4096

(67 p3 − 387 p2q − 1563 pq2 − 1042 q3

)t4

(p+ q)3/2

− 3

512

(110 p3 − 147 p2q − 1101 pq2 − 734 q3

)t2t1

2

(p+ q)3/2

− 3

32768

(1548 p4 + 9796 p3q − 7681 p2q2 − 34954 pq3 − 17477 q4

)t3

(p+ q)2

+1

1024

(40 p4 − 250 p3q + 63 p2q2 + 626 pq3 + 313 q4

)t1

3

(p+ q)2

+1

16384

(1052 p5 + 308 qp4 − 4561 q2p3 − 284 q3p2 + 4135 q4p+ 1654 q5

)t2

(p+ q)5/2

− 1

32768

(272 p6 − 236 p5q − 176 p4q2 + 115 p3q3 + 45 p2q4 − 15 pq5 − 5 q6

)t1

(p+ q)3

(A.7)

F 11 =

1

8t3 +

1

6t1

3 +1

12

(p+ 2 q) t2√p+ q

− 1

24

p2t1p+ q

,(A.8)

F 12 =

1

2t3t1

3 +5

8t5t1 +

3

16t3

2

+5

6

(p+ 2 q) t4t1√p+ q

+1

4

(p+ 2 q) t3t2√p+ q

+1

3

(p+ 2 q) t2t13

√p+ q

+1

8

(p2 + 12 pq + 12 q2

)t3t1

p+ q+

1

12

(p2 + 4 pq + 4 q2

)t2

2

p+ q+

1

6qt1

4

− 1

12

(p3 − p2q − 9 pq2 − 6 q3

)t1t2

(p+ q)3/2

− 1

48

q(2 p2 − pq − q2

)t1

2

p+ q

− 1

5760

p2q2

p+ q,

(A.9)


F 13 =

5

8t5t1

4 +3

2t3

2t13 +

35

16t7t1

2 +15

4t5t3t1 +

3

8t3

3 +105

128t9

+35

8

(p+ 2 q) t6t12

√p+ q

+3

4

(p+ 2 q) t32t2√

p+ q+

35

16

(p+ 2 q) t8√p+ q

+5

6

(p+ 2 q) t4t14

√p+ q

+ 2(p+ 2 q) t3t2t1

3

√p+ q

+5

2

(p+ 2 q) t5t2t1√p+ q

+ 5(p+ 2 q) t4t3t1√

p+ q

+10

3

(p2 + 4 pq + 4 q2

)t4t2t1

p+ q+

9

8

(p2 + 8 pq + 8 q2

)t3

2t1

p+ q

+5

48

(23 p2 + 140 pq + 140 q2

)t5t1

2

p+ q+

1

4

(p2 + 10 pq + 10 q2

)t3t1

4

p+ q

+7

288

(76 p2 + 391 pq + 391 q2

)t7

p+ q+

2

3

(p2 + 4 pq + 4 q2

)t2

2t13

p+ q

+1

2

(p2 + 4 pq + 4 q2

)t3t2

2

p+ q+

1

2

(p3 + 17 p2q + 45 pq2 + 30 q3

)t3t2t1

(p+ q)3/2

+1

12

(p3 + 79 p2q + 231 pq2 + 154 q3

)t4t1

2

(p+ q)3/2

+11

12

(p+ 2 q) qt2t14

√p+ q

+1

9

(p3 + 6 p2q + 12 pq2 + 8 q3

)t2

3

(p+ q)3/2

+7

192

(10 p3 + 167 p2q + 441 pq2 + 294 q3

)t6

(p+ q)3/2

− 1

16

(2 p4 − 2 p3q − 101 p2q2 − 198 pq3 − 99 q4

)t3t1

2

(p+ q)2+

5

24q2t1

5

− 1

384

(57 p4 − 236 p3q − 2769 p2q2 − 5066 pq3 − 2533 q4

)t5

(p+ q)2

− 1

6

(p4 − 2 p3q − 26 p2q2 − 48 pq3 − 24 q4

)t2

2t1

(p+ q)2

− 1

1440

(37 p5 + 520 p4q − 178 p3q2 − 5172 p2q3 − 7580 pq4 − 3032 q5

)t4

(p+ q)5/2

− 1

24

(7 p3 − 4 p2q − 54 pq2 − 36 q3

)qt2t1

2

(p+ q)3/2

− 1

144

(9 p2 − 8 pq − 8 q2

)q2t1

3

p+ q

+1

960

(7 p6 − 16 qp5 − 239 q2p4 − 170 p3q3 + 605 q4p2 + 828 pq5 + 276 q6

)t3

(p+ q)3

+1

2880

(21 p5 − 2 p4q − 120 p3q2 − 40 p2q3 + 60 pq4 + 24 q5

)qt2

(p+ q)5/2

+7

5760

p4q2t1

(p+ q)2.

(A.10)

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Email address: alexandrovsash at gmail.com

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http://dx.doi.org/10.1007/s00222-011-0352-5

http://dx.doi.org/10.4310/SDG.1990.v1.n1.a5



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