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The Hodge-FVH Correspondence

Si-Qi Liu, Di Yang, Youjin Zhang, Chunhui Zhou

Abstract

The Hodge-FVH correspondence establishes a relationship between the special cubicHodge integrals and an integrable hierarchy, which is called the fractional Volterra hierarchy.In this paper we prove this correspondence. As an application of this result, we prove a gapcondition for certain special cubic Hodge integrals and give an algorithm for computing thecoefficients that appear in the gap condition.

Contents

1 Introduction 1

2 Tau-functions of the FVH 5

3 The topological tau-function of the FVH 14

4 Virasoro constraints 27

5 Proof of Theorem 1.1 32

6 Proofs of Propositions 1.2, 1.3 and Theorem 1.4 37

1 Introduction

The study of the deep relationship between topology of the Deligne–Mumford moduli spaceof stable algebraic curves and integrable hierarchies started from Witten’s conjecture [35]. Itstates that the formal series ZWK(t; ǫ) ∈ Q((ǫ2))[[t0, t1, . . . ]] defined by

ZWK(t; ǫ) := exp

(∑

g≥0

ǫ2g−2∑

k≥0

∑

i1,...,ik≥0

ti1 · · · tikk!

∫

Mg,k

ψi11 · · ·ψikk

)

is a particular tau-function of the Korteweg–de Vries (KdV) hierarchy. Here, Mg,k denotes theDeligne–Mumford moduli space of stable algebraic curves of genus g with k distinct markedpoints, ψi denotes the first Chern class of the ith cotangent line bundle over Mg,k, i = 1, . . . , k,and t := (ti)i≥0 is the infinite vector of indeterminates. The first proof of Witten’s conjecturewas given by Kontsevich in [20], and several other proofs were given in [19, 22, 28, 31].

2010 Mathematics Subject Classification: 37K10, 53D45, 14N35

1

The Hodge integrals over Mg,k are some rational numbers defined by

∫

Mg,k

ψi11 · · ·ψikk λj1 · · · λjl,

where λj is the j-th Chern class of the Hodge bundle Eg,k on Mg,k, j = 0, . . . , g. For thedimension reason these integrals vanish unless

(1.1) i1 + · · · + ik + j1 + · · ·+ jl = 3g − 3 + k.

They reduce to the ψ-class intersection numbers when j1 = j2 = · · · = 0. In this paper weconsider the generating series of certain cubic Hodge integrals given by

(1.2) Zcubic(t; p, q, r; ǫ) = e∑

g≥0 ǫ2g−2Hg(t;p,q,r),

where Hg(t; p, q, r), g ≥ 0 are the genus g cubic Hodge potentials

(1.3) Hg(t; p, q, r) =∑

k≥0

∑

i1,...,ik≥0

ti1 · · · tikk!

∫

Mg,k

ψi11 · · ·ψikk Cg(−p)Cg(−q)Cg(−r).

Here, Cg(z) :=∑g

j=0 λjzj denotes the Chern polynomial of Eg,k. It was proved in [8] that the

formal series ucubic defined by

ucubic := ǫ2∂2t0 logZcubic(t; p, q, r; ǫ)

satisfies an integrable hierarchy of Hamiltonian PDEs, called the cubic Hodge hierarchy.The above cubic Hodge integrals given in (1.3) are called special if p, q, r satisfy the following

local Calabi–Yau condition:

(1.4)1

p+

1

q+

1

r= 0.

Geometric significance for these special cubic Hodge integrals is manifested in the localizationtechnique of computing Gromov–Witten invariants for toric three-folds [3, 16, 21, 30], and bythe Gopakumar–Marino–Vafa conjecture regarding the Chern–Simons/string duality [15, 27].

The conjectural Hodge-FVH correspondence [26], which is a generalization of Witten’sconjecture, states that the cubic Hodge hierarchy for the special cubic Hodge integrals isequivalent to the fractional Volterra hierarchy (FVH). The goal of this paper is to prove thisconjecture. To give a precise statement of the Hodge-FVH correspondence, let us introducesome notations. Denote

(1.5) I1 := kp | k ∈ Z>0 , I2 = kq | k ∈ Z>0 , I := I1 ∪ I2.

The set I will serve as the set of indices. Also introduce the numbers cµ, µ ∈ I by

cµ :=

(−µ/rµ/p

), µ ∈ I1, cµ :=

(−µ/rµ/q

), µ ∈ I2.(1.6)

2

Theorem 1.1 (Hodge-FVH correspondence) Let Λ := eǫ∂x. The formal series

(1.7) u(x, T ; ǫ) :=(Λ− 1

2r − Λ12r

)(Λ

12q − Λ

− 12q

)logZcubic

(t(x, T ); p, q, r;

√p+ q

pqǫ

)

satisfies the FVH. More precisely, denote L = Λ1p + euΛ− 1

q , then for µ ∈ I we have

(1.8) ǫ∂L

∂Tµ=

[(L−µ

r

)+, L

], µ ∈ I1,

−[(L−µ

r

)−, L

], µ ∈ I2,

where

ti(x, T ) :=1

pq

∑

µ∈I

µi+1cµTµ − 1 + xδi,0 + δi,1.(1.9)

Moreover, the formal series Z(x, T ; ǫ) ∈ C((ǫ2))[[x− 1, T ]] defined by

(1.10) Z(x, T ; ǫ) := exp

(p2q2

(p + q)ǫ2A

)Zcubic

(t(x, T ); p, q, r;

√p+ q

pqǫ

)

is a tau-function of the FVH, where A is an element in C[[x− 1, T ]] defined by

(1.11) A :=1

2p2q2

∑

µ1,µ2∈I

µ1µ2µ1 + µ2

cµ1cµ2Tµ1Tµ2 +1

pq

∑

µ∈I

(x− µ

µ+ 1

)cµTµ +

1

4− x.

We will give the definition of tau-function of the FVH in Section 2. Note that for the case(p, q, r) = ( 1

m ,1n ,− 1

m+n) with m,n ∈ N, the time variables Tµ are identified with the timevariables sk in [24] in the following way:

Tbk = sk, k ∈ (N− n)\ (0 ∪ ((m+ n)N− n)) ,

where

bα+(m+n)ℓ :=

−αn + ℓ, α = −(n− 1), . . . ,−1,αm + ℓ, α = 0, . . . ,m− 1.

In order to prove Theorem 1.1, we will first consider the rational case, i.e.

(1.12) p =1

m, q =

1

n, r = −1

h,

where m and n are coprime positive integers, and h := m+ n. The key idea of our proof is touse the uniqueness theorem of solutions to the loop equation already shown in [24]. We willshow that there exists a particular tau-function τtop of the FVH satisfying certain technicalconditions required by the uniqueness theorem of [24], and then show that τtop satisfies theVirasoro constraints, which leads to the loop equation, and thus to a proof of the theorem.Theorem 1.1 in the general case is proved by using a continuation argument. These proofs aregiven in Sections 3–5.

As applications of Theorem 1.1, we prove Propositions 1.2, 1.3 and Theorem 1.4 (see below)on properties of certain special cubic Hodge integrals. Here Proposition 1.2 and Theorem 1.4are presented in [24] without proof. For the presentation of Proposition 1.2, we denote

(1.13) Hg(x, T ) := Hg

(t(x, T ); p, q, r

), g ≥ 0,

where t(x, T ) is defined in (1.9).

3

Proposition 1.2 For the rational case, Hg(x, T ), g ≥ 1 have the following property:

H1(x, T )−σ1 − 1

24log x(1.14)

=∑

m≥1

∑

λ1,...,λm∈I

c1; λ1,...,λm(σ1, σ3)

m!xλ1+···+λm−mTλ1 · · ·Tλm ,

Hg(x, T )−Rg(σ1, σ3)

x2g−2(1.15)

=∑

m≥1

∑

λ1,...,λm∈I

cg; λ1,...,λm(σ1, σ3)

m!xλ1+···+λm−m−(2g−2)Tλ1 · · ·Tλm , g ≥ 2.

In these formulae, σ1, σ3 are related with m,n by

(1.16) σ1 =1

h− 1

m− 1

n, σ3 =

2

h3− 2

m3− 2

n3,

cg; λ1,...,λm(σ1, σ3), g ≥ 1 and Rg(σ1, σ3), g ≥ 2 are certain polynomials of σ1, σ3 satisfying thecondition

(1.17) degRg ≤ 3g − 3 with degσ1 = 1,deg σ3 = 3,

and the degree (3g − 3)-part of Rg(σ1, σ3) is given by

(1.18)(−1)g

2(2g − 2)!

|B2g||B2g−2|2g(2g − 2)

(σ313

− σ36

)g−1

,

where Bj denotes the jth Bernoulli number.

The proof of this proposition is given in Section 6.The coefficients Rg(σ1, σ3) appearing in the formula (1.15) are hard to compute. In [24] we

gave the expressions of R2 and R3. Based on the Hodge-FVH correspondence, in particularon the introduction of the previously mentioned tau-function τtop of the FVH, we will give anew algorithm for computing Rg. Indeed, the τtop corresponds to a solution utop(x, T ; ǫ) tothe FVH, whose initial value V (x; ǫ) = utop(x, T = 0; ǫ) ∈ Q[[x − 1; ǫ]] satisfies the followingdifference equation:

(1.19)∑

0≤α1≤···≤αm≤n

e∑m

j=1 V (x+αjmǫ−(j− 12)nǫ;ǫ) =

(m+ n

m

)x.

We will show in Section 3 that this equation has a unique solution in C[[x− 1; ǫ]]. We will alsoshow that this unique solution has the form

(1.20) V (x; ǫ) =1

mlog x+

∑

g≥1

ǫ2gPg(m,n)

x2g,

where Pg(m,n) ∈ Q[m,n,m−1]. Define a sequence of rational numbers Ck(m,n), k ≥ 0 by thegenerating function

(1.21)∑

k≥0

Ck(m,n)z2k =

nhz2

4 sinh(nz2

)sinh

(hz2

) = 1−n2 + h2

24z2+

7n4 + 10n2h2 + 7h4

5760z4+· · · .

4

These rational numbers have the following explicit expression:

(1.22) Ck(m,n) =

k∑

i=0

(1− 21−2i

) (1− 21−2k+2i

)

(2i)!(2k − 2i)!B2iB2k−2in

2ih2k−2i.

Proposition 1.3 The polynomials Rg(σ1, σ3) in Proposition 1.2 can be represented in termsof Pg(m,n) and Ck(m,n) as follows:

(1.23) Rg =(2g − 3)!

(mnh)g

(m

g∑

k=1

Cg−k(m,n)Pk(m,n)

(2k − 1)!− Cg(m,n)

), g ≥ 2.

By using the Hodge-FVH correspondence we also have the following theorem on the gapcondition satisfied by certain special cubic Hodge integrals.

Theorem 1.4 For the rational case satisfying the condition that one of p, q is equal to 1, thefree energies Hg(x, T ), g ≥ 1 satisfy the following gap condition:

H1(x, T )|TII=0 −σ1 − 1

24log x ∈ C[[x, TI]],(1.24)

Hg(x, T )|TII=0 −Rg(σ1, σ3)

x2g−2∈ C[[x, TI]], g ≥ 2.(1.25)

Here TI = (T1, T2, T3, . . . ) and TII = (Tµ)µ∈I\N.

The proof of the above theorem is given in Section 6.

Organization of the paper. In Section 2 we give two definitions of tau-functions of theFVH, and prove the equivalence of these definitions. In Sections 3–5 we prove Theorem 1.1. InSection 6, as applications of Theorem 1.1, we give a proof of Proposition 1.2 as well as a proofof the gap phenomenon for certain special cubic Hodge integrals described by Theorem 1.4.

Acknowledgements. We are grateful to Boris Dubrovin for his support of the work andvery helpful discussions. We would like to thank Shuai Guo, Yongbin Ruan and Don Zagier fortheir interests and helpful comments. This work is partially supported by NSFC No. 11771238,No. 11725104, No. 11671371. Part of the work of C. Zhou was done during his PhD studiesat Tsinghua University; he thanks Tsinghua University for excellent working conditions andfinancial supports.

2 Tau-functions of the FVH

In this section, we first recall the definition of the tau-function of an arbitrary solution u(x, T ; ǫ)to the FVH that is given in [26]. Then we give an alternative definition, following the onesfor the KP and 2D-Toda hierarchies [5, 33, 34], of the tau-function in terms of wave functionsof the integrable hierarchy, and show the equivalence of the two definitions. We will use thefirst definition of the tau-function in Section 3 to construct the topological tau-function τtop ofthe FVH, and to prove that it has the form of the genus expansion required for the derivation

5

of the loop equation [24] from the Virasoro constraints. The second definition will be used inSection 4 to prove that the topological tau-function τtop satisfies the Virasoro constraints.

Let us introduce some notations. We denote by Au the ring Au,0[ux, uxx, . . . ] of differentialpolynomials of u, where Au,0 denotes the ring of smooth functions of u. Note that the shiftoperator Λ = eǫ∂x acts on Au[[ǫ]]. Introduce a gradation deg on Au[[ǫ]], defined via the followingdegree assignments:

(2.1) deg ∂kxu = k, deg ǫ = −1,

where k ≥ 1. For convenience, we denote

Λ1 = Λ1q , Λ2 = Λ

1p , Λ3 = Λ

1p+ 1

q .

Then the operator L introduced in Theorem 1.1 has the form

(2.2) L = Λ2 + euΛ−11 .

We also extend the gradation deg to the difference operators of the form∑

j∈Z gjΛj3Λ

a withgj ∈ Au[[ǫ]], a ∈ C by assigning deg Λ = 0.

2.1 The definition of the tau-function

The following two-point correlation functions in Au[[ǫ]] were introduced in [26]:

Ωλ,µ =

Λ122

∑λ/pj=1

Λj3−1

Λ3−1

(resΛ3

(Λ−j3 L−λ

r

)resΛ3

(L−µ

r Λj3

)), λ ∈ I1,

Λ122

∑λ/qj=1

Λj3−1

Λ3−1

(resΛ3

(L−λ

r Λj3

)resΛ3

(Λ−j3 L−µ

r

)), λ ∈ I2,

(2.3)

where µ ∈ I, and resΛ3 means to take “residue” in the following sense:

(2.4) resΛ3

∑

i∈Z

aiΛi3 := a0.

In the above formula (2.3), L−µr ∈ Au[[ǫ]]((Λ

−13 )) when µ ∈ I1, and L−µ

r ∈ Au[[ǫ]]((Λ3)) whenµ ∈ I2. See in [26] for the precise definition of these fractional powers of L.

Proposition 2.1 ([26]) Let u = u(x, T ; ǫ) be an arbitrary solution in C [[x− 1, T ; ǫ]] to theFVH (1.8). Then the following formulae hold true for any λ, µ, ν ∈ I:

Ωλ,µ = Ωµ,λ,∂Ωλ,µ∂Tν

=∂Ωµ,ν∂Tλ

,(2.5)

(Λ3 − 1)Λ− 1

22 Ωλ,µ = ǫ

∂

∂TλresΛ3 L

−µr .(2.6)

It follows from Proposition 2.1 that for an arbitrary solution u = u(x, T ; ǫ) in C [[x− 1, T ; ǫ]]to the FVH, there exists τ = τ(x, T ; ǫ) ∈ C((ǫ))[[x − 1, T ]] satisfying the relations

u =

(Λ

123 − Λ

− 12

3

)(Λ

121 − Λ

− 12

1

)log τ,(2.7)

6

ǫ (Λ3 − 1) Λ− 1

22

∂ log τ

∂Tλ= resΛ3 L

−λr , λ ∈ I,(2.8)

ǫ2∂2 log τ

∂Tλ∂Tµ= Ωλ,µ, λ, µ ∈ I.(2.9)

We call τ the tau-function of the solution u to the FVH. Note that the tau-function τ is uniquelydetermined by (2.7)–(2.9) up to multiplying by a factor of the form

(2.10) exp

ax+ b+

∑

λ∈I

dλTλ

,

where a, b, dλ ∈ C((ǫ)) are arbitrary constant Laurent series in ǫ.

Lemma 2.2 The following relations hold true for λ, µ ∈ I:

resΛ3 L−λ/r − cλe

λu/p ∈ ǫAu[[ǫ]],(2.11)

Ωλ,µ −1

p+ q

λµ

λ+ µcλcµe

(λ+µ)u/p ∈ ǫAu[[ǫ]],(2.12)

deg resΛ3 L−λ/r = 0, degΩλ,µ = 0.(2.13)

Here the constants cλ, cµ are defined in (1.6).

Proof From the definition of the fractional powers of L given in [26] we have, for any k ∈ Z,

resΛ3 L−λ/rΛ−k

3 −( −λ/rλ/p − k

)e(λ/p−k)u ∈ ǫAu[[ǫ]], λ ∈ I1,(2.14)

resΛ3 L−λ/rΛ−k

3 −( −λ/rλ/q + k

)e(λ/p−k)u ∈ ǫAu[[ǫ]], λ ∈ I2,(2.15)

deg resΛ3 L−λ/rΛ−k

3 = 0, λ ∈ I.(2.16)

Then by using the definition (2.3) of Ωλ,µ we find, for any λ, µ ∈ I1, that degΩλ,µ = 0 and

Ωλ,µ −1

p+ q

λµ

λ+ µcλcµe

(λ+µ)u/p = Ωλ,µ −λ/p∑

j=1

j

( −λ/rλ/p − j

)( −µ/rµ/p+ j

)e(λ+µ)u/p ∈ ǫAu[[ǫ]].

In the above equality, the following elementary identity (cf. [26]) is used:

λ/p∑

j=1

j

( −λ/rλ/p− j

)( −µ/rµ/p+ j

)=

1

p+ q

λµ

λ+ µcλcµ, λ, µ ∈ I1.

This proves (2.11)–(2.13) for the case λ, µ ∈ I1. The proof for the rest cases is similar.

2.2 An alternative definition of the tau-function

The tau-function of a solution to the FVH can also be defined, following the approach of[5, 33, 34], via the introduction of wave functions for the FVH. To this end, we first introducethe dressing operators.

7

Lemma 2.3 For an arbitrary solution u = u(x, T ; ǫ) in C[[x−1, T ; ǫ]] to the FVH (1.8), thereexist difference operators Φ1,Φ2, called the dressing operators, of the form

(2.17) Φ1(x, T ; ǫ) =∑

i≥0

a1,i(x, T ; ǫ)Λ−i3 , Φ2(x, T ; ǫ) =

∑

i≥0

a2,i(x, T ; ǫ)Λi3

with a1,i, a2,i ∈ C((ǫ))[[x − 1, T ]], i ≥ 0 and a1,0(x, T ; ǫ) ≡ 1, such that

L = Φ1Λ2Φ−11 = Φ2Λ

−11 Φ−1

2 ,(2.18)

ǫ∂Φ1

∂Tµ= −

(Φ1Λ

µ/p3 Φ−1

1

)−Φ1, ǫ

∂Φ2

∂Tµ=(Φ1Λ

µ/p3 Φ−1

1

)+Φ2, µ ∈ I1,(2.19)

ǫ∂Φ1

∂Tµ= −

(Φ2Λ

−µ/q3 Φ−1

2

)−Φ1, ǫ

∂Φ2

∂Tµ=(Φ2Λ

−µ/q3 Φ−1

2

)+Φ2, µ ∈ I2.(2.20)

These two operators Φ1 and Φ2 are uniquely determined by u up to right-multiplications byoperators of the form

Φ1 7→ Φ1

(1 + c1,1Λ

−13 + c1,2Λ

−23 + · · ·

), Φ2 7→ Φ2

(c2,0 + c2,1Λ3 + c2,2Λ

23 + · · ·

),

where ci,j are arbitrary Laurent series of ǫ with constant coefficients.

Proof The equation L = Φ1Λ2Φ−11 is equivalent to the following recursion relations:

(Λ2 − 1) a1,1 = −eu,(Λ2 − 1) a1,i = −euΛ−1

1 a1,i−1, i ≥ 2.

On another hand, by using the FVH (1.8) one can check that equations (2.18)–(2.20) for Φ1

are compatible, from which we know the existence of Φ1. If we have another operator Φ′1 which

satisfies the same equations as for Φ1, then Φ−11 Φ′

1 satisfies the following two equations:

[Φ−11 Φ′

1, Λ2

]= 0, ǫ∂Tµ

(Φ−11 Φ′

1

)= 0, ∀µ ∈ I.

Hence Φ−11 Φ′

1 must have the form

Φ−11 Φ′

1 = 1 + c1,1Λ−13 + c1,2Λ

−23 + · · · .

The assertion of the lemma for Φ2 can be proved in a similar way.

Given a solution u = u(x, T ; ǫ) ∈ C[[x− 1, T ; ǫ]] to the FVH (1.8), we have the associatedwave functions ψ1, ψ2 defined by

(2.21) ψ1 := Φ1eϑ1 , ψ2 := Φ2e

ϑ2 ,

where Φ1, Φ2 are the dressing operators,

ϑ1 = ϑ1(x, T ; ǫ; z) := −rxǫlog z +

∑

µ∈J1

Tµǫz

µp +

1

2

∑

µ∈J3

Tµǫz

µp ,(2.22)

ϑ2 = ϑ2(x, T ; ǫ; z) := rx

ǫlog z −

∑

µ∈J2

Tµǫz

µq − 1

2

∑

µ∈J3

Tµǫz

µq ,(2.23)

8

and the index sets J1, J2, J3 are defined by

(2.24) J1 := I1\I2, J2 := I2\I1, J3 := I1 ∩ I2.

It is easy to verify that ψ1, ψ2 satisfy the following equations:

(2.25) Lψi = λiψi, ǫ∂ψi∂Tµ

= Pµψi, µ ∈ I, i = 1, 2.

Here λ1 = z−r/p, λ2 = z−r/q, and

(2.26) Pµ :=

(L−µ

r

)+, µ ∈ J1,

−(L−µ

r

)−, µ ∈ J2,

12

(L−µ

r

)+− 1

2

(L−µ

r

)−, µ ∈ J3.

We also define the dual wave functions ψ∗1 and ψ∗

2 by

(2.27) ψ∗1 :=

(Φ−11

)∗e−ϑ1 , ψ∗

2 :=(Φ−12

)∗e−ϑ2 .

Here we recall that Λ∗ := Λ−1, and for two difference operators A and B,

(2.28) (AB)∗ := B∗A∗.

Theorem 2.4 A formal series u ∈ C[[x−1, T ; ǫ]] satisfies the FVH if and only if the associatedwave functions and their duals satisfy the following bilinear equations for all ℓ ∈ Z:

resz ψ1(x, T ; ǫ; z)ψ∗1

(x− ℓ

rǫ, T ′; ǫ; z

)dzz

= resz ψ2(x, T ; ǫ; z)ψ∗2

(x− ℓ

rǫ, T ′; ǫ; z

)dzz.(2.29)

Lemma 2.5 Two operator-valued formal series of the form

D1(x, T ) ∈ C((Λ−13 , ǫ

))[[x− 1, T ]], D2(x, T ) ∈ C ((Λ3, ǫ)) [[x− 1, T ]]

are equal if and only if the following identity holds true for all ℓ ∈ Z:

reszD1(x, T ) (ψ1(x, T ; z)) Λℓ3

(ψ∗1(x, T

′; z)) dzz

(2.30)

= reszD2(x, T ) (ψ2(x, T ; z)) Λℓ3

(ψ∗2(x, T

′; z)) dzz.

The proofs of the above Theorem 2.4 and Lemma 2.5 are standard (cf. [1]).Let us now introduce the notion of tau-function via the wave functions. The following

notations will be used. Denote by φi and φ∗i , i = 1, 2 the following formal series:

φ1(x, T ; ǫ; z) := Φ1(x, T ; ǫ)|Λ3→z, φ2(x, T ; ǫ; z) := Φ2(x, T ; ǫ)|Λ3→z−1 ,(2.31)

φ∗1(x, T ; ǫ; z) := Φ∗1(x, T ; ǫ)|Λ3→z−1 , φ∗2(x, T ; ǫ; z) := Φ∗

2(x, T ; ǫ)|Λ3→z.(2.32)

Clearly, we have

φ1 =∑

i≥0

a1,i(x, T ; ǫ) z−i, φ2 =

∑

i≥0

a2,i(x, T ; ǫ) z−i.

9

For an arbitrary function f(T ), denote

f(T −

[z−1]1

)= exp

(−∑

µ∈I1

z−µ/p

µ/pǫ∂

∂Tµ

)f(T ),

f(T −

[z−1]2

)= exp

(−∑

µ∈I2

z−µ/q

µ/qǫ∂

∂Tµ

)f(T ).

Theorem 2.6 For an arbitrary solution u(x, T ; ǫ) in C[[x − 1, T ; ǫ]] to the FVH, there existsa choice of the dressing operators Φi, i = 1, 2 such that the associated formal series φi, φ

∗i ,

i = 1, 2 can be represented in terms of a certain formal power series τ ∈ C((ǫ))[[x − 1, T ]] asfollows:

φ1(x, T ; ǫ; z) =τ(x, T − [z−1]1; ǫ)

τ(x, T ; ǫ), φ2(x, T ; ǫ; z) =

τ(x− ǫ/r, T − [z−1]2; ǫ)

τ(x, T ; ǫ),(2.33)

φ∗1(x, T ; ǫ; z) =τ(x− ǫ/r, T + [z−1]1; ǫ)

τ(x− ǫ/r, T ; ǫ), φ∗2(x, T ; ǫ; z) =

τ(x, T + [z−1]2; ǫ)

τ(x− ǫ/r, T ; ǫ).(2.34)

We call (Φ1,Φ2, τ) a dressing triple for the FVH. We also call the formal power series τ theSato type tau-function of the solution u.

Proof We prove the theorem by employing the method used in [34]. Firstly, by using thebilinear equations (2.29) one can prove that

φ1(x, T ; ǫ; ζ)φ∗1

(x+

ǫ

r, T − [ζ−1]1 −

[ξ−1]1; ǫ; ζ

)(2.35)

= φ1(x, T ; ǫ; ξ)φ∗1

(x+

ǫ

r, T − [ζ−1]1 −

[ξ−1]1; ǫ; ξ

),

φ1(x, T ; ǫ; ζ)φ∗1

(x, T − [ζ−1]1 −

[ξ−1]2; ǫ; ζ

)(2.36)

= φ2(x, T ; ǫ; ξ)φ∗2

(x, T − [ζ−1]1 −

[ξ−1]2; ǫ; ξ

),

φ2(x, T ; ǫ; ζ)φ∗2

(x− ǫ

r, T − [ζ−1]2 −

[ξ−1]2; ǫ; ζ

)(2.37)

= φ2(x, T ; ǫ; ξ)φ∗2

(x− ǫ

r, T − [ζ−1]2 −

[ξ−1]2; ǫ; ξ

).

Comparing the coefficients of ζ0 of both sides of (2.35) and of (2.36) respectively, we obtain

(2.38) φ1(x, T ; ǫ; ξ)φ∗1

(x+

ǫ

r, T −

[ξ−1]1; ǫ; ξ

)= 1, φ2(x, T ; ǫ; ξ)φ

∗2

(x, T −

[ξ−1]2; ǫ; ξ

)= 1.

By using these two relations we can eliminate φ∗1 and φ∗2 in (2.35)–(2.37) and obtain

φ1(x, T −

[ξ−1]1; ǫ; ζ

)φ1 (x, T ; ǫ; ξ) = φ1

(x, T −

[ζ−1]1; ǫ; ξ

)φ1 (x, T ; ǫ; ζ) ,(2.39)

φ1

(x− ǫ

r, T −

[ξ−1]2; ǫ; ζ

)φ2 (x, T ; ǫ; ξ) = φ2

(x, T − [ζ−1]1; ǫ; ξ

)φ1 (x, T ; ǫ; ζ) ,(2.40)

φ2

(x− ǫ

r, T −

[ξ−1]2; ǫ; ζ

)φ2 (x, T ; ǫ; ξ) = φ2

(x− ǫ

r, T − [ζ−1]2; ǫ; ξ

)φ2 (x, T ; ǫ; ζ) .(2.41)

Equations (2.39) and (2.41) imply the existence of two formal series

τ1, τ2 ∈ C((ǫ))[[x − 1, T ]]

10

satisfying the equations

φ1(x, T ; ǫ; z) =τ1(x, T − [z−1]1; ǫ)

τ1(x, T ; ǫ), φ2(x, T ; ǫ; z) =

τ2(x− ǫ/r, T − [z−1]2; ǫ)

τ2(x, T ; ǫ).

Define f = log τ2 − log τ1, then we obtain from equation (2.40) that

f(x− ǫ

r, T −

[ξ−1]1− [ζ−1]2; ǫ

)+ f (x, T ; ǫ)

=f(x− ǫ

r, T − [ζ−1]2; ǫ

)+ f

(x, T −

[ξ−1]1; ǫ),

whose general solution in C((ǫ))[[x− 1, T ]] has the form

f(x, T ; ǫ) = f1(x, T ′′; ǫ

)+ f2

(T ′; ǫ

)+ f0(T

′′′; ǫ) + b(ǫ),

where

T ′ = (Tµ | µ ∈ J1) , T ′′ = (Tµ | µ ∈ J2) , T ′′′ = (Tµ | µ ∈ J3) ,

and f0 ∈ C((ǫ))[[T ′′′]] is linear in T ′′′,

f1(x, T ′′; ǫ

)∈ C((ǫ))[[x − 1, T ′′]], f2

(T ′; ǫ

)∈ C((ǫ))[[T ′]], b(ǫ) ∈ C((ǫ)).

Let τ = τ1ef1 , then we have

log φ1 (x, T ; ǫ; z) = log τ(x, T − [z−1]1; ǫ

)− log τ (x, T ; ǫ) ,

log φ2 (x, T ; ǫ; z) = log τ(x− ǫ

r, T − [z−1]2; ǫ

)− log τ (x, T ; ǫ)− logC (ǫ; z)

for some C(ǫ; z) ∈ C[[ǫ; z−1/q]].Finally, by replacing Φ2 with Φ2 C

(ǫ; Λ−1

3

), we find that Φ1, the new Φ2, and τ(x, T ; ǫ)

satisfy the equations given in (2.33). The equations given in (2.34) then follow from (2.38).The theorem is proved.

We note that for a fixed solution u ∈ C[[x − 1, T ; ǫ] the Sato type tau-function τ of thesolution is uniquely determined up to a factor of the form (2.10).

2.3 The equivalence of the two definitions of the tau-function

For an arbitrary solution u in C[[x− 1, T ; ǫ]] to the FVH, let τs be the tau-function defined inSection 2.2. We prove in this subsection the equivalence of τs with the tau-function defined inSection 2.1. In other words, we are to prove that τs satisfies the relations (2.7)–(2.9). Let usstart with the proof of the following lemma.

Lemma 2.7 The following formula holds true:

(2.42) ǫ(Λ3 − 1)∂ log τs∂Tµ

= resΛ3 L−µ/r, µ ∈ I.

11

Proof Let Φ1,Φ2 be the dressing operators for the FVH such that Φ1,Φ2, τs form a dressingtriple. Recall that Φ2 =

∑i≥0 a2,iΛ

i3. By using the second formula of (2.33) we obtain

(2.43) (Λ3 − 1) log τs(x, T ; ǫ) = log a2,0(x, T ; ǫ).

Also, it follows from the equations in (2.19), (2.20) that

(2.44) ǫ∂a2,0∂Tµ

=(resΛ3 L

−µ/r)a2,0.

The lemma is then proved by substituting (2.43) into (2.44).

Lemma 2.8 ∀λ, µ ∈ I, the following relations hold true:

ǫ2∂2 log τs∂Tλ∂Tµ

∈ Au[[ǫ]],(2.45)

deg ǫ2∂2 log τs∂Tλ∂Tµ

= 0.(2.46)

Proof Let ψ1, ψ2 be the wave functions associated to Φ1,Φ2 respectively. It follows from(2.21), (2.31) that

(2.47) ǫ∂Tµφ1

φ1= ǫ

∂Tµψ1

ψ1− ǫ∂Tµϑ1 =

Pµψ1

ψ1− ǫ∂Tµϑ1 = − 1

ψ1

(L−µ/r

)−ψ1, µ ∈ I.

Observe that the operator(L−µ/r

)−can be represented in the form

(2.48)(L−µ/r

)−= −

∑

k≥1

fµ,k(u, ux, . . . )Lkp/r,

where fµ,k = fµ,k(u, ux, . . . ; ǫ) ∈ Au[[ǫ]] are given by the following recursion relations:

(2.49) fµ,k = − resΛ3

(L−µ/rΛk3

)−k−1∑

j=1

fµ,j resΛ3

(Ljp/rΛk3

), k ≥ 1.

Then from Lemma 2.2 it follows that deg fµ,k = 0. By using the relation L−p/rψ1 = zψ1 wefind that the formula (2.47) can be written as

(2.50) ǫ∂Tµφ1

φ1=∑

k≥1

fµ,kz−k ∈ Au[[ǫ]][[z

−1]], µ ∈ I.

From Theorem 2.6 we know that

ǫ∂Tµφ1

φ1= −z−1ǫ2∂Tµ∂Tp log τs −

z−2

2

(ǫ2∂Tµ∂T2p log τs − ǫ3∂Tµ∂

2Tp log τs

)+ · · · .(2.51)

Then the relations (2.45)–(2.46) for λ ∈ I1, µ ∈ I follow from (2.50) and (2.51). The relations(2.45)–(2.46) for λ ∈ I2, µ ∈ I can be proved in a similar way. The lemma is proved.

12

Lemma 2.9 The Sato type tau-function τs satisfies the relations

(2.52) ǫ2∂2 log τs∂Tλ∂Tµ

− 1

p+ q

λµ

λ+ µcλcµe

(λ+µ)u/p ∈ ǫAu[[ǫ]], ∀λ, µ ∈ I.

Proof Consider the case λ = kp, µ = lp with k, l ≥ 1. From the formulae (2.14), (2.15) and(2.49), it follows that the differential polynomials fkp,l must satisfy

fkp,l − gkp,le(k+l)u ∈ ǫAu[[ǫ]],

where the constants gkp,l satisfy the recursion relation

gkp,l = −(−kp/rk + l

)−

l−1∑

j=1

(jp/r

l − j

)gkp,j, l ≥ 1.

Solving this recursion relation we obtain

(2.53) gkp,l = − p

p+ q

k

k + lckpclp.

Here the numbers cµ, µ ∈ I are defined in (1.6). Therefore, from (2.50) we have

(2.54) ǫ∂ log φ1∂Tkp

= −∑

l≥1

p

p+ q

k

k + lckpclpe

(k+l)uz−l +Gk

for some Gk = Gk(u, ux, . . . ; ǫ; z) ∈ ǫAu[[ǫ]][[z−1]] satisfying degGk = 0.

On another hand, from the definition of the Sato type tau-function, we know that thederivatives of log τs with respect to T ′s are Laurent series in ǫ with degree being greater orequal to −2. So from (2.33) it follows that

ǫ∂ log φ1∂Tkp

= ǫ

(e−

∑l≥1

1

lzlǫ ∂∂Tlp − 1

)∂ log τs∂Tkp

= −∑

l≥1

ǫ2

l

∂2 log τs∂Tkp∂Tlp

z−l + Gk

for some Gk = Gk(u, ux, . . . ; ǫ; z) ∈ ǫAu[[ǫ]][[z−1]] satisfying deg Gk = 0. By comparing this

formula with (2.54) we obtain

ǫ2∂2 log τs∂Tkp∂Tlp

− p

p+ q

kl

k + lckpclpe

(k+l)u ∈ ǫAu[[ǫ]].

Similarly, we can prove the cases when λ and µ take other values. The lemma is proved.

Proposition 2.10 The function τs

(x+ ǫ

2p , T ; ǫ)satisfies the defining relations (2.7)–(2.9) for

the tau-function of the FVH.

Proof From the equation (2.18) it follows that

u(x, T ; ǫ) =(1− Λ−1

1

)log a2,0(x, T ; ǫ).

13

Therefore, by using the formula (2.43) we know that τs

(x+ ǫ

2p , T ; ǫ)satisfies the equation (2.7),

i.e.

u(x, T ; ǫ) =(Λ1/23 − Λ

−1/23

)(Λ1/21 − Λ

−1/21

)log τs

(x+

ǫ

2p, T ; ǫ

).(2.55)

Let us note that the FVH (1.8) can be represented in the form

(2.56)∂u

∂Tµ= ǫ−1

(Λ1/21 − Λ

−1/21

)Λ−1/21 resΛ3 L

µh.

So by differentiating both sides of (2.55) with respect to Tµ and Tλ, and by using (2.6) weobtain

(Λ1/23 − Λ

−1/23

)(Λ1/21 − Λ

−1/21

)Ωλ,µ

= ǫ2(Λ1/23 − Λ

−1/23

)(Λ1/21 − Λ

−1/21

) ∂2 log τs(x+ ǫ2p , T ; ǫ)

∂Tλ∂Tµ, λ, µ ∈ I.

Combining with Lemma 2.8 we find that

∆λ,µ := ǫ2∂2 log τs(x+ ǫ

2p , T ; ǫ)

∂Tλ∂Tµ−Ωλ,µ ∈ Ker

(∂2x : Au[[ǫ]] → Au[[ǫ]]

).

Thus it follows from Lemmas 2.2, 2.8, 2.9 that deg∆λ,µ = 0 and ∆λ,µ ∈ ǫAu[[ǫ]], so ∆λ,µ mustbe zero. The proposition is then proved by using Lemma 2.7.

3 The topological tau-function of the FVH

In this section, we restrict our discussions to the rational case, i.e.

(3.1) p =1

m, q =

1

n, r = −1

h,

where m,n are coprime positive integers, and h = m+ n.In our previous work [24], we showed that the formal power series Z(x, T ; ǫ) defined in (1.10)

satisfies the following Virasoro constraints:

(3.2) LkZ(x, T ; ǫ) = 0, k ≥ 0,

where

L0 =∑

µ∈I

µTµ∂

∂Tµ+

x2

2mnhǫ2+σ124

− Γ(m)Γ(n)

Γ(1 + h)

∂

∂T1,(3.3)

Lk =∑

µ∈I

µTµ∂

∂Tµ+k+

x

mn

∂

∂Tk− Γ(m)Γ(n)

Γ(1 + h)

∂

∂T1+k(3.4)

+ǫ2

2m

∑

λ+µ=kλ,µ∈I1

∂2

∂Tλ∂Tµ+ǫ2

2n

∑

λ+µ=kλ,µ∈I2

∂2

∂Tλ∂Tµ, k ≥ 1,

14

and σ1 is defined in (1.16). From the definitions (1.2), (1.10) we know that Z(x, T ; ǫ) has thegenus expansion

logZ(x, T ; ǫ) =1

mnhǫ2A+

∑

g≥0

(mnh)g−1ǫ2g−2Hg(t(x, T )),(3.5)

where A and Hg(t(x, T )), g ≥ 0 are defined in (1.11) and (1.13) respectively. From [8] we know

that the genus zero part H0(t(x, T )) corresponds to a particular solution to the dispersionlessFVH, and that Hg(t(x, T )), g ≥ 1 satisfy certain technical conditions. Based on this we provedin [24] that the Virasoro constraints given in (3.2) lead to the so-called loop equation, whichprovides a way of determining Hg(t(x, T )), g ≥ 1 uniquely.

In order to prove Theorem 1.1, we are to construct in this section a particular tau-functionτtop of the FVH which has the genus expansion and satisfies the same technical conditions, andto prove in the next sections that it satisfies the Virasoro constraints Lkτtop = 0, k ≥ 0. ThenTheorem 1.1 in the rational case follows from the uniqueness theorem of the loop equationthat was proved in [24]. The theorem for the general case will be proved from the one for therational case by using a continuation argument.

The results of this section are given in the following theorem.

Theorem 3.1 For the rational case, there exists a nontrivial tau-function

τtop = τtop(x, T ; ǫ) ∈ C((ǫ2))[[x − 1, T ]]

of the FVH (1.8) satisfying the following two equations:

L0τtop = 0,(3.6)

K0τtop +τtop24

= 0,(3.7)

where L0 is defined in (3.3), and K0 is a linear operator given by

K0 :=∑

µ∈I

Tµ∂

∂Tµ+ x

∂

∂x+ ǫ

∂

∂ǫ− Γ(m)Γ(n)

Γ(1 + h)

∂

∂T1.(3.8)

Moreover, this tau-function possesses the genus expansion

(3.9) log τtop(x, T ; ǫ) =∑

g≥0

ǫ2g−2Fg(x, T )

for some Fg(x, T ) ∈ C[[x − 1, T ]], g ≥ 0. Furthermore, for g ≥ 1, there exist functionsFg = Fg(z0, . . . , z3g−2) such that

(3.10) Fg(x, T ) = Fg

(vtop, v

′top, . . . , v

(3g−2)top

)

and

F1 =1

24log z1 −

mh+ n2

24nhz0,(3.11)

∑

i≥1

izi∂Fg∂zi

= (2g − 2)Fg, Fg ∈ z−(4g−4)1 C[z1, . . . , z3g−2], g ≥ 2.(3.12)

Here vtop := ∂2xF0 and ′ = ∂x.

15

We note that the non-trivial tau-function τtop in the above theorem is uniquely determinedby (3.6)–(3.7) up to a constant factor that is irrelevant for the current study. Thus the iden-tity (3.10) for g = 1 should be understood as under the suitable choice of the constant factor.In other words, we use the g = 1 identity of (3.10) to determine the constant for τtop. We callthis τtop the topological tau-function of the FVH. Our construction of τtop will consist of fivesteps:Step I. To fix the particular solution utop(x, T ; ǫ) to the FVH, corresponding to τtop, by usingthe initial value problem for the FVH which is implied by the Virasoro constraint L0τtop = 0.Step II. To show that the solution utop(x, T ; ǫ) has the following form of genus expansion:

(3.13) utop(x, T ; ǫ) = vtop +∑

g≥1

ǫ2gAg

(v′top, v

′′top, . . . , v

(3g)top

),

where vtop(x, T ) = utop(x, T ; 0), v′top = ∂xvtop, v

(k)top = ∂kxvtop.

Step III. To show a certain transcendency of the function vtop.

Step IV. To show the existence of functions Fg(z0, z1, . . . , z3g−2) such that

(3.14) Ag

(v′top, v

′′top, . . . , v

(3g)top

)= ∂2xFg

(vtop, v

′top, v

′′top, . . . , v

(3g−2)top

), g ≥ 1.

Step V. To construct the tau function τtop satisfying the conditions (2.7)–(2.9), and show thatlog τtop has the genus expansion of the following form:

(3.15) log τtop =1

ǫ2F0(x, T ) +

∑

g≥1

ǫ2g−2Fg

(vtop, v

′top, . . . , v

(3g−2)top

),

where F0(x, T ) satisfies the relation vtop(x, T ) = ∂2xF0(x, T ).Proof of Theorem 3.1. Let us prove the theorem step by step as outlined above.Step I. To find the solution utop(x, T ; ǫ) to the FVH, let us fix its initial value utop(x, 0; ǫ)by using the condition (3.6). Assuming the existence of such a solution with the associated

tau-function τtop, we apply the operator ǫ(Λ3−1)Λ−1/22 to both sides of the equation

L0τtopτtop

= 0,and put T = 0, then we arrive at the equation

(3.16) ǫ(Λ3 − 1)Λ−1/22

(x2

2mnhǫ2− Γ(m)Γ(n)

Γ(1 + h)

∂ log τtop∂T1

)∣∣∣∣T=0

= 0.

Using the defining relation (2.8) for the tau-functions and the fact that

ǫ(Λ3 − 1)Λ−1/22 = ǫ

(hǫ∂x +

nh

2ǫ2∂2x +O

(ǫ3)),

we obtain from (3.16) the following equation:

(3.17)(resΛ3 L

h)∣∣∣T=0

=

(m+ n

m

)(x+

nǫ

2

).

It follows from (2.2) and (3.1) that

resΛ3 Lh =

∑

0≤α1≤···≤αm≤n

exp

m∑

j=1

utop (x+ ǫαjm− ǫ(j − 1)n, T ; ǫ)

.(3.18)

16

By using (3.17), (3.18) and by making the shift x → x − n2 ǫ we arrive at the equation (1.19)

for utop(x, 0; ǫ).Let us show that there exists a unique solution in C[[x− 1; ǫ]] to the equation (1.19). First

we observe that the equation (1.19) can be written in the form

(3.19) emV (x;ǫ)

(1 +

∑

k≥1

ǫk∑

l≥1

∑

j1,...,jl≥1j1+···+jl=k

Cj1,...,jl∂j1x V (x; ǫ) · · · ∂jlx V (x; ǫ)

)= x,

where Cj1,...,jl are constants depending onm,n. Then by substituting V (x; ǫ) =∑

k≥0 ǫkV [k](x)

into (3.19), we find that V [0](x) = 1m log x, and that V [k](x), k ≥ 1 can be uniquely solved in a

recursive way. Moreover, V [k](x) ∈ C[[x−1]] for k ≥ 0. This shows the existence and uniquenessof solution in C[[x− 1; ǫ]] to the equation (1.19). Combining this with the statement that theFVH (1.8) is integrable [26], we arrive at the following proposition.

Proposition 3.2 There is a unique solution

utop = utop(x, T ; ǫ) ∈ C[[x− 1, T ; ǫ]]

to the FVH satisfying the initial condition

(3.20) utop(x, 0; ǫ) = V (x; ǫ),

where V (x; ǫ) is the unique solution in C[[x − 1; ǫ]] to the equation (1.19). We call utop thetopological solution to the FVH.

We show next that V (x; ǫ) and utop(x, T ; ǫ) can be represented as power series of ǫ2. Tosee this let us introduce some notations. Let Y denote the set of partitions of non-negativeintegers, i.e.,

Y :=

(J1, J2, . . . ) ∈ ZN | J1 ≥ J2 ≥ · · · ≥ 0,

∑

i≥1

Ji <∞.

For a partition J = (J1, J2, . . . ) ∈ Y, denote by l(J) the number of non-zero components of J ,by |J | := J1 + J2 + . . . the weight of J , and by Yk the set of all partitions of weight k. Wehave the following lemma.

Lemma 3.3 For each λ ∈ I, we have

(3.21) Λ− 1

21 resΛ3 L

λh = cλeλmu

∑

g≥0

ǫ2gM[g]λ

(u′, · · · , u(2g)

),

where cλ is defined in (1.6),

(3.22) M[g]λ :=M

[g]λ

(u′, · · · , u(2g)

)=∑

J∈Y2g

aJ(λ)u(J) ∈ Au, g ≥ 0,

andu(k) := ∂kxu, u(J) := u(J1) · · · u(Jl(J)), k ≥ 0, J = (J1, J2, . . . ).

Moreover, the coefficient a∅(λ) = 1, and the coefficients aJ(λ), J ∈ Yeven are polynomials of λwith degrees less than or equal to l(J) + |J |/2.

17

Proof We first show that Λ− 1

21 resΛ3 L

λh ∈ Au[[ǫ2]] for each λ ∈ I. Indeed,

Λ− 1

21 resΛ3 L

λh = Λ− 1

21 resΛ3

(Λ−12 + eΛ1uΛ1

)λh(3.23)

= Λ121 resΛ3

(Λ−12 + euΛ1

)λh,(3.24)

which shows that Λ− 1

21 resΛ3 L

λh is invariant under the map ǫ 7→ −ǫ. Here, (3.23) is valid dueto the fact that

resΛ3

∑

i∈Z

αi Λi3 = resΛ3

∑

i∈Z

Λ−i3 αi,

and (3.24) is valid because of the fact that

resΛ3 Lλh∣∣u 7→Λ1u

= Λ1 resΛ3 Lλh.

For λ = 1/m, we have

(3.25) Λ−1/21 resΛ3 L

h/m =Λ1/23 − Λ

−1/23

Λ1/22 − Λ

−1/22

eu =h

meu(1 +

∑

g≥1

ǫ2gM[g]1/m

),

from which it follows that M[g]1/m ∈ C

[u′, . . . , u(2g)

]have the form (3.22). Note that Proposi-

tion 2.1 implies the following tau-symmetry property [8, 12]:

∂

∂T1/m

(Λ− 1

21 resΛ3 L

λh

)=

∂

∂Tλ

(Λ− 1

21 resΛ3 L

h/m

), ∀λ ∈ I.(3.26)

From this equation we arrive at a certain recursion relation for M[g]λ , g ≥ 1, and we find that

M[g]λ has the required form (3.22) with the coefficients aJ(λ) being polynomials of λ with degree

less than or equal to l(J) + |J |/2. The lemma is proved.

Lemma 3.4 The unique solution in Q[[x− 1; ǫ]] to the difference equation (1.19) can be rep-resented in the form

(3.27) V (x; ǫ) =1

mlog x+

∑

g≥1

ǫ2gPg(m,n)

x2g,

where Pg(m,n) ∈ Q[m,n,m−1

]for g ≥ 1.

Proof Note that the equation (1.19) can be written as

(3.28) resΛ3

(Λm + eV (x−nǫ

2;ǫ)Λ−n

)h=

(m+ n

m

)x.

From Lemma 3.3 it follows that this equation can also be represented in the form

(3.29) emV∑

g≥0

ǫ2gM[g]1

(V (1), . . . , V (2g)

)= x,

18

where the coefficients M[g]1 are obtained recursively by using the tau-symmetry condition (3.26)

with λ = 1. Then by substituting the expression (3.27) of V (x; ǫ) into (3.29) one can determinethe coefficients Pg(m,n) recursively, and show that they belong to Q[m,n,m−1]. The lemmais proved.

We will give the explicit expressions of some Pg(m,n) in Section 6.From the Lemmas 3.3, 3.4 we know that the right-hand side of (2.56) is in Au[[ǫ

2]], andthat the power series V (x; ǫ) is in C[[x− 1; ǫ2]], thus we arrive at the following corollary.

Corollary 3.5 The topological solution utop is an element of C[[x− 1, T ; ǫ2]].

Step II. Let us proceed to prove that utop(x, T ; ǫ) has the genus expansion (3.13) by using theso-called Euler–Lagrange equation.

Proposition 3.6 The formal power series u := utop satisfies the following equation:

(3.30)∑

λ∈I

λTλresΛ3Lλh +

x+ n2 ǫ

mn= 0,

where Tλ = Tλ − Γ(m)Γ(n)Γ(1+h) δλ,1.

Proof Define a power series F (x, T ; ǫ) ∈ C[[x− 1, T ; ǫ]] by

(3.31) F (x, T ; ǫ) :=∑

λ∈I

λTλ resΛ3 Lλhtop +

x+ n2 ǫ

mn,

where Ltop := Λ2 + eutopΛ−11 . We are to show F (x, T ; ǫ) ≡ 0. By using (2.5), (2.6), (2.56) and

(3.21) we obtain, after a lengthy calculation, that

∂F (x, T ; ǫ)

∂Tµ=∑

k≥0

∂(resΛ3 L

µhtop

)

∂u(k)∂kxF (x, T ; ǫ), ∀µ ∈ I.(3.32)

Here F (x, T ; ǫ) is a power series in C[[x− 1, T ; ǫ]] defined by

(3.33) F (x, T ; ǫ) := ǫ−1(1− Λ−1

1

)F (x, T ; ǫ) =

∑

λ∈I

λTλ∂utop∂Tλ

+1

m.

By taking T = 0 in (3.31) we obtain

F (x, 0; ǫ) = −Γ(m)Γ(n)

Γ(1 + h)resΛ3 L

htop

∣∣∣T=0

+x+ n

2 ǫ

mn.

Thus F (x, 0; ǫ) vanishes due to the fact that utop(x, 0; ǫ) = V (x; ǫ) satisfies the equation (3.28)(or equivalently (1.19)). So from (3.33) it follows that

(3.34) F (x, 0; ǫ) = ǫ−1(1− Λ−1

1

)F (x, 0; ǫ) = 0.

Then by mathematical induction we know from (3.32), (3.33) that

(3.35)∂iF

∂Tµ1 · · · ∂Tµi(x, 0; ǫ) = 0, ∀ i ≥ 0, µ1, . . . , µi ∈ I,

19

which leads to the vanishing of F (x, T ; ǫ). The proposition is proved.

We call equation (3.30) the Euler–Lagrange equation for the FVH.Define a power series vtop ∈ C[[x− 1, T ]] by

(3.36) vtop = vtop(x, T ) := utop(x, T ; ǫ = 0).

Then it satisfies the following dispersionless FVH:

(3.37)∂v

∂Tλ= λmncλe

λmv ∂v

∂x, λ ∈ I.

It follows from Proposition 3.6 that vtop also satisfies the following equation:

(3.38)∑

λ∈I

λcλTλeλmv +

x

mn= 0.

We call (3.38) the dispersionless Euler-Lagrange equation for the FVH.

Lemma 3.7 There exist functions

(3.39) Ag = Ag(z1, . . . , z3g) ∈ z−(4g−2)1 C [z1, z2, . . . , z3g] , g ≥ 1

satisfying the homogeneity condition

(3.40)∑

i≥1

izi∂Ag∂zi

= 2gAg, g ≥ 1,

such that utop(x, T ; ǫ) can be represented in the form

(3.41) utop(x, T ; ǫ) = vtop(x, T ) +∑

g≥1

ǫ2gAg

(v′top(x, T ), . . . , v

(3g)top (x, T )

).

Here v(k)top(x, T ) = ∂kxvtop(x, T ).

Proof According to Corollary 3.5, utop(x, T ; ǫ) can be represented in the form

(3.42) utop(x, T ; ǫ) =∑

g≥0

ǫ2gu[g]top(x, T ), u

[0]top(x, T ) = vtop(x, T ),

where u[g]top(x, T ) =: u

[g]top ∈ C[[x − 1, T ]]. The Euler–Lagrange equation (3.30) for u = utop is

then equivalent to (3.38) together with the following recursion relations for u[g]top, g ≥ 1:

u[g]top

∑

λ∈I

λ2cλTλeλmvtop(3.43)

=∑

J∈Yeven

∑

λ∈I

λcλaJ(λ)Tλeλmvtop

∑

n≥ℓ(J)

∑

0≤g1,...,gn≤g−1

g1+···+gn=g−|J|2

b[g]J ;g1,...,gn

n∏

i=1

∂Jix u[gi]top,

where b[g]J ;g1,...,gn

are some constants and aJ(λ) are the coefficients that appear in (3.22).

20

Define a sequence of power series (cf. [18]) by

(3.44) Qk :=∑

λ∈I

λk+1cλTλeλmvtop , k ≥ 1.

By taking the x-derivative of (3.38) and (3.44) we find that Qk, k ≥ 1 satisfy the relations

(3.45) Q1 = − 1

m2nv′top, Qk =

1

mv′top∂xQk−1, k ≥ 2.

So we have

(3.46) Qk ∈(v′top

)1−2kC

[v′top, . . . , v

(k)top

],∑

i≥1

iv(i)top

∂Qk

∂v(i)top

= −Qk, k ≥ 1.

When g = 1, we know from (3.43), (3.45) that u[1]top = A1(v

′top, v

′′top, v

′′′top), where

A1(z1, z2, z3) =nh

24

(z3z1

− z22z21

+ z2

).

It is obvious that A1 satisfies the conditions (3.39), (3.40). Assume that we have already foundthe functions

(3.47) Ag′ = Ag′(z1, . . . , z3g′) ∈ z−(4g′−2)1 C[z1, . . . , z3g′ ], g′ ≤ g − 1

satisfying the homogeneity condition

(3.48)∑

i≥1

izi∂Ag′

∂zi= 2g′Ag′ ,

and that u[g′]top := Ag′

(v(1)top, . . . , v

(3g′)top

)satisfy the equation (3.43) for g′ ≤ g − 1. According to

Lemma 3.3, aJ(λ) are polynomials of λ with degrees less than or equal to l(J) + |J |/2, so itfollows from (3.46) that

(3.49)∑

λ∈I

λcλTλaJ(λ)eλmvtop ∈

(v′top

)1−2l(J)−|J |C

[v′top, . . . , v

(l(J)+|J |/2)top

], ∀ J ∈ Yeven.

Thus by using (3.45)–(3.49) we know the existence of a unique solution u[g]top to the equa-

tion (3.43) that satisfies the properties required by the lemma. The lemma is proved.

Step III. Let us show that the formula (3.41) yields the quasi-trivial transformation of theFVH [12, 25]. Namely, for any solution v = v(x, T ) to the dispersionless FVH (3.37) withv′ 6≡ 0,

(3.50) u = v +∑

g≥1

ǫ2gAg

(v′, . . . , v(3g)

)

gives a solution to the FVH. To this end, we first prove the following lemma on the transcen-dency of the solution vtop to the dispersionless FVH (3.37).

21

Lemma 3.8 Let F (z0, z1, . . . , zN ) ∈ Az0,0[z1, . . . , zN ] be a polynomial in the indeterminatesz1, . . . , zN with coefficients depending smoothly on z0, which satisfies the condition

(3.51) F(vtop(x, T ), v

′top(x, T ), . . . , v

(N)top (x, T )

)= 0,

then F (z0, z1, . . . , zN ) ≡ 0.

Proof From the dispersionless FVH (3.37), it follows that

∂F

∂Tλ

(vtop, v

′top, . . . , v

(N)top

)= ncλ

N∑

j=0

∂F

∂zj

(vtop, v

′top, . . . , v

(N)top

)∂j+1x

(eλmvtop

), λ ∈ I.

Dividing the right-hand side of the above equation by eλmvtop we obtain a polynomial in λ withdegree N + 1. Since this polynomial vanishes for any λ ∈ I, all its coefficients must vanish.Thus we obtain

∂F

∂zj

(vtop, v

′top, . . . , v

(N)top

)= 0, j = 0, . . . , N.

For a similar reason we find that for all k ≥ 1, 0 ≤ j1, . . . , jk ≤ N ,

∂kF

∂zj1 · · · ∂zjk

(vtop, v

′top, . . . , v

(N)top

)= 0,

which lead to the vanishing of F . The lemma is proved.

Now let us prove that the function u defined by (3.50) satisfies the FVH. Indeed, for anyλ ∈ I, it follows from Theorem 1.2 of [23] the existence of a quasi-trivial transformation

u = v +∑

g≥1

ǫ2gAλ,g

(v, v′, . . . , v(Ng)

),

which transforms (3.37) to the ∂Tλ-flow of the FVH. Here

Aλ,g(z0, . . . , zNg

)∈ z

−N ′g

1 Az0,0

[z1, . . . , zNg

],

and Ng and N ′g are some positive integers. Since utop satisfies the FVH, from (3.41) it follows

thatAg

(v′top, . . . , v

(3g)top

)= Aλ,g

(vtop, v

′top, . . . , v

(Ng)top

), g ≥ 1.

Thus by using Lemma 3.8 we arrive at

Ag (z1, . . . , z3g) = Aλ,g(z0, z1, . . . , zNg

), ∀ g ≥ 1,

so (3.50) is the quasi-trivial transformation of the FVH.Step IV. We are to prove the existence of the functions Fg satisfying the equations in (3.14).To this end, we need to use the following lemma which can be proved by applying Theorem 1.2of [23] to the Hamiltonian structure of the FVH given in [26].

22

Lemma 3.9 The quasi-trivial transformation (3.14) transforms the Hamiltonian structure

∂v

∂Tλ= P0

(δH

[0]λ

δv(x)

),(3.52)

P0 =nh

m∂x, H

[0]λ =

cλλh

∫eλmvdx(3.53)

of the dispersionless FVH to the one of the FVH given by

∂u

∂Tλ= P

(δHλ

δu(x)

),(3.54)

P =(Λ3 − 1)

(1− Λ−1

1

)

ǫ(Λ2 − 1), Hλ =

1

λh

∫resΛ3 L

λhdx.(3.55)

It follows from the above Lemma that

(3.56) Pδ

δu

∫udx = P0

δ

δv

∫ (v +

∑

g≥1

ǫ2gAg

(v′, . . . , v(3g)

))dx,

so we haveδ

δv

∫Ag

(v′, . . . , v(3g)

)dx ≡ const, ∀ g ≥ 1.

By using the homogeneity condition (3.40) we know that the constants in the above formulamust vanish. Therefore, there exist functions

Bg = Bg(z0, . . . , z3g−1) ∈ Az0,0

[log(z1), z1, z

−11 , z2, . . . , z3g−1

], g ≥ 1

satisfying the relations

(3.57) Ag =∑

i≥0

zi+1∂Bg∂zi

,∑

i≥1

izi∂Bg∂zi

= (2g − 1)Bg.

By taking derivatives with respect to z0 on both sides of the above equations we obtain

(3.58)∑

i≥0

zi+1∂

∂zi

(∂Bg∂z0

)= 0,

∑

i≥1

izi∂

∂zi

(∂Bg∂z0

)= (2g − 1)

∂Bg∂z0

.

It follows that Bg does not depend on z0. Thus by using (3.39) and (3.57) we arrive at

Bg ∈ z−(4g−3)1 C[z1, . . . , z3g−1], g ≥ 1.

Now let us proceed to prove that Bg(v′, v′′, . . . , v(3g−1)

)can be represented as the x-

derivative of a certain function of v, v′, v′′, . . . . We first prove the following lemma.

Lemma 3.10 If F ∈ (v′)−N′Av,0

[v′, . . . , v(N)

]satisfies the equations

∫Feλmvv′dx = 0, ∀λ ∈ I,(3.59)

∑

i≥1

iv(i)∂F

∂v(i)= N ′′F(3.60)

for some positive integers N,N ′, N ′′, then F ≡ 0.

23

Proof For any λ ∈ I, a straightforward calculation shows that F satisfies the equation

(3.61)δ

δv

∫Feλmvv′dx =

N∑

k=0

(−1)k((δkF ) ∂

kx

(eλmvv′

)+(∂kxF

)δk

(eλmvv′

)),

where the operators δk are defined by

δk :=∑

i≥0

(−1)i(i+ k

k

)∂ix

∂

∂v(i+k), k ≥ 0.

The left-hand side of (3.61) vanishes due to the condition (3.59). Dividing the right-hand sideof (3.61) by eλmv , we obtain a polynomial in λ of degree N . The vanishing of this polynomialfor any λ ∈ I yields ∂F/∂v(k) = 0, k = 1, . . . , N . Thus by using the condition (3.60) weconclude that F ≡ 0. The lemma is proved.

Taking the derivative with respect to Tλ, λ ∈ I on both sides of (3.50) and integrating withrespect to x we find that

ǫ−1∂−1x

(1− Λ−1

1

)resΛ3 L

λh(3.62)

=ncλeλmv + ncλ

∑

i≥1

∂i+1x

(eλmv

) ∂

∂v(i)

∑

g≥1

ǫ2gBg

(v′, . . . , v(3g−1)

),

where the FVH (2.56) and its dispersionless limit (3.37) are used. Denote

∆λ := ǫ−1∂−1x

(1− Λ−1

1

)resΛ3 L

λh − ncλeλmv.

Then from Lemma 2.2 one can represent ∆λ in the form

∆λ =∑

k≥1

ǫ2k∆[k]λ (v, v′, . . . ).

Here, ∆[k]λ ∈ (v′)−mλ,kAv for some nonnegative integers mλ,k, and deg∆

[k]λ = 2k. It follows

from the formula (2.6) that

0 =

∫∂∆λ

∂Tµdx =

∫ ∑

k≥0

∂∆λ

∂v(k)∂k+1x (ncµe

µmv) dx = µmncµ

∫eµmvv′

(δ

δv

∫∆λdx

)dx.

Then by employing Lemma 3.10 we obtain

δ

δv

∫∆

[k]λ dx = 0, k ≥ 1,

which, together with (3.62), yields the relations

(3.63) 0 =

∫∆λdx = λmncλ

∫eλmvv′

δ

δv

∫ ∑

g≥1

ǫ2gBg

(v′, . . . , v(3g−1)

)dx

dx

for all λ ∈ I. By using again Lemma 3.10 we obtain

(3.64)δ

δv

∫Bg

(v′, . . . , v(3g−1)

)dx = 0, ∀ g ≥ 1.

Thus by employing (3.57) and (3.64) we arrive at the following lemma.

24

Lemma 3.11 There exist functions

F1(z0, z1) =nh

24log z1 +

nh

24z0,(3.65)

Fg(z1, . . . , z3g−2) ∈ z−(4g−4)1 C [z1, . . . , z3g−2] , g ≥ 2,(3.66)

satisfying the quasi-homogeneous condition

(3.67)∑

i≥1

izi∂Fg∂zi

= (2g − 2)Fg +nh

24δg,1, g ≥ 1,

such that the functions Ag given in Lemma 3.7 can be represented in the form

(3.68) Ag

(v′, . . . , v(3g)

)= ∂2xFg

(v, v′, . . . , v(3g−2)

).

Step V. Finally, let us prove the existence of the tau-function τtop. Following the approachof [7], define F0 = F0(x, T ) ∈ C[[x− 1, T ]] by

(3.69) F0 :=mn

2h

∑

λ,µ∈I

λµ

λ+ µcλcµTλTµe

(λ+µ)mvtop +x

h

∑

µ∈I

cµTµeµmvtop +

x2

2nhvtop.

By using the dispersionless Euler-Lagrange equation (3.38) we find that F0 satisfies

∂2xF0 =1

nhvtop,(3.70)

∂x∂TλF0 =cλheλmvtop , ∀λ ∈ I,(3.71)

∂Tλ∂TµF0 =mn

h

λµ

λ+ µcλcµe

(λ+µ)mvtop , ∀λ, µ ∈ I,(3.72)

∑

λ∈I

λTλ∂F0

∂Tλ+

x2

2mnh= 0,

∑

λ∈I

Tλ∂F0

∂Tλ+ x

∂F0

∂x= 2F0.(3.73)

From Lemmas 3.7, 3.11 we know the existence of functions

F1 = F1(z0, z1) :=1

24log z1 −

mh+ n2

24nhz0,(3.74)

Fg = Fg(z0, z1, . . . , z3g−2) ∈ z−(4g−4)1 C[z1, . . . , z3g−2], g ≥ 2,(3.75)

such that

utop =(Λ1/23 − Λ

−1/23

)(Λ1/21 − Λ

−1/21

)∑

g≥0

ǫ2g−2Fg,(3.76)

∑

i≥1

izi∂Fg∂zi

= (2g − 2)Fg +1

24δg,1, g ≥ 1,(3.77)

where Fg = Fg(x, T ) := Fg

(vtop, v

′top, . . . , v

(3g−2)top

)∈ C[[x − 1, T ]]. Clearly, Fg and Fg are

related by

(3.78) Fg =1

nh

g∑

k=1

Cg−k(m,n)∂2g−2kFk +

Cg(m,n)

nhz2g−2, ∂ :=

∑

i≥0

zi+1∂

∂zi,

25

where Ck(m,n), k ≥ 0 are defined in (1.21).Now let us define a formal power series τtop ∈ C((ǫ))[[x − 1, T ]] by the formula

(3.79) τtop = τtop(x, T ; ǫ) := e∑

g≥0 ǫ2g−2Fg .

Then (3.76) can be written as

(3.80) utop =(Λ1/23 − Λ

−1/23

)(Λ1/21 − Λ

−1/21

)log τtop.

From the identities (3.73) it follows that

(3.81)∑

λ∈I

λTλ∂vtop∂Tλ

+1

m= 0,

∑

λ∈I

Tλ∂vtop∂Tλ

+ x∂vtop∂x

= 0,

which yield

(3.82)∑

λ∈I

λTλ∂v

(i)top

∂Tλ+

1

mδi,0 = 0,

∑

λ∈I

Tλ∂v

(i)top

∂Tλ+ x

∂v(i)top

∂x= −iv(i)top.

So by using (3.75) and (3.77) we know that τtop satisfies the following conditions:

(3.83) L0τtop = 0, K0τtop +1

24τtop = 0,

where the operators L0 and K0 are defined by (3.3) and (3.8), respectively.Now it remains to prove that the power series τtop ∈ C((ǫ))[[x − 1, T ]] is a tau-function of

the FVH (1.8). For any given λ, µ ∈ I, define two power series in C[[x− 1, T ; ǫ]] by

∆λ := (Λ3 − 1) Λ− 1

22 ǫ

∂ log τtop∂Tλ

− resΛ3 Lλh∣∣u=utop

,

∆λ,µ := ǫ2∂2 log τtop∂Tλ∂Tµ

− Ωλ,µ∣∣u=utop

.

We are to show that

(3.84) ∆λ = 0, ∆λ,µ = 0.

Firstly, from the definition (3.79) of τtop, equations (3.71), (3.72) and Lemmas 2.2, 3.8 we knowthe existence of

∆∗λ = ∆∗

λ(z0, z1, . . . ), ∆∗λ,µ = ∆∗

λ,µ(z0, z1, . . . ) ∈ ǫAz0,0[z−11 , z1, z2, . . . ][[ǫ]](3.85)

satisfying∆λ = ∆∗

λ

(vtop, v

′top, . . .

), ∆λ,µ = ∆∗

λ,µ

(vtop, v

′top, . . .

).

Then by using Lemmas 2.2, 3.8, the second equality of (3.83) and the second identity of (3.82)we find that

(3.86)

∑

i≥1

izi∂

∂zi− ǫ

∂

∂ǫ

∆∗

λ = 0,

∑

i≥1

izi∂

∂zi− ǫ

∂

∂ǫ

∆∗

λ,µ = 0.

26

Next, by applying ǫ∂Tλ and ǫ2∂Tλ∂Tµ to both sides of (3.80) we obtain

∂x∆λ = 0, ∂2x∆λ,µ = 0.

Then by using Lemma 3.8 we arrive at

∂∆∗λ = 0, ∂2∆∗

λ,µ = 0,

where ∂ is defined in (3.78). Together with the formulae (3.85)–(3.86) this implies that

∆∗λ = 0, ∆∗

λ,µ = 0.

Hence the identities (3.84) hold true. It follows that the power series τtop satisfies the defini-tion (2.7)–(2.9) of the tau-function of the FVH. Theorem 3.1 is proved.

4 Virasoro constraints

The goal of this section is to show, following the approaches used in [1, 2, 32], that the topo-logical tau-function τtop of the FVH satisfies the Virasoro constraints Lkτtop = 0, k ≥ 0, wherethe operators Lk are defined in (3.3), (3.4).

We will first derive some useful formulae valid for arbitrary power series solutions to theFVH. Let u be a solution to the FVH in the ring C[[x − 1, T ; ǫ]], (Φ1,Φ2, τ) a dressing tripleassociated to u, and ψ1, ψ2 the wave functions corresponding to Φ1,Φ2. See their definitionsgiven in Section 2. Introduce the following notations:(i) Γ1 and Γ2 denote the following elements in C[Λ3,Λ

−13 , ǫ][[x− 1, T ]]:

Γ1 :=x

hǫΛ−13 +

m

ǫ

∑

µ∈J1

µTµΛµm−13 +

m

2ǫ

∑

µ∈J3

µTµΛµm−13 ,(4.1)

Γ2 := − x

hǫΛ3 −

n

ǫ

∑

µ∈J2

µTµΛ−µn+13 − n

2ǫ

∑

µ∈J3

µTµΛ−µn+13 .(4.2)

Here J1,J2,J3 are defined in (2.24).(ii) M1 and M2 denote the following difference operators:

(4.3) M1 := Φ1Γ1Φ−11 , M2 := Φ2Γ2Φ

−12 .

Clearly, M1 ∈ C((Λ−13 , ǫ))[[x − 1, T ]], M2 ∈ C((Λ3, ǫ))[[x − 1, T ]].

(iii) N1 and N2 denote the following generating series of difference operators:

N1 = N1(ξ, ζ) :=∑

k≥1

(ζ − ξ)k

(k − 1)!

∑

d∈Z

ξ−k−dMk−11 L(k+d−1)h/m,(4.4)

N2 = N2(ξ, ζ) :=∑

k≥1

(ζ − ξ)k

(k − 1)!

∑

d∈Z

ξ−k−dMk−12 L(k+d−1)h/n.(4.5)

We note that for k ∈ Z, Lk/m ∈ Au[[ǫ]]((Λ−13 )), Lk/n ∈ Au[[ǫ]]((Λ3)). It is then clear that

N1 ∈ C((Λ−13 , ǫ))[[x − 1, T ]][[ξ, ξ−1]][[ζ − ξ]],

27

N2 ∈ C((Λ3, ǫ))[[x − 1, T ]][[ξ, ξ−1]][[ζ − ξ]].

See [26] for the definition of the fractional powers of L.(iv) X1 and X2 denote the following two operators:

X1(ξ, ζ) = eϑ1(x,T ;ǫ;ζ)−ϑ1(x,T ;ǫ;ξ) exp

(∑

µ∈I1

(1

ξµm− 1

ζµm

)1

µmǫ∂

∂Tµ

),(4.6)

X2(ξ, ζ) = eϑ2(x,T ;ǫ;ζ)−ϑ2(x,T ;ǫ;ξ) exp

(∑

µ∈I2

(1

ζµn− 1

ξµn

)1

µnǫ∂

∂Tµ

),(4.7)

where ϑ1, ϑ2 are defined in (2.22), (2.23).(v) Denote by Yk, k ≥ 0 the following operators in C((ǫ))[[Λ3,Λ

−13 ]][[x− 1, T ]]:

(4.8) Yk :=1

mM1L

(k+1/m)h − 1

nM2L

(k+1/n)h − Γ(m)Γ(n)

Γ(1 + h)ǫL(k+1)h.

Lemma 4.1 The following formulae hold true:

N1(ξ, ζ)ψ1(x, T ; ǫ; z)(4.9)

=(ζ − ξ)∑

d∈Z

ξ−dzd−1X1(ξ, ζ)τs(x, T − [z−1]1; ǫ)

τs(x, T − [z−1]1; ǫ)ψ1(x, T ; ǫ; z),

N2(ξ, ζ)ψ2(x, T ; ǫ; z)(4.10)

=(ζ − ξ)∑

d∈Z

ξ−dzd−1X2(ξ, ζ)τs(x+ hǫ, T − [z−1]2; ǫ)

τs(x+ hǫ, T − [z−1]2; ǫ)ψ2(x, T ; ǫ; z).

Proof The generating series N1 can be written as

N1(ξ, ζ) := (ζ − ξ)e(ζ−ξ)M1∑

d∈Z

ξ−dL(d−1)h/m.

So we have

1

ζ − ξN1(ξ, ζ)ψ1(x, T ; ǫ; z) =

∑

d∈Z

ξ−dzd−1e(ζ−ξ)∂zψ1(x, T ; ǫ; z)

=∑

d∈Z

ξ−dzd−1eϑ1(x,T ;ǫ;z+ζ−ξ)τs(x, T − [(z + ζ − ξ)−1]1; ǫ)

τs(x, T ; ǫ)

=∑

d∈Z

ξ−dzd−1eϑ1(x,T ;ǫ;z+ζ−ξ)−ϑ1(x,T ;ǫ;z)τs(x, T − [(z + ζ − ξ)−1]1; ǫ)

τs(x, T − [z−1]1; ǫ)ψ1(x, T ; ǫ; z)

=∑

d∈Z

ξ−dzd−1eϑ1(x,T ;ǫ;ζ)−ϑ1(x,T ;ǫ;ξ)τs(x, T − [ζ−1]1; ǫ)

τs(x, T − [ξ−1]1; ǫ)ψ1(x, T ; ǫ; z)

=∑

d∈Z

ξ−dzd−1eϑ1(x,T ;ǫ;ζ)−ϑ1(x,T ;ǫ;ξ)τs(x, T − [z−1]1 − [ζ−1]1 + [ξ−1]1; ǫ)

τs(x, T − [z−1]1; ǫ)ψ1(x, T ; ǫ; z)

=∑

d∈Z

ξ−dzd−1X1(ξ, ζ)τs(x, T − [z−1]1; ǫ)

τs(x, T − [z−1]1; ǫ)ψ1(x, T ; ǫ; z).

28

Here, the following property of the delta-function is used:

f(z)∑

d∈Z

ξ−dzd−1 = f(ξ)∑

d∈Z

ξ−dzd−1, ∀ f(z) ∈ R[[z, z−1]],

where R is any given ring. So the formula (4.9) holds true. The formula (4.10) can be provedin a similar way. The lemma is proved.

Lemma 4.2 We have the following relations between the operators Yk and the Virasoro oper-ators Lk, k ≥ 0:

−(Yk)−ψ1

ψ1=

(e∑

µ∈I1

−z−µm

µmǫ ∂∂Tµ − 1

)Lkτsτs

,(4.11)

(Yk)+ψ2

ψ2=

(ehǫ∂x+

∑µ∈I2

z−µn

µnǫ ∂∂Tµ − 1

)Lkτsτs

,(4.12)

where

(4.13) Lk := Lk −x

2mnǫδk,0 −

hǫ

2mn

∂

∂Tk, k ≥ 0.

Proof From the definition (4.8) of Yk and the definitions (4.4), (4.5) of N1, N2 we obtain

−(Yk)− =− 1

2mresξ ξ

km+1(∂2ζN1(ξ, ζ)|ζ=ξ

)−+

1

2nresξ ξ

kn+1(∂2ζN2(ξ, ζ)|ζ=ξ

)−

(4.14)

+Γ(m)Γ(n)

Γ(1 + h)ǫ

(L(k+1)h

)−.

To show (4.11), we need to represent − 1ψ1

(Lkh

)−ψ1, − 1

ψ1(Ni)−ψ1, i = 1, 2 in terms of τs.

From (2.25), (2.31) and (2.33) it follows that

(4.15)−(Lkh

)−ψ1

ψ1=

(e−∑

µ∈I1

z−µm

µmǫ ∂∂Tµ − 1

)ǫ∂Tkτsτs

, k ≥ 1.

To calculate −(N1)−ψ1/ψ1 and −(N2)−ψ1/ψ1, we observe that N1 and N2 can be decomposedinto the following forms:

(4.16) N1 = f1

(Lh/m

)+ g1

(Lh/n

), N2 = f2

(Lh/m

)+ g2

(Lh/n

),

where f1(z), f2(z), g1(z) and g2(z) are defined by

f1(z) := −X1(ξ, ζ)τs(x, T − [z−1]1; ǫ)

τs(x, T − [z−1]1; ǫ)

1− ζ/z

1− ξ/z,(4.17)

g1(z) :=ζ

ξ

X1(ξ, ζ)τs(x+ hǫ, T − [z−1]2; ǫ)

τs(x+ hǫ, T − [z−1]2),(4.18)

f2(z) :=ζ

ξ

X2(ξ, ζ)τs(x, T − [z−1]1; ǫ)

τ(x, T − [z−1]1; ǫ),(4.19)

g2(z) := −X2(ξ, ζ)τs(x+ hǫ, T − [z−1]2; ǫ)

τs(z + hǫ, T − [z−1]2; ǫ)

1− ζ/z

1− ξ/z.(4.20)

29

Indeed, denote

D0 := e12ǫ

∑k≥1 TkL

kh ∈ C[Λ3,Λ−13 ][[T ; ǫ−1]].

Then by applying D0 on both sides of the bilinear equations (2.29), we obtain

resz zxhǫ e

1ǫ

∑µ∈I1

Tµzµm τs(x, T − [z−1]1; ǫ)

τs(x, T ; ǫ)ψ∗1(x+ ℓhǫ, T ′; ǫ; z)

dz

z(4.21)

= resz z− x

hǫ e1ǫ

∑µ∈J2

Tµzµn τs(x+ hǫ, T − [z−1]2; ǫ)

τs(x, T ; ǫ)ψ∗2(x+ ℓhǫ, T ′; ǫ; z)

dz

z, ∀ ℓ ∈ Z.

By applying the operator

τs(x, T − [ζ−1]1 +

[ξ−1]1; ǫ)

τs(x, T ; ǫ)X1(ξ, ζ)

on both sides of (4.21) we arrive at the identities

reszX1(ξ, ζ)τs

(x, T − [z−1]1; ǫ

)

τs (x, T − [z−1]1; ǫ)

1− z/ζ

1− z/ξ

(D0ψ1(x, T ; ǫ; z)

)ψ∗1(x+ ℓhǫ, T ′; ǫ; z)

dz

z(4.22)

= reszX1(ξ, ζ)τs

(x+ hǫ, T − [z−1]2; ǫ

)

τs (x+ hǫ, T − [z−1]2)

(D0ψ2(x, T ; ǫ; z)

)ψ∗2(x+ ℓhǫ, T ′; ǫ; z)

dz

z.

On another hand, by using the fact that

(ζ − ξ)∑

d∈Z

ξ−dzd−1 = −1− ζ/z

1− ξ/z+ζ

ξ

1− z/ζ

1− z/ξ,

we can rewrite equation (4.9) in the form

N1ψ1 = f1(z)ψ1 +ζ

ξ

X1(ξ, ζ)τs(x, T − [z−1]1; ǫ)

τs(x, T − [z−1]1; ǫ)

1− z/ζ

1− z/ξψ1.

So it follows from (4.22) and the equations Lh/mψ1 = zψ1, Lh/nψ2 = zψ2 that

resz

[(N1 − f1(L

h/m))D0ψ1(x, T ; ǫ; z)

]ψ∗1(x+ ℓhǫ, T ′; ǫ; z)

dz

z

=resz

[g1(L

h/n)D0ψ2(x, T ; ǫ; z)]ψ∗2(x+ ℓhǫ, T ′; ǫ; z)

dz

z, ∀ ℓ ∈ Z.

From the above equations and Lemma 2.5 we obtain the first identity of (4.16). The proof forthe second identity of (4.16) is similar.

By using (4.16) and the identities

[g1

(Lh/n

)]−= 0,

[g2

(Lh/n

)]−= 0

we find that

(N1)− = f1

(Lh/m

)− resΛ3 f1

(Lh/m

), (N2)− = f2

(Lh/m

)− resΛ3 f2

(Lh/m

).

30

Together with (4.17)–(4.20) we obtain

−(N1)− ψ1

ψ1=

(e−

∑µ∈I1

z−µm

µmǫ ∂∂Tµ − 1

)(X1(ξ, ζ)τs(x, T ; ǫ)

τs(x, T ; ǫ)+ N1

),(4.23)

−(N2)− ψ1

ψ1= −

(e−

∑µ∈I1

z−µm

µmǫ ∂∂Tµ − 1

)(ζ

ξ

X2(ξ, ζ)τs(x, T ; ǫ)

τs(x, T ; ǫ)+ N2

),(4.24)

where N1 and N2 are given by

N1 =

(e− 1

2

∑k≥1

ζkm−ξkm

kmzkm − 1

)X1(ξ, ζ)τs(x, T ; ǫ)

τs(x, T ; ǫ),

N2 =

(e− 1

2

∑k≥1

ζkn−ξkn

kmzkm − 1

)ζ

ξ

X2(ξ, ζ)τs(x, T ; ǫ)

τs(x, T ; ǫ).

Now by using (4.14) together with (4.15), (4.23) and (4.24) we arrive at

−(Yk)− ψ1

ψ1=

(e−

∑µ∈I1

z−µm

µmǫ ∂∂Tµ − 1

)[1

τs(x, T ; ǫ)

(1

2mresξ ξ

km+1∂2ζX1(ξ, ζ)∣∣∣ζ=ξ

+1

2nresξ ξ

kn+1∂2ζ

(ζ

ξX2(ξ, ζ)

)∣∣∣ζ=ξ

− Γ(m)Γ(n)

Γ(1 + h)

∂

∂Tk+1

)τs(x, T ; ǫ)

],

which leads to the formula (4.11) via a straightforward calculation. The proof of the for-mula (4.12) is similar. The lemma is proved.

We are ready to prove the following theorem on the Virasoro constraints of the topologicaltau-function τtop. In the proof of the theorem the notations ψ1, ψ2, Yk(k ≥ 0), etc. correspondto utop.

Theorem 4.3 The topological tau-function τtop satisfies the following Virasoro constraints:

(4.25) Lkτtop = 0, k ≥ 0.

Proof Recall from Proposition 2.10 that

τtop(x, T ; ǫ) = τs

(x+

mǫ

2, T ; ǫ

).

From Theorem 3.1 we know that L0τtop = 0, which is, by a shift x 7→ x− mǫ2 , equivalent to

(L0 −

x

2nhǫ+

m

8nh

)τs = 0.

Then it follows from Lemma 4.2 that

(4.26) − (Y0)− ψ1 = 0, (Y0)+ ψ2 = − 1

2mψ2.

Since − (Y0)− and (Y0)+ can be represented in the form

− (Y0)− =∑

k≤−1

γkLkh/m, (Y0)+ =

∑

k≥0

γkL−kh/n,

31

where γk ∈ C((ǫ))[[x − 1, T ]], k ∈ Z, the formulae in (4.26) imply that γk = − 12mδk,0, which

yields Y0 = − 12m . So it follows from (4.8) that

(4.27) Yk = Y0Lkh = − 1

2mLkh, ∀ k ≥ 1.

By using (4.27) together with Lemma 4.2 and the identities

−(Lkh

)−ψ1

ψ1=

(e∑

µ∈I1

z−µm

µmǫ ∂∂Tµ − 1

)ǫ∂Tkτsτs

,

(Lkh

)+ψ2

ψ2=

(ehǫ∂xe

∑µ∈I2

z−µn

µnǫ ∂∂Tµ − 1

)ǫ∂Tkτsτs

,

we obtain for arbitrary k ≥ 1 that

(e∑

µ∈I1z−µm

µmǫ ∂∂Tµ − 1

) (Lk + ǫ2m

∂∂Tk

)τs

τs= 0,

(ehǫ∂xe

∑µ∈I2

z−µn

µnǫ ∂∂Tµ − 1

) (Lk + ǫ2m

∂∂Tk

)τs

τs= 0.

This implies (Lk −

ǫ

2n

∂

∂Tk

)τs = Ck(ǫ) τs, Ck(ǫ) ∈ C[[ǫ]],

which, after making the shift x 7→ x+ mǫ2 , is equivalent to

Lkτtop = Ck(ǫ)τtop.

From the commutation relations[L0, Lk] = −kLk,

it follows that Ck(ǫ) = 0 for any k ≥ 1. The theorem is proved.

5 Proof of Theorem 1.1

Let us first prove Theorem 1.1 for the case p = 1/m, q = 1/n, r = −1/h, where m,n arecoprime positive integers and h = m+ n. To this end we will prove the following proposition,which generalizes part of the results of [9, 11].

Proposition 5.1 The following identity holds true:

(5.1) Z(x, T ; ǫ) = τtop(x, T ; ǫ),

where Z(x, T ; ǫ) and τtop(x, T ; ǫ) are defined in (1.10) and Theorem 3.1 respectively.

Proof By using the genus expansions (3.5), (3.9) and comparing the coefficients of ǫ2g−2 of thelogarithms of both sides of (5.1), we find that the identity (5.1) is equivalent to the followinginfinitely many identities:

(5.2) Fg(x, T ) = Fg(x, T ), g ≥ 0,

32

where Fg(x, T ) are defined by

(5.3) Fg(x, T ) := (mnh)g−1Hg

(t(x, T ); 1/m, 1/n,−1/h

)+ δg,0

A

mnh.

Let us first prove the identity (5.2) for g = 0 by using the procedure that is similar to theone used in [11]. Indeed, it is well known that H0(t; 1/m, 1/n,−1/h) has the following explicitexpression [7, 8]:

(5.4) H0(t; 1/m, 1/n,−1/h) =1

2

∑

i,j≥0

titjvH(t)

i+j+1

(i+ j + 1)i!j!, ti = ti − δi,1,

where vH(t) ∈ C[[t]] is determined by the following equation:

(5.5) vH(t) =∑

i≥0

tivH(t)

i

i!.

Under the time substitutions (1.9) we find that vH(t(x, T )) satisfies the equation

(5.6)∑

λ∈I

λcλTλeλvH(t(x,T )) +

x

mn= 0.

By comparing this equation with (3.38) and by noticing that the solution to (3.38) in C[[x−1, T ]]must be unique we arrive at the following simple but crucial identity:

(5.7) vtop(x, T ) =1

mvH(t(x, T )),

where we recall that vtop(x, T ) =: vtop is the topological solution to the dispersionless FVH.Substituting (1.9) and (5.7) into (5.4), and using (3.69), we find, after a lengthy but straight-forward calculation that is similar to the one given in [11], that

(5.8) F0(x, T ) = F0(x, T ).

Next we are to show that the power series Fg and Fg with g ≥ 1 satisfy the technicalconditions required by the uniqueness theorem proved in [24]. For Fg, this has been given inthe statement of Theorem 3.1: there exist functions Fg(z0, . . . , z3g−2), g ≥ 1 satisfying (3.11)and (3.12) such that

(5.9) Fg(x, T ) = Fg

(vtop, v

′top, . . . , v

(3g−2)top

), g ≥ 1.

For Fg, the technical conditions were proved in [8, 10, 12, 24]: there exist functions

Fg(z0, z1, . . . , z3g−2), g ≥ 1

such that

Fg(x, T ) = Fg

(mvtop(x, T ),mv

′top(x, T ), . . . ,mv

(3g−2)top (x, T )

), g ≥ 1.(5.10)

33

Moreover, these functions can be chosen to satisfy the conditions

F1 = F1(z0, z1) =1

24log z1 +

σ124z0,(5.11)

∑

i≥1

izi∂Fg∂zi

= (2g − 2)Fg, Fg ∈ z−(4g−4)1 C[z1, . . . , z3g−2], g ≥ 2.(5.12)

Here we use the relation (5.7) and the fact that ∂x = ∂t0 .From Theorem 4.3 and the results of [24] we know the validity of the following Virasoro

constraints:

Lkτtop = 0,(5.13)

LkZ = 0,(5.14)

where k ≥ 0, and the operators Lk are defined in (3.3) and (3.4). According to [24], theequations (5.4), (5.5), (5.10) and (5.14) imply that Fg, g ≥ 1 satisfy the loop equation of[24]. From [24] we also know that the solution to the loop equation is unique under theconditions (5.11), (5.12). On the other hand, by using the relations (5.7), (5.8), (3.10), (3.38)and (5.13) we know that Fg, g ≥ 1 satisfy the same loop equation. Then the fact that Fgalso satisfy the conditions (3.11) and (3.12) leads to the validity of the identity (5.1). Theproposition is proved.

Proposition 5.1 implies the validity of Theorem 1.1 for the rational case. Let us proceedto prove the theorem for the general case when p, q, r ∈ C and satisfy the local Calabi–Yaucondition. To this end, let us first prove some lemmas.

Lemma 5.2 The power series u(x, T ; ǫ) ∈ C[[x− 1, T ; ǫ]] defined by (1.7) satisfies a hierarchyof evolutionary PDEs of the form

(5.15)∂u

∂Tλ=

λ

pqcλe

λpu

u′ +

∑

g≥1

ǫ2g∑

J∈Y2g+1

Cλ,J(p, q)u(J)

, λ ∈ I,

where u(J) := ∂J1x u · · · ∂Jl(J)x u, and Cλ,J(p, q) ∈ C

[p, q, p−1, q−1, (p + q)−1

], J ∈ Y are homo-

geneous rational functions of p and q of degree 1 − |J |, which are understood as the resultingexpressions under the substitution λ = kp or λ = kq, viewing the positive integer k as a fixedparameter.

Proof From [8] we know that

(5.16) w := − ǫ2

pqr∂2t0 logZcubic

(t; p, q, r;

√p+ q

pqǫ

)

satisfies the Hodge hierarchy, which has the form

∂w

∂ti= Xi(w,w

′, . . . ), i ≥ 0

for some Xi(w,w′, . . . ) =: Xi ∈ Aw,0[w

′, w′′, . . . ][[ǫ]]. From (5.10)–(5.12) it follows that in thequasi-trivial transformation

w = v +∆w = v + ǫ2∂2t0 F1(v, v′) + ǫ4∂2t0 F2(v

′, v′′, v(3), v(4)) + . . .

34

the function ∆w = ∆w(v′, v′′, . . . ) does not depend on v. So we know from the quasi-trivialityof the Hodge hierarchy that the evolutionary equations for w = w(t(x, T )) have the forms

(5.17)∂w

∂Tλ=

λ

pqcλe

λw

w′ +

∑

g≥1

ǫ2g∑

J∈Y2g+1

Cλ,J(p, q)w(J)

, λ ∈ I,

where the coefficients Cλ,J(p, q) ∈ C[p, q, p−1, q−1, (p + q)−1].Introduce a gradation on C[p, q, p−1, q−1, (p+q)−1][[t]] via the following degree assignments:

deg p = deg q = 1, deg ti = i− 1, i ≥ 0.

Using the relations (1.1), we find that degHg(t; p, q, r) = 3g − 3. Consequently, by using (1.9)we obtain degw = −1 as well as

deg Cλ,J(p, q) = l(J)− |J |.

From (1.7) and (5.16) we see that u and w are related by a Miura-type transformation

(5.18) u = pw +2p2 + 2pq + q2

24pq2w′′ǫ2 + · · · .

Thus by using (5.17) and (5.18) we obtain the equations (5.15) together with

Cλ,J(p, q) ∈ C[p, q, p−1, q−1, (p + q)−1].

Finally, from (5.15) and the fact that deg u = 0, we find that Cλ,J(p, q) are homogeneousrational functions of p, q satisfying degCλ,J(p, q) = 1− |J |. The lemma is proved.

Lemma 5.3 The flows of the FVH (1.8) can be represented in the form

(5.19)∂u

∂Tλ=

λ

pqcλe

λpu

u′ +

∑

g≥1

ǫ2g∑

J∈Y2g+1

Cλ,J(p, q)u(J)

, λ ∈ I,

where Cλ,J(p, q) have the same properties as those of Cλ,J(p, q) which appear in Lemma 5.2.

Proof One can prove this lemma by induction based on the following identity:

∂

∂T1/m

(∂u

∂Tλ

)=

∂

∂Tλ

(∂u

∂T1/m

), ∀λ ∈ I.

We omit the details here.

For the reader’s convenience, we give a few examples of Cλ,J(p, q) as follows:

Ckp,(3)(p, q) =k

12

p+ q

pq2, Ckp,(2,1)(p, q) =

k(2k + 1)

12

p+ q

pq2,

Ckq,(3)(p, q) =k

12

p+ q

p2q, Ckq,(2,1)(p, q) =

k

12

(p+ q)(2kq + p)

p3q.

35

Now by using the validity of Theorem 1.1 for the rational case and by using Lemmas 5.2,5.3 we know that the equalities

Cλ,J

(1

m,1

n

)= Cλ,J

(1

m,1

n

), ∀ J ∈ Y

hold true for arbitrary coprime positive integers m and n. Then from the homogeneity ofCλ,J(p, q) and Cλ,J(p, q) it follows the validity of the identities

(5.20) Cλ,J(p, q) = Cλ,J(p, q), ∀ J ∈ Y

for arbitrary non-zero p, q ∈ C. Hence the power series u(x, T ; ǫ) given by the formula (1.7) isa solution to the FVH (1.8).

To complete the proof of Theorem 1.1, we need to show that the power series Z(x, T ; ǫ)given by the formula (1.10) is a tau-function of the FVH, i.e. to verify that it satisfies therelations (2.8) and (2.9). Let us introduce, for any given λ, µ ∈ I, the following power seriesin C[[x− 1, T ; ǫ]]:

∆λ := (Λ3 − 1) Λ− 1

22 ǫ

∂ logZ(x, T ; ǫ)

∂Tλ− resΛ3 L

−λ/r∣∣∣u=u(x,T ;ǫ)

,

∆λ,µ := ǫ2∂2 logZ(x, T ; ǫ)

∂Tλ∂Tµ− Ωλ,µ|u=u(x,T ;ǫ) ,

where u(x, T ; ǫ) is defined by (1.7). We are to show that

(5.21) ∆λ = 0, ∆λ,µ = 0.

Using the genus expansion of logZcubic given in [8] one can represent the two point correlationfunctions of Zcubic in the form

(5.22)p+ q

p2q2ǫ2∂2 logZcubic

∂ti∂tj

(t(x, T ); p, q, r;

√p+ q

pqǫ

)=

(v/p)i+j+1

(i+ j + 1)i!j!+Xi,j(v, v

′, . . . ; ǫ)

for i, j ≥ 0. Herev = v(x, T ) := u(x, T ; ǫ = 0) ∈ C[[x− 1, T ]],

and Xi,j(z0, z1, . . . ; ǫ) =: Xi,j ∈ ǫAz0,0

[z−11 , z1, z2, . . .

][[ǫ]] satisfy the equation

∑

k≥1

kzk∂Xi,j

∂zk− ǫ

∂Xi,j

∂ǫ= 0.

Then by using the definition (1.10) of Z(x, T ; ǫ) and the formulae (5.22) we arrive at

ǫ2∂2 logZ(x, T ; ǫ)

∂x∂Tλ= −rcλ +

∑

i≥0

∂ti∂Tλ

ǫ2∂2 logZcubic

∂t0∂ti

(t(x, T ); p, q, r;

√p+ q

pqǫ

)

= −rcλeλv/p + fλ(v, v′, . . . ; ǫ),

ǫ2∂2 logZ(x, T ; ǫ)

∂Tλ∂Tµ= − r

pq

λµ

λ+ µ+∑

i,j≥0

∂ti∂Tλ

∂tj∂Tµ

ǫ2∂2 logZcubic

∂ti∂tj

(t(x, T ); p, q, r;

√p+ q

pqǫ

)

36

=1

p+ q

λµ

λ+ µcλcµe

(λ+µ)v/p + fλ,µ(v, v′, . . . ; ǫ),

where fλ(z0, z1, . . . ; ǫ) =: fλ, fλ,µ(z0, z1, . . . ; ǫ) =: fλ,µ ∈ ǫAz0,0

[z−11 , z1, z2, . . .

][[ǫ]] satisfy the

equations ∑

k≥1

kzk∂fλ∂zk

− ǫ∂fλ∂ǫ

= 0,∑

k≥1

kzk∂fλ,µ∂zk

− ǫ∂fλ,µ∂ǫ

= 0.

Hence by using Lemma 2.2 we find that there exist elements

(5.23) ∆∗λ = ∆∗

λ(z0, z1, . . . ), ∆∗λ,µ = ∆∗

λ,µ(z0, z1, . . . ) ∈ ǫAz0,0

[z−11 , z1, z2, . . .

][[ǫ]]

satisfying the homogeneity conditions

(5.24)∑

i≥1

izi∂∆∗

λ

∂zi− ǫ

∂∆∗λ

∂ǫ= 0,

∑

i≥1

izi∂∆∗

λ,µ

∂zi− ǫ

∂∆∗λ,µ

∂ǫ= 0,

such that∆λ = ∆∗

λ(v, v′, . . . ), ∆λ,µ = ∆∗

λ,µ(v, v′, . . . ).

On the other hand, applying the operators ǫ∂Tλ and ǫ2∂Tλ∂Tµ to both sides of (1.7), and usingequations (2.6) and (2.56), we obtain

∂x∆λ = 0, ∂2x∆λ,µ = 0.

Then by using the transcendency of v(x, T ) we have the identities (cf. Lemma 3.8)

(5.25) ∂∆∗λ = 0, ∂2∆∗

λ,µ = 0,

where ∂ =∑

i≥0 zi+1∂∂zi

. Combining with (5.23), (5.24), we have ∆∗λ = 0 and ∆∗

λ,µ = 0. So therelations (5.21) hold true. Theorem 1.1 is proved.

6 Proofs of Propositions 1.2, 1.3 and Theorem 1.4

In this section, we first prove Proposition 1.2. Then we do some explicit calculations for thecoefficients Rg based on the Hodge–FVH correspondence, which leads to a proof of Proposi-tion 1.3. Finally, we give a proof of Theorem 1.4 for certain special cubic Hodge integrals.Proof of Proposition 1.2. In the rational case, we have p = 1/m, q = 1/n for some coprimepositive integers m,n. By using (3.6) and (3.7) as well as Theorem 5.1 we find that Z satisfies

∑

µ∈I

(1− µ)Tµ∂Z

∂Tµ+ x

∂Z

∂x+ ǫ

∂Z

∂ǫ− x2

2mnhǫ2Z − σ1

24Z = 0,

from which the properties (1.14), (1.15) follow immediately. The degree estimate degRg(σ1, σ3) ≤3g − 3 with g ≥ 2 follows from Theorem 1.3 of [8]. To show (1.18) we first note that for g ≥ 2,

Hg(t; p, q, r)(6.1)

=(−1)g−1∑

n≥0

∑

i1,...,in≥0

ti1 · · · tinn!

∫

Mg,n

ψi11 · · ·ψinn λgλg−1λg−2

(pgqg−1rg−2 + (p↔ q ↔ r)

)

37

+ (−1)g−1∑

n≥0

∑

i1,...,in≥0

ti1 · · · tinn!

∫

Mg,n

ψi11 · · ·ψinn λ3g−1 pg−1qg−1rg−1 + · · · ,

where

(p ↔ q ↔ r) := pgrg−1qg−2 + qgpg−1rg−2 + qgrg−1pg−2 + rgpg−1qg−2 + rgqg−1pg−2,

and the dots “· · · ” denotes the terms in Hg(t; p, q, r), as polynomials of p, q, r, having degreestrictly less than 3g− 3. From (1.12) and (1.16) it follows that the degrees defined for polyno-mials in p, q, r and the ones for polynomials in σ1, σ3 are consistent. As it was proved in [8],the following formula holds true for g ≥ 2:

∑

n≥0

∑

i1,...,in≥0

ti1 · · · tinn!

∫

Mg,n

ψi11 · · ·ψinn λgλg−1λg−2 =1

2(2g − 2)!

|B2g|2g

|B2g−2|2g − 2

∂2g−2v(t)

∂t2g−20

,

where v(t) := ∂2t0H0(t; p, q, r). Then the identity (1.18) follows from Mumford’s relation

λ3g−1 = 2λgλg−1λg−2

(see [29]) and the fact that

(6.2) v(t(x, T ))|T=0 = log x.

The proposition is proved.

We proceed to prove Proposition 1.3.Proof of Proposition 1.3. Recall that Lemma 3.4 states that the difference equation (1.19),i.e. the equation

(6.3)∑

0≤α1≤···≤αm≤n

e∑m

j=1 V (x+αjmǫ−(j− 12)nǫ;ǫ) =

(m+ n

m

)x,

has a unique solution of the form

(6.4) V (x; ǫ) =1

mlog x+

∑

g≥1

ǫ2gPg(m,n)

x2g.

We have shown in [24] that

(6.5) logZcubic

(t(x, 0);

1

m,1

n,−1

h; ǫ

)= ǫ−2H0(t(x, 0)) +

σ1 − 1

24log x+

∑

g≥2

ǫ2g−2 Rgx2g−2

,

where Rg = Rg(σ1, σ3) ∈ Q[σ1, σ3], g ≥ 2, and σ1, σ3 are given in (1.16). From Theorem 1.1 itfollows that

utop(x, T ; ǫ) =

(Λ

123 − Λ

− 12

3

)(Λ

121 − Λ

− 12

1

)logZcubic

(t(x, T );

1

m,1

n,−1

h; ǫ√mnh

).

So by using (3.20) we obtain

V (x; ǫ) =

(Λ

123 − Λ

− 12

3

)(Λ

121 − Λ

− 12

1

)logZcubic

(t(x, 0);

1

m,1

n,−1

h; ǫ√mnh

).(6.6)

The proposition then follows from (6.4)–(6.6).

38

Example 6.1 Let us apply Proposition 1.3 to compute Rg for g = 2, 3, 4. We first derive the

expressions of the differential polynomials M[g]1 = M

[g]1 (u′, . . . , u(2g)) defined in (3.21), (3.22)

for g = 2, 3, 4. By using (3.26) we obtain

M[1]1 =

mn(2mh− n)

24u′′ +

m3n(h+ 1)

24(u′)2,

M[2]1 =

mn(24m2nh2 − 4mh(2m2 + 2mn+ 7n2) + 7n3

)

5760u(4)

+m3n(h+ 1)

(12mnh− 4m2 − 6mn− 9n2

)

1440u′u′′′

+m2n

(36m2nh2 − 4mh(3m2 − 2mn+ 8n2)− 12m3 − 8m2n− 12mn2 + 5n3

)

5760(u′′)2

+m4n(h+ 1)

(22mnh− (8m2 + 13n2)− 4h

)

2880(u′)2u′′

+m5n(h+ 1)

(5mnh− (2m2 − 3mn+ 2n2)− 2h

)

5760(u′)4.

Since the expressions for M[3]1 and M

[4]1 are quite long, we omit them here. By using (3.27)

and (3.29) we obtain

P1 =nh(m− 1)

24m,

P2 =− nh(m− 1)

960m

(4mnh+ 3m2 − 3mn− 3n2 −m

),

P3 =nh(m− 1)

72576m

(8m2n2h2 + 4mnh

(44m2 − 17mn − 17n2

)+ 95m4 − 75m3n

−48m2n2 + 54mn3 + 27n4 −m(49m2 − 13mn − 13n2

)+m2 +m

).

Here we also omit the long expression of P4. Then it follows from Proposition 1.3 that

R2 = − 1

1440+

13σ15760

− 7σ215760

+1

17280

(σ31 −

σ32

),

R3 =1

181440− 107σ1

362880+

145σ21290304

− 31σ31161280

−(

31

1088640− 113σ1

4354560

)(σ31 −

σ32

)−(σ31 − σ3/2

)2

13063680,

which coincide with the ones given in [24]. In the same way we also obtain the followingexpression of R4:

R4 =211

10886400+

σ148600

− 1193σ214354560

+18629σ3158060800

− 127σ411290240

+

(83

3870720− 1657σ1

32659200+

6469σ21261273600

)(σ31 −

σ32

)

+

(17

65318400+

247σ11567641600

)(σ31 −

σ32

)2−(σ31 − σ3/2

)3

2351462400.

39

Let us now consider the case when one of p, q is taken to be 1 and give a proof of Theorem 1.4.It suffices to consider the case with p = 1. So in what follows we assume that

(6.7) (p, q, r) =

(1, q,− q

q + 1

).

We note that in this case the parameter 1/q is identified with the framing parameter f . Formore details about the framing, see for example [3].

We need the following lemma.

Lemma 6.2 For the rational case with p = 1, the initial value V (x; ǫ) for utop(x, T ; ǫ) has thefollowing explicit expression:

(6.8) V (x; ǫ) ≡ log x.

Proof In the present case, the equation (1.19) reads

n∑

i=0

Λi−n/2eV = (n+ 1)x,

whose unique solution in C[[x− 1; ǫ]] is clearly given by V = log x. The lemma is proved.

By employing Lemma 6.2 and Proposition 1.3 one can obtain the following explicit expres-sions for the polynomials Rg:

R1 = − 1

24

((q + 1) + (q + 1)−1

),(6.9)

Rg = −(2g − 3)!

(B2g +

g∑

i=1

Bg−iBg+i((q + 1)i + (q + 1)−i

)), g ≥ 2,(6.10)

where Bi :=1−21−i

i! Bi, i ≥ 0. By using (1.3), (1.9) and (1.15) we know that Rg, g ≥ 1 arerelated to the cubic Hodge integrals by the formula

Rg =∑

J∈Y

(−1)ℓ(J)

m(J)!

∫

Mg,ℓ(J)

Cg(−1) Cg(−q) Cg(

q

1 + q

) ℓ(J)∏

i=1

ψJi+1i .(6.11)

Therefore, the formulae (6.9), (6.10) yield relations for the Hodge integrals. These relationsspecialize to (1.18) of [9] when q = 1.

Now let us prove Theorem 1.4.Proof of Theorem 1.4. By using (5.3) we know that it suffices to show that (mnh)1−gFg(x, T ),g ≥ 1 possess the property given by (1.24), (1.25). According to Theorem 1.1, eF(x,T ;ǫ) is thetopological tau-function of the FVH. Due to the symmetry with respect to the interchange ofp, q in Fg(x, T ), we only need to prove the theorem for the case p = 1.

Denote F(x, TI; ǫ) := F(x, T ; ǫ)|TII=0, and introduce the k-point correlators for F(x, TI; ǫ)as follows:

(6.12) Ci1,...,ik(x, ǫ) := ǫk∂kF(x, TI; ǫ)

∂Ti1 . . . ∂Tik

∣∣∣∣TI=0

, k ≥ 1, i1, . . . , ik ≥ 1.

40

From Proposition 1.2 we see that in order to prove the theorem, we only need to show thatall these correlators belong to C[[x, ǫ]]. To this end, let us first look at the two-point correla-tors. Using the relation (2.9) between the tau-function of the FVH and two-point correlationfunctions Ωλ,µ, we obtain

Ci1,i2(x, ǫ) = Ωi1,i2 |T=0, i1, i2 ≥ 1.

From the definition (2.3) with p = 1 it follows that Ωi,j are polynomials of Λ12+ℓeutop , ℓ ∈ Z.

Then by using Lemma 6.2 we arrive at the fact that the 2-point correlators Ci1,i2(x, ǫ) ∈ C[[x, ǫ]].Now let us introduce the notations

Ωλ1,...,λk = ǫk−2∂Tλ1 . . . ∂Tλk−2Ωλk−1,λk , k ≥ 3.

By using (2.56) and by induction on k we know that Ωi1,...,ik with k ≥ 2 are all polynomials

of Λ12+ℓeutop , ℓ ∈ Z. So from Lemma 6.2 it follows that Ci1,...,ik(x, ǫ) ∈ C[[x, ǫ]] for all k ≥ 3.

It remains to show that the 1-point correlators also belong to C[[x, ǫ]]. To this end, let us usethe equation (3.6). Dividing this equation by Z = τtop, taking its derivative with respect toTi, and then putting T = 0, we find that Ci(x, ǫ) ∈ C[[x, ǫ]], i ≥ 1. The theorem is proved.

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Si-Qi Liu, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. Chinae-mail: liusq@tsinghua.edu.cn

Di Yang, School of Mathematical Sciences, University of Science and Technology of China,Hefei 230026, P.R. Chinae-mail: diyang@ustc.edu.cn

Youjin Zhang, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. Chinae-mail: youjin@tsinghua.edu.cn

Chunhui Zhou, School of Mathematical Sciences, University of Science and Technology of China,Hefei 230026, P.R. China

e-mail: zhouch@ustc.edu.cn

43