Frobenius structure of An singularities

and (n + 1)KdV hierarchies

Niek de Kleijn

August 20, 2013

Abstract

We study the relation between Frobenius structures and integrable hi-erarchies using a specific well known example. We discuss general Frobe-nius structures and the specific Frobenius structure related to the singular-ity of An type. Most notably we prove that there is a Frobenius structure,in the axiomatic sense introduced in [1], related to such singularities. Wefind explicitely flat and canonical coordinates of this Frobenius manifold.We introduce the quantization formalism and use it to derive the vertexoperator form of the (n + 1)KdV hierarchies. Finally we discuss Given-tal’s formula for a total descendant potential related to the An singularityand follow Givental’s argumentation [2] to prove that this constitutes atau-function of the (n + 1)KdV hierarchy. To do this we use the essen-tial concepts of period vectors (as given in [3]) and the phase form (asintroduced by Givental in [2]).

Contents

0 Introduction -1

I Frobenius Structures 1

1 Definition of a Frobenius Structure 1

2 Deformed Connection and Canonical Coordinates 6

3 Frobenius Structure of An Singularity 13

4 Flat coordinates of An Singularity 15

5 Multiplication and Euler Vector Field of An Singularity 20

6 Deformed Flat and Canonical Coordinates of An Singularity 24

II KdV hierarchies 29

7 Quantization Formalism and Operators on the Fock Space 30

8 (n+ 1)KdV Hierarchy and Vertex Operators 34

III Solving the (n+ 1)KdV Hierarchy 40

9 Givental’s Formula 40

10 Period Vectors 41

11 The symplectic operators Rt and St 45

12 Phase Factors 49

13 Commutation with Vertex Operators 51

14 Ancestor Potential and Regularity 57

15 Conclusion and Discussion 62

0 Introduction

Purpose of the Thesis

The theory of Frobenius manifolds has been shown to be both wide spread anddeep [1, 3, 4, 5, 6, 7, 8, 9]. Consequently there are a great many different waysto view the general elements of this construction. It has been proposed [4] thatthe most natural frame for Frobenius manifolds is in the theory of integrablehierarchies. This claim is backed up by the realization that there is a Frobeniusmanifold related to every integrable hierarchy of a certain type. However it isstill an open question what this means for other classes of Frobenius structures[4, 10].

This relation was prompted by the discovery of Witten and Kontsevich of therelation between the topology of the moduli spaces of stable complex curvesand the Korteweg-de Vries hierarchy [11, 12]. This relation is seen when weexpress the integrable hierarchy in terms of Hirota quadratic equations andtau-functions. Which leaves the open question of framing the relation in a wayapplicable to general hierarchies expressed in Lax form [10], since the HQE-form of integrable hierarchies is only shown for a specific subset. In this casewe can also see the direct relation to the theory of Gromow-Witten invariantsof a compact Kahler manifold. It is shown that the generating function of suchGW-invariants in the case of the simplest manifold (that of a point) is in fact atau-function of the KdV hierarchy.

At this point we find again the theory of Frobenius manifolds related to thetheory of GW-invariants. Namely when we view the canonical moduli space re-lated to a topological field theory, this is a Frobenius manifold [5]. By Givental’sformula we can relate to any semisimple Frobenius manifold a total descendantpotential D. This is done by acting on this generating function of GW-invariants(or a product of them) with certain (quantized) operators related in an essen-tial way to the Frobenius manifold [13, 2]. Givental conjectures that this totaldescendant potential is again a generating function of GW-invariants of thecorresponding (calibrated) Frobenius structure [13]. This sets up nicely to com-plete the correspondence by yielding again a tau-function of the correspondingintegrable hierarchy.

After this general discussion of the ideas related to Frobenius structures letus get to the point of this thesis. The first examples of Frobenius structuresare not found in the theory of topological/cohomological field theories, butrather in the setting of singularity theory [14, 8, 9, 15]. It is shown that theWitten-Kontsevich tau-function mentioned above is in fact related, as the totaldescendant potential, to the singularity of A1 type [2, 11, 12]. Thus by Givental’sformula we find a natural correspondence of this total descendant potential ofthe A1 singularity to a total descendant potential related to the An singularity.On the other hand there is also the connection of the KdV hierarchy to the(n+ 1)KdV hierarchies (in HQE form) [2], which we will show below.

To study the general relation of Frobenius manifolds, integrable hierarchies andtotal descendant potentials as tau-functions we will focus on this specific ex-ample as laid out in [2]. This will lead to the identification of certain major

-1

players in the construction which would need to be generalized to the case ofan arbitrary Frobenius manifold related to an integrable hierarchy.

Organization of the Thesis

The thesis is presented as three parts, divided according to the general natureof the subjects discussed in them. The first part, entitled Frobenius structures,starts with 2 sections discussing the basic elements of Frobenius structures, thatwe will need. We begin with the axiomatic definitions of Frobenius algebras andmanifolds as presented in [1]. In section 2 the most powerful tools, specificallythe deformed connection and canonical coordinates, will be discussed. Thisshould give a good picture of the general theory of Frobenius manifolds.

The rest of Part I is devoted to the Frobenius structure related to the Ansingularity. This will be defined and discussed quite explicitely. All elementsof the first two sections will be shown explicitely. Additionally a few elementsof the structure that are specific to the An singularity will be discussed. Inthis part the essential concept of oscillatory integrals will be introduced. Mostimportantly we prove theorem 7, showing that we actually have to do with aFrobenius structure as defined in [1] and in fact one related to a solution to theWitten-Dijkgraaf-Verlinde-Verlinde equations.

Part II will set us up for the discussion on integrable hierarchies and totaldescendant potentials by introducing first the quantization formalism. Thisformalism is essential for the formulation of Givental’s formula and the vertexoperator form of the (n + 1)KdV hierarchies. The rest of part II will be usedto derive this form of the (n+ 1)KdV hierarchies starting from a HQE form ofthe Kadomtsev-Petviashvili hierarchy. Most important here is the derivation ofthe regularity condition (8.18).

Lastly in part III we will define all ingredients of Givental’s formula in a handson approach, i.e. in the setting of the Frobenius structure related to the Ansingularity. In this part also the essential concepts including period vectors andphase forms are introduced. Most importantly we prove theorem 20 followingGivental’s argumentation in [2].

Acknowledgement

I would like to take the time here to thank Sergey Shadrin for his excellentsupervision and many meetings. Also I thank Sergey Shadrin, Petya Dunin-Barkowski and Alexandr Buryak for their inspiring lectures that led me to thesubject of this thesis. Lastly I would like to thank all fellow students of mathe-matics/mathematical physics that have been ready to help me in some way orother during the writing of this thesis, most notably I thank Felix Wierstra andEddie Nijholt.

0

Part I

Frobenius Structures

1 Definition of a Frobenius Structure

To discuss the Frobenius structure corresponding to the singularity of An type,we will first need to know what a Frobenius structure is exactly. First of all, weshall need to give a definition of Frobenius structures. This involves two mainingredients: Frobenius algebras, which we will define and discuss first, and acomplex manifold, which we introduce subsequently. We will follow most ofthe conventions adopted in [1, 5] and refer to these as well for a more in-depthdiscussion of the general structure of a Frobenius manifold. We will concludethis section by giving an example of Frobenius structures. Namely those gener-ated by solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (or simply WDVV)equations. This will provide us with a substantial intuition. Moreover it canbe shown that a specific type of Frobenius structure is always ismorphic to onegenerated in this way [1].

Frobenius Algebra

A Frobenius algebra (A, 1, < ·, · >) consists of an associative commutative unitalalgebra over C equipped with a symmetric non-degenerate bilinear form < ·, · >such that

< a · b, c >=< a, b · c > ∀a, b, c ∈ A (1.1)

this property is called invariance of < ·, · >.

A Frobenius algebra A is called graded if there is a linear operator Q : A → Aand a number d ∈ C such that

Q(a · b) = Q(a) · b+ a ·Q(b) (1.2)

and< Q(a), b > + < a,Q(b) >= d < a, b > (1.3)

∀a, b ∈ A. Then Q is called the grading operator and d is called the charge ofthe Frobenius algebra.

Furthermore we can consider a graded Frobenius algebra A over (in the sense ofa module) a graded associative commutative ring R. In this case we have twograding operators QA : A→ A and QR : R→ R satisfying

QR(αβ) = QR(α)β + αQR(β) ∀α, β ∈ R (1.4)

QA(ab) = QA(a)b+ aQA(b) ∀a, b ∈ A (1.5)

QA(αb) = QR(α)b+ αQA(b) ∀α ∈ R,∀b ∈ A (1.6)

and

QR(< a, b >) + d < a, b >=< QA(a), b > + < a,QA(b) > ∀a, b ∈ A, (1.7)

again we call d ∈ C the charge of the Frobenius algebra.

1

Frobenius Manifold

A Frobenius manifold (also called a Frobenius structure) consists of a complexmanifold M equipped with a structure of Frobenius algebra (TtM, 1t, < ·, · >t)on the tangent spaces ∀t ∈ M . This Frobenius structure depends analyticallyon t. Concretely this last comment means that for (E1, . . . , En) a local frame,where n = Dim(M), the functions γkij(t), defined by

Eit · Ejt =

n∑k=1

γkij(t)Ekt ,

and the functions < Eit , Ejt >t should be analytic for all i, j, k = 1, . . . , n.

Moreover the following 4 axioms should be satisfied.

1

First of all the bilinear form < ·, · >t, which is already assumed symmetricnon-degenerate and invariant in the sense of Frobenius algebra, should be flat.Stated otherwise, there should be locally a system of coordinates (t1, . . . , tn) onM such that the matrix ηij =< ∂ti , ∂tj >t does not depend on the point t ∈M .These local coordinates are called flat coordinates.

2

Denoting by e ∈ X (M) the vector field such that et = 1t for all t ∈ M and by∇ the Levi-Civita connection corresponding to < ·, · >t, we should have

∇e = 0. (1.8)

Note that e is analytic since we assumed the algebra structure to vary analyti-cally.

3

By the invariance of < ·, · > and commutativity of the Frobenius algebras wehave naturally the fully symmetric trilinear form c on TM defined by

c(u, v, w) :=< u · v, w > ∀u, v, w ∈ TM (1.9)

The corresponding four-linear form

C(z, u, v, w) := (∇zc)(u, v, w) ∀z, u, v, w ∈ TM (1.10)

should also be fully symmetric.

4

Note that the space X (M) of vector fields on M has acquired the structure ofa Frobenius algebra over the ring F(M) of functions on M . There should bea fixed linear vector field E ∈ X (M), i.e. ∇∇E = 0, which we call the Eulerfield. Then the operators

QF(M) := E (1.11)

2

andQX (M) := Id+ adE (1.12)

should introduce the structure of a graded Frobenius algebra over a graded ringof some charge d ∈ C on X (M).

WDVV Equations

As an example and general source of intuition, let us consider the Witten-Dijkgraaf-Verlinde-Verlinde equations. The WDVV-equations are firstly a sys-tem of equations concerning a function Φ(t1, . . . , tn) (shorthand Φ(t)) of somecomplex variables (t1, . . . , tn), for n ∈ N, and a symmetric non-degenerate n×nmatrix ηαβ . Namely

∂α∂β∂λΦ(t)ηλµ∂µ∂γ∂δΦ(t) = ∂δ∂β∂λΦ(t)ηλµ∂µ∂γ∂αΦ(t) (1.13)

for all α, β, γ, δ = 1, . . . , n where we use the Einstein summation convention(in fact we will use it everywhere in the rest of this section) and we denote∂α := ∂

∂tα . More elements are involved, but we will introduce these when theyare needed.

For any α, β, γ = 1, . . . , n we define the functions cγαβ(t) by

cγαβ(t) := ηγδ∂δ∂α∂βΦ(t) (1.14)

We can use these functions to define the algebras At =< e1, . . . , en >C. Wheremultiplication is given by

eα · eβ = cγαβ(t)eγ , (1.15)

for all α, β = 1, . . . , n. Multiplication of the WDVV equation by ηεα (using thesummation convention) and substitution of the structure constants yields

cλγδcελβ = cλβδc

ελγ . (1.16)

Thus we see that these algebras are associative. Obviously these algebras arealso commutative.

We require a marked variable ti such that

∂α∂β∂iΦ(t) = ηαβ (1.17)

where the matrix ηαβ is inverse to ηαβ . This matrix defines a symmetric non-degenerate bilinear form < ·, · > on each algebra At by

< eα, eβ >:= ηαβ . (1.18)

This form is clearly invariant in the sense of Frobenius algebra since

< eα · eβ , eγ >= ηδε∂ε∂α∂βΦ(t)ηδγ = ∂α∂β∂γΦ(t), (1.19)

for all α, β, γ = 1, . . . , n. Note that we now have the vecor ei, which is the unitin every algebra At.

3

Finally we require the quasi-homogeneity condition concerning the constantsd1, . . . , dn, df ∈ C, which is given by

Φ(cd1t1, . . . , cdntn) = cdfΦ(t). (1.20)

We can restate this condition using a version of Euler’s homogeneous functiontheorem. So define the Euler vector field E (this explains the name) given by

E =

n∑α=1

dαtα∂α. (1.21)

Then we should haveLEΦ(t) = dfΦ(t) (1.22)

up to second order terms (since this will not have an effect on the algebras,which only depend on the third derivatives of Φ). Also we impose the extraconditions that di = 1 and dα 6= 0 for all α = 1, . . . , n.

Remark 1 In fact the extra condition dα 6= 0 for all α = 1, . . . , n is notneeded, but true for the Frobenius structure corresponding to the An singularity.Dropping this condition would imply the more complicated Euler vector field:

E =

n∑α=1

(dαtα + rα)∂α, with rα 6= 0 only if dα = 0. This case is included in [1,

5].

Often in the literature it is preferred to use the constants

qα = 1− dα and d = 3− df . (1.23)

In terms of these constants we have the following relation.

Lemma 1(qα + qβ − d)ηαβ = 0, (1.24)

for all α, β = 1, . . . , n. Here no summation is implied.

Proof:

The proof follows from equation (1.22) by taking the triple derivative ∂α∂β∂ion both sides. The right hand side then simply reads (3 − d)ηαβ by equation(1.17). For the left hand side we find

ηαβ + dβηαβ + dαηαβ +

n∑γ=1

dγtγ∂γηαβ = (3− qα − qβ)ηαβ

since η does not depend on t. equating these two expressions then yields therelation.

Now it is already clear how to relate a structure of Frobenius manifold to thesesolutions. The coordinates tα are our flat coordinates and they form a complex

4

manifold M . We define the algebra structure on the tangent spaces by theisomorphism At ' TtM given by eα 7→ ∂α. Obviously condition 1 is met andsince e = ∂i corresponds to a flat coordinate we have also condition 2. All thatis left is to show the conditions 3 and 4. For condition 3 simply note that

∂µcλαβ = ∂αc

λµβ , (1.25)

for all α, β, λ, µ.

As for condition 4, we let QR = E and QA = Id+adE . Now identifying eα = ∂αwe have

QR(fg) = E(fg) = E(f)g + fE(g) = fQR(g) +QR(f)g,

QA(feβ) = feβ + [E, feβ ] = feβ + E(f)eβ − f∑δ

∂βdδtδ∂δ =

= fQA(eβ) +QR(f)eβ ,

QA(eα · eβ) = QA(cγαβeγ) = QR(cγαβ)eγ + cγαβQA(eγ) =

= eα · eβ + E(cγαβ)eγ + cγαβ [E, ∂γ ] =

= (1−dα−dβ)(eα ·eβ)+(3−d)cγαβeγ−∑ε

dεηεγ∂ε∂α∂βΦ∂γ−cγαβ∂γ

∑δ

dδtδ∂δ =

= (2−dα−dβ)(eα·eβ)+∑γ,ε

(2−d−dε−dγ)ηεγ∂ε∂α∂βΦ∂γ = (2−dα−dβ)(eα·eβ) =

= eα · eβ − eα ·

(∑δ

∂βdδtδ∂δ

)+ eα · eβ −

(∑δ

∂αdδtδ∂δ

)· eβ =

eα · eβ + eα · [E, eβ ] + eα · eβ + [E, eα] · eβ = eαQA(eβ) +QA(eα)eβ ,

QR(< eα, eβ >) + d < eα, eβ >= (qα + qβ) < eα, eβ >=

= 2 < eα, eβ > +∑δ

< −∂αdδtδ∂δ, eβ > +∑δ

< eα,−∂βdδtδ∂δ >=

= 2 < eα, eβ > + < [E, eα], eβ > + < eα, [E, eβ ] >=

=< QA(eα), eβ > + < eα, QA(eβ) >,

for all α, β and all f, g ∈ F(M). Here we used lemma 1 and (1.22).

Now we see that every solution of the WDVV equations defines a Frobeniusstructure and by lemma 2.1 of [1] also locally every Frobenius manifold with adiagonalizable ∇E is decribed by a solution to the WDVV equations.

5

2 Deformed Connection and Canonical Coordi-nates

Let us now explore the definition of Frobenius structures above. We will dothis by defining two of the most useful tools related to Frobenius structures.Namely the deformed connection ∇ and the canonical coordinates. We will alsoshow some defining qualities of these tools. For a more detailed exposition see[1, 16]. Most notably the deformed connection offers an alternative definitionof Frobenius manifolds [17]. Every Frobenius manifold in this section will beassumed to be obtainable from a solution to the WDVV equations, i.e. ∇E isassumed diagonalizable. This assumption is made merely for convenience andclarity. It should be clear how to adapt statements and proofs made in thissection to incorporate the general case. Moreover we will see that the Frobeniusstructure related to the An singularity has diagonalizable ∇E. Thus the resultsof this secion will be perfectly applicable to later sections.

Deformed Connection

From the definition of Frobenius manifold we get automatically the Levi-Civitaconnection ∇, we will use this and combine it with the multiplication in thetangent bundle to form the deformed connection. In fact we will define a pencilof connections depending on some parameter z ∈ C.

Suppose M is a Frobenius manifold and ∇ is the corresponding Levi-Civitaconnection. We define the deformed connection ∇ by

∇vw := ∇vw + zv · w, (2.1)

where as mentioned z ∈ C, v, w ∈ X (M). Now we can extend this to a mero-morphic connection on the product manifold M × C by defining what happensin the added dimension. So on M × C we define the deformed connection by

0 = ∇ud

dz= ∇ d

dz

d

dz(2.2)

and

∇ ddzv = ∂zv + E · v − 1

zµv, (2.3)

where µ =df−1

2 − ∇E, u, v ∈ X (M × C) and u, v have zero component alongC. Where we mean that

µv =df − 1

2v −∇vE.

A natural question is wether this deformed connection is again flat. Althoughwe will not prove it explicitely in the general case, it should become clear thatthe answer is positive, mainly by virtue of the 4 axioms stated above. In fact,one could alternatively define a Frobenius structure using the metric, algebrastructure and deformed connection. Instead of the 4 axioms we would merelyask flatness of the deformed connection [17]. Substantial intuition shall be givento this point. Since, to verify the Frobenius structure of the An singularity, we

6

will show flatness of the deformed connection to check some of the axioms ofsection 1. In the general case, the deformed connection is shown to be flatexactly by associativity of the algebra structure and these axioms. This is doneexplicitely in [1]. To show that the deformed connection is flat it is sufficientto prove that a coordinate system exists that gives rise to a frame of horizontalsections. The following lemma gives a set of equations that should hold for sucha coordinate system. In the remainder of this section we will denote by ti the flatcoordinates for i = 1, . . . , n, where n = Dim(M), and we will denote ∂i := ∂

∂ti.

Additionally we will define the structure constants ckij(t) for i, j, k = 1, . . . , n by

∂i|t · ∂j |t =

n∑k=1

ckij(t)∂k|t, (2.4)

for any t ∈M .

Lemma 2 For a function f ∈ F(M) such that the derivatives ξi := ∂if satisfy

∂iξj = z

n∑k=1

ckijξk (2.5)

and

∂zξj =

n∑k,l=1

dktkclkjξl −

ξjz

(qj −d

2), (2.6)

for any i, j = 1, . . . , n, the one-form df is horizontal with respect to the deformedconnection, i.e. ∇df = 0.

Proof:

To find these equations we will simply have to apply the deformed connectionto df . So let us find the Christoffel symbols Γ, for any combination of sub- andsuperscripts, for the deformed connection. Note first of all that by equations(2.2) we have Γkiz = 0 = Γkzz for any i, k = 1, . . . , n, i.e. we need not worryabout the dz part of df . Secondly we have

∇∂i∂j = z

n∑k=1

ckij∂k

so Γkij = zckij and Γzij = 0, for any i, j, k = 1, . . . , n. Additionally

∇ ddz∂k =

n∑i,j=1

(ditic

jik −

δjkz

(qj −d

2)

)∂j

so Γjzk = − δjkz (qk − d2 ) +

n∑i=1

diticjik and Γzzk = 0 for k = 1, . . . , n. Thus we find

∇∂idf =

n∑j=1

(∂iξj − z

n∑k=1

ckijξk

)dtj (2.7)

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and

∇ ddz

df =

n∑j=1

∂zξj − n∑i,k=1

ditickijξk +

ξjz

(qj −d

2)

dtj (2.8)

Thus equating all the independent parts to 0 we find exactly equations (2.5)and (2.6). Lemma is proved.

The above equations might seem a bit messy. However they are a lot easier tointerpret when put in the following form:

∂iξ = z(∂i•)ξ (2.9)

and

∂zξ = (E − 1

zµ)ξ, (2.10)

where (∂i•) is a matrix with entries (∂i•)jk = ckij , Eij =

n∑k=1

dktkcjki and ξ is a

vector with entries ξi = ∂if for i, j, k = 1, . . . , n.

Let us now turn to the new matrix introduced in the definition of the deformedconnection. It is diagonalizable by diagonalizability of ∇E. Furthermore wehave the following lemma.

Lemma 3 µ is anti-adjoint, i.e.

< µv,w > + < v, µw >= 0 ∀v, w ∈ TM. (2.11)

Proof:

The proof comes down to the relation given in lemma 1. By linearity it issufficient to show that

< µ∂i, ∂j > + < ∂i, µ∂j >= 0 ∀i, j = 1, . . . , n.

In terms of this basis we know that µ is diagonal and the entries on the diagonalare given above so we find that

< µ∂i, ∂j > + < ∂i, µ∂j >= (qi + qj − d) < ∂i, ∂j >

then lemma 1 concludes the proof.

Canonical Coordinates

We return to the situation of a Frobenius manifold M (as opposed to M×C). AFrobenius manifold is said to be semisimple if the tangent spaces are semisimplein the sense of Frobenius algebras. A Frobenius algebra A is called semisimple ifit contains no nilpotent elements, i.e. no elements 0 6= a ∈ A such that ∃m ∈ Nsuch that

am = 0. (2.12)

In fact the semisimplicity condition is a local one. Thus it makes sense to calla point t ∈ M semisimple if TtM is a semisimple algebra. Then there exists

8

in fact a semisimple neighborhood of t (i.e. such that all the points in thisneighborhood are semisimple). This makes sense since the algebra structureshould vary analytically. In a semisimple neighborhood there exists anothernatural system of coordinates called canonical. Where the flat and deformedflat coordinates are natural in the geometric sense, these canonical coordinatesare natural by virtue of the algebraic properties of their generated frame.

The canonical coordinates are a system of coordinates such that the correspond-ing frame of the tangent bundle is orthogonal and has the structure constantsδijδik for multiplication of the vector fields corresponding to the ith and jthcanonical coordinate. To show existence of this coordinate system we will firstneed to prove the following lemma about semisimple Frobenius algebras. Wewill follow the path laid out in [1].

Lemma 4 Any finite-dimensional Frobenius Algebra A is isomorphic to anorthogonal direct sum

A ' A0 ⊕ C⊕ . . .⊕ C (2.13)

where A0 is a nilpotent algebra

Proof:

Let λ0, λ1, . . . , λk ∈ A∗ be the roots of A where λ0 ≡ 0. So the λi are linearfunctionals such that for any a ∈ A the eigenvalues of the multiplication by aare λ0(a), . . . , λk(a). Let

Aj :=⋂a∈A

Ker((a·)− λj(a)I)pj(a) (2.14)

for j = 0, 1, . . . , k, I the identity matrix and pj(a) the Riesz-index correspondingto the linear map (a·) and the eigenvalue λj(a). So in fact Aj is the intersectionof all the generalized eigenspaces corresponding to the linear maps (a·) andeigenvalues λj(a) for all a ∈ A. In particular we see that A0 consists of thenilpotent elements of A. So A is semisimple if and only if A0 = 0. It is anestablished fact of linear algebra that Ai ∩Aj = 0 for i 6= j [18]. So clearly

A =

k⊕j=0

Aj . (2.15)

Moreover Ai ·Aj = 0 whenever i 6= j since these lie in the intersection. Thuswe have λj(Ai) = 0 for i 6= j.

Now, since we have invariance of the bilinear form, it is left to prove that theAi are one-dimensional for i 6= 0. For i 6= 0 let 0 6= vi ∈ Ai be a vector suchthat

vvi = λi(v)vi

for all v ∈ A, i.e. an eigenvector. Suppose λi(vi) = 0 then

v2i = 0,

which would make vi nilpotent. However A0∩Ai = 0 so λi(vi) 6= 0. Put then

πi =vi

λi(vi). (2.16)

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So we have π2i = πi. Suppose wi ∈ Ai is another eigenvector . Then

wi = λi(πi)wi = πiwi = wiπi = λi(wi)πi,

so wi is proportional to πi. Suppose that ui ∈ Ai, then there is n ∈ N such that

(πi − λi(πi))nui = 0.

Note that (πi − λi(πi))2 = 1− πi so we have either (πi − λi(πi))n = πi − λi(πi)or (πi − λi(πi))n = λi(πi)− πi, depending on the parity of n. So in either casewe have

λi(ui)πi = πiui = ui.

So again we find that ui is proportional to πi. Thus in fact all the vectors inAi are proportional to πi. In conclusion we have the subalgebra A0 of nilpotentelements and the subalgebras Ai which are generated by the πi with πiπj =δijπi.

Now let us move on to the case of a semisimple Frobenius structure.

Proposition 5 For M a Frobenius manifold we have, near a semisimple pointt ∈M , a system of local coordinates (u1, . . . un), where n = Dim(M), such that

∂ui · ∂uj = δij∂ui , (2.17)

for all i, j = 1 . . . , n. These coordinates are called canonical coordinates.

Proof:

Since the algebra structure on the tangent spaces varies analytically, we have,by the previous lemma, a frame of idempotents π1, . . . , πn with πi · πj = δijπi,for all i, j = 1, . . . , n. Thus it is sufficient to show that the lie brackets [πi, πj ]vanish for all i, j = 1, . . . , n. To show this we will use the deformed connection.Define the coefficients Γkij and fkij by

∇πiπj =

n∑k=1

Γkijπk (2.18)

and

[πi, πj ] =

n∑k=1

fkijπk, (2.19)

for all i, j = 1, . . . , n. Now from flatness of the connection we find that

0 = ∇πi∇πjπk − ∇πj ∇πiπk − ∇[πi,πj ]πk =

=

n∑l,p=1

ΓljkΓpil − ΓlikΓpjl − flijΓ

plk

πp+

+z

n∑l=1

(Γljkδil − Γlikδjl + Γlikδjk − Γljkδik − f lijδkl)πl+

10

+z2(δjkδij − δikδij)πi, (2.20)

for any i, j, k = 1, . . . , n. Thus, equating the linear parts in z, we find that

Γljkδil + Γlkiδkj − Γlkiδjl − Γlkjδki = f lijδkl, (2.21)

for any i, j, k, l = 1, . . . , n. So setting l = k yields

fkij = 0.

Lastly we have, of course, the freedom to call these coordinates whatever wewant. We call them canonical.

Now we can recast all the elements that make up the Frobenius structure in thefollowing form.

Theorem 6 Let u1(t), . . . , un(t) be the eigenvalues of the operator of multipli-cation by the Euler vector field. They form a system of local coordinates near asemisimple point t ∈M and in these coordinates we have

∂ui · ∂uj = δij∂ui (2.22)

e =

n∑i=1

∂ui (2.23)

E =

n∑i=1

ui∂ui (2.24)

and

< ·, · >=

n∑i=1

hidu2i , (2.25)

where e is the unit vector field, E is the euler vector field and

hi = ∂ui

n∑j,k=1

< ∂uk , ∂j > tj , (2.26)

with (t1, . . . , tn) the flat coordinates.

Proof:

The lemma takes care of the existence of a coordinate system that satisfies (2.22)and thus also (2.23). So the first thing we should prove is that the coordinatesthat generate the frame (π1, . . . , πn) in the proof of the lemma are indeed theeigenvalues of the operator of multiplication by the Euler vector field. Recallthe grading operators (1.11) and (1.12). Then (1.5) implies

LE(a · b)− LE(a) · b− a · LE(b) = a · b, (2.27)

for all a, b ∈ X (M). With a = πi, b = πj and i 6= j this yields

LE(πi) · πj + LE(πj)πi = 0.

11

Mulitplying this through by πj then yields

LE(πi)πj = 0,

which implies LE(πi) = λiπi, for some λi ∈ F(M) since it should hold for allj 6= i. Now applying (2.27) with a = b = πi yields

λiπi − 2λiπi = πi,

which shows that λi = −1. Recall that the lemma showed that (π1, . . . , πn) isthe frame generated by some coordinates (u1, . . . , un). In these coordinates wecould write E =

∑ni=1Ei(u)∂ui , then LE(πi) = −πi yields

∂Ei∂uj

= δij . (2.28)

So we see that after a translation in the coordinates if necessary (which will notaffect the corresponding frame), we find (2.24).

The equation (2.25) follows nearly immediately. First of all note that, writingagain πi = ∂ui , we find

< πi, πj >=< e, πi · πj >=< e, δijπi >

which yields

< ·, · >=

n∑i=1

< e, πi > du2i ,

since (π1, . . . , πn) forms an orthogonal frame. Then we find

< e, πi >=

n∑k=1

< ∂uk , ∂ui >=

n∑k,j=1

< ∂uk , ∂j > dtj(∂ui) =

∂ui

n∑k,j=1

< ∂uk , ∂j > tj = hi.

Corollary 2 Note that all the points t ∈M where (E(t)•) has n distinct eigen-values are semisimple. Note also that we can find our canonical coordinatessimply by finding these eigenvalues. Lastly we can recast the equations relatedto the deformed connection in terms of the new coordinates. In these coordinatesfor instance E will just be a diagonal matrix. This is done and the results arestudied in for instance [1]. We will however not need this complete analysis.

Note that the frame generated by the canonical coordinates is automaticallyorthogonal. Then it is clear that to every frame generated by canonical co-ordinates there is also related an orthonormal frame with structure constants√< ∂ui , ∂ui >δijδik where the ui represent the corresponding canonical coordi-

nates.

12

3 Frobenius Structure of An Singularity

Now we are ready to define and discuss the Frobenius structure associated toa singularity of An type. By such a singularity we mean the germ of functionsf : C→ C that preserve 0 and have an n-fold degenerate critical point at 0. Todefine the Frobenius structure related to this singularity we will be working witha certain representative. We will state in this section all the basic elements thatmake up the Frobenius structure. However, we will not prove that the conditions1-4 of section 1 are met. This will be done by first exploring the concepts offlat coordinates, deformed flat coordinates and the Euler field in this explicitcase. Finally in section 6 we will prove that we have in fact been discussing aFrobenius structure, i.e. show that the conditions 1-4 are met. We will start bydefining the structure at the point 0 in our manifold (in all coordinate systemswe will use). This specific structure in the point 0 (which is not semisimple)will be used in the next part to discuss the (n+ 1)KdV hierarchies.

Local Algebra as Frobenius Algebra

Let [f ] : (C, 0)→ (C, 0) be the germ of a holomorphic function at a singularityof An type, for some n ∈ N. We pick the representative f(x) = xn+1. Note thatthis is a choice of preference.

Let H := C[x]/〈fx〉 =< 1, x, x2, . . . , xn−1 >C denote the local algebra of thesingularity. We define on H the bilinear form < ·, · > by

< v,w >= Resx=∞

v(x)w(x)dx

fx(3.1)

for v(x), w(x) ∈ H, which is clearly well defined. Note that by commutativity ofH this form is symmetric and for u, v, w ∈ H we have < u · v, w >=< u, v ·w >.

If v ∈ H is such that ∀w ∈ H we have < v,w >= 0, say v =∑ni=1 vix

n−i

with vi ∈ C ∀i, then < v,−(n + 1)xj−1 >= vj for j = 1, 2, . . . , n. This showsv = 0 and thus that the form is non-degenerate. So this gives (H, 1, < ·, · >)the structure of a Frobenius algebra.

Parameter Space as Frobenius Manifold

Now we would like to extend this structure of Frobenius algebra to the tangentspaces of the parameter space T of the miniversal deformation F (x, τ) of f ,thus giving T the structure of a Frobenius manifold. To clarify, the miniversaldeformation is given by n complex numbers τ = (τ1, . . . , τn) as

F (x, τ) = xn+1 + τ1xn−1 + . . .+ τn.

Note that we have F (x, 0) = f . By parameter space we mean the space of suchcomplex numbers. So in fact T ' Cn and to every point τ = (τ1, . . . , τn) ∈ Twe relate a deformation F (x, τ). So we automatically have a coordinate systemon T .

We begin at τ = 0 by identifying the basis ∂τi |0ni=1 of T0T with the basis[xn−i]ni=1 of H by sending ∂τi |0 to [xn−i] and pulling back < ·, · > under

13

this identification. We denote this pullback by < ·, · >0 (as we will in generaldenote the bilinear form on any TτT by < ·, · >τ ). In terms of the basis of T0Tmentioned above we have for all i, j = 1, . . . , n

< ∂τi |0, ∂τj |0 >0= −δi+j,n+1

n+ 1= Resx=∞

(∂τiF |0)(∂τjF |0)

∂xF |0dx, (3.2)

where |0 refers to setting τ = 0. We extend this form by defining

< ∂τi |τ , ∂τj |τ >τ= Resx=∞

(∂τiF |τ )(∂τjF |τ )

∂xF |τdx, (3.3)

for all i, j = 1, . . . , n.

The coordinates τ1, . . . , τn are natural in the sense of simplicity of the expressionfor F in terms of these coordinates. Therefore we will call them the naturalcoordinates. However they are obviously not generally flat and we will showmomentarily that they are also not canonical. If (a1, . . . , an) is another (local)system of coordinates on T then, since ∂ai =

∑nj=1 ∂aiτj∂τj , we have

< ∂ai , ∂aj >τ=<

n∑p=1

∂aiτp∂τp ,

n∑p=1

∂ajτp∂τp >τ=

=

n∑p,q=1

(∂aiτp|τ )(∂ajτq|τ ) < ∂τp , ∂τq >τ=

=

n∑p,q=1

(∂aiτp|τ )(∂ajτq|τ ) Resx=∞

(∂τpF |τ )(∂τqF |τ )

∂xF |τdx =

= Resx=∞

(∂aiF |τ )(∂ajF |τ )

∂xF |τdx, (3.4)

for all i, j = 1, . . . n. So we actually have an expression for the metric in anycoordinate system.

Now we also have naturally an algebra structure on the tangent spaces to T ,which agrees with the algebra structure on H, by pulling back the algebrastructure on C[x]/〈∂xF |τ 〉 under the isomorphism ∂τi |τ 7→ [∂τiF |τ ]. Note that

< ∂τi · ∂τj , ∂τk >τ= Resx=∞

(∂τiF |τ )(∂τjF |τ )(∂τkF |τ )

∂xF |τdx, (3.5)

for all i, j, k = 1, . . . , n, which shows that this metric is invariant in the sense ofFrobenius algebras. Also clearly the metric is symmetric.

To summarize we have now the manifold T of parameters τ of the deformationF (x, τ) = xn+1 +τ1x

n−1 +. . .+τn, i.e. every point is a specific deformation of f .We define an algebra structure on the tangent spaces TτT by pulling back thealgebra structure on C[x]/〈∂xF 〉 under the isomorphism of vector spaces givenby sending vτ ∈ TτT to [vτ (F )] ∈ C[x]/〈∂xF |τ 〉. Then we define the bilinear

form < ·, · >τ on T by < vτ , wτ >τ= Resx=∞(vτ (F ))(wτ (F ))

∂xF |τ dx. Now we want

14

to show that this defines a Frobenius manifold with diagonalizable ∇E. Thenthere is an underlying solution of the WDVV equations. To do this we will needto show that the form < ·, · >τ (which we will call metric from now on) is flat.The flat coordinates will then be a lot more convenient to show that T satisfiesthe other axioms to be a Frobenius manifold. The following theorem will beproved at the end of section 6.

Theorem 7 The parameter space T of miniversal deformations of the singu-larity f(x) = xn+1 for n ∈ N together with the algebra structure on the tangentspaces given by vτ · wτ = zτ for some vτ , wτ ∈ TτT and zτ ∈ TτT such that(vτ (F ))(wτ (F )) = zτ (F ) + G(x, τ)∂xF |τ (where G is just some polynomial inx and τ and zτ (F ) is not dividable by (∂xF )|τ in C[x]) and the bilinear form

< ·, · >τ given by < vτ , wτ >= Resx=∞(vτ (F ))(wτ (F ))

∂xF |τ dx is a Frobenius manifold.

In other words the construction given in this section defines a Frobenius mani-fold. First we will devote a section to the flat coordinates.

4 Flat coordinates of An Singularity

By now we have all the main ingredients of the Frobenius structure. Namelythe manifold, metric and algebra structure. However we have not yet shownthat these satisfy all the requirements. Even though the natural coordinatesare natural, they are not the easiest to work with to explore the other aspectsof the Frobenius structure. In sections 1 and 2 most discussions were framedin the flat coordinate system. This system is arguably the most useful to workwith on any Frobenius manifold. Finding the flat coordinates in the case of theAn singularity is not the easiest task however. In this section we will give anabstract definition of the coordinates and show that we can always find them,at least recursively, expressed in terms of the natural coordinates. Subsequentlywe will explore the relation between the flat and natural coordinates. Of coursewe will also show flatness of the coordinates.

Definition of the Flat Coordinates

Consider again the An singularity f(x) = xn+1. This function has the miniversaldeformation F (x, τ) = xn+1 + τ1x

n−1 + . . .+ τn for any (τ1, . . . , τn). To definethe flat coordinates (t1, . . . , tn) on the parameter space of these deformations

we first consider the Laurent series k = (F (x, τ))1

n+1 . Since at any point in theparameter space of F , i.e. for any choice of τ , F (x, τ) is a polynomial of ordern+ 1 in x we know that

(F (x, τ))1

n+1 = x+ b1x−1 + b2x

−2 . . . =: k (4.1)

for some infinite sequence bi in C such that

kn+1 = F (x, τ) = xn+1 + τ1xn−1 . . .+ τn. (4.2)

We will show how to express the τi in the bi by finding n sets S(n,i) for 1 ≤ i ≤ n,which we define in the next section.

15

Finally we define the flat coordinates ti as solutions of the equation

x = k +1

n+ 1

(t1k

+t2k2

+ . . .+tnkn

)+

1

O(kn+1)(4.3)

(So we have some part of order smaller then −n in k, we denote this fact in thefollowing by adding O(k−n−1)) at any point in the parameter space of F (x, τ).We will show later that this is in fact well defined since one can express the flatcoordinates in terms of natural coordinates recursively and vice versa.

The Sets S(n,i)

Consider the following algebraic problem. Fix n, i ∈ N with 0 < i ≤ n. Wewant to find all combinations of numbers nj ∈ N ∪ 0 for j = 0, 1, 2, . . . suchthat

n− i = n0 −∞∑j=1

jnj (4.4)

and

n+ 1 =

∞∑j=0

nj . (4.5)

Suppose n0, n1, n2, . . . solves this problem then it is clear that n0 6= 0 since ifn0 = 0 then since n + 1 = n1 + n2 + . . . we know that not all the nj are 0 forj > 0 so −

∑∞j=1 jnj < 0 ≤ n− i ∀0 < i ≤ n. So at least n0 6= 0. Moreover the

second equality in the problem tells us that only finitely many of the nj can benon-zero. Now if n0, n1, n2, . . . solves this problem we define the non-zero-tupleform of this solution to be the (m+ 1)-tuple of elements in Z2

((n0, 0), (nj1 , j1), (nj2 , j2), . . . , (njm , jm))

where m is the number of non-zero nj with j 6= 0 and jk runs over the indicesof these non-zero nj . For example, suppose i = 3 (and n is higher than 3) thenn0 = n−1 and n1 = 2 solves the problem. Then the non-zero-tuple form of thissolution is ((n− 1, 0), (2, 1)). Now we simply define

S(n,i) = non-zero-tuple forms of solutions of the above problem with n and i.(4.6)

Now we see that by equating the coefficients of equal powers of x in kn+1 andF (x, τ) we have

τi =∑

((n0,0),(nj1 ,j1),...,(njm ,jm))∈S(n,i)

(n+ 1

n0, nj1 . . . , njm

) m∏k=1

bnjkjk

, (4.7)

for all i = 1, . . . , n.

Finding the Flat Coordinates

Now we can find the flat coordinates recursively and thus show that they arewell defined. First of all we need to find the way that the natural coordinatesdepend on the bi.

16

Lemma 8 For all 0 < i ≤ n we have that τi in F (x; τ) does not depend on bjfor j > i.

Proof:

We will show that in the problem at the beginning of the last section nj = 0 ifj > i. Suppose there is a solution n0, n1, . . . with nj 6= 0 for a j > i, then since

nj ∈ N ∪ 0 we have nj ≥ 1. Then n0 = n+ 1−∞∑k=1

nk ≤ n+ 1− nj ≤ n.

Then we have n0−∞∑k=1

knk ≤ n0−jnj ≤ n0−j ≤ n−j < n−i. This contradiction

proves the lemma.

So in fact τi can only depend on the bj with j ≤ i, but then we can also expressthe bj using τj and bk with k < j. So in fact we can express bj using the τkwith k ≤ j.

Suppose

x = k +1

n+ 1

(t1k

+t2k2

+ . . .+tnkn

)+O(k−n−1)

then

x−1 =1

k + 1n+1 ( t1k + t2

k2 + . . .+ tnkn ) +O(k−n−1)

=

=1

k

1

1 + (( t1n+1

1k2 + . . .+ tn

n+11

kn+1 ) +O(k−n−2))= (4.8)

1k (1− ( t1

n+11k2 + . . .+ tn

n+11

kn+1 ) + ( t1n+1

1k2 + . . .+ tn

n+11

kn+1 )2 − . . .+O(k−n−2))

To show that the flat coordinates are well defined we will give a recursive wayof finding them. Going by the degree of k in the expression

x+ b1x−1 + b2x

−2 . . . = k

after we replace x = k+ 1n+1 ( t1k + t2

k2 + . . .+ tnkn ) +O(k−n−1). Then we see that

in the term x−j the highest power of k is −j. However in the term x we see thatthe highest degree of k in which tj appears is −j. So we see that we can expressfirst t1 in terms of only b1. Then using this answer we can express t2 in termsof b1 and b2 and so on. Moreover using the above Lemma and its consequencewe can express tj using only the τi with i ≤ j. This shows the coordinates arewell defined and gives us a way to express the natural coordinates in terms ofthe flat coordinates.

Example 1 Let us set n = 4 to do an example. So we haveF (x, τ) = x5 + τ1x

3 + τ2x2 + τ3x + τ4. First we determine the sets S(4,i), for

i = 1, 2, 3, 4. We haveS(4,1) = ((4, 0), (1, 1));

S(4,2) = ((4, 0), (1, 2));

17

S(4,3) = ((4, 0), (1, 3)), ((3, 0), (2, 1));

S(4,4) = ((4, 0), (1, 4)), ((3, 0), (1, 1), (1, 2)).

So the formula above givesτ1 = 5b1;

τ2 = 5b2;

τ3 = 5b3 + 10b21;

τ4 = 5b4 + 20b1b2.

Now let us express the flat coordinates in terms of the bj. So first we look at thecoefficient of k−1 this gives t1

5 + b1 = 0. Going through the degrees −1 throughto −4 yields

t15

+ b1 = 0;

t25

+ b2 = 0;

t35− b1

t15

+ b3 = 0;

t45− b1

t25− b2

t15

+ b4 = 0.

Now we can solve starting from the top to get an expression of the τi in termsof the flat coordinates. This yields

τ1 = −t1;

τ2 = −t2;

τ3 =t215− t3;

τ4 =2

5t1t2 − t4.

Relation between Natural and Flat Coordinates

The example shows a very easy relation between τ1 and t1. It shows the samerelation between τ2 and t2. In fact the dependence of τi on ti seems ratherstraightforward so we can guess the following lemma from the example.

Lemma 9 ∂tiτi = −1 for all i = 1, . . . , n.

Proof:

First of all we will prove that the relation for the power −i of k in the procedureof writing t in terms of b gives an equation of the following form

tin+ 1

+ bi +G(b1, . . . , bi−1) = 0

for some function G. Again we get this equation by substituting the expression

for x in terms of t and k in k = x +

∞∑i=1

bix−i and equating like powers of k.

18

Since the only power of k present on the left hand side is 1 this basically meansthat we find the coefficient of k to the power −i on the right hand side andequate it to 0. Indeed the first term k−i we encounter is in x and has coefficienttin+1 . In the other powers of x the order of k that has coefficient involving tiwill always be lower then −i so the only term involving ti is ti

n+1 . The highest

order of k in x−j is k−j so indeed no bj with j > i appear in the expression andbi only appears by adding it once. So the above expression holds.

Secondly we want to show that a similar expression holds for τ namely

τi = H(b1, . . . , bi−1) + (n+ 1)bi

for some function H. We already know that τi does not depend on bj with j > i.So what is left is to show this specific dependence on bi. Now if we look at theexpression (4.7) we see that we need to prove that there is no non-zero-tuplein S(n,i) that has a term (ni, i) with ni > 1 and apart from the non-zero-tuple((n, 0), (1, i)) there is no non-zero-tuple with a term (1, i). This is quite easy toprove.

Suppose we have a solution with ni > 1 then

n+ 1 =

∞∑j=0

nj ≥ n0 + n1 > n0 + 1

so n > n0, but then

n− i = n0 +

∞∑j=1

jnj ≤ n0 − ini < n− i,

which is absurd. Thus such a solution does not exist. Suppose we have a solutionwith ni = 1 then we have n ≥ n0 and n − i ≤ n0 − i which shows that n0 = nand thus yields the non-zero-tuple ((n, 0), (1, i)).

Thus when we define τ in terms of t we get τi = −ti + J(t1, . . . ti−1) whereJ(t1, . . . , ti−1) = H(b1, . . . , bi−1) − (n + 1)G(b1, . . . , bi−1) with the b’s in termsof the t’s and we remember that bj does not depend on tp if p > j. Then thelemma is proved.

Corollary 3 This shows also that τ1 = −t1 in general.

Flatness

Now, to justify calling these new coordinates flat, let us explicitely calculate< ∂i, ∂j > for 1 ≤ i, j ≤ n, where from now on we denote ∂i = ∂ti and wesuppress the subscript τ if it is immaterial. Recall that the metric was given bythe formula

< ∂i, ∂j >= Resx=∞

(∂iF )(∂jF )

∂xFdx

To calculate this we switch to the variable k and use the identities

∂jF = ∂jkn+1 = (∂jx)(∂xk

n+1) and (∂iF )dx = (∂ix)∂xkn+1dx = (∂ix)dkn+1

19

to obtain< ∂i, ∂j >= (n+ 1) Res

k=∞(∂ix)(∂jx)kndk =

= (n+ 1) Resk=∞

(1

(n+ 1)ki+O(k−n−1)

)(1

(n+ 1)kj+O(k−n−1)

)kndk =

= Resk=∞

(1

(n+ 1)ki+j+O(k−n−1−min(i,j))

)kndk.

Now we know that 1 ≤ i, j so n−n− 1−min(i, j) < −1 thus only the first partcan contribute to the residue. This yields finally

< ∂i, ∂j >=−δi+j,n+1

n+ 1, (4.9)

which is indeed independent of the point t ∈ T .

5 Multiplication and Euler Vector Field of An

Singularity

We would like to prove that T is a Frobenius manifold and in fact one that canbe defined by a solution of the WDVV equations. To do this we will have todefine the Euler vector field E and look a little closer at the structure constantsckij for TT with the basis given by the flat coordinates.

The Polynomials Kij and Relations

We defined the multiplication on TT by identifying the basis ∂τini=1 in TτTwith the basis [∂τiF |τ ] in C[x]/〈∂xF 〉. We can easily adapt this constructionto find the structure constants corresponding to the frame generated by the flatcoordinates, as pointed out above. We will now elaborate on this constructionand find a useful relation in the following lemma.

Lemma 10 Denoting by ∂i the derivative with respect to the flat coordinate ti,there are, for all i, j = 1, . . . , n, polynomials Kij(x, t) satisfying

(∂iF )(∂jF ) =

n∑p=1

cpij(t)∂pF +Kij∂xF (5.1)

and∂i∂jF = ∂xKij (5.2)

where ∂i · ∂j =

n∑p=1

cpij(t)∂p.

Proof:

The coefficients cpij(t) that show up in these equations come from the multipli-

cation ∂i · ∂j =∑np=1 c

pij(t)∂p in each of the algebras TtT . They are defined

by identifying ∂i with [∂iF |t] in the algebra C[x]/〈∂xF |t〉, going from ∂τi as in

20

(3.4) shows this easily. So in fact the first equation follows by definition of thismultiplication since

∂i · ∂j −n∑p=1

cpij(t)∂p = 0 (5.3)

so [(∂iF )(∂jF )−

n∑p=1

cpij(t)∂pF

]= [0] (5.4)

i.e. there is a uniquely defined polynomial Kij satisfying the first of the aboveequations. So the real trick is in the second equation. By looking at the possibledegrees in x of the polynomials in the first equations we see already that

Kij =

n−1∑p=0

K(p)ij (t)xp.

We can also write F (x, τ) = xn+1 + τ1xn−1 + . . . + τn, plugging these into the

second equation and equating the coefficients of different degrees of x we seethat we need to prove that

∂i∂jτa = (n+ 1− a)K(n+1−a)ij ∀a = 2, . . . , n (5.5)

and that ∂i∂jτ1 = 0. However since τ1 = −t1 by corollary 3 we already knowthat ∂i∂jτ1 = 0 ∀i, j = 1, . . . , n. To prove the rest we first express τa as aresidue and switch to the variable k

τa = − Resx=∞

F (x, τ)x(a−n−1)dx =

= − 1

a− nResk=∞

kn+1dxa−n =n+ 1

a− nResk=∞

knxa−ndk (5.6)

where kn+1 = F (x, t) as before, i.e. x = k + 1n+1 ( t1k + . . .+ tn

kn ) +O(k−n−1).

The next step is to consider the second derivatives ∂i∂jτa. This yields

∂i∂jτa = (n+ 1) Resk=∞

kn((a−n− 1)xa−n−2(∂ix)(∂jx) +xa−n−1∂i∂jx)dk, (5.7)

since we have expressed x as a function of k and the flat coordinates. Nowconsider the second summand in this residue. From the above expression ofx we see that the highest power of k in xa−n−1 is a − n − 1 and the highestpossible power of k in ∂i∂jx is −n− 1 so the highest possible power of k in thissecond summand is n+ a− n− 1− n− 1 = a− n− 2 ≤ −2 < −1 so it does notcontribute to the residue and we have

∂i∂jτa = Resk=∞

(a− n− 1)xa−n−2(∂ix)(∂jx)dkn+1 =

= Resx=∞

(a− n− 1)xa−n−2(∂ix)(∂jx)dF. (5.8)

21

Now somehow we will need to relate this to Kij i.e. we need to relate it toderivatives of F (x, t) with respect to the flat coordinates. Luckily we have∂ixdF = ∂ix∂xFdx = ∂iFdx. So

∂i∂jτa = Resx=∞

(a− n− 1)xa−n−2 (∂iF )(∂jF )

∂xFdx =

= Resx=∞

(a− n− 1)xa−n−2

(Kij +

n∑p=1

cpij(t)∂pF

∂xF

)dx. (5.9)

Let us consider this second summand again, the highest degree of x in ∂xF isn; in the sum the highest degree of x is n− 1 (namely in ∂1F ). So the highestdegree of x in the second summand is a−n−2+n−1−n = a−n−3 ≤ −3 < −1so again the second summand doesn’t contribute to the residue and we have

∂i∂jτa = Resx=∞

(a− n− 1)xa−n−2

(n−1∑p=0

K(p)ij x

p

)dx =

= −(a− n− 1)K(n+1−a)ij = (n+ 1− a)K

(n+1−a)ij , (5.10)

so the lemma is proved.

Constants of Homogeniety

To show that T is a Frobenius manifold we will have to fix a linear vector fieldE on it, such that (1.11) and (1.12) define a grading on the Frobenius algebraX (T ). Since the coordinates (t1, . . . , tn) are flat we see that for any choice(d1, . . . , dn) ∈ Cn

E =

n∑i=1

diti∂i (5.11)

has ∇∇E = 0. Since for flat coordinates we have ∇∂k = ∂k for ∇ the Levi-

Civita connection. By which is meant that if v =n∑i=1

vi∂i and w =n∑i=1

wi∂i

then

∇wv =

n∑i,j=1

wi(∂ivj)∂j .

Moreover ∇∂kE = dk∂k so in the basis given by the flat coordinates ∇E =diag(d1, . . . , dn) thus ∇E is indeed diagonalized. Now let us imagine for amoment that T is a Frobenius manifold that is induced by a solution Φ tothe WDVV equations. Then, since our function F is related to Φ by equation(5.1), we would like to figure out consistent constants di and df of homogeneity.Such that equation (1.22) holds up to second order terms in the flat coordinatesti. We might then just as well find the degrees of homogeneity for the thirdderivatives cijk = ∂i∂j∂kΦ. These offer us the relation to F by the equationdefining multiplication. So, if we assign the degrees di to ti for all i = 1, . . . , nand df to Φ, then we should assign to cijk the degree df −di−dj−dk. Actuallyin the definition of multiplication (5.1) we find ckij =

∑nl=1 η

klcijl, where ηkl is

22

inverse to ηkl =< ∂k, ∂l >=−δk+l,n+1

n+1 , i.e. ηkl = −(n+ 1)δk+l,n+1. Then we seethat

ckij = −(n+ 1)∂i∂j∂n+1−kΦ (5.12)

and thus ckij is of degree df − di − dj − dn+1−k.

Then we can find the constants d1, . . . , dn, df by requiring

F (cdxx, cd1t1, . . . , cdntn) = cF (x, t). (5.13)

The expression (4.2) for F shows that we need to assign to x the degree dx = 1n+1

and equation (4.3) shows we need to assign the same degree to k. This equationalso shows that di − idx = dx and thus that di = (i + 1)dx = i+1

n+1 , for all

i = 1, . . . , n. Since the degree of the total tiki should equal the degree of x and

we have already deduced that the degree of k equals that of x.

Using equations (5.1) and (5.2) we can now do a check and find out the degreedf . First of all since F has degree 1 we know that ∂iF has degree 1− di = n−i

n+1

so the term on the left hand side has total degree 2n−i−jn+1 . Both terms on the

right hand side should also have this degree. Now ∂i∂jF = ∂xKij so denotingby dij the degree of Kij we have

1− di − dj = dij − dx

and this yields dij = n−i−jn+1 . So the term on the right hand side of (5.1) with

the factor Kij has degree dij + 1−dx = 2n−i−jn+1 , which checks out. Now all that

is left is to find df . To do this we simply solve the equation

df − di − dj − dn+1−k + 1− dk =2n− i− jn+ 1

for df . This yields df = 2n+4n+1 . So

di =i+ 1

n+ 1and df =

2n+ 4

n+ 1∀i = 1, . . . , n. (5.14)

Remark 4 Note that we have also obtained the homogeneity condition

x

n+ 1∂xF (x, t) +

n∑i=1

i+ 1

n+ 1ti∂iF (x, t) = F (x, t) (5.15)

and we have found that explicitely

E =

n∑i=1

i+ 1

n+ 1ti∂i (5.16)

and

LEΦ =2n+ 4

n+ 1Φ (5.17)

up to second order terms. In terms of the alternative constants given in (1.23)we find qi = n−i

n+1 and d = n−1n+1 as in [1] (after renumbering the flat coordinates).

Remark 5 In fact finding these constants results from assigning to F the degree1. We will see that this yields an E that satisfies condition 4 in section 1.

23

6 Deformed Flat and Canonical Coordinates ofAn Singularity

In this section we will show explicitely that the deformed connection defined insection 2 is flat for the Frobenius structure corresponding to the An singularity.We will do this by finding a solution to the system of equations (2.9) and (2.10).This will subsequently allow us to prove that our structure does in fact satisfythe definition of a Frobenius structure, i.e. we will prove theorem 7. At thispoint we will have 3 natural coordinate systems: the natural coordinates definedin the deformation, the flat coordinates and the deformed flat coordinates. Soin this section we will also finish this list by finding the canonical coordinatesfor the An singularity.

Deformed Flat Coordinates

Consider the oscillatory integrals

JB = (−2πz)−12

∫B

ezF (x,t)dx (6.1)

where B is a cycle in some relative homology group, which ensures that theintegrand is integrable over B. We will specify the relative homology later. Fornow, we just want to prove that the exterior derivatives of these functions arehorizontal with respect to the deformed connection ∇, whenever the cycles aresuch that the integrals make sense. To show this it is sufficient to show that theequations (2.5) and (2.6) hold with ξj = ∂j(−2π)

12JB .

We can use the polynomials Kij and Stokes’ theorem to show the first equality

∂iξj = ∂i

(z

12

∫B

(∂jF )ezFdx

)=

= z12

∫B

(∂i∂jF )ezFdx+ z32

∫B

(∂iF )(∂jF )ezFdx =

= z12

∫B

(∂xKij)ezFdx+ z

32

∫B

(Kij∂xF +

n∑k=1

ckij(t)∂kF )ezFdx =

= z

n∑k=1

ckij(t)∂k(z−12

∫B

ezFdx) + z12

∫B

∂x(KijezF )dx = z

n∑k=1

ckij(t)ξk (6.2)

so indeed the oscillatory integrals satisfy the first equation. In the last equalitywe used the fact that the second integral vanishes by virtue of the behaviour ofthe exponent near the end points of the cycle B. In other words we used theproperties of the relative homology which will be defined below.

To show the second equation we will first compute z∂zJB so using equation(5.15) we get

z∂zJB = −1

2JB + (−2π)−

12 z

12

∫B

FezFdx =

24

= −1

2JB + (−2π)−

12 z

12

∫B

(x

1 + n∂xF +

n∑i=1

diti∂iF

)ezFdx =

= −1

2JB +

n∑i=1

diti∂iJB + (−2πz)−12

∫B

x

1 + n∂xe

zFdx =

= −1

2JB +

n∑i=1

diti∂iJB + (−2πz)−12

∫B

∂x

(x

1 + nezF)

dx− 1

n+ 1JB =

=d− 2

2JB +

n∑i=1

diti∂iJB (6.3)

since 12 + 1

1+n = n+32n+2 = 2−d

2 . In the last equality we used again the properties ofthe relative homology defined below, to show that the third summand vanishes.

Then we see that indeed

∂zξj =1

z

(d− 2

2ξj + djξj +

n∑i=1

diti∂iξj

)=

=

n∑i,k=1

ditickijξk −

ξjz

(qj −

d

2

), (6.4)

which is the second equation.

Now let us turn to the cycle B. We need to choose it such that the exponent inthe definition (6.1) oscillates at the end points. So let

Y := limM→∞

H1(C, x|Re zF (x, t) < −M). (6.5)

Then for any cycle B ∈ Y ' Zn we can assure that indeed the integrals withderivatives of x as integrands vanish and thus that ∂jJB satisfies the equations(2.5) and (2.6).

So let us construct once and for all a basis Bini=1 of Y. This then yields abasis dJBini=1∪dz of horizontal one-forms which shows that ∇ is flat. Withthe deformed flat coordinates JBi .

Denote by Vλ,t the Milnor fiber

Vλ,t := x ∈ C|F (x, t) = λ, (6.6)

for λ ∈ C and t ∈ T . Then, for i = 1, . . . , n, we define the cycle Bi as follows.For any semisimple t ∈ T we have, as will be shown below, n critical values(u1, . . . , un) of the function F (x, t), with some ordering. First we pick a pathfrom λ = ui to zλ → −∞, where all the other (isolated) critical values areavoided. In every Milnor fiber along this path we pick a cycle that vanishes asλ approaches ui. We call their union Bi. For an in-depth discussion of theseconstructions we refer to [14].

25

Proof of Theorem 7

In this section we will complete the proof of theorem seven. We have nowconstructed explicitely the flat coordinates and thus already shown that themetric is flat. Also non-degeneracy of the metric shows from the matrix< ∂i, ∂j >=

−δi+j,n+1

n+1 which is clearly not singular (and anti-diagonal). We hadalready shown invariance by equation (3.5) and symmetry is clear. Thus thefirst condition of section 1 is satisfied.

Now ∂nF = ∂n(xn+1 +τ1xn−1 + . . .+τn) = ∂nτn = −1, since τi does not depend

on tj with j > i and we had already deduced that ∂iτi = −1 for all i = 1, . . . , n.So we find that for all i = 1, . . . , n we have

(∂nF )(∂iF ) = −∂iF

so we see from (5.1) that ckni = −δik = ckin, for all i, k = 1, . . . , n. So

(−∂n) · v = v = v · (−∂n) ∀v ∈ TtT for all t ∈ T , (6.7)

i.e. (−∂n) is the unit vector field. Now it is clear that indeed ∇(−∂n) = 0, sincetn is a flat coordinate, and thus the second condition of section 1 is satisfied.

Next we need to prove symmetry of the four-linear form C defined in equation(1.10). By linearity it is sufficient to show symmetry on a basis and since wealready have symmetry of the three-linear form c defined in equation (1.9) weonly need to show that

∂m < ∂i · ∂j , ∂k >= ∂i < ∂m · ∂j , ∂k >,

since ∇∂p = ∂p. Now since ∂i · ∂j = cpij∂p and < ∂i, ∂j >=−δi+j,n+1

n+1 it issufficient to show that

∂mckij = ∂ic

kmj . (6.8)

This makes sense since if T is a Frobenius manifold it is one that has an under-lying solution to the WDVV equations and in that case the structure constantsare just third derivatives and should definitely satisfy this equation. We havenot yet proved that T is a Frobenius manifold however, but this is where the de-formed connection comes in. Since we have shown that the deformed connectionis flat, we have in general that

∇v∇wu− ∇w∇vu− ∇[v,w]u = 0 ∀v, w, u ∈ X (T × C). (6.9)

Substituting v = ∂i, w = ∂j and u = ∂k then yields

∇∂i(z∂j · ∂k)− ∇∂j (z∂i · ∂k) = 0,

since [∂i, ∂j ] = 0. This then yields

z

n∑l=1

(∂icljk − ∂jclik)∂l + z2

n∑l,m=1

(cljkcmil − clikcmjl )∂m = 0. (6.10)

So equating the coefficients of every basis vector and like powers of z to 0 wefind exactly the equations (6.8) and the equations of associativity of the algebras

26

(1.16), which we already knew. In any case this proves that the four-linear termC is indeed symmetric for the parameter space of the An singularity. Thus thethird condition of section 1 is satisfied.

The last axiom asks for a linear vector field and, since we want to show also thatthere is an underlying solution of the WDVV equations, we also want it to bediagonalizable. Of course we have already shown that E in equation (5.11) hasboth these features. All that is left to be shown is that the equations (1.4)-(1.7)hold for the grading operators deifned in (1.11) and (1.12). The first equality(1.4) holds. It is simply equivalent to the Leibniz rule for the vector field E.Similarly we have

QA(fv) = fv + [E, fv] = fv + E(f)v + f [E, v] = QR(f)v + fQA(v),

immediately for all f ∈ F(T ) and v ∈ X (T ). So (1.5) holds as well.

Now note that, in the setting of lemma 2, we should have ∂i∂zξj = ∂z∂iξj , for alli, j = 1, . . . , n. Equating the different powers in z we find again the condition3 and we see that µ should not depend on the point t ∈ T . The relevantinformation concerning the grading operators comes for the terms independentof z however. This yields

n∑k=1

ckijξk −n∑

k,l=1

ckijδkl

(2− d

2− dk

)ξl =

n∑k,l=1

(∂idktk)clkjξl +

n∑k,l=1

dktk(∂iclkj)ξl −

∑k,l

δkjclikξl

(2− d

2− dj

), (6.11)

for any i, j = 1, . . . , n. Then using the symmetry of the four-linear form C, i.e.condition 3, we find

n∑k,l=1

dktk(∂kclij) =

n∑k=1

(1− di − dj + dk)ckij . (6.12)

So we find that indeed

QA(∂i · ∂j) =

n∑k=1

QR(ckij)∂k + ckijQA(∂k) =

=

n∑l,k=1

dltl∂lckij∂k +

n∑k=1

ckij∂k + ckij [E, ∂k] =

= (2− di − dj)∂i · ∂j = 2∂i · ∂j + ∂i · [E, ∂j ] + [E, ∂i] · ∂j =

= ∂i ·QA(∂j) +QA(∂i) · ∂j ,

so (1.6) holds. Finally we have

< QA(∂i), ∂j > + < ∂i, QA(∂j) >= (2− di − dj)−δi+j,n+1

n+ 1=

27

=

(2− i+ 1

n+ 1− n+ 1− i+ 1

n+ 1

)−δi+j,n+1

n+ 1= d < ∂i, ∂j >=

= QR(< ∂i, ∂j >) + d < ∂i, ∂j >

So we see that all 4 conditions are met and thus T is indeed a Frobenius manifoldwith an underlying solution to the WDVV equations. Theorem 7 is proved.

Remark 6 To prove theorem 7 we actually used only the metric and the flatdeformed connection. Thus it is possible to give an equivalent definition ofFrobenius manifold. In this sense a Frobenius manifold is a manifold with aflat metric and a pencil of connections that extends in the sense of equations(2.1)-(2.3) to a flat connection on the direct product M × C. This definition isused in [17].

Canonical Coordinates of An Singularity

As was noted in section 2 the eigenvalues of the operation of multiplication bythe Euler vector field serve as canonical coordinates. Now we claim that thesemisimple points of T are the points in which F has no degenerate criticalvalues. So the points in which F has n distinct non-degenerate critical values.This fact is expressed in the following lemma.

Lemma 11 The eigenvalues of E in a semisimple point t ∈ T are the criticalvalues of F (x, t).

Proof:

Suppose v is an eigenvector at the point t ∈ T then

Ev = λv for some λ ∈ C.

In other words we have E · v = λv. We can translate this to the condition

[LEF |t] · [LvF |t] = λ[LvF |t]

in the space C[x]/〈∂xF |t〉. Now in general we have the homogeneity condition(5.15) which yields

[F |t −x

n+ 1∂xF |t] · [LvF |t] = λ[LvF |t],

but [∂xF |t] = [0]. So we arrive at the equation

[F |t − λ] · [LvF |t] = [0]. (6.13)

So in fact(F |t − λ)LvF |t = K(x)∂xF |t,

for some K(x) ∈ C[x]. At a critical point xi with i = 1, . . . , n the right handside of this equation vanishes. Thus we have either λ = F (xi, t) or LvF |t = 0.However the degree of the right hand side in x is n while the degree of LvF |tin x is not higher than n − 1. Thus there is at least 1 critical point such thatLvF |t 6= 0.

28

So we see that every eigenvector of E has a critical value as eigenvalue. Nowsuppose v 6= w are both eigenvectors of E . Suppose the critical points of F |tare x1, . . . , xn and the critical value corresponding to v is F (xi, t). Then weneed to show that the above argument does not fail for w. So we need toshow that LwF |t does not vanish for a different critical value from xi. But if(LwF |t)|xj = 0 for all xj but xi and the same goes for LvF |t then we find v = w.This contradiction shows that for every one of the distinct eigenvectors vi of Ethere is also a distinct critical point that is not a root of LviF |t.

This shows that the coordinate system (u1, . . . , un) of critical values is the canon-ical coordinate system. So let us go on to calculate the metric for the frame(∂u1

, . . . , ∂un) generated by the canoncical coordinates. Note that we can ex-pand ∂xF |t for any semisimple t ∈ T around the critical point xi correspondingto the critical value ui. In fact since F |t is a polynomial we will again find apolynomial. We have (dropping for now the |t since it is immaterial)

∂xF = ∂xF |xi + ∆i(x− xi) +O((x− xi)2), (6.14)

where ∆i denotes the Hessian of F in the point xi. However we can drop thefirst term since xi is a critical point. On the other hand we can expand F itselfin the same way, which yields

F = F |xi + ∂xF |xi(x− xi) +1

2∆i(x− xi)2 +O((x− xi)3). (6.15)

Note that this time the second summand vanishes while we have F |xi = ui. Nowsince the ui are the canonical coordinate system we have the multiplication

∂ui · ∂uj = δij∂ui .

So we have by the definition of the multiplication that

(∂uiF )(∂ujF ) = δij∂uiF +Nij∂xF, (6.16)

for some polynomial Nij ∈ C[x]. Finally we have

< ∂ui , ∂uj >= Resx=∞

δij∂uiF +Nij∂xF

∂xFdx =

δij Resx=∞

1 + ∂ui(12∆i(x− xi)2 +O((x− xi)3))

∆i(x− xi) +O((x− xi)2)dx = −δij∆−1

i (6.17)

Thus we can define the normalized frame √−∆i∂uini=1. Then let Ψ(τ) denote

the transition matrix from the frame of flat coordinates to the orthonormalframe, i.e.

n∑j=1

[Ψ]ij∂j =√−∆i∂ui . (6.18)

Obviously this transformation matrix could be defined for any (semisimple)Frobenius structure. However we will not need many properties for the rest.For a complete exposition of Ψ in the general case see [1].

29

Part II

KdV hierarchies

7 Quantization Formalism and Operators on theFock Space

We will need the quantization formalism, in order to give a formulation of the(n+1)KdV hierarchies and Givental’s formula for the total descendant potential.This will allow us to define the Fock space and quantized operators on it. Thedescendant potential will be found as an element in the Fock space, which isobtained from another element by action of certain operators. In this section wewill first give a definition of the Fock space and subsequently we will show howto quantize linear and quadratic Hamiltonians and certain operators to act onthis Fock space. For this section we will return to the situation in the beginningof section 3, i.e. we restrict ourselves to the point 0 ∈ T .

Quantization Formalism

As before let H = C[x]/〈fx〉 denote the local algebra of the singularity. Weknow that the residue pairing < ·, · >0 is a non-degenerate symmetric bilinearform on H. Now let H := H((z−1)) denote the space of Laurent series in oneindeterminate z−1 with coefficients in H. Let Ω denote the bilinear form on Hdefined by

Ω(f,g) =1

2πi

∮< f(−z),g(z) >0 dz ∀f,g ∈ H. (7.1)

Note that, since < ·, · >0 is symmetric, the change of variable z 7→ −z in theintegral shows that for all f,g ∈ H we have

Ω(f,g) = −Ω(g, f).

Moreover if Ω(h, ·) = 0 for some h ∈ H then we would need < h(−z), · >0= 0,thus by non-degeneracy of this last form we see that Ω is non-degenerate andthus that Ω is a symplectic form on H.

Consider the polarization H = H+ ⊕ H− of H into the Lagrangian subspacesH+ = H[z] and H− = z−1H[[z−1]]. This identifies (H,Ω) with the cotangentbundle T ∗H+. Then let qi be a coordinate system on H+ and pi the dualcoordinate system on H− such that Ω =

∑i pi ∧ qi, i.e. they form a Darboux

coordinate system on T ∗H+. Suppose we have the orthonormal basis e1, . . . enfor H, then we have for a general element f ∈ H that

f =∑k≥0

n∑α=1

eαqα,kzk +

∑k≥0

n∑α=1

eαpα,k(−z)−k−1. (7.2)

The quantization formalism will associate to any constant, linear and quadraticfunction G on (H,Ω) a differential operator G of order smaller then 3. Thisoperator will act on functions on H+. We will call the space of such functionsthe Fock space.

30

The quantization G of a constant, linear or quadratic function on (H,Ω) iscarried out by the recipe

qi :=qi√~

pi :=√~∂qi (7.3)

and(qiqj) :=

qiqj~

(qipj) := qi∂qj (pipj) := ~∂qi∂qj , (7.4)

where√~ is just some formal variable. The quantization is thus a representation

of the Heisenberg algebra of constant and linear Hamiltonians. However it isonly a projective representation of the Lie algebra of quadratic Hamiltonians,i.e. we have

F,G = [F , G] ∀ constant and linear Hamiltonians F,G (7.5)

F,G = [F , G] + C(F,G) ∀ quadratic Hamiltonians F,G, (7.6)

where ·, · denotes the Poisson bracket corresponding to Ω, [·, ·] denotes thecommutator bracket and C is a cocycle characterized by the properties

C(pipj , qiqj) = 1 if i 6= j, C(p2i , q

2j ) = 2 (7.7)

and C = 0 on all other pairs of quadratic Darboux monomials [2]. To reiterate,these differential operators act on formal functions (with coefficients dependingon (√~)±1) on H+. The space of such functions is called the Fock space. Most

notably the total descendant potential will be viewed as an element of the Fockspace.

Twisted Loop Group

Let us now turn to quantizations of different (rather specific) operators. Let usdefine the loop group LGL(H) as the space of GL(H) valued formal functionsin z. These then constitute tranformations on the space H and it is natural torestrict our attention to those elements of the loop group that preserve Ω. Thisyields the following definition of the twisted loop group

L(2)GL(H) := U ∈ LGL(H)| UT (−z)U(z) = 1, (7.8)

where the transposition is with respect to < ·, · >0 and the 1 represents theidenity matrix in GL(H). This space has corresponding Lie algebra L(2)End(H).We can make a split of this space into the subalgebras of upper and lowertriangular tranformations, g+ respectively g−, given by

g± := u(z) =∑k>0

ukz±k| uk ∈ End(H), uT (−z) + u(z) = 0 (7.9)

We can quantize the elements of the twisted loop group as follows. For U ∈L(2)GL(H) we define U = explnM. So we express first the operator U as eu

for some u ∈ L(2)End(H). Subsequently we express this as a linear or quadratichamiltonian Hu given by

Hu(f) =1

2Ω(f, uf), (7.10)

31

for f ∈ H. Then we exponentiate the result of the procedure set out in (7.3) and(7.4). For future reference let us set out the general quantization of elementsof the lower, respectively upper, triangular matrices. We do this by giving thequantization of the corresponding elements of the Lie subalgebras g±.

Explicit calculation yields for s =∑k>0

skz−k that

s =∑p>0

n∑α,β=1

(sp)αβ

(−1)p−1∑i≥0

qα,i+p∂β,i +1

2~∑

i+j=p−1

(−1)iqα,iqβ,j

,

(7.11)where (sp)αβ =< eα, speβ >0 for the orthonormal basis and

∂β,i = ∂∂qβ,i

. Similarly we have for r =∑k>0

rkzk that

r =∑p>0

n∑α,β=1

(rp)αβ

(−1)p−1∑i≥0

qα,i∂β,i+p +~2

∑i+j=p−1

(−1)j+1∂α,i∂β,j

,

(7.12)where again (rp)αβ =< eα, rpeβ >0.

Action on the Fock Space

We will need to know how the quantized symplectic operators given by suchupper and lower triangular elements act on the Fock space. Thus we needto find a way to split the operators into a multiplication by elements in H+

and derivations. For the following discussion see also [13, 19]. Let us startby describing this procedure for the lower triangular operators. So supposeS ∈ L(2)GL(H) such that

S(z) = es(z) where s ∈ g−,

the above discussion shows that

lnS =∑p>0

n∑α,β=1

(sp)αβ

(−1)p−1∑i≥0

qα,i+p∂β,i +1

2~∑

i+j=p−1

(−1)iqα,iqβ,j

(7.13)

and

lnS−1 =∑p>0

n∑α,β=1

(sp)αβ

(−1)p∑i≥0

qα,i+p∂β,i +1

2~∑

i+j=p−1

(−1)i+1qα,iqβ,j

,

(7.14)

where s =∑k>0

skz−k as before. First we will write this as the product of two

exponents using the Campbell-Baker-Hausdorff formula. So denoting

Y =1

2~∑i,j≥0

n∑α,β=1

(si+j+1)αβ(−1)i+1qα,iqβ,j (7.15)

32

and

X =∑p>0

∑i≥0

n∑α,β=1

(−1)(sp)αβqβ,i+p∂α,i, (7.16)

were we used the fact that sTp = (−1)p−1sp since s ∈ g−, we require Z such that

eX+Y = eXeZ .

Now note that since

[qα,i+p∂β,i, qγ,sqε,r] = qα,i+p(δβ,γδi,sqε,r + δβ,εδi,rqγ,s), (7.17)

for all α, β, γ, ε = 1, . . . , n and i, r, s, p ≥ 0, we see that [X,Y ] ∈ Ker [Y, ·].Where we view [Y, ·] as a linear operator on the Heisenberg Lie algebra givenby the quantization formalism. In this case the CBH-formula yields

Z =−e−adX + 1

adXY =

∑p≥0

(−1)p(adX)p

(p+ 1)!Y =

1

2~∑

(p≥0)

∑(a+b=p)

∑(f1,...,fa≥0)

∑(g1,...,gb≥0)

∑(i,j≥0)

n∑(α,β=1)

(pa

)(p+ 1)!

(−1)i+1+f1+...+fa+b·

·(sfa · · · sf1si+j+1sg1 · · · sgb)αβqα,i+f1+...+faqβ,j+g1+···+gb , (7.18)

where we set s0 = 0. In most respects a horrible formula to work with of course.However note that we could give Z in the form

2~Z = W (q,q) =∑k,l≥0

< Wklqk, ql > . (7.19)

Now the quadratic form W gives us a description for Z and we see that W isdefined by ∑

k,l

Wkl

wkzl=St(w)S(z)− 1

w−1 + z−1, (7.20)

by expanding the right hand side and comparing to (7.18). Here 1 signifies theidentity matrix.

Suppose that G(q) is an element of the Fock space. Then we see now that

S−1G(q) = eXeZG(q) = eW (q,q)

2~ eXG(q) = eW (q,q)

2~ G([Sq]+), (7.21)

where we denote by [Sq]+ the truncation of negative powers in z of S(z)q(z).

We can describe the upper triangular operators in a similar fashion. So supposeR ∈ L(2)GL(H) such that

R(z) = er(z) where r ∈ g+.

Denoting this time

33

Y =~2

∑i,j≥0

n∑α,β=1

(ri+j+1)αβ(−1)j∂α,i∂β,j (7.22)

and

X =∑p>0

∑i≥0

n∑α,β=1

(−1)(rp)αβqβ,i∂α,i+p, (7.23)

where we used again that rTp = (−1)p−1rp since r ∈ g+. As before we have[X,Y ] ∈ Ker[Y, ·], since

[qα,i∂β,i+p, ∂γ,s∂ε,r] = −(δα,γδi,s∂ε,r + δβ,εδi,r∂γ,s)∂β,i+p, (7.24)

for all α, β, γ, ε = 1, . . . , n and i, r, s, p ≥ 0. Thus we find this time that Z equals

~2

∑(p≥0)

∑(a+b=p)

∑(f1,...,fa≥0)

∑(g1,...,gb≥0)

∑(i,j≥0)

n∑(α,β=1)

(pa

)(p+ 1)!

(−1)j+f1+...+fa+a·

·(rgb · · · rg1si+j+1rf1 · · · rfa)αβ∂α,i+g1+...+gbqβ,j+f1+···+fa , (7.25)

where we set r0 = 0. Again we express Z using a quadratic form such that

2

~Z = V (∂, ∂) =

∑k,l≥0

< qk, Vklql > . (7.26)

This time V is defined by∑k,l

Vklwkzl =

I −R(w)RT (z)

w + z. (7.27)

So for G as before we have this time

R−1G(q) = e~V (∂,∂)

2 G(Rq). (7.28)

From the action of these inverse operators we can infer the action of S and R.

8 (n+ 1)KdV Hierarchy and Vertex Operators

To show how the Frobenius structure related to the An singularity allows one toturn a solution to the KdV hierarchy into a solution of the (n+1)KdV hierarchywe will need to put these hierarchies in a specific form. In fact there are manyways to view integrable hierarchies in general [10, 4]. We will start from theansatz of the Kadomtsev-Petviashvili (or simply KP) hierarchy given by Hirotaquadratic equations in vertex operator form [20]. We will subsequently showhow to translate these equations to the frame of the quantization formalism andthe Fock space in the case of the KdV and (n+ 1)KdV hierarchies. To do thiswe will also use certain elements of the Frobenius structure in the point 0 ∈ T ,as we did in the last section.

34

From KP to (n+ 1)KdV Hierarchy

We start from the KP hierarchy in the form given in [2]. Thus a function Φ(x),which is assumed to be of the form

Φ(x) = exp∞∑g=0

~g−1φ(g)(x), (8.1)

is a solution to the KP hierarchy if

Resξ=∞

dξ exp∑j>0

ξj√~

(x′j − x′′j ) exp−∑j>0

ξ−j

j

√~(∂x′j − ∂x′′j )Φ(x′)Φ(x′′) = 0

(8.2)where x = (x1, x2, . . .) is a vector variable. By definition, solutions of the(n+1)KdV hierarchy are those solutions of the KP hierarchy that do not dependon the variables xj with j ≡ 0 mod (n + 1). A more convenient interpretationis available however. First of all the change of variables

xj =x′j + x′′j

2, yj =

x′j − x′′j2

, ∂xj = ∂x′j + ∂x′′j and ∂yj = ∂x′j − ∂x′′j (8.3)

yields the alternative version of the KP hierarchy

Resξ=∞

dξ exp2∑j>0

ξj√~yj exp−

∑j>0

ξ−j

j

√~∂yjΦ(x + y)Φ(x− y) = 0. (8.4)

Expanding this equation in y then yields an infinite system of partial differentialequations which is another form of the KP hierarchy. This would yield theresidue of some infinite series in y where the coefficients of monomials ym arelaurent series in ξ−1. So for each of these Laurent series there is a highest powerof ξ depending on m.

2KdV

Let us look specifically at the 2KdV hierarchy (also just called KdV hierarchy).So we are looking at solutions of KP that do not depend on xj for j ≡ 0 mod 2.Thus we can omit the derivatives ∂y2k , but note that we cannot omit the mul-tiplications by y2k. This yields the following form of the KdV hierarchy

Resξ=∞

dξ S(ξ2) exp2∑j odd

ξj√~yj exp−

∑j odd

ξ−j

j

√~∂yjΦ(x + y)Φ(x− y) = 0,

where the sums run only over positive odd values of j and S(ξ2) = e2∑k>0

ξ2k√~y2k .

Now since Φ does not depend on xj for j odd we can treat S as an arbitrary

function in ξ2 (and√~). This shows that vanishing of the residue above is

equivalent to the coefficients of odd (positive) powers of ξ−1 in the coefficientsof ym in the expansion of the one-form

dξ exp2∑j odd

ξj√~yj exp−

∑j odd

ξ−j

j

√~∂yjΦ(x + y)Φ(x− y)

35

vanishing. Since every one of these coefficients can be made to equal the co-efficient of ym in the residue by picking a suitable S. We can then state thatΦ(xodd) is a solution to the KdV hierarchy if and only if the one-form abovehas no odd non-positive powers of ξ. By substracting the one-form with theopposite sign in the exponents we see that the resulting form cannot have anynegative even powers of ξ. This gives us the final form of the KdV hierarchy.Namely, a function Φ(xodd) solves the KdV hierarchy if and only if the form∑±±dξ exp±

∑j odd

ξj√~

(x′j − x′′j ) exp∓∑j odd

ξ−j

j

√~(∂x′j − ∂x′′j )Φ(x′)Φ(x′′)

(8.5)is regular in ξ2. By regular in ξ2 we mean exactly that in the expansion of theform in y the Laurent series that form coefficients to the different powers of yhave no negative powers in ξ2. So they should be polynomials in ξ2.

Now let T (t) denote the Witten-Kontsevich tau-function, where t = t(z) =t0 + t1z + t2z

2 + . . .. Then we define

DA1(q) := T (t), where q(z) := t(z)− z. (8.6)

Witten’s conjecture which was proved by Kontsevich [11, 12] now says that DA1

satisfies the condition (8.5) after the substitution

qk =(2k + 1)!!

2k√

2x2k+1 (8.7)

for k ∈ N∪0, which yields also ∂x2k+1= (2k+1)!!

2k√

2∂qk . Substituting this in (8.5)

we see that the exponents are quantized linear Hamiltonians in the Heisenbergalgebra as given in section 7. Clearly we can seperate the parts depending onq’ and q”. Moreover we can use the notation where λ = ξ2 such that we cansum over the roots of λ instead of over ±, this yields∑

√λ=±ξ

dλ√λ

[Γ−(λ)DA1(q’)][Γ+(λ)DA1

(q”)]

is regular in λ, where

Γ±(λ) = exp∓∑k<0

1√2

(d

dλ)kλ−

12 q−k−1 exp±

∑k≥0

(−1)k√2

(d

dλ)kλ−

12 pk.

Here we have omitted the accents on q and p, but it should be clear where theaccents are meant to be. In any case we will soon get rid of the accents. The term

( ddλ

)kλ−12 for k < 0 signifies a function fk(λ) such that ( d

dλ)−kfk(λ) = λ−

12 ,

i.e. fk(λ) = 12k

λ−k−12

(−2k−1)!! . Now we can get rid of the accents by realizing that the

linear Hamiltonians that appear in the exponents Γ± can be encoded by theirHamiltonian vector fields. Then by the standard relation q = ∂pf and p = −∂qffor the Hamiltonian vector fields of the Hamiltonian f in Darboux coordinates,we see in general that qk 7→ −

√2(−z)−1−k and pk 7→

√2zk. Thus the following

formal expression ∑√λ=±ξ

dλ√λ

[Γ−(λ)DA1(q’)][Γ+(λ)DA1

(q”)], (8.8)

36

should be regular in λ. Here we have

Γ±(λ) = exp±∑k<0

(d

dλ)kλ−

12 (−z)k exp±

∑k≥0

(d

dλ)kλ−

12 (−z)k. (8.9)

This is what we will call the vertex operator form of the KdV hierarchy. TheΓ± are what is called vertex operators in this context. Note that DA1 is anelement in the Fock space and the vertex operators act on it.

(n+ 1)KdV

Now we would like to find vertex operators for the (n+ 1)KdV hierarchy as wehave found for the KdV hierarchy. It is apparent that we could simply write

Γ± = exp±∑k∈Z

(d

dλ)kλ−

12 (−z)k.

Thus we should generalize the role of λ−12 and its (anti-)derivatives. We can

write this as the integral

λ−12 =

∫α

1

x= 2

∫α

dx

dx2

over the one-point cycle α = [xα] such that λ = x2α and η := dx

dx2is a 0-form

such that η ∧ dx2 = dx. This corresponds to λ = f(xα) and η = dxdf

for f the

A1 singularity. This leads to the following definition of the vertex operators forthe (n+ 1)KdV hierarchy

Γα = exp∑k∈Z

I(k)α (λ)(−z)k (8.10)

where the I(k)α are functions with values in the local algebra of the An singularity,

which are called period vectors, and α ∈ f−1(λ) is a one-point cycle. Just asfor their analogue in the 2KdV hierarchy we will define the period vectors asconsecutive (anti-)derivatives of each other

d

dλI(k)α (λ) = I(k+1)

α (λ). (8.11)

In the next part we will need a generalization of the period vectors and therewe will specify how exactly to fix the anti-derivatives. In this case it should beclear from the expression (8.14) below what has been done. Otherwise we referto appendix A.

From now on we will omit the subscript α for the period vectors when it iswarranted. We define I(0) using the non-degenerate form < ·, ·, >0 by

< I(0)β (λ), [φ] >0= −

∫β

φ(x)dx

df(x), (8.12)

where [φ] ∈ H, β ⊂ f−1(λ). Since the form is non-degenerate this defines theperiod vector uniquely up to monodromy of the cycle.

37

Now let us apply this definition to the An singularity. So we pick a one-pointcycle α = [xα] such that λ = xn+1

α . This yields

< I(0)α , [xn−i] >0= −

∫α

xn−idx

dxn+1=−λ

−in+1

n+ 1, (8.13)

so we have I(0)α =

n∑i=1

[xi−1]λ−in+1 . This follows from the bilinear form

< [xi], [xj ] >0=−δi+j,n−1

n+ 1.

So we find ∑k∈Z

I(k)(−z)k =

∑k∈Z

n∑i=1

[xi−1]zk

(n+ 1)k

∏∞r=0[i+ r(n+ 1)]∏∞r=k[i+ r(n+ 1)]

λ−i+k(n+1)n+1 . (8.14)

Now we can apply the same procedure in the general case as we did in the caseof the KdV hierarchy. First we reverse the encoding sending the operators p, qto their Hamiltonian vector fields. We see that

Ω(√n+ 1[xn−i]z−k−1,

√n+ 1[xi−1]zk) = (−1)k (8.15)

while the other elements in this sum are orthogonal. So we encode qi,k 7→−√n+ 1(−z)−k−1[xn−i] and pi,k 7→

√n+ 1[xi−1]zk. By substituting this and

subsequently making the substitution

qi,k =i(i+ (n+ 1)(i+ 2(n+ 1) · · · (1 + k(n+ 1))

(n+ 1)k√n+ 1

xk(n+1)+i (8.16)

and the consequence for p we find∑k∈Z

I(k)(−z)k =

= −∑k≥0

n∑i=1

λi+k(n+1)n+1

√~

xk(n+1)+i +∑k≥0

n∑i=1

λ−i+k(n+1)n+1

i+ k(n+ 1)

√~∂xk(n+1)+i

. (8.17)

So in fact we have∑k∈Z

I(k)(−z)k = −∑

j∈(Z−nZ)≥0

ξj√~xj +

∑j∈(Z−nZ)≥0

ξ−j

j

√~∂xj ,

where ξn+1 = λ. Comparing this to the KP hierarchy (8.2) we can constructthe (n+ 1)KdV hierarchy in vertex operator form in the same way as we did forthe case n = 2 (so it is related to the A1 singularity). This yields the followingstatement. A function D(q) satisfies the (n+ 1)KdV hierarchy if and only if∑

α

λ−nn+1 dλ(Γ−αD)(q’)(ΓαD))(q”), (8.18)

38

is regular in λ. Here the sum is over the different one-point cycles α = [xα(λ)]

such that λ = xn+1α . So the sum is taken over the n + 1 different roots λ

1n+1

differing by roots of unity and

Γα = exp∑k∈Z

I(k)α (λ)(−z)k. (8.19)

Equation (8.18) is the form of the KdV hierarchy that we will be working with.It has been shown by Kontsevich that DA1 satisfies the equation for n = 1.The rest of this thesis will be dedicated to showing that in fact the (n+ 1)KdVhierarchy splits up in a particular way in n copies of the KdV hierarchy.

39

Part III

Solving the (n + 1)KdV Hierarchy

9 Givental’s Formula

In this last part we will show how the solution to the KdV hierarchy togetherwith Givental’s formula for the total descendant potential corresponding to asemisimple Frobenius structure yields a solution to the (n+1)KdV hierarchy [2,7, 13]. We will do this in a hands on approach, i.e. we will give the formula andits ingredients strictly in the setting of the An singularity. Subsequently we willshow why this indeed yields a solution to the (n + 1)KdV hierarchy followingthe reasoning made in [2] apart from a few minor adjustments. Let us begin bystating Givental’s formula

D(q) := C(t)S−1t Ψ(t)Rt

exp(

U

z)

n∏i=1

DA1(qi). (9.1)

Here C(t) signifies an invertible matrix which in our case will be reduced to aconstant matrix, St and Rt are certain lower respectively upper triangular quan-tized symplectic operators, which will be defined below. We have the matrixU which is diagonal with the eigenvalues of multiplication by the Euler vectorfield on the diagonal. Recall that Ψ is the basis tranformation matrix fromthe frame of flat coordinates to the frame of normalized canonical coordinates.Furthermore we have

(q1, . . . ,qn) = Ψ−1q ∈ Cn[z]. (9.2)

Here we are referring to the coordinates of q ∈ H[z] in terms of the normalizedcanonical coordinates in TtT identified with H = T0T via the metric. Thequantization Ψ is viewed in the sense that for G(q) an arbitrary element of theFock space we have ΨG(q) = G(Ψ−1q). Lastly t ∈ T is a semisimple point.Givental proves in [13] that this total descendant potential D is independent ofthe point t ∈ T (in fact this is the reason for C(t)). We will first spend a fewsections in defining and discussing the ingredients of this formula.

Important Remark

We need to remark here the nature of the operators St and Rt. Above it ismentioned that these are lower respectively upper triangular operators. So wehave

St(z) = 1 + S1z−1 + S2z

−2 + . . . (9.3)

andRt(z) = 1 +R1z +R2z

2 + . . . , (9.4)

where 1 denotes the identity matrix and St and Rt are the quantized versions.The parameter z is a coordinate on C and we could switch to the coordinatesystem w = 1

z . This would seem to change the nature of these operators.However, we need to consider the action on functions in the Fock space. Herewe have a clear description of the action of lower and upper triangular operators

40

given by (7.21) and (7.28). In discussing the elements corresponding to theFrobenius structure of the An singularity, for instance the deformed connection,we have used the conventions of [1], which are clearest in this setting. Howeverwhen discussing the quantization and symplectic operators it is more useful touse the convention of [2]. Practically this means that we will switch from onecoordinate system on C to the other when discussing these parts. This willensure that the nature of the symplectic operators is always expressed clearly.Whenever such a change of coordinate systems is implied we will also note thisdirectly.

Note then that in Givental’s formula above the term in the exponent is Uz in the

coordinate system that expresses the nature of the symplectic operators. Fur-thermore the coordinate system for the vertex operators in (8.19) also matchesthat of (9.3) and (9.4). On the other hand the coordinate used in (6.1) isopposite to that used in (9.3) and (9.4).

10 Period Vectors

To show the solution of the (n + 1)KdV hierarchy that arises from the Ansingularity and the Witten-Kontsevich tau-function we shall need to use theFrobenius structure considered in part I. First of all we will want to generalizethe definition of vertex operators to the rest of the parameter space of thesingularity. To do this we simply need to generalize the definition of the periodvectors along any cycles in the Milnor fibers of the singularity [14]. In this sectionwe will define the period vectors and vector fields related to the oscillatingintegrals. We will show that, in the case that the cycle in the definition of theperiod vector is of a certain type we can write the vector fields related to theoscillating integrals as a version of Laplace transform of the period vectors. Wewill derive some useful relations for both the oscillating integrals and the periodvectors. In this entire section we might suppress the indices referring to cycleswhen it is appropriate.

Period Vectors on TWe want to expand the definitions given in the previous part to find periodvectors as vectorvalued functions that assign to each point t ∈ T and λ ∈ Ca vector I

(k)β (λ, t) ∈ TtT , such that I

(k)β (λ, 0) = I

(k)β (λ). They appear in the

definition of the vertex operators and related objects like the phase factors whichwill be introduced in another section. Here β is some cycle in the milnor fiber

Vλ,t = x ∈ C|F (x, t) = λ

that appears in the vertex operator Γβ and k ∈ Z. We will specify the cyclebelow.

Let us define then the period vectors for two cases which we will both use.First suppose α is a one point cycle in the milnor fiber Vλ,t (thus α depends onthe point (t, λ) ∈ T × C). Then we have the following definition of the periodvectors, which matches the one given in [3].

41

Definition 1 The period vectors are defined by

< I(−p)α (λ, t), ∂i >t= −∂i

∫α

d−1

((λ− F (x, t))p

p!dx

)(10.1)

andI(p)α (λ, t) = ∂pλI

(0)α (λ, t) (10.2)

for all p ≥ 0.

Here d−1 signifies the linear operator acting on forms on C by the rule

d−1(xkdx) =xk+1

k + 1.

As one would expect given the notation. Note that the period vectors might bemultiple valued depending on the monodromy of α.

Note that by non-degeneracy of the form < ·, · > the period vectors are indeeddefined above. Moreover

< I(0)α , ∂i >0= −∂i

∫α

d−1dx = −∫α

∂ix = −∫α

∂iF

∂xF,

since we have ∂ix∂xF = ∂iF . Now, since ∂iF |0 = xn−i and 1∂xF∧ dF = dx,

this shows that the definition agrees with (8.12).

We expand this definition to include period vectors corresponding to 1-cycles(relative to milnor fibers) β(t, λ) ∈ H1(C, Vλ,t;Z) as follows.

Definition 2 These period vectors are defined by

< I(−p)β (λ, t), ∂i >t= −∂i

∫β

(λ− F (x, t))p

p!dx (10.3)

andI

(p)β (λ, t) = ∂pλI

(0)β (λ, t) (10.4)

for all p ≥ 0.

Note that by Stokes’ theorem we have obviously

I(k)β = I(k)

α2− I(k)

α1, (10.5)

where α1 and α2 are the endpoints of β, such that β starts at α1.

The following lemma summarizes the most valuable properties of the periodvectors.

Lemma 12 We have for I the following useful relations:

∂λI(k) = I(k+1) ∂iI

(k) = −(∂i•)∂nI(k)

∂λI(k) = ∂nI

(k) (λ∂λ + E)I(k) = (−µ− k − 1

2)I(k), (10.6)

for any k ∈ Z.

42

Proof:

Note that we have omitted the cycles in the above statements. In fact they arevalid for any appropraite 0- or 1-cycle. To prove this, note that we will onlyhave to prove it explicitely for the one-point cycles, by virtue of equation (10.6).

Suppose then that α = [xα(λ)] such that F (xα, t) = λ. Note that, by definition,we have the upper left identity for k ≥ 0. So suppose that p > 0. To prove theupper left identity we need to show that

∂λI(−p)α = I(−p+1)

α .

Let Gp(λ, t, x) be the function such that

< I(−p)α , ∂i >= −∂i

∫α

Gp, (10.7)

for all i = 1, . . . , n. So we have

dGp =(λ− F )p

p!dx.

Then we have, for all i = 1, . . . , n,

< ∂λI(−p)α , ∂i >= −∂λ∂i

∫α

Gp = −∂i∂λ(Gp|xα(λ)) =

= −∂i(∂λGp)|xα − ∂i(∂xGp)|xα∂λxα =

= −∂i∫α

∂λGp = −∂i∫α

d−1(∂λ∂xGpdx) =

= −∂i∫α

d−1

((λ− F )p−1

(p− 1)!dx

)=< I(−p+1)

α , ∂i >, (10.8)

where we used the chain rule and, in the fourth equality, the fact that

(∂xGp)|xα =(λ− F (xα, t))

p

p!= 0,

since F (xα, t) = λ. So by non-degeneracy of the bilinear form this proves theupper left identity.

Similarly, for the lower left identity we have

−∂i∂n∫α

d−1

((λ− F )p

p!dx

)= −∂i

∫α

d−1

((λ− F )p−1

(p− 1)!dx

), (10.9)

for all i = 1, . . . , n. Where we used that −∂nF = 1. Thus we have

< ∂nI(−p)α , ∂i >=< I(1−p)

α , ∂i > .

Thus, in light of the upper left identity we see that the lower left identity holdsfor any k ∈ Z.

Then the upper right identity can be restated as

∂iI(k) = −(∂i•)∂λI(k). (10.10)

43

Note that, by virtue of the upper left identity, it is sufficient to prove it fork ≤ 2. Now we have(

−∂i∂j −n∑k=1

ckij∂k∂λ

)∫α

d−1

((λ− F )p

p!dx

)=

∫α

d−1

[((λ− F )p−1

(p− 1)!∂i∂jF −

(λ− F )p−2

(p− 2)!∂iF∂jF +

(λ− F )p−2

(p− 2)!

n∑k=1

ckij∂kF

)dx

]=

∫α

d−1

[((λ− F )p−1

(p− 1)!∂xKij −

(λ− F )p−2

(p− 2)!Kij∂xF

)dx

]=

∫α

(λ− F )p−1

(p− 1)!Kij = 0,

for all i, j = 1, . . . , n and p ≥ 2. So we see that, for all i, j = 1, . . . , n and p ≥ 2,we have

0 =< ∂iI(−p)α , ∂j > + < ∂λI

(−p)α , ∂i · ∂j >=

=< ∂iI(−p)α , ∂j > + < ∂i · (∂λI(−p)

α ), ∂j >=< (∂i + (∂i•)∂λ)I(−p)α , ∂j >,

which proves the upper right identity.

Finally let us turn to the lower right identity. Note that it is yet again enoughto prove the identity for k < 0, since, by virtue of the upper left identity, itthen follows for all higher superscripts. So suppose again p > 0. We have forall i = 1, . . . , n

< λ∂λI(−p)α , ∂i >= −∂iλ

∫α

d−1

((λ− F )p−1

(p− 1)!dx

)=

= p < I(−p)α , ∂i > −∂i

∫α

d−1

((λ− F )p−1

(p− 1)!Fdx

)=

=< pI(−p)α , ∂i > −∂i

∫α

d−1

((λ− F )p−1

(p− 1)!

(x∂xF

n+ 1+ EF

)dx

)=

=< pI(−p)α , ∂i > − < EI(−p)

α , ∂i > +[∂i, E]

∫α

d−1

((λ− F )p

p!dx

)+

+1

n+ 1< I(−p)

α , ∂i > +∂i

∫α

d−1

(∂xx(λ− F )p

p!(n+ 1)dx

)=

=< (p− E +1

n+ 1− di)I(−p)

α , ∂i >,

where we used the upper left identity in (10.7) and (5.15). On the other handwe have for all i = 1, . . . , n

< (−µ− 1

2)I(−p)α , ∂i >=< I(−p)

α , µ∂i > −1

2< I(−p)

α , ∂i >=

= (µi −1

2) < I(−p)

α , ∂i >= (1

n+ 1− di) < I(−p)

α , ∂i >,

44

where we used (2.11) and (5.14). Combining these two we see that indeed forall i = 1, . . . , n we have

< (λ∂λ+E)I(−p)α , ∂i >=< (p+

1

n+ 1−di)I(−p)

α , ∂i >=< (−µ+p− 1

2)I(−p)α , ∂i >,

which concludes the lemma.

Apart from these identities we will need an expansion of certain period vectors

I(0)βi

, for i = 1, . . . , n, in what follows. Here the 1-cycle βi is related to the cyclesappearing in the definition of the deformed flat coordinates. Namely βi is thatcycle which vanishes as λ nears ui along the path that defines Bi. We will callthese cycles the vanishing cycles [2, 14].

For any i = 1, . . . , n we see, from expression (6.15), that, in a neighborhood ofthe critical point xi, the equation F (x, t) = λ has two solutions (see also [3])

xi,± = xi ±1√∆i

√2(λ− ui) +O(

√2(λ− ui)

2). (10.11)

So we see that∫βi

dx = xi,+ − xi,− =2√∆i

√2(λ− ui) +O(

√2(λ− ui)

2),

for all i = 1, . . . , n. Which yields finally

< I(0)βi, ∂j >=

∂jui√∆i

2√2(λ− ui)

(1 + . . .), (10.12)

for all i, j = 1, . . . , n and the dots signify a power series in 2(λ− ui).

11 The symplectic operators Rt and St

The symplectic operator Rt

The relation between the cycles βi and Bi mentioned above warrants a de-scription of certain vector fields JBi related to the oscillating integrals JBi as

a version of Laplace transform of the period vectors I(0)βi

. We will need this

relation to discuss the quantized symplectic operator Rt in Givental’s formula.Moreover the vector fields JB are used to define this operator in the first place.So let us define the vector fields JB and derive the main properties.

Definition 3 For all B ∈ Y we define JB by

< JB , ∂i >= ∂iJB for all i = 1, . . . , n (11.1)

Here JB(z, t) ∈ TtT is a vector valued function on C × T . We will show mo-mentarily how these vector fields are related to the ingredients of Givental’sformula.

The following lemma summarizes the most useful properties of the vector fieldsJ .

45

Lemma 13 The vector fields J defined above satisfy the following relations:

∂iJ = z(∂i•)J (z∂z − E)J = µJ, (11.2)

for all i = 1, . . . , n.

Proof:

To show the above equations it is enough, by non-degeneracy of < ·, · >, toshow that

< ∂iJ, ∂j >=< z(∂i·)J, ∂j > < (z∂z − E)J, ∂j >=< µJ, ∂j >

for all i, j = 1, . . . , n.

Starting with the first equation we have

< ∂iJ, ∂j >= ∂i < J, ∂j >= ∂i∂jJ = zn∑k=1

ckij∂kJ =

= z < J,

n∑k=1

ckij∂k >= z < J, ∂i · ∂j >=< z(∂i·)J, ∂j >,

where we used invariance of the bilinear form and (6.2). So this shows the firstequation.

To prove the second equation we compute

< µJ, ∂j >=< J,−µj∂j >= z(−µjz∂jJ ) = z(∂z∂jJ −

n∑i,k=1

ditickij∂kJ ) =

= z∂z∂jJ −n∑i=1

diti∂i∂jJ =< (z∂z − E)J, ∂j >,

where we used the result of lemma 3 and (6.4). This shows the second equation.

Now according to [2, 1] the symplectic operator Rt is uniquely determined byits asymptotics. Moreover this asymptotic behaviour, near the critical points,is given by matrix solutions J to the system of equations (11.1) and (11.2), inthe following way

(−z)J ∼ ΨRtezU (11.3)

So the aymptotics of a matrix with colums given by the vectors fields JBi willgive us a big part of Givental’s formula. So let us explore the asymptotics.

First of all note that we have

JBi = (−2πz)−12

∫ ∞ui

ezλI(0)βi

dλ, (11.4)

for all i = 1, . . . , n, where the integration may be carried out over any ray inthe half plane where Re zλ < 0. Here we used the correspondence of the cycles

46

Bi and βi, which should be clear from this relation. So when we extend thecompact cycle βi from the critical value to infinity we get the non-compact cycleBi. Now to see this we simply calculate

(−2πz)−12 <

∫ ∞ui

ezλI(0)βi

dλ, ∂j >= (−2πz)−12

∫ ∞ui

ezλ(∂j

∫βi

dx)dλ =

= (−2πz)−12 ∂j

∫ ∞ui

∫βi

ezλdxdλ = ∂jJBi ,

for any i, j = 1, . . . , n. Where we used Fubini’s theorem and the specific defi-nition of βi. Since the cycle βi extends to Bi when we integrate from ui to ∞and since the first cycle is in the Milnor fiber we have λ = F in the integral.This shows that equation (11.4) is indeed correct.

This relation allows us to study the asymptotics of J simply by using the ex-pansion (10.12) for I. Now we can use this expansion to obtain an expressionfor the components of I in the canonical frame. Namely

[I(0)βi

]j =< I(0)βi,√−∆j∂uj >=

n∑a=1

Ψja < I(0)βi, ∂a >=

n∑a=1

Ψja

(δia +

∑k>0

Akai[2(λ− ui)]k)

2√2(λ− ui)

, (11.5)

for all i, j = 1, . . . , n and some series of matrices Ak. Now we simply computethe integral, which yields

2(−2πz)−12

∫ ∞ui

ezλ[2(λ− ui)]k−12 dλ =

(−z)−k−1ezui(2π−12 )

∫ ∞−∞

e−x2

2 x2kdx = (−z)−1−k(2k − 1)!!ezui , (11.6)

for all i = 1, . . . , n and k ∈ Z≥0, where we used the substitution 2(λ−ui) = −x2

2z .So we see that

(−z)[JBi ]j ∼n∑a=1

Ψja

(δai +

∑k>0

(2k − 1)!!Akai(−z)−k)ezui , (11.7)

for all i = 1, . . . , n. Thus when we switch to the coordinates in which

R = 1 +R1z +R2z2 + . . .

we see thatRaik = (−1)k(2k − 1)!!Akai. (11.8)

Although this might not be a complete definition of Rt it will be quite sufficient,as will be shown below.

47

The symplectic operator St

Recall the equations (2.9) and (2.10) satisfied by deformed horizontal sections,i.e. those given by the oscillatory integrals. These will give us the definition ofthe St matrix. Namely St is the symplectic operator that sends solutions φ of(

∂z +µ

z

)φ = 0 (11.9)

to solutions of (2.10) near z = 0. So for such solutions φ we have near z = 0(∂z − E +

µ

z

)Stφ = 0. (11.10)

Note that in this coordinate St(z) = 1 + S1z + S2z2 + . . .. Existence and

uniqueness of this operator is shown in [1, 13]. Dubrovin shows that the gaugetransformation St can be found in the case that µ is not resonant. By non-resonance we mean that

µi − µj /∈ Z,

for all i, j = 1, . . . , n. This clearly holds in the case of the Frobenius structurecorresponding to our An singularity. In the general case there would be a slightadaption of the statements made below [1]. Furthermore a choice of St is calleda calibration of the Frobenius manifold. In fact the operator St can be used todefine the deformed connection and thus the Frobenius structure [13].

Lemma 14 The operator St(z) is a fundamental solution to

∂iS = z(∂i•)S, (11.11)

for all i = 1, . . . , n. Moreover we have

[St, µ] = (z∂z − E)St. (11.12)

Proof:

We find easily the fundamental matrix solution z−µ of (11.9). Now we hadalready found solutions to (2.10) given by the oscillatory integrals. Thus wehave here a fundamental matrix solution P with entries Pij = ∂iJBj . Then wesee that near z = 0 we have

P = Stz−µC, (11.13)

for some invertible matrix C determined by the ordering of the bases ∂ini=1

and Bini=1. This then immediately implies (11.11). To find (11.12) we simplycompute

0 =(∂z +

µ

z− E

)Stz−µ = (∂zSt)z

−µ − Stµz−µ−1 + µStz−µ−1 − EStz−µ =

= ((z∂z − E)St + [µ, St]) z−µ−1,

where we used (11.11). This proves the lemma.

48

12 Phase Factors

We have now sufficiently defined all ingredients of Givental’s formula apart fromC(t), which we shall come to in section 14. To prove that D in (9.1) satisfiesthe regularity condition of (8.18) we need to commute the operators presentin Givental’s formula past the vertex operators (8.19). For any semisimplet ∈ T this will yield eventually n independent copies of the KdV hierarchy (8.8)(including DA1

), which we know is regular in λ. However, apart from thesewe will also find some extra factors called phase factors. Givental introducedthe phase form [2], which can be used to express most of the phase factors. Inthe end this will allow us to show that certain phase factors are holomorphic inλ and thus regular. In this section we will define and discuss the phase form,mainly to show some properties that we need in the final section.

Suppose β is a cycle such that the corresponding period vector is defined.

Definition 4 The phase form Wβ corresponding to the cycle β is given by

Wβ(λ, t) := − < I(0)β , dI

(−1)β >, (12.1)

where d is to be taken as on F(T ).

Note that we haveWβ = − < I

(0)β ,dI

(−1)β >=

= −n∑i=1

< I(0)β , ∂iI

(−1)β > dti =

n∑i=1

< I(0)β , ∂i · I(0)

β > dti, (12.2)

where we used the upper right identity of (10.6). These phase forms have certaininteresting properties as shown in [2, 7]. We will use this section to show theproperties that we need. For a more complete discussion see the referencesabove.

Firstly note that the phase form is closed, as a form on T , since

dWβ =

n∑i,j=1

∂j < I(0)β , ∂i · I(0)

β > dtj ∧ dti =

=

n∑i,j=1

(< ∂jI(0)β , ∂i ·I(0)β > + < I

(0)β , ∂i ·(∂jI(0)β ) > + < I

(0)β , (∂j(∂i•))I(0)β >)dtj∧dti =

n∑i,j=1

(< ∂jI(0)β , ∂i · I(0)β > + < ∂iI

(0)β , ∂j · I(0)β > + < I

(0)β , (∂j(∂i•))I(0)β >)dtj ∧ dti = 0,

(12.3)

where we used the symmetry and invariance of < ·, · >, the consequence∂j(∂i·) = ∂i(∂j ·) of (6.8) and the fact that dtj ∧ dti = −dti ∧ dtj . Since thismeans that what is inside the brackets is symmetric under interchanging i andj while dtj ∧ dti is anti-symmetric.

Secondly by the lower left equation in lemma 12 we see that the phase form isinvariant under ∂λ − ∂n. In other words we have

L∂λ−∂nWβ = (i∂λ−∂n d + d i∂λ−∂n)Wβ =

49

= d < I(0)β , I

(0)β > +i∂λ−∂n(

n∑i=1

∂λ < I(0)β , ∂i · I(0)

β > dλ ∧ dti) =

= d < I(0)β , I

(0)β > +

n∑i=1

∂λ < I(0)β , ∂i · I(0)

β > dti − ∂λ < I(0)β , I

(0)β > dλ =

= d < I(0)β , I

(0)β > −2

n∑i=1

< I(0)β , ∂iI

(0)β > dti − ∂λ < I

(0)β , I

(0)β > dλ = 0 (12.4)

Here we see d in the Cartan formula as on F(C × T ) . This shows that wecan express the phase form as strictly dependent on the point in the parameterspace. So define

Wβ(t) = Wβ(0, t) (12.5)

then we haveWβ(λ, t) =Wβ(t− λ1), (12.6)

where −1 ∈ T such that F (x,−1) = f(x)−1. So −1 = (0, . . . , 0,−1) in naturalcoordinates and −1 = (0, . . . , 0, 1) in flat coordinates.

In the next section we will need the relation between the phase forms andintersection indices of the corresponding cycles. First of all we introduce abilinear form on the cotangent spaces, see also [1, 15]. We define the intersectionform (·, ·) by

(ω1, ω2) = iE(ω1 · ω2), (12.7)

for all ω1, ω2 ∈ T ∗T . Here the multiplication is pulled back from the tangentbundle using the (non-degenerate) metric and iE signifies the interior product.So suppose I1, I2 ∈ TT such that < I1, · >= ω1 and < I2, · >= ω2, then wehave

(ω1, ω2) =< I1 · I2, E >,

since ω1 · ω2 =< I1 · I2, · > by definition.

Now it is proven in [15] and [14] that this intersection form is proportional tothe intersection index in homology carried over to the cotangent bundle whenthe self-intersection of vanishing cycles is normalized to +2. For details werefer to the references given above, since most elements of this disussion are notrelevant to the task at hand. So we restrict ourselves to the comment that, since

iE(Wβ) =< I(0)β · I

(0)β , E >, we see that

iE(Wβ) = −(β, β), (12.8)

where (·, ·) denotes the intersection index. Which in trun implies that∫ −λ1−1

Wβ = −(β, β)

∫ λ

1

dξ

ξ, (12.9)

for a proof see [2, 7, 15].

Note also that we could in fact define the phase form for two different cycles.So we introduce the polarized phase form Wα,β in the obvious sense. In this

50

case we also have the identities dWα,β = 0 and L∂λ−∂nWα,β = 0. So we define

again Wα,β(t) = Wα,β(0, t). Then we have again the identities

Wα,β(λ, t) =Wα,β(t− λ1) and iE(Wα,β) = −(α, β).

Now note that the period vectors are not necessarily single valued, thus thesame thing goes for the phase form. In fact this depends on the monodromyof the cycles corresponding to the phase form. However integrating the phaseforms over certain paths will lead to a well defined number. More precisely, wedefine the discrimimant Σ corresponding to the the An singularity as the subsetof points (t, λ) ∈ T × C where F (x, t) = λ has less than n solutions. Then if αis a cycle that is invariant under the monodromy around a loop in T × C− Σ,this means the phase form Wα is single valued along this loop so we can define(in this case) the period

∮Wα [2].

We will need one more fact about the phase form. Namely that the period∮γWα, for α a cycle that has integer intersection indices with the vanishing

cycles, is an integer multiple of 2π√−1. For a proof see [2]. Note that this only

makes sense in the case that α is invariant under the mondromy around γ.

13 Commutation with Vertex Operators

In this section we will finally show how to commute the ingredients of Givental’sformula past the vertex operators. This will set us up completely to completethe proof of regularity of D in the sense of (8.18).

Commutation of St

We have already made the beginning to being able to commute the operator Stin the previous part. Recall the action of quantized lower triangular symplecticoperators as given in (7.21). Suppose G(q) is an arbitrary element of the Fockspace. Then (7.21) implies easily that

StG(q) = e−12~W ([S−1

t q]+,[S−1t q]+)G([S−1

t q]+), (13.1)

with W and the notation [S−1t q]+ as in part II. Note however that this W is

related to St by (7.20) and thus depends also on the point t ∈ T .

Now suppose f ∈ H is an arbitary element. Then we have

f =∑k∈Z

fkzk, (13.2)

where fk ∈ H for all k ∈ Z. Note that, in the case that fk = (−1)kI(k)α (λ, 0) for

all k ∈ Z we have ef = Γα.

However, let us first inspect the general case. Denote by f+ and f− the non-negative respectively the negative powers of f in z, i.e. f+ =

∑k≥0 fkz

k and

f− = f− f+. Then we can split the quantization of ef by the rule

ef = ef+ef− . (13.3)

51

Then we can use (7.21) and (13.1) to see that

StefS−1t G = e

12W (f+,f+)eStfG, (13.4)

by a computation similar to that leading to equation (7.21), we use again theCBH-formula.

Then to find out how we can commute St past the vertex operators we have twotasks. We should compute W (f+, f+) and the quantization of Sf, when

f = ln Γα =∑k∈Z

I(k)α (λ, 0)(−z)k.

Both these computations will follow in some way from the following importanttheorem. We will prove it by combining the proofs of Givental and Milanov [2,3].

Theorem 15 Let f(λ, t) =∑k∈Z

I(k)α (λ, t)(−z)k, for some cycle such that the

corresponding period vectors are defined. Then we have

f(λ, t) = Stf(λ, 0). (13.5)

Remark 7 We mean this in the following way. As shown above (for the k = 0)case, the period vectors expand as Laurent series in λ (with fractional exponents)near λ = ∞. The maximum exponent in I(k) tends to −∞ as k tends to ∞.We can expand Sf in z. The coefficients of such an expansion, which are infintesums themselves, converge in the 1

λ -adic sense.

Proof:

In fact most of the work to prove this theorem has already been done in lemma’s12 and 14. Note that by lemma 12 we have

∂if = z−1(∂i•)f, ∂λf = ∂nf and (z∂z + λ∂λ + E)f = (−µ− 1

2)f, (13.6)

for all i = 1, . . . , n. Now, denoting f0(λ) := f(λ, 0), we have

∂if0 = 0, ∂λf0 = −z−1f0 and (z∂z + λ∂λ)f0 = (−µ− 1

2)f0. (13.7)

Now we can combine this with the identities in lemma 14 to see that f = Stf0satisfies (13.6). Note that when we use the identities of lemma 14 we need toswitch to the coordinate that expresses the nature of St again.

Note from (5.16), and in fact the general theory, that E|t=0 = 0. Then we seefrom the defining property for St, given in (11.9) and (11.10), that S|t=0 = 1,the identity matrix. This shows that

Stf0|t=0 = f0 = f|t=0.

Thus since they are equal in t = 0 and they are both solutions to (13.6) thelemma is proved.

52

So we have found Stf. Then let us compute the quadratic form W (f+, f+). Todo this we shall first need to study the quadratic form W more closely. Recallthe defining equations (7.19) and (7.20). Recall that since W is related to Stwe can view W (q,q) as dependent on t ∈ T . Now since St|t=0 = 1 we see fromequation (7.20) that W |t=0 = 0. Thus we are able to express the quadratic formas

W (q,q) =

∫ t

0

n∑i=1

∂iW (q,q)dti.

To make use of this we need to determine ∂iW for i = 1, . . . , n.

Let us start by differentiating (7.20), this yields

∂i

(STt (w)St(z)− Iw−1 + z−1

)=

(∂iSt(w))TSt(z) + STt (w)∂iSt(z)

w−1 + z−1=

=w−1(∂i · St(w))TSt(z) + z−1STt (w)∂i · St(z)

w−1 + z−1= ST (w)∂i · S(z),

for all i = 1, . . . , n. Here we used the identity (11.11), which reads as

∂iS = z−1∂i · S,

for all i = 1, . . . , n, in the coordinate that shows the nature of S (which is usedin part II and thus in (7.20)). We also used the fact that, by invariance of< ·, · >, we have that (∂i·)T = (∂i·) for all i = 1, . . . , n. So we see that

∂iW (q,q) =< [Sq]0, ∂i · [Sq]0 >, (13.8)

where [Sq]0 = [Sq]+|z=0.

Now by theorem 15 we see that for f =∑k∈Z I

(k)β (λ, 0)(−z)k we have

[Stf+]0 = [Stf]0 = I(0)β (λ, t).

So we see that

W (f+, f+) =

∫ t

0

n∑i=1

< I(0)β , ∂i · I(0)

β > dti =

∫ t

0

Wβ , (13.9)

for any cycle β such that the period vectors are defined. Note that the phaseform is not necessarily singlevalued and the integral may depend on the pathfrom 0 to t. We will however not specify the path, instead we will assume thatwhenever the ending points are the same in an integral the path is also thesame. Now note that we can use the properties of the phase form in (12.6) and(12.9) to write

W (f+, f+) =

∫ (λ,t)

(λ,0)

Wβ =

∫ t−λ1

−λ1Wβ =

=

∫ t−λ1

−1Wβ −

∫ −λ1−1

Wβ =

∫ t−λ1

−1Wβ + (β, β)

∫ λ

1

dξ

ξ. (13.10)

Now in the interest of clarity in what follows we give the following definition,which we have already been hinting at.

53

Definition 5

Γαt (λ) := exp∑k<0

I(k)α (λ, t)(−z)k exp

∑k≥0

I(k)α (λ, t)(−z)k. (13.11)

Note then that Γα = Γα0 . Using this notation we have the following propositionwhich determines the commutation of St with the vertex operators. We will ingeneral call any Γαt a vertex operator.

Proposition 16

StΓα0 S−1t = e

∫ t−λ1−1

Wβ2 +(β,β)

∫ λ1

dξ2ξ Γαt (13.12)

Proof:

The proof is given by substituting f =∑k∈Z I

(k)α (λ, 0)(−z)k in (13.4) and using

theorem 15 to calculate Stf and W (f+, f+). This is done above.

Commutation of Rt

Now let us turn our attention to the other ingredients of Givental’s formula, mostimportantly the quantized symplectic operator Rt. Proposition 16 shows thatwe will have to learn how to commute this operator past the vertex operatorsΓαt at a semi-simple point t ∈ T . Now we have a description of the relationbetween the period vectors and Rt, given in (11.5) and (11.8). However thisrelates only the period vectors corresponding to the n vanishing cycles βi. Thuswe will first need to write the vertex operators corresponding to the one-pointcycles α in terms of the vertex operators corresponding to the vanishing cycles.We will do this explicitely in the next section. In this section we will stick tothe general case of a cycle that can be written as the sum of a vanishing cycleand a monodromy invariant cycle. We will also discuss the commutation of Rtwith vertex operators corresponding to vanishing cycles.

So, let β be a vanishing cycle vanishing at the point (u, t) and let α′ be a cyclethat is invariant under the monodromy group around this point. Now definethe cycle α by

α := (α, β)β

2+ α′. (13.13)

Note that we have actually just defined α = cβ2 + α′ and by virtue of thenormalization of (β, β) this yields automatically c = (α, β). Then we have thefollowing proposition.

Proposition 17

Γαt = eKΓα′

t Γβ2t , (13.14)

where K = (α, β)

∫ t−u1

t−λ1W β

2 ,α′ .

Proof:

54

We will follow the proof set out in [2]. Denote h =∑k∈Z I

(k)α (−z)k, f =

(α, β)∑k∈Z I

(k)β2

(−z)k and g =∑k∈Z I

(k)α′ (−z)k and define f±,h± and g± as

in (13.3). Then we have

Γαt = eh = eh−eh+ = eg−+f−eg++f+ = e[f−,g+]egef = eΩ(f−,g+)Γα

′

t Γβ2t ,

since the quantization is a representation of the Heisenberg Lie algebra forlinear Hamiltonians as mentioned before. So all that is left is to show thatK = Ω(f−,g+). Clearly we have

Ω(f−,g+) = (α, β)∑k≥0

(−1)k < I(−1−k)β2

, I(k)α‘ > .

Meanwhile we find by repeated integration by parts that∫ λ

u

< I(0)β2

, I(0)α′ > dξ =< I

(−1)β2

, I(0)α′ > |

λu −

∫ λ

u

< I(−1)β2

, I(−1)α′ > dξ = . . . =

=

m−1∑k=0

(−1)k < I(−1−k)β2

, I(k)α′ > |

λu + (−1)m

∫ λ

u

< I(−m)β2

, I(m)α′ > dξ, (13.15)

for all m ∈ N. Now note that, since α′ is invariant under the monodromy at the

point u, I(k)α′ is holomorphic at this point for all k ∈ Z. Additionally we see from

equation (10.12) that [I(−1−k)β2

]j ∼ (λ− u)k+ 12 (1 + . . .). These two observations

show that the integral in (13.15) is o((λ − u)m−12 ) so it tends to 0 as m tends

to ∞. Now note that the second observation also shows that I−1−kβ2

(u, t) = 0.

So we find that

K = (α, β)

∫ λ

u

< I(0)β2

, I(0)α′ > dξ = (α, β)

∫ (λ,t)

(u,t)

< I(0)β2

,dI(−1)α′ > −W β

2 ,α′ =

= −(α, β)

∫ t−λ1

t−u1W β

2 ,α′ , (13.16)

which proves the proposition

Now let us turn our attention to commuting ΨRteUz past the vertex operators

corresponding to the vanishing cycles. As in the case of the lower triangular

operator we have for N = ΨRteUz that N−1efN is proportional to eN−1f, where

f is as in (13.2). So let us start by finding this exponent.

Theorem 18 Near λ = ui we have∑k∈Z

(−z)kI(k)βi

= ΨRteUz

√−∆i∂ui

√2∑k∈Z

(−z)k(d

dλ)kλ−

12 , (13.17)

for all i = 1, . . . , n. In the coordinate that expresses the nature of Rt.

Remark 8 Similar to theorem 15 we need an explanation of how to view theseinfinite series. In fact this remark is the same as in theorem 15 except that theconvergence is now in the

√λ− ui-adic sense.

55

Proof:

Every statement in this proof will hold for i = 1, . . . , n. We have in fact donemost of the work to prove this theorem already in deriving (11.5) and (11.8).This will be apparent once we deal with the right most part of equation (13.17).

Consider the series∑k∈Z z

kI(k)βi

(λ+ ui), where we omit the other coordinate t.Taylor expanding this around λ near ui yields

∑k∈Z

zkI(k)βi

(λ+ ui) =∑k∈Z

zk

(I

(k)βi

(λ) + I(k+1)βi

(λ)ui +I

(k+2)βi

2!u2i + . . .

)

=

( ∞∑n=0

unin!zn

)(∑k∈Z

zkI(k)βi

(λ)

)= e

uiz

∑k∈Z

zkI(k)βi

(λ). (13.18)

Now the equations (11.5), (11.8) and (13.18) simply amount to (13.17).

Now let us turn to the constant of proportionality. Note that we need not worryabout the Ψ part in this case, since it is simply a basis transformation. We willtreat first the constant of proportionality corresponding to Rt. Just as in (13.4)we find from (7.28) that

R−1t efRt = e−

12V (f−,f−)eR−1f, (13.19)

where f and f− are as in (13.3). So we will need to compute V (f−, f−) for

f− =∑k≥0

I(−k−1)β ∂qk .

We will use a method comparable to the one which was used to determineW (f+, f+). First of all note that

∂λV (f−, f−) =∑k,l≥0

< I(−k)∂qk , [V(k−1)l + Vk(l−1)]I(−l)∂ql >=

=< I(0)β , I

(0)β > − <

∑k≥0

RTk I(−k)β ,

∑l≥0

RTl I(−l)β >, (13.20)

where we used the upper left identity of lemma 12 and equation (7.27). Nowlet us fix a vanishing cycle βi. Then we can use theorem 18 to see that∑

k≥0

RTk I(−k)βi

=∑k≥0

RTk∑l≥0

(−1)lRl(d

dλ)−l−k

2√−∆i∂ui√

2(λ− ui)=

2√−∆i∂ui√

2(λ− ui),

(13.21)where we also used the condition in (7.8).

Now we can expand the vector coefficients f−k = I(−1−k)βi

using (10.12) to find

that [f−k]i ∼ (λ−ui)k+ 12 (1+ . . .). So we see that at λ = ui we have V (f−, f−) =

0. Thus we find finally that

V (f−, f−) =

∫ λ

ui

(< I

(0)βi, I

(0)βi

> − 2

ξ − ui

)dξ =

56

=

∫ t−ui1

t−λ1

(Wβi −

2dtnτn − ui − tn

). (13.22)

Now we have found all the tools to commute ΨRteUz past the vertex operators

corresponding to vanishing cycles.

Proposition 19 For all vanishing cycles βi, with i = 1, . . . , n, we have

(ΨRteUz )−1e−

12ViΓ

± βi2t (ΨRte

Uz ) = e−

12vi(. . .1⊗ (Γ±)(i) ⊗ 1 . . .), (13.23)

where we use the notation of [2]. Thus the (i) means the ith position in the ten-sor product. Meanwhile every jth position only acts on those function dependenton qj, for all j = 1, . . . , n. Furthermore

Vi =

∫ t−ui1

t−λ1

(W βi

2

− dtn2(τn − ui − tn)

)(13.24)

and

vi =

∫ λ

λ−ui

dξ

2ξ. (13.25)

Proof:

Note that, as with proposition 16, most of the work has already been done.This time we still need to figure out the commutation of the quantized operator

e(Uz ). Note however that this is just a special case of proposition 16. In this

case we have the effect that√−∆i∂ui√2(λ−ui)

is transformed into√−∆i∂ui√

2λ. While the

corresponding phase factor is equal to λ−uiλ = e−2vi . Now the branch of the

square root is specified by the path taken in vi. Then combining this with(13.17), (13.19) and (13.22) yields (13.23).

14 Ancestor Potential and Regularity

We have assembled all the ingredients of Givental’s formula and have figuredout the commutation with vertex operators. All that is left to do is show theregularity of (8.18) with D as in (9.1). To show this we will introduce theancestor potential as in [13]. This will show us how to deal with the invertiblematrix C(t) and how to implement the propositions 16 and 19. The proof ofregularity then consists merely of dealing with the phase factors.

Ancestor Potential

We have assumed the solution to the (n + 1)KdV hierarchy to be of the form(8.1). Combining this with Givental’s formula Kontsevich and Manin proved[21, 13] that we can rewrite the total descendant potential D in terms of theancestor potential At at a point t ∈ T as follows

D = eF1(t)S−1

t At. (14.1)

57

Here At is of the form

At(~, t) := exp∞∑g=0

~g−1Fgt , (14.2)

where Fgt is the genus g ancestor potential [13]. Now we see this ancestorpotential as an element of the Fock space by using again the construction in(8.6). Furthermore F 1(t) is the genus 1 Gromow-Witten potential. In stayingwith our hands on approach let us merely state what these concepts mean inour specific case. We can see from (9.1) that (14.1) is a reasonable restatementand the exponent is related to the invertible matrix C(t).

To be completely clear of all the ingredients of Givental’s formula note that thisinvertible matrix is related to the Rt matrix. Up to a constant it is defined by

C(t) := exp

1

2

∫ t n∑i=1

Rii1 (u)dui

. (14.3)

In fact this factor is needed to compensate for the cocyle C arising from thequantization procedure to make the entire expression D independent of thepoint t used in the definition [13]. Now we can simply define

F 1(t) :=1

48

n∑i=1

ln ∆i(t) + lnC(t), (14.4)

so that we have the expression (14.1) with

At(q) = Ψ(t)RteUz

n∏i=1

DA1(qi)∆

− 148

t . (14.5)

This is desirable since it was shown in [22, 15] that the function F 1 is constant,thus we won’t have to worry about it at all.

This means that we can rewrite the condition (8.18) in terms of the ancestorpotential using the commutation of St past the vertex operators. Before wedo so however let us note one more fact about the phase factors this produces.Note that∫ −λ1

−1Wα =

∫ λ

1

−1

n+ 1

n∑i=1

xi−1

xnxn−i

xndxn+1 = − n

n+ 1lnλ, (14.6)

for α in the level where xn+1 = −τn. Here we simply used the definitionof the phase form and the period vectors. Now we see by the remark aboutproportionality made in section 12 that for these cycles n

n+1 = (α, α). Thus inthe expression (8.18)

λ−nn+1 = λ−(α,α).

Using the Ancestor potential we can rephrase (8.18) in the following way∑α

λ−(α,α)dλ(Γ−αD)(q’)(ΓαD))(q”) =

58

=∑α

λ−(α,α)dλ(Γ−α0 eF1(t)S−1

t At)(q’)(Γα0 eF 1(t)S−1

t At)(q”) =

e2F 1(t)S−1t

∑α

λ−(α,α)e∫ t−λ1−1

Wα+(α,α)∫ λ1

dξξ dλ(Γ−αt At)(q′)(Γαt At)(q′′), (14.7)

note that the quantized symplectic operator S−1 acts on both terms (in Given-tal’s notation one could write S−1

t ⊗ S−1t ).

Regularity

Now we are ready to prove the regularity of D in the sense of (8.18). So thissection will be devoted to proving the following theorem.

Theorem 20 For D as in (9.1) the one-form[∑α

Γα0 ⊗ Γα0λ−(α,α)dλ

]D ⊗D (14.8)

is regular in λ. Here the tensor product is meant as in [2], i.e. the left slotcorresponds to the variable q′ and the right one to q′′. Furthermore the one-point cycles α are as in (8.18).

Proof:

We will follow the proof set out in [2]. Note that due to the previous discussionwe can rephrase the theorem as follows. The expression (14.8) is regular in λ ifand only if [∑

α

Γ−αt ⊗ Γαt e∫ t−λ1−1

Wα+(α,α)∫ λ1

dξξ λ−(α,α)dλ

]At ⊗At (14.9)

is regular in λ for t ∈ T semisimple. Note that this means it is immediatelyregular for all semisimple points in T . Now since the one-point cycles α form anorbit under the monodromy group we see that he expression (14.9) is invariantunder this monodromy. Then according to the discussion in [2] with regard tothe elements of the Fock space (see also Appendix B) we merely have to provethat the meromorphic (in λ) form (14.9) has no poles. The function has possiblepoles at the critical points. Thus the theorem will we proved by showing thatthere are in fact no poles at the distinct critical points u1, . . . , un. These aregiven by the canonical coordinates for the point t ∈ T .

So let us fix a critical value λ = ui. There will be two one-point cycles α± suchthat βi = α+−α−. Then note that all the other one-point cycles α in (14.9) areinvariant under the monodromy around ui and thus the corresponding periodvectors and phase factors will be holomorphic at the point λ = ui. So we willonly need to worry about the summands concerning the cycles α± in (14.9).

Denote α′ = α−+α+

2 , then α′ is clearly invariant under monodromy around ui.Note that

α± = ±βi2

+ α′, (14.10)

59

which shows that I(k)α± = I

(k)

± βi2+ I

(k)α′ for all k ∈ Z. Where again I

(k)α′ is holomor-

phic at λ = ui for all k ∈ Z. Overmore we can use proposition 17 to write

Γα±t = e±KiΓα

′

t Γ± βi2t , (14.11)

where Ki =

∫ t−ui1

t−λ1W βi

2 ,α′ . Then the summands in (14.9), that are not neces-

sarily holomorphic at λ = ui, add up to

Γ−α′

t ⊗ Γα′

t

[(∑±L±(λ)Γ

∓ βi2t ⊗ Γ

± βi2t

)At ⊗At

]dλ, (14.12)

where the functions L± are the combination of phase factors from (14.9) and(14.11).

Now we are in a position to apply proposition 19, by virtue of (14.5). Notethat in the tensor product notation of proposition 19 we now get simply some

factor of holomorphic functions Γ±α′

t DA1(q′j)DA1

(q′′j ) for j 6= i and the possiblynon-holomorphic factor is that which corresponds to the variable qi. Namely∑

√λ=±ξ

l±(λ)dλ√λ

[Γ−DA1(q′i)][Γ

+DA1(q′′i )], (14.13)

where the functions l± are given by the phase factors L±, those coming fromproposition 19 and the extraction of 1√

λ. Thus the theorem will be proved if we

can show that the phase factors l± are identical and holomorphic, since in thatcase (14.13) is regular by Witten’s conjecture.

So finally we turn our attention to the accumulated phase factors. We have

ln l± =

∫ t−λ1

−1Wα± + (α±, α±)

∫ λ

1

dξ

ξ− lnλ(α±,α±) + ln

√λ−

∫ λ

1

dξ

2ξ

±2

∫ t−ui1

t−λ1W βi

2 ,α′ +

∫ t−ui1

t−λ1(Wβi −

dtn2(τn − ui − tn)

)−∫ 1

λ−ui

dξ

2ξ. (14.14)

Here the first two summands come from the commutation of S−1t , the third

summand comes from equation (8.18) and (14.6), the fourth summand comesfrom the extraction of 1√

λto get (14.13), the fifth and last two summands come

from the commutation of ΨRteUz and finally the sixth summand comes from

(14.11). Note thatWα± =W βi

2

+Wα′ ± 2W βi2 ,α

′ , (14.15)

by (14.10). So we haveln l± =

=

∫ t−λ1

−1Wα′ + (α±, α±)

∮γ±

dξ

ξ+

∮γ′±

dξ

2ξ± 2

∫ t−ui1

−1W βi

2 ,α′ +

∫ t−(ui+1)1

−1W βi

2

+

∫ t−ui1

t−(ui+1)1

(W βi2

− dtn2(τn − ui − tn)

) +

∫ τn−λ

τn−(ui+1)

dtn2(τn − ui − tn)

−∫ 1

λ−ui

dξ

2ξ,

(14.16)

60

where the contours γ± and γ′± come from combining the second and third re-spectively the fourth and fifth summands in (14.14). Apart from the last termthe other terms are caused by applying (14.15) and changing the endpoints. Theconstant ui + 1 is chosen exactly such that the last two integrals cancel eachother. Now note that the contours γ± and γ′± as well as the fourth throughsixth summands may depend on the cycle α±, but they do not depend on λ.Thus all the λ dependence comes from the first summand. Note however thathere the cycle α′ is involved, which is invariant under the monodromy aoundui, and therefore the first term is holomorphic at λ = ui. Thus we see that thephase factors l± are at least holomorphic. So all that is left to show is that theycoincide near λ = ui.

So let us find the constant of proportionality between l+ and l−. We have

ln l+ − ln l− = (α−, α−)

∮γ+−γ−

dξ

ξ+ 4

∫ t−λ1

−1W βi

2 ,α′ +

∮γ′+−γ′−

dξ

2ξ=

= (α−, α−)

∮γ+−γ−

dξ

ξ+

∫ t−λ1

−1(Wα+

−Wα−) +

∮γ′+−γ′−

dξ

2ξ, (14.17)

where we used that 4W βi2 ,α

′ =Wα+−Wα− , since α′ = α++α−2 and βi

2 = α+−α−2 .

Suppose γ1 is a loop around λ = 0 that makes exactly enough turns inside theline −λ1 such that α− transported along this loop becomes α+. This loop existssince α+ and α− belong to the same orbit of the classical monodromy group.Then by (12.9) the first summand in (14.17) equals

∮γ1Wα− .

For ε ∈ R>0 let γ2(ε) be the path starting at −1 and approaching t− ui1 alongthe same path as in the second summand, but stopping a (small) distance εaway. Then let γ3(ε) be a loop of size ε that goes around the discriminant Σ(along which the middle integral in (14.17) is taken) near t− ui1. So the cycleα+ become α− when it is transported along γ3(ε). Thus we see that

limε→0

∫γ2(ε)γ3(ε)γ−1

2 (ε)

Wα+ =

∫ t−ui1

−1(Wα+ −Wα−) + lim

ε→0

∫γ3(ε)

Wα+ .

Now let us write again Wα+= W βi

2

+ 2W βi2 ,α

′ +Wα′ . Then near the point

t− ui1 we see from the expansion (10.12) that the first summand contains the

term dλ2(λ−ui) , while the rest is either analytic at λ = ui or has a singularity of

the form dλ√λ−ui

. So we find that

limε→0

∫γ3(ε)

Wα+ =

∮dλ

2(λ− ui)= π√−1,

where the last contour is around λ = ui. So this part coincides with the lastsummand of (14.17).

Combining these last two paragraphs we see that

ln l+− ln l− =

∮γ1

Wα−+

∮γ2γ3γ

−12

Wα+ +2π√−1 = 2π

√−1+

∮γ1γ2γ3γ

−12

Wα− =

61

= 2π√−1(a+ 1), (14.18)

for some a ∈ Z. Where the last equality follows from the fact that α− is invariantunder the monodromy along the loop γ1γ2γ3γ

−12 and the remark at the end of

section 12, which is proven in proposition 1 of [2]. Note that we have simplychosen some ε > 0 since the argument doesn’t actually depend on ε. Then wesee that by (14.18) we have l+ = l− and the theorem is proved.

15 Conclusion and Discussion

Now we have seen a (reasonably) complete exposition of the relation betweenintegrable hierarchies and Frobenius structures in the case of the An singularity.The most interesting question now is which elements of this exposition will beeasily translated to other pairs of Frobenius structures and integrable hierar-chies. To give an idea of the efforts in answering this question and the furtherscope of the subject involving Frobenius structures and integrable hierarchieswe will quickly summarize the main points made above and point out the Ansingularity specific elements and possible generalizations.

Quick Summary

Let us begin in the situation after section 7. So we already have the quantizationformalism in place. Then we see that naively the hierarchy is defined simply bythe vertex operators Γα and the cycles α. Certainly if we look at the form (14.8).The vertex operators are simply explained using the period vectors, which arereadily available from the Frobenius structure.

To do this however we also need to specify the allowed cycles α and note thatin (14.8) we sum over a full orbit of the monodromy group. So we see that weimmediately need the miniversal deformation of the singularity. Actually at thispoint we only need it in the point 0 ∈ T , so specifically we need the singularityitself. Then we simply define the cycles using the Milnor fibers Vλ,0.

Note that this in fact constitutes a choice of representative for the singularity.For instance in [2] a different representative is chosen which leads to slightlydifferent definitions considering the vertex operators and quantization, since< ·, · > and thus Ω are affected. Note also that the period vectors are imme-diately multiple valued and ramified along the discriminant by definition of theone-point cycles α.

Next we need to find the operators Rt and St and their commutation past thevertex operators. Here we depend heavily on the deformed connection. Fora definition of the Rt operator we depend on the oscillatory integrals, whichsatisfy the system of equations (11.1) and (11.2). Thus we need a specific typeof deformed flat coordinates. In fact if we consider the commutation of theRt matrix past the vertex operators corresponding to vanishing cycles, we seethat the crucial fact is given in the Laplace transform (11.4) and the expansion(10.12). In fact most of the argument of theorem 20 comes down to these tworelations, since this allows us to split the (n+ 1)KdV hierarchy into n copies ofthe KdV hierarchy.

62

Note that the two main ingredients defining the oscillatory integrals (and thusthe asymptotical approximation which yields Rt and its commutation) are theminversal deformation F and the basis of cycles Bini=1. Where we use theminiversal deformation both explicitely in the definition (6.1) and in the defini-tion of the basis of cycles Bini=1.

Now the commutation of the quantized symplectic operators is evident consid-ering the remarks above. Although for the commutation of Rt we also rely onthe split (14.10) and thus the vanishing cycles. These are defined using thebasis of cycles Bini=1. The most important factors at this point are then thephase factors and thus the phase form. Here we use again the definition of theperiod vectors. However we also use the specific properties of the one-point andvanishing cycles.

Specific and General Elements

Let us try and pinpoint which elements used to prove theorem 20 are specificto the An singularity. First of all we should note that, although we have con-sequently used the specific An singularity, the construction is easily generalizedto include the other simple singularities [3, 7, 9, 15]. The Frobenius structurearising from these singularities was shown by K. Saito [9] and relies upon theexistence of a primitive form ω (proved by M. Saito [15]). In the case of the Ansingularity this primitive form is simply dx.

However let us see which elements can be defined also in the setting of anarbitrary Frobenius structure. Obviously everything that was discussed in thefirst two sections falls into this category. When we look at the definition of theSt operator we see that this merely depends on the deformed connection. Thusit is also readily available in the general structure. Moreover the Rt operator isavailable in this sense. However not immediately in the way that we defined, orused, it. For this we would need an analogue of the miniversal deformation todefine the oscillatory integrals and the basis of cycles Bini=1.

Dubrovin shows in [1] that such an analogue, which he calls superpotential,exists for a certain type of semisimple Frobenius manifold. It is also shownthere that the analogue of the period vectors I(0) can be defined, in the senseof a solution to the equations in lemma 12. It seems then that all the elementsare available in such a Frobenius structure, if a suitable primitive form can befound. It would need to be checked whether sufficient cycles Bi are availableand whether the phase form has the same nice properties in the general case.Another object that was used quite a lot in the construction of the vertexoperators form of the integrable hierarchies and the proof of theorem 20 is themonodromy group. In the case of the An singularity this simply corresponds tothe different solutions to F (x, t) = λ. However the mondromy group can alsobe defined in the case of an arbitrary Frobenius manifold [5].

It seems indeed that the discussion of Forbenius manifolds is deeply intertwinedwith the discussion of integrable hierarchies. Both theories have many inter-esting aspects in their own right. However this case of the An singularity andof simple singularities in general does show that there is a very natural andimmediate connection between both subjects.

63

Appendix A

In section 8 it was mentioned that there is some freedom in the way to definethe period vectors in the point 0 ∈ T . In fact this freedom is avaiblable atan arbitrary point as well. Mainly we are concerned with the period vectorssatisfying the upper right identity in lemma 12. Since we need to define I(k)

for any k ∈ Z we have two options. Either we define a specific superscript, I(0)

seems to make most sense, and a consistent way of taking the anti-derivative∂pλ for p < 0. Or we proceed as we did in section 10, i.e. by defining explicitelyall period vectors with negative superscripts and proving the relation. We havechosen to follow the definitions of [3], in [2] the first option is used. In thisAppendix we will briefly summarize this method. This method uses the conceptsof stabilization of the singularity [14].

Stabilization

The process of stabilization is in particular a way to fix certain anti-derivates ofthe period vectors. However it is also a meaningful construction in its own right.The idea is to increase the dimension of the domain of the singularity f whileminimally affecting the structure. For a detailed explanation of stabilization seealso [14].

To reiterate let f : (C, 0)→ (C, 0) be the germ of a holomorphic function withan isolated critical point at the origin. A stabilization of f is the germ of theholomorphic function f defined as

f(x, y1, . . . , ym) = f(x) +

m∑i=1

y2i for some m ∈ N, (15.1)

thus f : (Cm+1, 0) → (C, 0) has again an isolated singularity at the origin.Moreover the multiplicity of the singularity of f is equal to that of f . It turnsout that most properties of the singularity are either equal or very similar tothose properties of the stabilization.

If F (x, τ) is the deformation of f then F (x, τ) +

m∑i=1

y2i is the corresponding

deformation of the stabilization. If the degenerate isolated singularity splitsinto multiple non-degenerate singularities in the deformation then the same willhappen in the deformation of the stabilization. Note also that the structure ofthe local algebra

C[x, y1, . . . , yn]/〈fx, fy1 , . . . , fyn〉

of the stabilization is essentially the same as the structure of the local algebraof the singularity. The primitive form and residue metric of the stabilizationshould be obvious.

Anti-derivatives

Now we need to slightly adapt the definition in (8.12) to account for the stabi-lization. Mainly to take care of the anti-derivatives. To do this note that wecan move to a stabilization of the singulartiy f . So suppose k < 0 then we will

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pass to the stabilization with dimension m = 2l + 1, i.e. we add 2l variables,such that −l < k. The main reason for considering only an odd dimension isthat this means certain intersection forms are equal for the stabilization and theoriginal singularity [14]. Then we have in that case

< I(0)β (λ), [φ] >0= −(

1

2π

d

dλ)l∫β

φ(x)dx1 ∧ . . . ∧ dxm

df(x), (15.2)

thus to define the I(k) for k < 0 we just pass to the dimension m = 2l + 1 with−l < k. Now the derivatives will cancel and thus the anti-derivatives are fixed.We include the factor 1

(2π)lto make the left hand side of (15.2) independent of

l. Since for β in f−1(λ)∩Cm and f = x21 + x2

2 + . . .+ x2m the A1 singularity we

get that

∫β

dx

df= (2π)l(2l − 1)!!λl−

12 [2].

Appendix B

In proving theorem 20 and in fact in most arguments made in that section thereis an open question. Since we have been discussing a lot of infinite series weneed now to adress the meaning of the formulas. This will in fact completethe proof of theorem 20 and as mentioned before the discussion is also seen insection 8 of [2]. We will omit most proofs in this section, they can be found inthe references however.

We start by defining the nature of functions such as the Witten-Kontsevichtau-function.

Definition 6 We define an symptotical function as an expression of the form

exp∑g≥0

~g−1F (g)(t), (15.3)

where F (g) is a formal function on the space of polynomials t = t0 + t1z + . . .with vector coefficients tk =

∑nα=1 t

αk eα ∈ H for all k ≥ 0. Here eαni=1 forms

on orthonormal basis of H.

Then we will call such an asymptotical function tame if

∂α1tk1

. . . ∂αrtkrF(g)|t=0 = 0, (15.4)

whenever k1 + . . . + kr > 3g − 3 + r. Particularly we note that the Witten-Kontsevich tau-function is tame. This can be seen by the definition (not in-cluded above)

T (t) = exp∑g≥0

~g−1∑m≥0

1

m!

∫Mg,m

t(ψ1) ∧ . . . ∧ t(ψm),

whereMg,m is the (compact) Mumford-Deligne moduli space of genus g stablecomplex curves with m marked points and ψi for i = 1, . . . ,m are the psi-classes,i.e. the first Chern classes of the universal cotangent line bundles. Since we haveDimCMg,m = 3g − 3 + m. To such an asymptotical function we associate an

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(asymptotical) element in the Fock space via the dilaton shift q(z) = t(z) − zas we did before. This is then an asymptotical function of q around the shiftedorigin q = −z1 for 1 ∈ H the unit.

Now to use this tameness of functions in the proof of theorem 20 we will haveto see what it has to do with the vertex operator form of the (n + 1)KdVhierarchies. So we refer to the vertex operators form (8.18) of the (n + 1)KdVhierarchies. Suppose G is an asymptotical function. Then we see that, by virtueof the change of coordinates as given in (8.3) (after switching back to x′ andx′′) we have

(Γ−βt ⊗ Γβt )(G(x′)⊗ G(x′′) =

= exp2∑k≥0

< I(−1−k)β ,

qk√~> exp−

∑k≥0

(−1)kI(k)β

√~∂qkG(x + q)G(x− q),

(15.5)for suitable β (and we favor the notation q over y). Here we follow roughly thereasoning which arived us at (8.4). Using the genus expansion (15.3) we canrewrite (15.5) as

exp

2∑k≥0

< I(−1−k)β ,

qk√~> +

∑g≥0

~g−1∑±F (g)(χ±)

, (15.6)

whereχ± = x± q∓

√~∑k≥0

I(k)β (−z)k.

Now the crucial proposition, which is proved by Givental in [2] using the aboveexpression and the tameness condition, is the following.

Proposition 21 Suppose G as in (15.3) is a tame asymptotical function of x.

Then the expression (15.6) divided by exp 2F(0)(x)~ expands into power series

in ~, x and q√~ whose coefficients depend polynomially on finitely many period

vectors.

This allows us to view the regularity condition of (8.18) as a condition of singl-evaluedness of the terms involving the period vectors (including those that arein the phase forms). This is what is meant by the comments in the beginningof the proof of theorem 20.

This would conclude the proof as long asD is in fact a tame asymptotical elementof the Fock space. Or in fact we need merely tameness of the asymptoticalelement At for some semisimple t ∈ T . Using a construction based on summingover connected graphs [2, 19] it is proved that for tame asymptotical functions Gwe have that R−1G is again tame asymptotical for symplectic upper triangularoperators R.

Since we need only the tameness of At we need not worry about the lower trian-gular operator St. However there is another lower triangular operator included

in the definition of At. Namely the operator exp Uz . In general operators ofthis form do not behave as nicely as the upper triangular operators. Thus wewill need the n copies of the (dilaton shifted) Witten-Kontsevich tau-function

66

to be exp Uz -stable. Here a tame asymptotical function G is called T -stable ifTG is again tame. We have this however since we have

1

zDA1

= 0

by virtue of the Witten-Kontsevich tau-function satisfiying the string equation[11]. This shows that, since every one of the n factors of DA1 gets acted on by

exp uz for u the corresponding critical value, each of the n factors of DA1 in(9.1) gets preserved and thus At is tame.

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