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A_{n-1} singularities and nKdV hierarchies

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arXiv:math/0209205v2 [math.AG] 28 May 2003

An?1 SINGULARITIES AND nKDV HIERARCHIES
ALEXANDER GIVENTAL Abstract. According to a conjecture of E. Witten [21] proved by M. Kontsevich [13], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally generalized under the name the total descendent potential in the theory of Gromov – Witten invariants of symplectic manifolds. The papers [5, 4] contain two equivalent constructions, motivated by some results in Gromov – Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito’s Frobenius structure [17] on the miniversal deformation of the An?1 -singularity, the total descendent potential is a tau-function of the nKdV hierarchy. We derive this result from a more general construction for solutions of the nKdV hierarchy from n ? 1 solutions of the KdV hierarchy.

1. Introduction: Singularities and Frobenius structures. First examples of Frobenius structures were discovered by K. Saito [17] in the context of singularity theory. We begin with a brief overview of very few basic elements of his (rather sophisticated) construction and refer to [10] for further details. Let f : Cm , 0 → C, 0 be the germ of a holomorphic function at an isolated critical point of multiplicity N . We will assume for simplicity that f is weightedhomogeneous. Let T be the parameter space of its miniversal deformation F (x, τ ). Tangent spaces to T are naturally equipped with the algebra structure: Tτ T = C[x]/(Fx (·, τ )). Pick a holomorphic weighted-homogeneous volume form ωτ on Cm possibly depending on the parameters τ . Then the Hessians ?(x) of critical points x ∈ crit(F (·, τ )) become well-de?ned. The corresponding residue paring (φ, ψ )τ =
x∈crit(F (·,τ ))

φ(x)ψ (x) ?(x)

is known to de?ne a non-degenerate symmetric bilinear form on Tτ T which depends analytically on τ , extends across the bifurcation hypersurface without singularities and thus makes Tτ T Frobenius algebras. The key point of K.Saito’s theory is that there exists (according to a theorem of M. Saito, see [10]) a choice of ω (called primitive) that makes the family of Frobenius algebras a Frobenius structure. The latter means certain integrability property which will be recalled lated when needed. We refer to [3, 15] for a detailed account of numerous manifestations of the property — such as ?atness of the metric (·, ·) for example.
Research is partially supported by NSF Grant DMS-0072658.
1

2

ALEXANDER GIVENTAL

In the case of simple singularities a weighted-homogeneous volume form ω coincides with dx1 ∧ ... ∧ dxm (up to a non-zero constant factor which in fact does not a?ect the metric (·, ·) ) and therefore ω is primitive. In the example An?1 we set m = 1, f (x) = xn /n, F (x, τ ) = xn /n + τ1 xn?2 + ... + τn?1 , ω = dx. The basis {?τi } in T0 T is identi?ed with the basis xn?1?i of the local algebra H = C[x]/(xn?1 ), and the residue pairing in this basis takes the form (?τi , ?τj )0 = δi+j,n?1 . By the general theory, the following residue metric is ?at: (?τi , ?τj )τ = Resx=∞ x2n?2?i?j dx . F ′ (x, τ )

In Gromov – Witten theory, intersection indices in moduli spaces of genus-0 pseudo-holomorphic curves in a given compact symplectic manifold de?ne a Frobenius structure on the cohomology space of the manifold. What is the structure behind intersection theory in spaces of higher genus pseudo-holomorphic curves, and is it possible to recover the totality of higher genus Gromov – Witten invariants from the Frobenius structure? While the answer to the ?rst question is yet unknown, the answer to the second one seems to be positive in the semisimple case. According to [5] the total descendent potential corresponding to a semisimple Frobenius manifold can be de?ned by the formula
N

(1)

? τ exp(U/z )? ??1 Ψ(τ ) R D(q) = C (τ ) S τ

i=1

DA1 (qi ).

The ingredients of the formula will be explained later in the context of singularity theory. Roughly, the function ln D is supposed to have the form of “a genus exg ?1 (g ) pansion” F (q) where F (g) depend on the sequence q of vector variables q0 , q1 , q2 , ... taking values in the local algebra H of the singularity. The Taylor coef?cients of F (g) are to play the role of genus-g Gromov – Witten invariants and their gravitational descendents. The product term in (1) is the tensor product of N copies of the total descendent potential for the A1 -singularity (which is a tau-function of the KdV hierarchy and is discussed in Section 3). The product is considered as an “element of a Fock space”. The S , R and exp(U/z ) are elements of a certain group (of loops in the variable z ) acting on the elements of the Fock space via some “quantization” representation ?. The loops S (z ), R(z ) and exp(U/z ) (as well as C and Ψ which are a non-zero normalizing constant and an invertible matrix) are de?ned in terms of the Frobenius structure and in the case of singularities allow convenient descriptions via oscillatory integrals and their asymptotics. The ingredients of the formula depend on a choice of the point τ ∈ T which has to be semisimple, i. e. the function F (·, τ ) must have N non-degenerate critical points. For example, U is the diagonal matrix of the critical values of F (·, τ ). As it is explained in [5], the resulting function D does not depend on τ , satis?es the so called 3g ? 2-jet condition, Virasoro constraints and has the correct (in the sense of [3]) genus-0 part F (0) . 1 In this paper, we will prove that in the case of An?1 -singularities, the function (1) is a tau-function of the nKdV-hierarchy (Theorem 1). In Section 2, we describe the quantization formalism underlying (1). The KP, KdV and nKdV-hierarchies are described in Section 3 in terms of the so called vertex operators of the in?nite dimensional Lie algebra theory [11]. In Section 4, we reconcile the notations of representation theory and singularity theory and state
1According to a result from [4], a function with these properties, when exists, is unique.

An?1 SINGULARITIES AND nKDV HIERARCHIES

3

Theorem 1. In Sections 5 and 6, we study conjugations of the vertex operators by the operators S and R. The corresponding Theorems 2 and 3 are the technical heart of the paper and provide surprisingly simple and general formulations in terms of singularity theory. In Section 7, we show how various central constants (somewhat neglected in the preceeding computations) are governed by a certain multiple-valued closed 1-form W on the complement to the discriminant. The form W appears to be a new object in singularity theory, and its properties play a key role in the proof of Theorem 1. In Section 8, we discuss in detail the “Fock spaces” intertwined by the operators S and R and describe analyticity properties of our vertex operators. In Section 9, we state and prove Theorem 4 which interprets the formula (1) as a device transforming some solutions of the KdV-hierarchy into solutions of the nKdV-hierarchy (and which contains Theorem 1 as a special case). Relationships with “Wn -gravity theory” are discussed in Section 10 (Theorem 5). The appendix, included mostly for aesthetic considerations, contains a direct treatment of genus-0 consequences of Theorem 1. Slightly generalizing the methods of the present paper, one can prove that the total descendent potential (1) corresponding to an ADE-singularity satis?es an integrable hierarchy described explicitly in terms of vertex operators and very similar to the famous hierarchy of Kac – Wakimoto [12] constructed via representation theory of loop Lie algebras. We will return to this subject in [9] Acknowledgments. Substantial part of the paper was written during our stay at IHES (Paris) and MPI (Bonn) in Summer ’02. We would like to thank these institutions for hospitality, and the National Science Foundation — for ?nancial support. We are also thankful to E. Frenkel and P. Pribik for their interest and stimulating discussions, to A. Schwarz for consultations on Wn -gravity, and especially to T. Milanov for several useful observations. 2. The quantization formalism. Consider the local algebra H = C[x]/(fx ) as a vector space with a non-degenerate symmetric bilinear form de?ned by the residue pairing (a, b)0 = Resx=0 a(x)b(x)dx1 ∧ ... ∧ dxm /fx1 ...fxm . Let H = H ((z ?1 )) denote the space of Laurent series in one indeterminate z ?1 with coe?cients in H . We equip H with the even symplectic form (2) ?(f , g) = 1 2πi (f (?z ), g(z ))0 dz = ??(g, f ).

The polarization H = H+ ⊕ H? de?ned by the lagrangian subspaces H+ = H [z ], H? = z ?1 H [[z ?1 ]] identi?es (H, ?) with the cotangent bundle T ? H+ . Then the standard quantization convention associates to constant, linear and quadratic ? of order ≤ 2 acting on functions hamiltonians G on (H, ?) di?erential operators G on H+ . More precisely, let {qα } be a coordinate system on H+ and {pα } — the dual coordinate system on H? so that the symplectic structure in these coordinates assumes the Darboux form ? = α pα ∧ qα . For example, when H is the standard one-dimensional Euclidean space then (3) f= qk z k + pk (?z )?1?k

4

ALEXANDER GIVENTAL

is such a coordinate system. In a Darboux coordinate system the quantization convention reads √ √ ?/?qα , (4) (qα )? := qα / , (pα )? := (5) (qα qβ )? := qα qβ , (qα pβ )? := qα ? ?2 , (pα pβ )? := . ?qβ ?qα ?qβ

where {·, ·} is the Poisson bracket, [·, ·] is the commutator, and C is a cocycle characterized by the properties that
2 C (pα pβ , qα qβ ) = 1 if α = β, C (p2 α , qα ) = 2 ,

The quantization is a representation of the Heisenberg algebra of constant and linear hamiltonians, but it is only a projective representation of the Lie algebra of quadratic hamiltonians on H to the Lie algebra of di?erential operators. For quadratic hamiltonians F and G we have ?, G ? ] + C (F, G) {F, G}? = [F

and C = 0 on all other pairs of quadratic Darboux monomials. The di?erential operators act on formal functions (with coe?cients depending on ±1/2 ) on the space H+ of vector-polynomials q = q0 + q1 z + q2 z 2 + ... with the coe?cients q0 , q1 , q2 ... ∈ H . We will often refer to such functions as elements of the Fock space. Consider now linear operators on H which preserve the symplectic structure and commute with multiplication by z . They form a twisted version of the loop group LGL(H ). It consists of the loops M (z ) satisfying M t (?z )M (z ) = 1 where t means ? are transposition with respect to the inner product (·, ·)0 . Quantized operators M ? de?ned as exp(ln M )? (though the domain of M in the “Fock space” may depend ? and R ? in the formula (1) are of this nature. Moreover, the on M ). The operators S loops S (z ) and R(z ) are triangular in the sense that S (z ) = 1 + S1 z ?1 + S2 z ?2 + ... and R(z ) = 1 + R1 z + R2 z 2 + .... 3. Example: KP and KdV hierarchies. The goal of this section is to reconcile the conventional theory of integrable hierarchies with the quantization formalism of the previous section in the example of KdV (i.e. 2KdV) hierarchy. The nKdV hierarchies will be treated in this paper as “reductions modulo n” of the KP hierarchy. The KP hierarchy has an abstract description as a sequence of commuting ?ows on the semi-in?nite grassmannian with the time variables x1 , x2 , x3 , .... The “bosonic-fermionic correspondence” identi?es the space of semi-in?nite forms with the symmetric algebra C[x] in the variables x = (x1 , x2 , x3 , ...). Under the Pl¨ ucker embedding, points of the grassmannian are transformed into 1-dimensional subspaces spanned by certain functions of x, and the KP ?ows are de?ned tautologically as time translations. The equations of the KP hierarchy thus assume the form of Hirota quadratic equations describing the image of the grassmannian under the Pl¨ ucker embedding. It will be convenient for us to use the following vertex operator construction of the Hirota quadratic equations. According to [11], Ch. 14, a function Φ(x) (which g ?1 (g ) we will assume to have the form exp φ (x)) satis?es the KP hierarchy i? (6) Resζ =∞ dζ e
j>0 ′′ ζ j (x ′ j ?x j )/



e

?

j>0

ζ ?j j



(?x′ ??x′′ )
j j

Φ(x′ )Φ(x′′ ) = 0.

An?1 SINGULARITIES AND nKDV HIERARCHIES

5

The equation is interpreted in the following way. The change
′′ ′ ′′ ? ?x′′ + ?x′′ , ?yj = ?x′ xj = (x′ j + xj )/2, yj = (xj ? xj )/2, ?xj = ?x′ j j j j

transforms the equation (6) into (7) Resζ =∞ dζ e2
j>0

√ ζ j yj /

e?

j>0

ζ ?j j



?yj

Φ(x + y)Φ(x ? y) = 0.

Expanding in y yields an in?nite system of equations on partial derivatives of Φ(x) which is an abstract form of the KP hierarchy. Note that prior to extracting the residue, the expansion of (7) in y is an in?nite series with the property that the coe?cient at each monomial ym is a Laurent series in ζ ?1 , i.e. the powers of ζ are bounded from above by a constant depending on m. We should therefore think of the expressions in (6),(7) as expansions near ζ = ∞. Below we call such an expression regular in ζ if it contains no negative powers of ζ , i.e. the coe?cient at each monomial ym is a polynomial. By de?nition, solutions of the nKdV hierarchy (also called Gelfand – Dickey or Wn -hierarchy) are those solutions of the KP hierarchy which do not depend on xj with j ≡ 0 mod n. For n = 2 we obtain the KdV hierarchy whose solutions depend therefore only on xodd and do not depend on xeven . Note that the derivations ?y2k in (7) can be omitted while the multiplications by y2k cannot. Thinking of exp 2 k>0 ζ 2k y2k as an arbitrary function of ζ 2 and symmetrizing (7) over the Galois group Z2 of the covering ζ → ζ 2 , we arrive at the following description of the KdV hierarchy: a function Φ(xodd ) satis?es the KdV hierarchy if and only if the following differential 1-form is regular in ζ 2 : (8)
±

±dζ e±

j odd

′′ ζ j (x ′ j ?x j )/



e

?

j odd

ζ ?j j



(?x′ ??x′′ )
j j

Φ(x′ )Φ(x′′ ).

The Witten-Kontsevich tau-function is de?ned as (9) T (t) = exp
∞ g=0 g ?1

1 m! m=0



Mg,m

t(ψ1 ) ∧ ... ∧ t(ψm ),

where Mg,m are the Deligne – Mumford moduli spaces of stable genus g compact complex curves with m marked points, ψi are the 1-st Chern classes of the universal cotangent line bundles (formed by the cotangent lines to the curves at the i-th marked points) over Mg,m , and t is a polynomial t(z ) = t0 + t1 z + t2 z 2 + .... It is known (see for instance [21]) that T satis?es the string and dilaton equations ?t0 T ?
∞ k=0

tk+1 ?tk T =

t2 0 T , 3?t1 T ? 2



k=0

1 (2k + 1)tk ?tk T = ? T . 8

At t0 = 0 the genus-g part of ln T depends only on t1 , ..., t3g?2 (for dimensional reasons). This implies that T is well-de?ned at least as a formal function of, say, , t0 / , t1 , t2 , .... Note however that the vector ?elds on the LHS of the string and dilaton equations become linear homogeneous after the change of variables qk = tk ? δk,1 called the dilaton shift. We de?ne an element in the Fock space by (10) DA1 (q) := T (t), where q(z ) := t(z ) ? z. Thus DA1 is well-de?ned as a formal function near the shifted origin q (z ) = ?z .

6

ALEXANDER GIVENTAL

According to Witten’s conjecture [21] proved by Kontsevich [13] the function DA1 satis?es the KdV hierarchy (8) after the substitution qk = (2k + 1)!!x2k+1 , k = 0, 1, 2, .... We also have ?x2k+1 = (2k + 1)!!?qk . We are going to rewrite (8) in terms of Section 2. The exponents in (8) are elements of the Heisenberg Lie algebra and are quantizations of linear hamiltonians in the symplectic space H. We will encode the hamiltonians by the corresponding (constant) hamiltonian vector ?elds. The standard relationship q ˙ = hp , p ˙ = ?hq between hamiltonians h and their vector ?elds dictates the following correspondence between the Darboux coordinates (3) as linear functions on H and vectors in H: pk → z k , qk → ?(?z )?1?k , k = 0, 1, 2, .... Using the notation λ = ζ 2 /2, we can rewrite the KdV hierarchy (8) for DA1 in the form dλ (Γ? (λ)DA1 )(q′ ) (Γ+ (λ)DA1 )(q′′ ) √ is regular in λ, (11) √ λ 2λ=±ζ √ where the sum is taken over the two values of 2λ, and (12) Γ± (λ) := e±
?1/2 d k k<0 ( dλ ) (2λ)

(?z )k



?1/2 d k k≥0 ( dλ ) (2λ)

(?z )k

.

We will informally refer to (11,12) as the KdV hierarchy for the total descendent potential DA1 . 4. The vertex operators for nKdV. Returning to the setting of Section 1, we introduce vertex operators associated with cycles vanishing at isolated critical points in a fashion generalizing the role of (2λ)?1/2 =
[x]:x2 /2=λ

dx/d(x2 /2)

in (12). More precisely, the operators will have the form (13) where
(k ) Iβ (k )

Γβ = e

k<0

Iβ (λ)(?z )k

(k)

e

k ≥0

Iβ (λ)(?z )k

(k)

,

are vector functions with values in H which are consecutive derivatives
(k+1)

of one another, dIβ /dλ = Iβ , and are de?ned as follows. Let f be a weighted-homogeneous singularity with the local algebra H and with the residue pairing (·, ·)0 de?ned by the volume form ω0 = dx1 ∧ ... ∧ dxm . We will always assume that the number of variables m = 2l + 1 is odd, that the monomials φ1 , ..., φN = 1 ∈ C[x] represent a basis in H , and that the spectrum deg(φ1 ω ), ..., deg(φN ω ) contains no integers. For [φ] ∈ H represented by a linear combination φ of the monomials φi , we put 1 d l dx1 ∧ ... ∧ dxm (0) ( Iβ (λ), [φ] )0 := ( ) , φ(x) 2π dλ df (x) ? 1 β ?f (λ) where β is a middle-dimensional cycle in the Milnor ?ber f ?1 (λ). 2 This de?nes (0) Iβ (λ) as a vector-function with homogeneous components of non-integer degrees, and we extend the de?nition to Iβ
m = 2l + 1 variables, we have
(k )

by the obvious derivations and anti-derivations

2When β is the vanishing cycle f ?1 (λ) ∩ Rm of the A singularity f = (x2 + ... + x2 )/2 in 1 m 1
β

dx/df = σ2l λl?1/2 where σ2l = 2(2π )l /(2l ? 1)!! is the volume

of the unit 2l-dimensional sphere. The factor 1/(2π )l in the de?nition of I (0) makes therefore (0) Iβ = 2λ?1/2 independent on l.

An?1 SINGULARITIES AND nKDV HIERARCHIES

7

in λ. This determines the vertex operator unambiguously up to the classical monodromy of the cycle. In the special case of An?1 singularities, Γβ are closely related to the vertex operators of the nKdV hierarchy. Put l = 0, f = xn /n, φi = xn?1?i , i = 1, ..., n ? 1. We take the cycle β to be one point x = (nλ)1/n at the level f ?1 (λ) and denote this cycle α. Then
(0) (Iα , [φi ])0 =

xn?1?i
α

dx = (nλ)?i/n . dxn /n

Equivalently,

(0) Iα

=

n?1 i?1 ](nλ)?i/n . i=1 [x n?1 i=1 k∈Z

This implies


k ∈Z

(k ) Iα (?z )k =

[xi?1 ]z k (nλ)?(i+kn)/n

(i + rn)/

∞ r =k

(i + rn).

r =0

The double sum contains exactly one summand with each power i + kn of ζ = (nλ)1/n not divisible by n. Let us compare the coe?cients at ζ ?j and ζ j . For j = i + kn we have ?j = n ? i + (?1 ? k )n. The corresponding vectors [xi?1 ]z k and [xn?i?1 ]z ?1?k in H = H ((z ?1 )) have the symplectic inner product (?1)k (while any other pairs are ?-orthogonal). The corresponding factorial products multiply to (?1)k+1 /(i + kn). Let ?/?qi,k denote the elements in the Heisenberg algebra (acting on the Fock space of functions on H+ ) which correspond to the vectors [xi?1 ]z k in H. The above computation means that the change (14)
(k )

qi,k = i(i + n)(i + 2n)...(i + kn)xi+kn

transforms k∈Z Iα (?z )k into ? j<0 xj ζ j + j>0 ?xj ζ ?j /j where j ∈ Z\nZ. Comparing with (6) we see that the change (14) transforms solutions of the nKdV hierarchy into functions D satisfying the condition (15)
α

(Γ?α D)(q′ ) (Γα D)(q′′ ) λ(1?n)/n dλ is regular in λ.

The sum here is taken over all the n values of λ1/n which correspond to the onepoint cycles α. In particular, the coe?cients λ(1?n)/n in di?erent summands di?er by appropriate n-th roots of unity (rather than coincide). Our goal in this paper is to prove the following theorem. Theorem 1. The total descendent potential DAn?1 of the An?1 -singularity de?ned by the formula (1) (as explained in [5]) satis?es (15) and therefore is transformed by the change (14) into a tau-function of the nKdV hierarchy. 5. From descendents to ancestors. According to the de?nition (1) the function D in Theorem 1 has the form D = where Aτ is some other element of the Fock space depending on τ ∈ T and called in [5] the total ancestor potential (and F(1) is a function of τ called the genus-1 Gromov-Witten potential which will be described in the next section and which actually vanishes in the case of simple singularities). Replacing ?D and Γ±α — with its conjugation S ?Γ±α S ? ?1 in (15) the function D with A = S we obtain a reformulation of Theorem 1 in terms of the ancestor potential. Let
(1) ??1 Aτ e F (τ ) S τ

8

ALEXANDER GIVENTAL

?Γβ S ??1 , ?rst formally, and then in the actual setting of singularity us compute S theory. A quantized “lower-triangular” symplectic operator S (z ) = 1+S1 z ?1 +S2 z ?2 +... acts on elements of the Fock space by the formula (Proposition 5.3 in [5]) ??1 G )(q) = eW (q,q)/2 G ([S q]+ ), (S where [S q]+ is the truncation of negative powers of z in S (z )q(z ), and the quadratic form W (q, q) = (Wkl qk , ql ) is de?ned by (16)
k,l≥0

S t (w)S (z ) ? 1 Wkl := . wk z l w ?1 + z ?1
?1

Respectively, For f ∈ H [[z, z ?1 ]], let (ef )? := ef? ef+ be the corresponding element in the Heisenberg group. The previous formulas show that (17) ? (ef ) ??1 G = eW (f+ ,f+ )/2 (eS f )? G . S ?S ?G )(q) = e?W ([S (S
? q ] + , [ S ? 1 q ] + ) /2 ?

G ([S ?1 q]+ ).

We are returning to the Frobenius structure on the parameter space T of a miniversal deformation of a (weighted - homogeneous) singularity. Consider the complex oscillating integral JB (τ ) = (?2πz )?m/2 lim eF (x,τ )/z ω.
B 3

Here B is a non-compact cycle from the relative homology group
M →∞

Hm (Cm , {x : Re(F (x, τ )/z ) ≤ ?M }) ? ZN .

We will assume that ω is primitive and use the notation ?1 , ..., ?N for partial derivative with respect to a ?at (and weighted - homogeneous) coordinate system (t1 , ..., tN ) of the residue metric. Saito’s theory of primitive forms guarantees that the di?erential equations for JB in ?at coordinates assume the following form: z?i ?j J =
k

ak ij ?k J , where ?i ? ?j =

ak ij (τ )?k

k

is the multiplication on the tangent spaces Tτ T . In particular, the linear pencil of connections on the cotangent bundle ? := d ? z ?1 (?i ?)t dti

is ?at for any z = 0 (since (?j JB )dtj provide a basis of ?-?at sections). The integrability of ? is a key axiom in the de?nition of Frobenius structures [3]. 4 The oscillating integral also satis?es the following homogeneity condition: (z?z + (deg ti )ti ?i ) z?j J = ??j z?j J ,

3The present description of the oscillating integral is accurate only for subdeformation τ ∈ T lower of f by terms of degrees lower than deg f = 1. Our excuses are that (i) such τ will su?ce for all our goals and (ii) T lower = T for An?1 and other simple singularities. 4Note that the operators ? ? are self-adjoint with respect to the metric. Identifying the tangent i and cotangent spaces via the metric, we get ? = d ? z ?1 (?i ?)dti , while the natural adjoint connection on the tangent bundle reads d + z ?1 (?i ?) dti .

An?1 SINGULARITIES AND nKDV HIERARCHIES

9

where ??j = deg(?j F ) + deg(ω ) ? m/2, j = 1, ..., N , is the spectrum of the singularity symmetric about 0. One can extend therefore the connection ? to the z -direction by ??z := ?z + ?/z + (E ?)t /z 2 , where E = (deg ti )ti ?i is the Euler ?eld and ? = diag(?1 , ..., ?N ) is the Hodge grading operator (anti-symmetric with respect to the metric and diagonal in a graded basis). The extended connection is ?at (since z?i JB dti provide a basis of ?at sections) and can be considered as an isomonodromic family of connections in z ∈ C\0 depending on the parameter τ ∈ T . Identifying the T ? T with T T via the metric, we obtain the connection operator ?z ? ?/z + (E ?)/z 2 . The connection is regular at z = ∞. At τ = 0 it turns into ?z ? ?/z .

In particular, the basis of ?at sections for the extended connection de?ned by the complex oscillating integrals z?i JB near z = ∞ has the form Sτ (z )z ? C where C is a constant invertible matrix depending on the basis of cycles B. This implies that S is a fundamental solution to z?i S = ?i ? S, i = 1, ..., N , and satis?es the homogeneity condition (z?z + E )S = ?S ? S?. A choice of the series solution S with these properties and satisfying the asymptotical condition S (∞) = 1 and the symplectic condition S t (?z )S (z ) = 1 is called in [5] calibration of the corresponding Frobenius structure. In general calibration is not unique (and may depend on ?nitely many constants) unless there is no integers among the spectral di?erences ?i ? ?j . It is therefore unique in the case of simple singularities. For more general weighted - homogeneous singularities a canonical choice is speci?ed by the condition that Sτ =0 = 1. 5 Let us consider now period vectors Iβ of λ and τ with values in H de?ned by the integrals over vanishing cycles β ∈ Hm?1 (Vτ (λ)) in the Milnor ?bers Vλ,τ = {x ∈ Cn : F (x, τ ) = λ}. We keep the notation m = 2l + 1 and other hypotheses of (k ) Section 4 and de?ne the period vectors Iβ by 6
l+ k (Iβ (λ, τ ), ?i ) := ?(2π )?l ?λ ?i (k ) Iβ (k ) β ?Vλ,τ l+ k d?1 ω = (2π )?l ?λ (k )

De?nition. The operator Sτ (z ) = 1 + S1 z ?1 + S2 z ?2 + ... is de?ned as a gauge transformation in the twisted loop group (i.e. S t (?z )S (z ) = 1) which transforms near z = ∞ the connection operator ?z ? ?/z + (E ?)/z 2 at the parameter value τ ∈ T into the connection operator ?z ? ?/z .

(?i F )
β

ω . dF

The vector-valued functions are multiple-valued and rami?ed along the discriminant where Vλ,τ becomes singular. We refer to [1] for a standard description of the re?ection monodromy group for the cycle β and the integrals. When τ = 0, (k ) the vector-functions Iβ specialize to those of the previous section. Theorem 2. Let f (λ, τ ) =
(k ) k k∈Z Iβ (λ, τ )(?z ) .

Then f (λ, τ ) = Sτ (z )f (λ, 0).

Remark. The integrals I (k) (λ) expand near λ = ∞ into Laurent series (with fractional exponents), and the maximal exponent in I (k) tends to ?∞ as k → ∞.
5This makes the descendent potential a special case of the ancestor potential A = S ?τ D with τ ?0 D is well-de?ned (see Section 9). τ = 0 provided that S 6Here d?1 ω is any m ? 1-form whose di?erential in x equals ω . In the second equality we assume for simplicity that ω is independent of τ as is the case for simple singularities.

10

ALEXANDER GIVENTAL

?l and Respectively, coe?cients in a z -series of the form S f with S = l≥ 0 S l z ( k + l) (k ) k , converge in the 1/λ-adic f = I (?z ) , which are in?nite sums l≥0 ±Sl I sense.

Proof. The period vectors Iβ are related to the oscillating integrals JB by (a version of) the Laplace transform and satisfy the di?erential equations (we remind that ω is primitive, ?i are ?at and ?N is the unit element in the Frobenius algebra (Tτ T , ?) so that ?N F = 1): ?i I = (?i ?)?N I, ?N I = ??λ I, (λ?λ + E )I = (? ? k ? 1/2)I. The equations determine the solution unambiguously from an initial condition (the specialization to τ = 0 will su?ce). Also by de?nition ?λ I (k) = I (k+1) . In terms of the generating function f = I (k) (?z )k the equations read: (18) ?i f = z ?1 (?i ?)f , ?N f + ?λ f = 0, (z?z + λ?λ + E )f = (? ? 1/2)f . ?i f0 = 0, ?λ f0 = ?z ?1 f0 , (z?z + λ?λ )f0 = (? ? 1/2)f0 . The specialization f0 = f (λ, 0) satis?es respectively Combining this with the equations for Sτ (i.e. ?i S = z ?1 ?i ? S and (z?z + E )S = ?S ? S?) we ?nd that f = Sτ f0 satis?es (18). Since f (λ, τ ) and Sτ f0 coincide at τ = 0 by de?nition, the result follows. 6. Stationary phase asymptotics. Consider the vectors ?elds JB on T de?ned by the oscillating integrals JB (τ ) via the formula (J, ?j ) = z?j J . As we discussed in Section 5, when B runs a basis in the appropriate homology group ZN , the vector ?elds form a fundamental solution to the system (19) ?i J = z ?1 (?i ?)J, (?z + (E ?)/z 2 )J = ?J. Now we choose τ semisimple, i.e. require the function F (·, τ ) to have N nondegenerate critical points x(i) . We denote ui the corresponding critical values (they form a local coordinate system on T called canonical) and denote ?i the Hessians of F (·, τ ) at the critical points with respect to the primitive volume form ω . Next, we construct a basis of cycles B1 , ..., BN as follows: in the levels Vλ,τ varying over an in?nite path from λ = ui toward λ/z → ?∞ avoiding other critical values, take a parallel family of cycles vanishing as λ approaches ui and declare their union in Cm to be Bi . In fact many details do not matter here since we are going to replace the oscillating integrals JBi by their stationary phase asymptotics near ui . In this way we get an asymptotical fundamental solution to the same system (19). The asymptotical solution has the form J ? Ψ Rτ (z ) exp(U/z ) where: ? U = diag(u1 , ..., uN ), ? √ Ψ(τ ) is the transition matrix from the basis {?j } in Tτ T to the basis ?i ?/?ui orthonormal with respect to the residue metric, 7 and ? Rτ (z ) = 1 + R1 z + R2 z 2 + ... is a formal power series with matrix coe?cients depending on τ .
7Note that the residue metric in the canonical coordinates assumes the form

(k )

Respectively the matrix Ψ satis?es the orthogonality condition therefore [Ψ?1 ]j i =
a a (?a , ?i )Ψj

a b a,b Ψi (?a , ?b )Ψj

1 2 ?? j (duj ) . = δi,j , and

= ?j

?1/2

?i uj .

An?1 SINGULARITIES AND nKDV HIERARCHIES

11

According to [6] (Proposition, part (d)) an asymptotical solution of this form to the system (19) is unique and automatically satis?es the symplectic condition Rt (?z )R(z ) = 1. According to the de?nition of the total descendent potential (1) given in [5] the data Ψ, R, U in (1) come from this unique asymptotical solution and thus coincide with the corresponding ingredients of the stationary phase asymptotics J ? ΨR(z ) exp(U/z ) described above. The coe?cient C in the formula (1) is de?ned (uniquely up to a non-zero constant factor) in terms of the diagonal entries of the matrix R1 (see [5]): C (τ ) := exp 1 2
τ ii R1 (u)dui .

The genus-1 Gromov-Witten potential F(1) of a semisimple Frobenius structure mentioned in the previous section is de?ned (up to an additive constant) by F(1) (τ ) := 1 48 ln ?i (τ ) + ln C (τ ).
i

As it is shown, for instance, in [7], the function F(1) is constant in the case of A2 -singularity. Using Hartogs’ principle one can derive from this (see, for example, [10]) that for arbitrary singularity it extends analytically from semisimple points τ through the caustic. In particular, F(1) is constant (as a regular function on T of zero homogeneity degree) in the case of all simple singularities. To complete the description of the formula (1), we note that (q1 , ..., qN ) = Ψ?1 q ∈ CN [z ] is the coordinate expression for q ∈ H [z ] in terms of our orthonormal basis in Tτ T identi?ed with H = T0 T via the ?at metric (·, ·). We have therefore the ancestor potential de?ned by the formula
N

(20)

? τ e(U/z)? Aτ (q) = Ψ(τ ) R

i=1

DA1 (qi )?i

?1/48

(τ ),

U/z and our next goal is to learn how to commute the vertex operators Γβ . τ past ΨRe f ?1 ? ? In fact, the conjugation J (e )?J of an element of the Heisenberg group by a ?1 quantized symplectic transformation is proportional to (eJ f )?. We postpone the discussion of the proportionality coe?cient and compute J ?1 f . Let βi be the cycle in H 2l (Vλ,τ ) vanishing as λ → ui along the same path as the one participating in the de?nition of the non-compact cycle Bi . The vector JBi ( l) of oscillating integrals is expressed via Iβi by the “Laplace transform” along the path:

where the dots mean power series in 2(λ ? ui ). In components, we ?nd (21) [Iβi ]j =
a (0) ai Ψj a δ + k>0 k Aai k [2(λ ? ui )]

(note that I (k) (ui , τ ) = 0 for k < 0). Near the critical value λ = ui we have the expansion ?j ui 2 (0) (Iβi , ?j ) = √ (1 + ...) ?i 2(λ ? ui ) 2 2(λ ? ui ) .

(?z )?l JBi (τ ) = √ ?2πz

?∞ ui

( ? l) eλ/z Iβi (λ, τ )dλ = √

1 ?2πz

∞ ui

eλ/z Iβi (λ, τ )

(0)

12

ALEXANDER GIVENTAL

Using the change of variables λ ? ui = ?zx2 /2 we compute (22) (23) 2 √ ?2πz
?∞ 0

(?z )k+1/2 ui /z e eλ/z [2(λ ? ui )]k?1/2 dλ = √ ?2πz

∞ ?∞ ui /z

e ?x .

2

/2 2k

x dx

= (?z )k (2k ? 1)!! e

Thus the asymptotics of JBi assumes the form [JBi ]j ?
ai Ψj a δ + a k (2k ? 1)!! Aai k (?z )

eui /z ,

k>0

αi and therefore Rk = (?1)k (2k ? 1)!! Aαi k . Substituting this into (21) and combining with the Taylor formula eu/z k∈Z z k I (k) (λ) = k∈Z z k I (k) (λ + u), we arrive at the following result. k Ψ R(z ) eU/z 1i I Theorem 3. Near λ = ui we have k∈Z (?z ) Iβi = √ where 1i = ?i ?/?ui is the i-th unit coordinate vector in CN and I(z, λ) := d k 2 k∈Z (?z )k ( dλ ) (2λ)?1/2 . (k )

Remark. Note that coe?cients of a z -series of the form Rf , where R = l≥0 Rl z l ( k ? l) . They converge in the and f = I (k) (?z )k , are in?nite sums l≥0 ±Rl I √ (k ) expands near λ = ui into a Laurent series in √λ ? ui -adic sense as long as I λ ? ui such that the lowest exponent tends to ∞ as k → ?∞. 7. The phase factors Let us introduce the phase 1-form
N

(24)

? β (λ, τ ) := ?(I (0) (λ, τ ), dI (?1) (λ, τ )) = W β β

i=1

(Iβ , ?i ? Iβ ) dti .

(0)

(0)

? α,β = It depends quadraticly on the cycle β , and we will occasionally denote W (?1) (0) ?(Iα , dIβ ) its polarization which is symmetric and bilinear in α, β . The phase form is, generally speaking, multiple-valued and is rami?ed along the discriminant where λ is a critical value of F (·, τ ). We discuss below some basic properties of the phase form. 8 ? β and the polarizations are closed since ?i (?j ?) = ?j (?i ?). 1. Both W ? is determined by the 2. The phase form is invariant under ?λ + ?N , i. e. W ? (0, τ ) via W ? (λ, τ ) = W (τ ? λ1). restriction W (τ ) := W 3. Let E = deg(ti )ti ?i be the Euler vector ?eld. Then iE Wα,β = ? α, β . Here α, β is the intersection index normalized in such a way that the self - intersection of a vanishing cycle equals +2. Indeed, (a, E ? b) is known to be proportional to ? the intersection form carried over to the cotangent spaces Tτ T by the di?erential of ?1 the period map τ → [d ω ] de?ned by the primitive form ω (see, for instance, [10]). According to [20], the proportionality coe?cient is independent of the singularity and can be computed in an example.
8T. Milanov has found an elegant description of the phase form in terms of the Frobenius multiplication on the cotangent bundle. WE refer to [9] for details and for explicit formulas in terms of the root systems in the case of ADE-singularities.

An?1 SINGULARITIES AND nKDV HIERARCHIES

13

The property of W means that exp Wβ is homogeneous of degree ? β, β . For example, when α is a 1-point cycle in the level xn /n = ?τn?1 of the An?1 singularity, we have (25)
?λ1 τ n ? 1 =? 1 nλ

Wα =

n?1

xn =n i=1

xi?1 xn?1?i xn 1?n d ( ? )= ln λ. xn?1 xn?1 n n

Note that (n ? 1)/n is the self-intersection index of α projected to the reduced homology group. 4. Suppose that a cycle α is invariant under the monodromy along a loop in the complement to the discriminant. Then the phase form Wα is single-valued along the loop, and we can talk about the period Wα . When a small loop γ = β 2 goes twice around the discriminant near a non-singular point, then the monodromy is trivial, and (26) Wα = ?2πi α, β 2 ,

where β is the cycle vanishing at the corresponding critical point. Indeed, α = α, β β/2 + α′ , where α′ , β = 0. Let λ = u be the critical value. √ (0) (0) Then Iα′ is analytic at λ = ui , and Iβ expands in λ ? u as in (21). This implies that Wα′ ,β and Wα′ ,α′ vanish, while Wβ/2,β/2 = ?2πi (as in (25) with n = 2). Obviously, the same is true for any conjugation δβ 2 δ ?1 (which itself is the square of δβδ ?1 ). Proposition 1. In the case of a simple singularity, suppose that a cycle α has integer intersection indices with vanishing cycles and is invariant under the monodromy along some loop γ . Then the corresponding period Wα is an integer multiple of 2πi. Proof. If a transformation from a ?nite re?ection group preserves some vector, then it can be written as a composition of re?ections in hyperplanes containing the vector. On the other hand, the (monodromy) re?ection group of a simple singularity is known to coincide with the quotient of corresponding Artin’s braid group (i. e. the fundamental group of the complement to the discriminant) by the normal subgroup generated by the squares of standard generators. Thus the loop 2 2 ′ ′ ′ γ can be written as the composition γ = β1 ...βr β1 ...βs , where βi , βi are “small” loops around non-singular points of the discriminant, and the monodromy along b′ i ′ preserves α. The loops βi have zero contributions to the period γ Wα (since α is orthogonal to the corresponding vanishing cycles), while the periods δ?1 β 2 δ Wα = i 2 are integer multiples of 2πi. β 2 Wα = ?2πi α, βi
i

We will show now how central constants in various commutation relations between vertex operators and symplectic transformations are expressed in terms of the phase form. In the situation of Theorem 1, let us compute the factor eW (f+ ,f+ )/2 de?ned by the formulas (16, 17). Di?erentiating (16) and using ?i S (z ) = z ?1 S (z ) and (?i ?)t = ?i ? we ?nd ?i W (q, q) = ([S q]0 , ?i ? [S q]0 ) where [S q]0 denotes the zero mode in S (z )q(z ). Since S |τ =0 = 1, we see from (16) that W |τ =0 = 0 and conclude
τ

W (q, q) =
0

([S q]0 , ?i ? [S q]0 )dti .

14

ALEXANDER GIVENTAL

[Sτ q]0 = Iβ (λ, τ ) and therefore the exponent W (f+ , f+ ) in (17) can be written as
τ

The di?erential 1-form here is closed and the integral does not depend on the path connecting the origin 0 ∈ T with τ ∈ T (at least when q ∈ H [z ]). (k ) We apply the formula to q = k≥0 (?z )k Iβ (λ, 0). According to Theorem 2,
(0)

(27)

W (f+ , f+ ) =
0 i

(Iβ (λ, t), ?i ? Iβ (λ, t))dti =

(0)

(0)

τ 0

?β W

This integral may depend on the path (in the complement of the discriminant) (0) which determines the branch of the multiple-valued vector-function Iβ . Slightly abusing notation, we indicate the end-points in such integrals but suppress the name of the path. However we always assume that in di?erent integrals the path is the same whenever the end-points are the same. Also we choose ?1 ∈ T , (de?ned by F (x, ?1) = f (x) ? 1) for the base point. Rewriting the integral via W W (f+ , f+ ) = computing the second integral as
?λ1 ?1 λ τ ?λ1 ?1

Wβ ?

?λ1

?1

Wβ ,
λ 1

Wβ = ?

1

(Iβ (ξ, 0), Iβ (ξ, 0))dξ = ? β, β

(0)

(0)

dξ , ξ

and combining this with Theorem 2, we arrive at the following conclusions. Proposition 2. Introduce the vertex operator (28) Then we have ?τ e? S
β,β
λ 1

Γβ τ (λ) = e

k<0

Iβ (λ,τ )(?z )k

(k)

e

k ≥0

Iβ (λ,τ )(?z )k

(k)

.

dξ/2ξ

? ?1 Γβ 0 (λ) Sτ = e

τ ?λ 1 ?1

W β /2

Γβ τ (λ).

The weights λ(1?n)/n in the formulation of Theorem 1 coincide with λ? α,α for λ the 1-point cycles α and di?er from exp(? α, α 1 dξ/ξ ) by the corresponding n-th roots of unity (as explained before the formulation of Theorem 1). Corollary. In the case of An?1 -singularities an element D of the Fock space satis?es the nKdV hierarchy (15) if and only if for some — and then for all — ?τ D satisfy the condition τ ∈ T the corresponding elements Aτ = S τ ?λ 1 λ Wα ? α,α 1 dξ/ξ dλ ?α ′′ ?1 (29) (Γτ Aτ )(q′ ) (Γα is regular in λ. τ Aτ )(q ) e λ α,α α Remark. As was explained in Section 3, the regularity condition refers to expansions into Laurent series in λ?1 , and in particular the multiple-valued functions (k ) (Iα , [φi ]) and Wα should be understood as series expansions λ?i ?1/2?k (a0 + a1 λ?1 + ...) and respectively ? α, α ln λ + b1 λ?1 + b2 λ?2 + ... near λ = ∞.

Let us return now to the situation of Theorem 3. ? ?1 on elements of According to [5], Proposition 7.3, the action of the operator R the Fock space is given by the formula ? ?1 G )(q) = (e V (?,? )/2 G )(Rq) (R

An?1 SINGULARITIES AND nKDV HIERARCHIES

15

where (Rq)(z ) = R(z )q(z ), and the “Laplacian” V (?, ? ) = by 1 ? R(w)Rt (z ) . Vkl wk z l = w+z This easily implies where f? = When f =
(?k ) Iβ , k,l≥0

(?qk , Vkl ?ql ) is de?ned

we ?nd

(k ) k k∈Z Iβ (?z ) , 9 2 ?λ V f? =

2 ?1 /2 ? ?1 (ef )?R ? = eV f? R (eR f )?, ?1?k (f?1?k , qk ) is interpreted as a linear function of q. k≥0 (?1)

we have f? = ((Iβ
(?k )

(Iβ

(?1?k)

, qk ). Using ?λ Iβ
( ? l)

(?1?k)

=

(30) (31)

k,l≥0 (0)

, ·), [Vk?1,l + Vk,l?1 ](Iβ
t Rk Iβ (?k )

, ·)) ).

= (Iβ , Iβ ) ? (

(0)

,

t Rl Iβ

( ? l)

2 Also V f? = 0 at λ = ui since f?k ? (λ ? ui )k+1/2 (1i + ...) vanish at λ = ui . Thus 2 V f? = λ

Let us assume now that β is a vanishing cycle βi . By Theorem 3, d 2 1i 2 1i t t (?k ) Rk (?1)l Rl ( )?l?k Iβi = Rk = . dλ 2(λ ? ui ) 2(λ ? ui )
(0) (0)

ui

(Iβi (ξ, τ ), Iβi (ξ, τ )) ?

Note that near ξ = ui both integrals diverge, but in the same way, so that the di?erence converges. The integral can be rewritten as
2 V f? = τ ?ui 1 τ ?λ1

2 (ξ ? ui )

dξ.

Wβi ?

Finally, conjugation of vertex operators by e(U/z)? is a special case of √ Proposition 2 and has the following e?ect: 1i / 2(λ ? ui ) is transformed to 1i / 2λ, and the ) /2 corresponding factor eW (f+ ,f+√ is equal to (λ?ui )/λ. Note that the correspondence between the branches of the · depends on the choice of a path connecting λ ? ui with λ. We summarize. Proposition 3. Let βi be one of the vanishing cycles. Put (32) Then ? (U/z)? ΨRe
?1 τ ?ui 1 τ ?λ1

2dtN τN ? ui ? tN

.

Wi :=

Wβi /2 ?
±β i / 2

dtN 2(τN ? ui ? tN )

λ

,

wi =
λ?ui

dξ 2ξ

e?Wi /2 Γτ

? (U/z)? = e?wi /2 ΨRe

...1 ? (Γ± )(i) ? 1... ,

where Γ± are the vertex operators (12), and the subscript (i) indicates the i-th position in the tensor product. Remark. The integration path in the de?nition of √ as the one that √wi is the same √ determines the branch of · under the translation λ ? ui → λ.
9We slightly abuse notation by identifying T T with CN by Ψ?1 and denoting in the same τ way (·, ·) the metric on Tτ T , and the standard inner products on CN and CN ? .

16

ALEXANDER GIVENTAL

Now let us consider a cycle α represented as the sum c β/2 + α′ where β is the cycle vanishing over the point (λ, τ ) = (u, τ ) on the discriminant, and α′ is a cycle invariant under the local monodromy near this point (so that c = α, β ). Proposition 4. For the vertex operators (28) we have
c Γα τ = e
τ ?u1 τ ?λ 1

Wβ/2,α′

cβ/2 Γα . τ Γτ



K α Proof. It is clear that Γα . The proportionality coe?cient eK arises τ = e Γτ Γτ ? ( l) (k ) ?+ , where f = c (?z )k Iβ/2 and g = (?z )l Iα′ . from commuting ef? across eg The constant K is equal therefore to the symplectic inner product ?(f? , g+ ). One easily ?nds (?1?k) (k) K=c , Iα′ ). (?1)k (Iβ/2



cβ/2

On the other hand, consecutive integration by parts yields
λ u

k ≥0

(Iβ/2 , Iα′ ) dξ =
(?1?k) Iβ/2

(0)

(0)

m? 1 k=0

(?1)k (Iβ/2

(?1?k)

m , Iα′ )|λ u + (?1)

(k )

λ u

(Iβ/2 , Iα′ ) dξ.
(k )

(? m)

(m)

Note that

holomorphic at λ = u. Thus the last integral is o(λ ? u)m?1/2 and hence tends to 0 as m → ∞. We conclude that
λ

? (λ ? u)k+1/2 (1i + ...) and vanish at λ = u, while Iα′ are
τ ?λ1

K=c
u

(Iβ/2 (ξ, τ ), Iα′ (ξ, τ )) dξ = ?c

(0)

(0)

τ ?u1

Wβ/2,α′ .

8. Asymptotical elements of the Fock space Various expressions with quantized symplectic transformations and vertex operators contain numerous in?nite sums, and we have to discuss now precise meaning of our formulas. By an asymptotical function we will mean an expression of the form exp
g ≥0 (g ) g ?1

F (g) (t),

where F is a formal function on the space H [t] of polynomials t(z ) = t0 + t1 z + t2 z 2 + ... with vector coe?cients tk = α tα k φα ∈ H . We will say that an asymptotical function is tame if ? ? | F (g) = 0 whenever k1 + ... + kr > 3g ? 3 + r. α1 ... r t=0 ?tk1 ?tα kr In particular, each F (g) is a formal series Fa,b (t0 )a (t1 )b of t0 , t1 with the coe?cients which are polynomials on t2 , ..., t3g?2+|a| . The Witten – Kontsevich tau-function is tame (as well as ancestor potentials [5] in Gromov – Witten theory are — because dimC Mg,r = 3g ? 3 + r). An asymptotical function is identi?ed with an asymptotical element in the Fock space (in the formalism of Section 2) via the dilaton shift q(z ) = t(z ) ? z and becomes therefore an asymptotical function of q (tame or not) with respect to the shifted origin q = ?z . The notation ?z := (?1)z is the only place where we use that the space H contains a distinguished non-zero vector 1.
(g )

An?1 SINGULARITIES AND nKDV HIERARCHIES

17

Proposition 5. Let R be an upper-triangular element of the twisted loop group, i. e. R(z ) = 1 + R1 z + R2 z 2 + ..., and Rt (?z )R(z ) = 1. Then the action of the ? on tame asymptotical elements of the Fock space is well-de?ned quantized operator R and yields tame asymptotical elements. ? ?1 on an asymptotical function Proof. As mentioned in Section 7 the action of R G takes the form ? ?1 G )(t) = (e (R
V (?,? )/2

G )(Rt + γ ), where γ (z ) = z ? R(z )z.

The operation ln G → ln(e V (?,? )/2 G ) can be described in terms of summation over connected graphs with vertex contributions de?ned by partial derivatives of g ?1 (g ) ln G := F , and edge factors given by the coe?cients Vkl of the “Laplacian” 10 V (?, ? ). In order to check that ln(e V (?,? )/2 G ) is tame, let us examine the αr 1 contribution of a connected graph with E edges into a Taylor coe?cient at tα k1 ...tkr . Let ? g (v ) be the genus of a vertex v , e(v ) — the number of edges incident to the vertex ( e(v ) = 2E ), ? l(v ) — the total sum of the indices in the derivatives Vkl ?tk ?tl applied to the vertex v ( l(v ) =: L), ? r(v ) — the number of marked points in v ( r(v ) = r), ? k (v ) — the total sum of the indices among k1 , ..., kr attributed to the vertex ( k (v ) = k1 + ... + kr =: K ). The total genus g of the graph (i.e. the power of to which the graph contributes) is determined by the formula g ? 1 = (g (v ) ? 1) + E . We see that g ≥ 0 since g (v ) ≥ 0 and E ? v 1 ≥ ?pr0f ile51. Since G is tame, the contribution of the graph vanishes unless k (v ) ≤ 3g (v ) ? 3 + e(v ) + r(v ) ? l(v ) for each v . Summing up we ?nd K≤3 (g (v ) ? 1) + 2E + r = 3g ? 3 + r ? L ? E ≤ 3g ? 3 + r.

Thus the required condition is satis?ed. Moreover, the number of edges of the graph and the indices in the edge factors Vk,l are bounded (L + E ≤ 3g ? 3 + r). Thus ln(e V (?,? )/2 G ) is well-de?ned since there are only ?nitely many terms of each genus g and degree r. The substitution of R(z )t(z ) instead of t(z ) preserves the above conclusions since the multiplication by R = R0 + R1 z + R2 z 2 + ... does not decrease the indices k1 , ..., kr (determined by the degree in z ). Finally, the series z ? R(z )z starts with z 2 since R0 = 1. Therefore the dilaton shift t(z ) → t(z ) + z ? R(z )z is also a well-de?ned operation in the class of tame asymptotical functions.
?1 ?τ As it is mentioned in Section 5, lower-triangular operators S act on an asympW (q , q )/ ? = [S q]+ totical element G of the Fock space by e G ([S q]+ ). The change q ?0 = ?1 = means ? t(z ) = [Sτ (z )t(z )]+ ? τ or, in components, t Sk (τ )tk ? τ , t ? Sk tk+1 , ... Suppose that ln G is a formal function of t and is therefore de?ned in the formal neighborhood of ? t = 0. When t(z ) is a polynomial, ? t = Sτ t ? τ = 0 10We are not going to enter here a detailed discussion of the Wick formula underlying the graph summation technique. However the reader may track the origin of the “graphical” interpretation of the operator R back to [6].

18

ALEXANDER GIVENTAL

??1 G ) a well-de?ned formal function means t0 = τ, t1 = t2 = ... = 0. This makes ln(S τ of t0 ? τ, t1 , t2 , ... (and ). ? with S (z ) = S0 + S1 z ?1 + S2 z ?2 + ... do not preserve the The operators S class of tame functions. In particular this applies to the rightmost operator in ? exp(U/z )?. Yet the formula (20) for the ancestor potential makes sense and ΨR de?nes a tame asymptotical function Aτ because the operators exp(u/z )? preserve DA1 . Indeed, the string equation for the Witten – Kontsevich tau-function coincides with (1/z )?DA1 = 0. More generally, let us call a tame asymptotical function G T -stable, if T G is also tame. Let G be exp(U/z )?-stable for all diagonal matrices U . 11 Then (1) ?1 ?τ exp(U (τ )/z )?G are well-de?ned and tame, while S ?τ Aτ are deeF Aτ := Ψτ R ?ned as asymptotical functions of t(z ) ? τ . Moreover, according to Theorem 7.1 ?1/48 ??1 Sτ Aτ does not depend on τ and in [5], the asymptotical element D := ?i is therefore well-de?ned as an asymptotical function of (t0 , t1 , ...) in the formal neighborhood of (τ, 0, ...) with any semisimple τ . Let us examine now the regularity condition (see Corollary to Proposition 2 of Section 7) in the description of integrable hierarchies via vertex operators. The β β ′ ′′ action of the vertex operators of the form Γ? τ ? Γτ on functions G (x ) ? G (x ) is described more explicitly (see (6), (7)) as composition of translations and multiplications: ? ? ? ? √ q k (?1?k) (k ) ?qk ? G (x + q)G (x ? q). (33) exp ?2 (Iβ , √ )? exp ?? (?1)k Iβ
k ≥0 k ≥0 (k )

The coe?cient Iβ (λ, τ ) can be represented near λ = ∞ by an in?nite series in fractional powers λν with the exponents ν from the union of N arithmetical sequences ?i ? 1/2 ? k + Z? . As we remarked in Section 8, the phase factors exp Wβ also expand into such series with ν ∈ ? β, β + Z? . The formulation that a vertex operator expression like (29) is regular in λ instructs us to expand (33) into a q-series. In fact (33) is manifestly invariant under the classical monodromy operator. As a result, the coe?cient at a given monomial qm expands into a Laurent series in λ?1 (k ) (since the coe?cient at each power λν depends only on ?nitely many Iβ ). The regularity condition, by de?nition, means that the coe?cients at negative powers of λ vanish (so that the Laurent series in λ?1 is a polynomial in λ). g ?1 (g ) On the other hand, recalling F ) and √ the genus expansion G = exp( using the notation Qk := qk / , we can rewrite (33) as (34) ? ?? ? The functions F we see that the cancel out. exp ?2 (Iβ
(?1?k)

, Qk ) +

g ?1

k ≥0

g ≥0

±

(g )

?1

are formal series of x. Rewriting the exponent as a series in -term 2F (0) (x) does not depend on λ and all the ?1/2 -terms

F (g ) ? x ±



Q?



k ≥0

Iβ (?z )k ??

(k )

11These requirements are satis?ed, for example, if G = D (q )...D (q ) where D are obtained 1 1 i N N from DA1 by translations q → q + α, where α(z ) = a0 + a1 z + a2 z 2 + ... is a vector-polynomial (or even a series) with coe?cients which are formal -series such that a0 and a1 are smaller than 1 in the -adic norm (and ak → 0 in this norm as k → ∞).

An?1 SINGULARITIES AND nKDV HIERARCHIES

19

g ?1 (g ) Proposition 6. Suppose that G = exp F is a tame asymptotical func(0) tion of x. Then (33) divided by exp(2F (x)/ ) expands into a power series in √ (k ) , x and Q whose coe?cients depend polynomially on ?nitely many Iβ each.

Proof. Recall that expansions of each F (g) (x) with as power series in x0 , x1 have coe?cients which depend only on ?nitely √ many x2 , x3 , .... Note that each (0) (1) Iβ , Iβ in (34) brings with itself an extra . We conclude that modulo high √ powers of the exponent of (34) is a series in Q, x whose coe?cients depend (k ) polynomially on ?nitely many Iβ each. Subtracting the singular term 2F (0) (x)/ and exponentiating does not alter this conclusion. Proposition 6 means, that the regularity requirement, when applied to tame asymptotical functions, can be understood not only as a statement about expansions near λ = ∞, but also as the property of analytic functions of λ (the polynomial (k ) expressions of Iβ and of the phase factors exp Wβ ) to be single-valued polynomial functions of λ. Finally, it is worth reiterating here some of our remarks from Sections 5 and 6 about conjugations of vertex operators by quantized elements of the twisted loop group: ? ?1 Γβ S ?τ by lower-triangular elements is well-de?ned via ? the conjugation S
τ τ (k )

the expansion of k Iβ (λ, τ )(?z )k as a series near λ = ∞, ? ?1 Γβ i R ? τ by upper-triangular elements is well-de?ned in ? the conjugation R τ τ terms of expansions near the critical value λ = ui , and ? the conjugation by exp(ui /z )? acts on k I (k) (λ)(?z )k as the translation λ → λ + ui ; it is applied in our computations only to the vertex operator de?ned by the analytic functions I (k) (λ) = (d/dλ)k (λ ? ui )?1/2 . 9. From nKdV to n ? 1 KdV

We prove here Theorem 1 as a special case (with D1 = ... = Dn?1 = DA1 ) of a more general result which yields a solution of the nKdV hierarchy from n ? 1 solutions of the KdV hierarchy. Theorem 4. Suppose that asymptotical functions Di (qi ), i = 1, ..., n ? 1, are tame and stable with respect to the string ?ows e(ui /z)? . Let us assume that the ingredients C, S, Ψ, R and U of the formula (1) correspond to the Frobenius structure of the An?1 -singularity. Then (35) ??1 Ψ(τ )R ?τ e(U (τ )/z)? D := C (τ )S τ
n?1 i=1

Di (qi )

satis?es the equations of the nKdV-hierarchy: ? ? (36) ?
1-point cycle α α α ? Γ? 0 ? Γ0 λ α,α

dλ? D ? D is regular in λ.

20

ALEXANDER GIVENTAL

Proof. Similarly to Corollary from Theorem 2 and Proposition 2, it is su?cient (1) ?τ D satis?es the condition: to prove that Aτ := e?F S (37) ? ? for at least one value of τ . We choose τ to be generic (so that F (·, τ ) is a Morse function) and prove (37) as follows. In view of Proposition 6 we can interpret (37) in terms of analytic functions in λ (rather than series in 1/λ). Since all the n one-point cycles α form an orbit of the monodromy group of the An?1 -singularity, one can argue that (37) is invariant under the whole monodromy group (and not only the classical monodromy operator). Thus (37) is meromorphic with possible poles at the distinct critical values u1 , ..., un?1 . The regularity property will follow if we prove that there are no poles at λ = ui . Let β = α+ ? α? be the cycle vanishing at λ = ui , and α± are two of the n one-point cycles. If α = α± , then α is invariant under the monodromy around ui , (k ) the corresponding vector-functions Iα are holomorphic at λ = ui , and therefore the phase factor and respectively the whole summand in (37) with the index α is holomorphic at λ = ui as well. When α = α± , we have α = ±β/2+ α′ where α′ = (α+ + α? )/2 is invariant under (k ) (k ) (k ) the monodromy around ui . Thus Iα± = I±β/2 + Iα′ where the second summand is holomorphic at λ = ui . We have therefore (38)
±β/2 ± , = e ±K Γα Γα τ τ Γτ


?

1-point cycle α

α α Γ? τ ? Γτ e

τ ?λ 1 ?1

Wα + α,α

λ dξ 1 ξ

λ?

α,α

dλ? Aτ ? Aτ is regular in λ

where the proportionality coe?cient e±K is described by Proposition 4 (with u = ui , and c = ±1). Thus the two summands in (37) with α = α± add up to (39)
α ? Γα Γ? τ τ
′ ′

±

β/2 β/2 Aτ ? Aτ ? Γ± C± (λ)Γ? τ τ

dλ,

where C± are some phase factors combined from (37) and (38). Let us now recall that ? (U/z)? Aτ = ΨRe and apply Theorem 3. We see that ? (U/z)? applied ? the square bracket in (39) has the form of the operator ΨRe to a product Fi of n ? 1 functions in n ? 1 di?erent groups of variables ′′ (q′ i , qi ), ′′ ? the factors Fi corresponding to i with βi = β are equal to Di (q′ i )Di (qi ), ? the factor corresponding to βi = β has the form (40)
√ 2λ=±ζ

Di (qi )Di

?1/48

dλ + ′′ c± (λ)(Γ? Di )(q′ i ) (Γ Di )(qi ) √ , λ

√ ? the phase factors c± / λ come from C± and from the phase factors described by Proposition 3.

An?1 SINGULARITIES AND nKDV HIERARCHIES

21

√ We assume that the factors λ here di?er by the sign (rather than coincide) the same way as in (11) (or (15) when n = 2). We claim that near λ = ui the functions c± (λ) coincide, are single-valued and analytic. In order to justify the claim, let us compute the phase factors explicitly. We have: (41) (42) ln c± = ±2
τ ?λ1 λ

?1 τ ?ui 1

Wα± + α± , α±
τ ?ui 1 τ ?λ1

1

dξ ? ln λ ξ

α± ,α±

√ + ln λ ?
1

λ 1

dξ 2ξ dξ . 2ξ

τ ?λ1

Wβi /2,α′ +

(Wβi /2 ?

dtN )? 2(τN ? ui ? tN )

λ?ui

Using bi-linearity of the phase form W with respect to the cycles α± = ±βi /2+ α′ we rewrite: (43) (44) ±2 (45)
τ ?ui 1 ?1

ln c± =

τ ?λ1 ?1

Wα′ + α± , α±

γ±

dξ + ξ

′ γ±

dξ 2ξ dtN ) 2(τN ? ui ? tN )
1 λ?ui

Wβi /2,α′ +

τ ?(ui +1)1 ?1

Wβi /2 + +
τ N ?λ

τ ?ui 1 τ ?(ui +1)1

(Wβi /2 ?

τN ?(ui +1)

dtN ? 2(τN ? ui ? tN )

dξ . 2ξ

The constant ui + 1 is chosen to make the integrals in (45) cancel exactly. The ′ contours γ± , γ± in (43) as well as all terms in (44) may depend on the cycle α± but are independent of λ, while the ?rst integral in (43) is a function of λ analytic at λ = ui and independent of the cycle. This implies that the phase factors c± are proportional to each other and are analytic near λ = ui . Let us show that the proportionality coe?cient equals 1. Since βi /2 = (α+ ? α? )/2 and α′ = (α+ + α? )/2, we have 4Wβi /2,α′ = Wα+ ? Wα? and therefore (46) ln c+ ? ln c? = α? , α?
γ+ ?γ ?

dξ + ξ

τ ?ui 1 ?1

(Wα+ ? Wα? ) +

′ ?γ ′ γ+ ?

dξ . 2ξ

Note that the one-point cycles α± belong to the same orbit of the classical monodromy (i.e. the cyclic group of the Coxeter transformation) and therefore the ?rst integral in (46) can be interpreted as γ1 Wα? where the loop γ1 makes several turns about λ = 0 inside the line ?λ1 so that α? transported along the loop becomes α+ in the end. Let γ2 (ε) denote the path starting at ?1 and approaching the point τ ?ui 1 on the discriminant (as in the second term in (46)) but stopping a small distance ε away from it. Let γ3 (ε) be a loop of size ε going around the discriminant near τ ? ui 1 (so that α+ transported along γ3 becomes α? in the end). The integral Wα+ along ?1 the path γ2 (ε)γ3 (ε)γ2 (ε) does not depend on ε (for homotopy reasons). In the limit ε → 0 it spits out the middle term of (46) plus limε→0 γ3 (ε) Wα+ . Writing Wα+ = Wβi /2 +2Wβi /2,α′ + Wα′ near τ ? ui 1 we see that the ?rst summand contains the term dλ/2(λ ? ui ), and the rest is either analytic at λ = ui or has a singularity

22

ALEXANDER GIVENTAL

√ like (analytic function) × dλ/ λ ? ui . This implies that
ε→0

lim

γ3 (ε)

Wα+ =

√ dλ = π ?1, 2(λ ? ui )

which coincides with the last integral in (46). We conclude that (46) can be inter?1 preted as the period of Wα? along the loop γ1 γ2 γ3 γ2 . The cycle α? is invariant under the monodromy along this loop. According to Proposition 1 the period is an √ integer multiple of 2π ?1. Thus c+ = c? . The proof of Theorem 4 is now completed as follows. Since Di satisfy the KdV hierarchy, we conclude that (40) is regular in λ. This implies that Fi , and hence (39) is single-valued near λ = ui and has no pole at λ = ui . Since the other ingredients of (37) are also holomorphic at λ = ui , we ?nd that (37) is regular at λ = ui . In particular, (37) is invariant with respect to the whole monodromy group (regardless of reliability of the previously mentioned abstract argument) and is regular in λ. 10. Some applications Due to Theorem 1 the total descendent potential DAn?1 de?ned by (1) satis?es the nKdV-hierarchy and is therefore “a tau-function”. In addition it satis?es the string equation (1/z )?DAn?1 = 0 (due to [5]). Solutions of the nKdV-hierarchies satisfying the string equation have been studied in the literature (see for instance [21, 18]) under the name Wn -gravity. By de?nition, the tau-functions in the Wn gravity theory are formal functions of the variables t0 , t1 , t2 , ... ∈ H . Our functions DAn?1 , to the contrary, are known to expand in formal series near semisimple t0 . It is our present goal to identify DAn with the tau-function singled out in the theory of Wn -gravity, and in particular — to establish analyticity of the total descendent potential at t0 = 0. It will be convenient for us to use another form of the nKdV-hierarchy based on the concept of Baker functions. Let exp g≥0 g?1 F (g) (t) be an asymptotical function in a formal neighborhood of t = 0. Given an asymptotical function G , the corresponding Baker function [19] (or wave function [11]) is de?ned as √ √ (?1?k) ,qk )/ (k ) G (q + Iα (?z )k )/G (q). bG = (Γα G )/G = e? k≥0 (Iα
k ≥0

Here Γ is the vertex operator (13) corresponding to a one-point cycle α. The Baker function can be understood as a q -series bG =
(m)

α

bG qm

(m)

with coe?cients bG which are Laurent series of ζ ?1 = λ?1/n (whose coe?cients, √ in their turn, are Laurent series in ). Let C√ ((ζ ?1 )) be the space of all such Laurent series, and let VG denote the subspace spanned over C√ [λ] by the coef(m)

?cients bG . According to the grassmannian description [19] of the KP-hierarchy, G satis?es the nKdV-hierarchy if and only if VG belongs to the principal cell of the semi-in?nite grassmannian (i.e. projects isomorphically onto C√ [ζ ] along ζ ?1 C√ [[ζ ?1 ]]).

An?1 SINGULARITIES AND nKDV HIERARCHIES

23

On the other hand, conjugation of Γα (λ)λ? yields Γα (λ ? u)(λ ? u)? α,α /2 and therefore (47)

α,α /2

by the string ?ow exp(u/z )?

In particular the string ?ow exp(u/z )? acts on the vertex operator expression in (15) by translation λ → λ ? u and therefore preserves the regularity requirement in (15). Thus the string ?ow is a symmetry of the nKdV-hierarchy. Moreover, let Gτ be the total descendent potential DAn?1 considered as an asymptotical function n?1 in the formal variable t = q ? τ + z , where τ = i=1 τi [φi ] ∈ H is a semisimple point (and z represents the dilaton shift). The invariance of DAn?1 with respect to the string ?ow can be restated via the Baker function bGτ as invariance of the space VGτ with respect to the operator A := α, α (τ, Iα (λ)) (1, Iα (λ)) d √ √ ? + ? . dλ 2λ
?1/2 (?1) (?2) [(τ, Iα ) ? (1, Iα )]} Γα e?τ ?z?1 . (0) (?1)

[(1/z )?, Γα ] = d/dλ ? α, α /2λ.

The ?rst two terms here come from (47) and the others come from Note that the space VGτ contains v := bGτ |q=0 which is a power series in ζ ?1 with the constant term 1. Such series form a group acting on the semi-in?nite grassmannian via multiplication. The invariance of VGτ with respect to A is equivalent to invariance of U := v ?1 VGτ relative to B = v ?1 Av . We have (48) B= (1, Iα (λ)) (τ, Iα (λ)) α, α d √ √ + + + ? dλ 2λ
(?1) (0) k ≥0

e?τ ?z?1 Γα = exp{

(?1)k

(fk , Iα √

(k+1)

)

,

where the linear function k≥0 (?1)k (fk , qk ) of q is the di?erential of ln Gτ at t = 0. The space U is a free C√ [λ]-module of rank n. It contains the series 1 and hence contains all λm and all B k (1). Note that the ζ ?1 -series B k (1) starts with (?1) ζ k (since (1, Iα ) ? λ1/n ) and therefore 1, B (1), ..., B n?1 (1) form a basis in U , n while B (1) = ?n/2 nλ + k>0 ak λ1?k/n . The coe?cients a1 , ..., an are uniquely determined by τ . Representing B n (1) ? ?n/2 nλ as a linear combination of the basis vectors, we obtain a system of equations for f0 , f1 , .... It is straightforward to see that the system is triangular and unambiguously determines all fk via τ . We ?nd that the space U and respectively VGτ is unique for each τ . Due to the correspondence between semi-in?nite subspaces, Baker functions and tau-functions (see for instance [19] or Exercises 14.44 – 14.47 in [11]) we conclude that the asymptotic function Gτ is completely characterized up to a scalar factor as a formal solution to the nKdV-hierarchy near q = τ ? z satisfying the string equation. Let us consider now the tau-function function Gτ corresponding to the (nonsemisimple) τ = 0. Existence of the function and of the corresponding space VG0 follows from the results of [18] (or from the above argument which is a slight variation on the theme of [18] anyway). Note that the corresponding operator (49) is homogeneous (of degree ?1) with respect to the grading deg λ = 1, deg = 2 + 2/n. This implies that the basis Ak (v ), k = 0, ..., n ? 1, in VG0 and respectively the tau-function G0 are homogeneous in the appropriate sense. More explicitly, g ?1 (g ) ln G0 has the form F (t), where F (g) are formal series of ti,k , i = 1, ..., n ? A = d/dλ ?
?1/2

(nλ)1/n ? (n ? 1)/2nλ

24

ALEXANDER GIVENTAL

1, k = 0, 1, 2, ... homogeneous of degree (1 ? g )(2 + 2/n) with respect to the grading deg ti,k = (i + 1)/n ? k . This follows from the famous fact [19] that the √ ?ows of ?ti,k ) the KP-hierarchy (which in our notation are represented by the derivations kn+n?i correspond in the grassmannian (and √description to the multiplication by ζ ? deg ti,k ). hence (kn + n ? i) deg ζ = deg By de?nition, the asymptotical function G0 is “the tau-function of the Wn -gravity theory” and, according to a conjecture of E. Witten [21], coincides with the total descendent potential in the intersection theory (developed in [16]) on moduli spaces of complex curves equipped with n-spin structures. Consider now the formal homogeneous function ln G0 as a power series in , t1 , t2 , ... with coe?cients (which are therefore also homogeneous) depending on t0 = (ti,0 , ..., tn?1,0 ) . Since all the components of t0 have positive degrees, we conclude that each coe?cient is polynomial in t0 . Thus translations G0 (t0 + τ, t1 , t2 , ...) are well-de?ned and yield asymptotical functions satisfying the same conditions — the nKdV-hierarchy and the string equation — as Gτ (t0 , t1 , t2 , ...). The previous uniqueness argument now implies Gτ (t) = G0 (t + τ ) for all τ ∈ H . We have proved the following result. Theorem 5. The total descendent potential DAn?1 of the An?1 -singularity coincides with the tau-function G0 introduced in the Wn -gravity theory. Corollaries. (1) The total descendent potential DAn?1 of the An?1 -singularity (which is an asymptotical function of t0 , t1 , ... de?ned in a formal neighborhood of (t0 , 0, ...) with semisimple t0 ) extends across the caustic to arbitrary t0 ∈ H . ?τ DAn?1 are well-de?ned for all τ ∈ H . (2) The ancestor potentials Aτ = S (3) The descendent potential DAn?1 = A0 and is tame. (4) The Gromov – Witten potentials F(g) of the An?1 -singularity (de?ned by g ?1 (g ) F (τ ) := ln Aτ |t=0 ) are polynomial functions of τ ∈ H of weighted degree (1 ? g )(2 + 2/n) and therefore vanish for g > 1. Appendix: Dispersionless limit In the “dispersionless limit” → 0 Theorem 1 implies that the genus-0 descendent potential F (0) of Saito’s Frobenius structure on the miniversal deformation of the An?1 -singularity satis?es the dispersionless nKdV-hierarchy. We give here a more direct proof of this fact using only the general theory of nKdV-hierarchies and the results of Section 5. No doubt, this relationship between the dispersionless nKdV hierarchies and An?1 -singularities has been known for quite a while (see for instance, [3, 14]), but we are not so sure about the following lemmas. Let us recall from Section 3 that an asymptotical function Φ(x) = e is said to satisfy the KP-hierarchy if (50) Resζ =∞ dζ e2
j>0 g ?1

φ(g) (x)

ζ j yj /



e?

j>0

ζ ?j j



?yj

Φ(x + y)Φ(x ? y) = 0.

Lemma 1. A function φ(0) satis?es the dispersionless limit of the KP-hierarchy (0) (nKdV-hierarchy) if and only if for each q the function exp[(d2 )(x)/2 ], where qφ (0) 2 at q, satis?es the KPdq φ is the quadratic form of the 2nd di?erential of φ hierarchy (the nKdV-hierarchy respectively).

An?1 SINGULARITIES AND nKDV HIERARCHIES

25

Proof. In order√ to pass to the limit → 0, divide (50) by Φ2 (x), put Y := y/ (and respectively ?y = ?Y ) and expand √ √ Y ) Φ(x ? Y) Φ(x + = eW (Y )+O( ) , Φ(x) Φ(x)



(0) where W (Y ) is the quadratic form d2 . Taking = 0 results in a closed system of xφ equations for φ(0) which, by de?nition, is the dispersionless KP-hierarchy. Namely, φ(x) satis?es the di?erential equations of the dispersionless hierarchy if for all x the quadratic di?erential W = d2 x φ satis?es the system of algebraic equations

(51)

Resζ =∞ dζ e2

j>0

ζ j Yj

e?

j>0

ζ ?j j

?Yj

e W (Y ) = 0 .

It is an observation of T. Milanov that the condition (51) for a quadratic form W is equivalent to (50) for the corresponding Gaussian distribution Φ(x) = eW (x)/2 . (One may in fact take = 1 everywhere.) Solutions of the (dispersionless) nKdV-hierarchy are those solutions of the (dispersionless) KP-hierarchy which do not depend on xi with i divisible by n. Remark. The form (50) of the KP-hierarchy is stronger than the usual system of dynamical equations for the function u := (ln Φ)xx (here x = x1 ). For example, the KdV-hierarchy in the form uxi = (Li (u, ux , ux x, ...))x is automatically satis?ed by any Φ = exp W/2 since u is constant. We will see in a moment that this is not at all the case for the algebraic system (51). Which Gaussian distributions e ij Wij xi xj /2 (we put = 1) satisfy the KP- and nKdV-hierarchies? Consider the corresponding Baker function bW (x) := e
ζ j xj ?

e

ζ ?j ?xj /j W (x)/2

e

= bW (0)e

i

x i (ζ i ?

j

Wij ζ ?j /j )

,

where bW (0) = exp( ij Wij ζ ?i?j /2ij ). Let VW denote the subspace in C((ζ ?1 )) spanned by the Taylor coe?cients of the normalized Baker function bW (x)/bW (0). Lemma 2. A Gaussian distribution exp W/2 satis?es the KP-hierarchy (nKdVhierarchy) if and only if the corresponding normalized Baker function generates a semi-in?nite subspace VW which is a subring (respectively a C[ζ n ]-subalgebra) in C((ζ ?1 )). Proof. Indeed, the subspace VW being semi-in?nite (which is necessary and su?cient for a function to satisfy the KP-hierarchy) means that the Laurent series 1, ζ ? W1j ζ ?j /j, ζ 2 ? W2j ζ ?j /j, ...

form a basis in VW . Taylor coe?cients of the normalized Baker function are arbitrary products of these series which therefore have to be in VW . Solutions to nKdV-hierarchy correspond to semi-in?nite subspaces invariant with respect to multiplication by ζ n . . Note that (i) the above basis in the space VW is canonical in the sense that it is obtained by lifting the basis 1, ζ, ζ 2 , ... from C[ζ ] to VW along ζ ?1 C[[ζ ?1 ]], (ii) the Baker function of a space V from the principal cell of the semi-in?nite grassmannian is normalized i? the 1st element in the canonical basis is 1,

26

ALEXANDER GIVENTAL

(iii) the rest of the basis determines the coe?cients Wij unambiguously, which establishes a 1–1 correspondence between Gaussian distributions satisfying the KPhierarchy and semi-in?nite subrings V ? C((ζ ?1 )) from the principal cell of the grassmannian, (iv) VW = C[x], where x := ζ ? W1j ζ ?j /j , and W1j , j = 1, 2, ..., are arbitrary numbers. Corollary. Gaussian distributions satisfying the nKdV-hierarchy are in 1–1 correspondence with equations of the form (52) xn + τ1 xn?2 + ... + τn?1 = λ,

parameterized by τ = (τ1 , ..., τn?1 ). Proof. When VW = C[x] corresponds to a solution of the nKdV-hierarchy, we must have ζ n ∈ C[x] and therefore ζ n = xn + τ0 xn?1 + ... + τn?1 for some τ0 , ..., τn?1 . On the other hand, W1j must vanish for all j divisible by n. Since all n solutions to the equation have the form x(?ζ ) where ? runs through the nth roots of 1, we conclude that the sum ?τ0 of all the n solutions vanishes. Vice versa, solving the equation for x by perturbation theory near x|τ =0 = λ1/n yields a series x = ζ + j ≥0 wj (τ )ζ ?j in ζ = λ1/n . Since the sum of all the n solutions x(?ζ ) equals 0, we conclude that wj = 0 for all j divisible by n. The semi-in?nite subspace C[x(ζ )] ? C((ζ )) is invariant under the multiplication by ζ n = λ due to (52). The genus-0 descendent potential F (0) of a Frobenius manifold (constructed in [3]) can be described (due to Proposition 5.3 and Corollary 5.4 in [5]) in terms of the function W discussed in Section 7:
τ

(53)

Wτ (q, q) =
0

([St q]0 , ?i ? [St q]0 )dti

Namely, let us regard W/2 as a family of functions in τ ∈ H depending (quadraticly) on the parameter q ∈ H+ = H [z ]. Then F (0) is the critical value function for this family. More precisely, the critical points τ are given by the equations ([Sτ q]0 , ?i ? [Sτ q]0 ) = 0 for all i. This is equivalent to [Sτ q]0 ? [Sτ q]0 = 0 and is satis?ed whenever [Sτ q]0 = 0. Recall that [S q]0 = q0 + S1 q1 + S2 q2 + ... where S = 1 + S1 z ?1 + ..., q = q0 + q1 z + .... When q(z ) = t0 ? z , we have [Sτ q]0 = t0 ? τ and ?nd a critical point τ = t0 . In general the equation [Sτ q]0 = 0 has a unique solution τ (t) de?ned by perturbation theory as a formal function of t = q + z (dilaton shift). Then F (0) (t) = Wτ (t) /2. (0) In fact the quadratic di?erential d2 coincides with the quadratic form Wτ (t) . tF In particular, it depends only on the critical point τ (rather than the parameter value t). 12 We are going to show that in the case of An?1 -singularities the Gaussian
12Moreover, according to [2, 8], Frobenius structures equipped with the genus-0 descendent potentials have the following axiomatic characterization. Let L denote the (germ at ?z of a) Lagrangian section in T ? H+ de?ned as the graph of dF (0) (subject to the dilaton shift). Identifying T ? H+ with (H, ?) by means of the standard polarization H = H+ ⊕ H? , we may regard L as a Lagrangian submanifold in H = H ((z ?1 )). Then L is a cone with the vertex at the origin and such that L intersects its tangent spaces L along zL. In particular, L is swept by the spaces zL varying in dim L/zL = dim H -parametric family, and the tangent spaces to L along each zL are constant and coincide with L.

An?1 SINGULARITIES AND nKDV HIERARCHIES

27

distributions eWτ /2 de?ned by (53) satisfy the nKdV-hierarchy — of course, modulo the rescaling (14): qi,k = i(i + n)...(i + kn)xi+kn . Lemma 3. The normalized Baker function of the Gaussian distribution eWτ /2 corresponding to (53) is equal to ? ? ?? (54) exp ?? ?Sτ (z )q(z ),
k ≥0 (?1?k) Iα (λ, τ )(?z )?1?k ?? .

Proof. By de?nition, the Baker function bWτ is e?Wτ /2 Γα eWτ /2 which after normalization and at = 1 becomes (55) e?
(?1?k) (λ,0),qk ) k≥0 (Iα

eWτ (

k ≥0 ( ? z )

k (k) Iα (λ,0),q)

.

Theorem 2 from Section 5 says that Sτ (z )
(k ) (?z )k Iα (λ, 0) = (m) (k ) (λ, τ ). (?z )k Iα

= ?a ∧ Iα and dS = a ∧ S/z where a = (?i ?)dti = On the other hand, dIα at . In particular d[S q]m = a ∧ [S q]m+1 . Therefore computing the second exponent in (55) from (53) and integrating by parts we ?nd
τ 0 (0) (Iα (λ, t), a(t) ∧ [St q]0 ) = τ 0 τ 0 (0) (a ∧ Iα , [S q]0 ) = ? τ 0 ?1 (dIα , [S q]0 ) =

(m?1)

?1 ?(Iα , [S q]0 )|τ 0 +

?1 (Iα , a ∧ [S q]1 ) = ... = ?

k ≥0

(?1?k) (Iα , [S q]k )|τ 0.

(The integral term eventually disappears because q is a polynomial in z .) The value at the lower limit t = 0 cancels with the ?rst exponent in (55), and the value at t = τ coincides with (54). Corollary. The vector space VWτ corresponding to the normalized Baker func(?1?k) tion (54) is spanned by 1 and by the components (Iα (λ, τ ), [φi ]) of the period (m) maps Iα with m < 0. The components of Iα are periods of the di?erential 0-forms x, x2 /2, ..., n?1 x /(n ? 1) on the level sets (56)
(?1)

xn + τ1 xn?2 + ... + τn?1 = λ n in the miniversal deformation of the An?1 -singularity. In the case when α is a onepoint cycle (i.e. x), the C[λ]-module generated by 1, x, x2 , ..., xn?1 is a subring in C((λ1/n )) due to (56). It remains to show therefore that this subring coincides with (m) VWτ , i.e. that it contains all components of Iα for m < ?1. Thus the following lemma completes the proof. Lemma 4. The period maps Iα
(m)

satisfy the equation

(m?1) (m) (? + 1/2 ? m)Iα = (λ ? E ?)Iα

where E = i (deg τi )τi ?τi is the Euler ?eld and ? + 1/2 is the spectral matrix, i.e. the diagonal matrix with entries 1/n, 2/n, ..., n ? 1/n.

28

ALEXANDER GIVENTAL

Proof. In view of the equations ?i I = (?i ?)?n?1 I and ?n?1 I = ??λ I satis?ed by (k ) (k ) all Iα , the lemma is a reformulation of the homogeneity condition (λ?λ + E )Iα = (k ) (? ? 1/2 ? k )Iα discussed in Section 5.

Remark. According to a uniqueness result of Dubrovin and Zhang [4], the total descendent potential of a semisimple Frobenius manifold is completely characterized as an asymptotical function exp g?1 F (g) which satis?es (i) the Virasoro constraints, (ii) the so-called 3g ? 2-jet condition, and (iii) whose genus-0 part F (0) coincides with the genus-0 descendent potential of the Frobenius manifold (constructed in [3]). According to [5], the function DAn?1 satis?es (i),(ii),(iii) and thus would coincide with the tau-function G0 of the Wn -gravity theory (see Section 10), if G0 were shown to satisfy (i),(ii),(iii) as well. In fact, the Virasoro constraints for G0 are well-known (see for instance [18]) and follow from the invariance of the corresponding semi-in?nite subspace V0 ? C√h ((λ?1/n )) under the operators λm A (where A is given by (49)). It is plausible (although at the moment we don’t know a direct proof of this) that the 3g ? 2-jet property, which is equivalent to the ancestor ?τ G0 being tame for all τ ∈ H , can be derived from a Lax-type potentials Aτ := S description of the nKdV-hierarchy. Thus, since the results of this Appendix imply (iii), this would give another proof of Theorems 1 and 5. Also, Dubrovin and Zhang have informed the author that (yet another?) proof of these results can be obtained on the basis of their axiomatic theory of integrable hierarchies [4].

An?1 SINGULARITIES AND nKDV HIERARCHIES

29

References
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