Abstract. Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finitedimensional Lie algebra, and W its Weyl group. The set of ginvariants in the basic representation of the affine Kac-Moody algebraĝ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F . Using period integrals, we construct an analytic continuation of the twisted representation of F . Our construction yields a global object, which may be called a W -twisted representation of F . Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W-algebra.1. Introduction 1.1. Motivation from Gromov-Witten theory. Recall that the Gromov -Witten (GW) invariants of a projective manifold X consist of correlators (1.1) τ k 1 (v 1 ), . . . , τ kn (v n ) g,n,dwhere v 1 , . . . , v n ∈ H * (X; C) are cohomology classes and the enumerative meaning of the correlator is the following. Let C 1 , . . . , C n be n cycles in X in a sufficiently generic position that are Poincaré dual to v 1 , . . . , v n , respectively. Then the GW invariant (1.1) counts the number of genus-g, degree-d holomorphic curves in X that are tangent (in an appropriate sense) to the cycles C i with multiplicities k i . For the precise definition we refer to [103,75,8,83]. After A. Givental [57], we organize the GW invariants in a generating series D X called the total descendant potential of X and defined as follows. Choose a basis {v i } N i=1of the vector (super)space H = H * (X; C) and let t k = τ k 1 (t k 1 ), . . . , τ kn (t kn ) g,n,d , where t = (t 0 , t 1 , . . . ) = (t i k ) and the definition of the correlator is extended multi-linearly in its arguments. The function D X is interpreted as a formal power series in the variables t i k with coefficients formal Laurent series in whose coefficients are elements of the Novikov ring C [Q].When X is a point and hence d = 0, the potential D pt (also known as the partition function of pure gravity) is a generating function for certain intersection numbers on the Deligne-Mumford moduli space of Riemann surfaces M g,n . It was conjectured by Witten [103] and proved by Kontsevich [74] that D pt is a tau-function for the KdV hierarchy of soliton equations. (We refer to [19,102] for excellent introductions to soliton equations.) In addition, D pt satisfies one more constraint called the string equation, which together with the KdV hierarchy determines uniquely D pt (see [103]). It was observed in [22,54,69] that the taufunction of KdV satisfying the string equation is characterized as the unique solution of L n D pt = 0 for n ≥ −1, where L n are certain differential operators representing the Virasoro algebra. This means that D pt is a highest-weight vector for the Virasoro algebra and in addition satisfies the string equation L −1 D pt = 0.One of the fundamental open questions in Gromov-Witten theory is the Virasoro conjecture suggested by S. Katz and the physicists Eguchi, Hori, Xiong, and...

Let M k,m be the space of Laurent polynomials in one variable x k + t 1 x k−1 + . . . t k+m x −m , where k, m ≥ 1 are fixed integers and t k+m = 0. According to B. Dubrovin [11], M k,m can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendent and ancestor potentials of M k,m (defined as in [16]) satisfy Hirota quadratic equations (HQE for short).Let C k,m be the orbifold obtained from P 1 by cutting small discs D 1 ∼ = {|z| ≤ ǫ} and D 2 ∼ = {|z −1 | ≤ ǫ} around z = 0 and z = ∞ and gluing back the orbifolds D 1 /Z k and D 2 /Z m in the obvious way. We show that the orbifold quantum cohomology of C k,m coincides with M k,m as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendent (respectively the ancestor) potential of M k,m is a generating function for the descendent (respectively ancestor) orbifold Gromov-Witten invariants of C k,m .There is a certain similarity between our HQE and the Lax operators of the Extended bi-graded Toda hierarchy, introduced by G. Carlet in [7]. Therefore, it is plausible that our HQE characterize the tau-functions of this hierarchy and we expect that the Extended bi-graded Toda hierarchy governs the Gromov-Witten theory of C k,m .

The paper [9] gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In [10], it is proved that the total descendent potential corresponding to K. Saito's Frobenius structure on the parameter space of the miniversal deformation of the A n−1 -singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A, D, E-type. Our description of the hierarchy is parallel to the vertex operator construction of Kac -Wakimoto [15] except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac -Wakimoto theory are studied on a case-by-case basis and remain, generally speaking, unknown.Let H = H((z −1 )) be the space of Laurent series f (z) = k∈Z f k z k in one indeterminate z −1 (i.e. finite in the direction of positive k) with vector coefficients f k ∈ H.

Abstract. The Extended Toda Hierarchy (shortly ETH) was introduce by E. Getzler [Ge] and independently by Y. Zhang [Z] in order to describe an integrable hierarchy which governs the Gromov-Witten invariants of CP 1 . The Lax type presentation of the ETH was given in [CDZ]. In this paper we give a description of the ETH in terms of tau-functions and Hirota Quadratic Equations (known also as Hirota Bilinear Equations). A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line.