Let G be a connected semi-simple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice ([28], [29]) and the flag manifold G/B.

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Abstract. Let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system |C | for any g and n satisfying C · C = 2g + 2n − 2. When g = 0, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary g in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to P 2 blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus g constrained to g points are also given in terms of quasi-modular forms. Contents

This paper outlines the construction of invariants of Hamiltonian group actions on symplectic manifolds. The invariants are derived from the solutions of a nonlinear rst order elliptic partial dierential equation involving the Cauchy-Riemann operator, the curvature, and the moment map (see (17) below). They are related to the Gromov invariants of the reduced spaces. Our motivation arises from the proof of the Atiyah-Floer conjecture in [17, 18, 19] which deals with the relation between holomorphic curves ! M S in the moduli space M S of at connections over a Riemann surface S and anti-self-dual instantons over the 4-manifold S. In [3] Atiyah and Bott interpret the space M S as a symplectic quotient of the space A S of connections on S by the action of the group G S of gauge transformations. A moment's thought shows that the various terms in the anti-self-duality equations over S (see equation (64) below) can be interpreted symplectically. Hence they should give rise to meaningful equations in a context where the space A S is replaced by a nite dimensional symplectic manifold M and the gauge group G S by a compact Lie group G with a Hamiltonian action on M . In this paper 2 we show how the resulting equations give rise to invariants of Hamiltonian group actions. The same adiabatic limit argument as in [19] then leads to a correspondence between these invariants and the Gromov{Witten invariants of the quotient M==G (Conjecture 3.6). This correspondence is the subject of the PhD thesis [27] of the second author. In Section 2 we review the relevant background material about Hamiltonian group actions, gauge theory, equivariant cohomology, and holomorphic curves in symplectic quotients. The heart of this paper is Section 3, where we discuss the equations and the...

In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians G(k,n) to some Gorenstein toric Fano varieties P(k,n) with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections X ⊂ G(k,n) of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational

For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus ≤ 1 Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology. 1

Abstract. We found an explicit description of all GL(n, R)-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds. 1.

In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F(n1,...,nl,n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F(n1,...,nl,n) to a certain Gorenstein toric Fano variety P(n1,...,nl,n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P(n1,...,nl,n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P(n1,...,nl,n).

We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of U(gl(N)) in terms of first order differential operators in Givental variables. The construction of this representation turns out to be closely connected with the integral representation based on the factorized Gauss decomposition. We also reveal the recursive structure of the Givental representation and provide the connection with the Baxter Q-operator formalism. Finally the generalization of the integral representation to the infinite and periodic quantum Toda wave functions is discussed.

Abstract. We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.