Abstract not found

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 study the space of stability conditions on the non-compact Calabi-Yau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of P 2. We give some evidence from mirror symmetry for this conjecture. 1.

We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the b Γ-class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau-Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan’s crepant resolution conjecture [66].

Abstract. We suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories. The first theory is the twisted sigma model of a toric variety in the infinite volume limit (the A–model). The second theory is an intermediate model, which we call the I–model. The equivalence between the A–model and the I–model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the T–duality. On the other hand, the I–model is closely related to the twisted Landau-Ginzburg model (the B–model) that is mirror dual to the A–model. Thus, the mirror symmetry is realized in two steps, via the I–model. In particular, we obtain a natural interpretation of the superpotential of the Landau-Ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety. We also relate the cohomology of the supercharges of the I–model to the chiral de Rham complex and the quantum cohomology of the underlying toric variety.

Abstract. We consider Laplace transforms of the Picard-Fuchs differential equations of Calabi-Yau hypersurfaces and calculate their Stokes matrices. We also introduce two different types of Laplace transforms of Gel’fand-Kapranov-Zelevinski hypergeometric systems. 1. Introduction – Laplace

math.AG/0401367

6. Remarks on the semisimple case 43

Main mathematical applications of Frobenius manifolds are in the theory of Gromov- Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories. 1991 MS Classification 32G34, 35Q15, 35Q53, 20F55, 53B50 WDVV equations of associativity is the problem of finding of a quasihomogeneous, up to at most quadratic polynomial, function F(t) of the variables t = (t 1,..., t n) and of a constant nondegenerate symmetric matrix ( η αβ) such that the following combinations of the third derivatives c γ αβ (t): = ηγǫ ∂ǫ∂α∂βF(t) for any t are structure constants of an asociative algebra At = span(e1,..., en), eα · eβ = c γ αβ (t)eγ, α, β = 1,...,n with the unity e = e1 (summation w.r.t. repeated indices

Abstract. We study certain differential rings over the moduli space of Calabi-Yau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasi-modular forms due to Kaneko-Zagier[23]. 1.