This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These GromovWitten type invariants depend on a Calabi-Yau A∞ category, which plays the role of the target in ordinary Gromov-Witten theory. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the u...

In the approach to Gromov–Witten theory developed by Givental, genus-zero Gromov–Witten invariants of a manifold X are encoded by a Lagrangian cone in a certain infinite-dimensional symplectic vector space. We give a construction of this cone, in the spirit of S-equivariant Floer theory, in terms of S-equivariant Gromov–Witten theory of X × P. This gives a conceptual understanding of the “dilat...

The abelian-nonabelian correspondence outlined in [BCFK08] gives a broad conjectural relationship between (twisted) Gromov-Witten invariants of related GIT quotients. This paper proves a case of the correspondence explicitly relating genus zero mpointed Gromov-Witten invariants of Grassmannians Gr(2, n) and products of projective space P ×P. Computation of the twisted Gromov-Witten invariants o...

In this paper, usng the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve.

The Virasoro conjecture predicts that the generating function of Gromov-Witten invariants is annihilated by infinitely many differential operators which form a half branch of the Virasoro algebra. This conjecture was proposed by Eguchi, Hori and Xiong [EHX2] and also by S. Katz [Ka] (see also [EJX]). It provides a powerful tool in the computation of Gromov-Witten invariants. In [LT], the author...

We represent stationary descendant Gromov-Witten invariants of projective space, up to explicit combinatorial factors, by polynomials. One application gives the asymptotic behaviour of large degree behaviour of stationary descendant Gromov-Witten invariants in terms of intersection numbers over the moduli space of curves. We also show that primary Gromov-Witten invariants are ”virtual” stationa...

We extend several notions and results from the classical Patterson-Sullivan theory to setting of Anosov subgroups higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove equality between Hausdorff dimensions flag limit sets, computed respect a suitable Gromov (pre-)metric manifold, critical exponents subgroups.