Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112Hui-Wen LinTaida Institute of Mathematical Sciences (TIMS), National Taiwan University, Taipei 106Chin-Lung WangTaida Institute of Mathematical Sciences (TIMS), National Taiwan University, Taipei 106

For projective conifold transitions between Calabi-Yau threefolds X and Y, with X close to Y in the moduli, we show that the combined information provided by the A model (Gromov--Witten theory in all genera) and B model (variation of Hodge structures) on X, linked along the vanishing cycles, determines the corresponding combined information on Y. Similar result holds in the reverse direction when linked with the exceptional curves.

Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112Feng QuDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112

We give an effective algorithm to compute the Euler characteristics χ(\mbar_{1,n}, \otimes_{i=1}^n L_i^{d_i}). In addition, we give a simple proof of Pandharipande's vanishing theorem H^j (\mbar_{0,n}, \otimes_{i=1}^n L_i^{d_i})=0 for j≥1,di≥0.

Moonshine, the Monster, and Related Topics: Joint Research Conference on Moonshine, the Monster, and Related Topics, June 18-23, 1994, Mount Holyoke College, South Hadley, Massachusetts, 193, 237, 1996

1. INDEX THEORY, ELLIPTIC CURVES AND LOOP GROUPS One can look at elliptic genus from several different points of view; from index theory, from representation theory of Kac-Moody affine Lie algebras or from the theory of elliptic functions and modular forms. Each of them shows us some quite different interesting features of ellitic genus. On the other hand we can also combine the forces of these three different mathematical fields to derive many interesting results in topology such as rigidity, divisibility and vanishing of topological invariants..

Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112R. PandharipandeDepartment of Mathematics, Princeton University, Princeton, New Jersey, 08540

A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K-theory is proved. A new linear equivalence in the Picard group of the moduli space of genus 0 stable maps relating the pull-backs of line bundles from the target via different markings is used for the reconstruction result. Examples of calculations in quantum cohomology and quantum K-theory are given.

Honglu FanDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112

Given two equivariant vector bundles over an algebraic GKM manifold with the same equivariant Chern classes, we show that the genus zero equivariant Gromov--Witten theory of their projective bundles are naturally isomorphic.