Universal equations for Gromov-Witten invariants

Xiaobo Liu
University of Notre Dame

Relations in tautological rings of moduli space of stable curves
produce partial differential equations for Gromov-Witten invariants for
all compact symplectic manifolds. Such equations do not depend on the
target manifolds, therefore are called universal equations. When the
target manifolds have semisimple quantum cohomology, we believe that
higher genus Gromov-Witten invariants are completely determined by
universal equations. This can be proved up to genus-2.
Without the assumption of semisimplicity, we still can get lot of
information from universal equations. In particular, we believe that the
Virasoro conjecture can be reduced to an SL(2) symmetry for all projective
varieties. Again this can be proved for genus-1 and genus-2 cases.

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