| Affiliation: | (1) Department of Mathematics, BrighamYoung University, Provo, UT 84602, USA;(2) Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA;(3) Department of Mathematics, University of Oregon, Eugene, OR 974003, USA |
| Abstract: | We define and study r-spin Gromov-Witten invariants and r-spin quantum cohomology of a projective variety V, where r≥2 is an integer. The main element of the construction is the space  of r-spin maps, the stable maps into a variety V from n-pointed algebraic curves of genus g with the additional data of an r-spin structure on the curve. We prove that is a Deligne-Mumford stack and use it to define the r-spin Gromov-Witten classes of V. We show that these classes yield a cohomological field theory (CohFT) which is isomorphic to the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. Restricting to genus zero, we obtain the notion of an r-spin quantum cohomology of V, whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of V and the rth Gelfand-Dickey hierarchy (or, equivalently, the Ar−1 singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the ψ classes in the definition of gravitational descendants.Research of the first author was partially supported by NSA grant number MDA904-99-1-0039Research of the second author was partially supported by NSF grant number DMS-9803427Research of the third author was partially supported by NSF grant DMS-0104397 |
