Stable Spin Maps, Gromov-Witten Invariants, and Quantum Cohomology

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dc.creatorJarvis, Tyler J.
dc.creatorKimura, Takashi
dc.creatorVaintrob, Arkady
dc.date2000-12-20
dc.date.accessioned2026-07-07T06:26:35Z
dc.date.available2026-07-07T06:26:35Z
dc.descriptionWe introduce the stack of r-spin maps. These are 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 this stack is a Deligne-Mumford stack, and we define analogs of the Gromov-Witten classes associated to these spaces. We show that these classes yield a cohomological field theory (CohFT) that is the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. When r=2, our construction gives the usual Gromov-Witten invariants of V. 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 r-th Gelfand-Dickey hierarchy (or, equivalently, the A_{r-1} singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the psi-classes in the definition of gravitational descendants. Finally, we compute the small phase space potential function when r=3 and V=CP^1.
dc.description37 pages, 4 figures. AMSLaTeX 2.0. Uses Paul Taylor's diagrams.tex
dc.identifierhttps://arxiv.org/abs/math/0012210
dc.identifierhttp://arxiv.org/abs/math/0012210
dc.identifierCommun. Math. Phys. 259(3), 511--543 (2005).
dc.identifierdoi:10.1007/s00220-005-1389-3
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/97151
dc.subjectAlgebraic Geometry
dc.subjectDifferential Geometry
dc.subjectQuantum Algebra
dc.subjectPrimary: 14N35, 53D45. Secondary: 14H10
dc.titleStable Spin Maps, Gromov-Witten Invariants, and Quantum Cohomology
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