Convergence and Analytic Decomposition of Quantum Cohomology of Toric Bundles
Yuki Koto

TL;DR
This paper proves the convergence of the equivariant big quantum cohomology of toric bundles, building on Brown's mirror theorem, and constructs an analytic decomposition of the quantum D-module of the total space.
Contribution
It establishes the convergence of quantum cohomology for toric bundles and provides a decomposition of the quantum D-module based on the base space's properties.
Findings
Quantum cohomology of E converges if that of B converges.
Quantum D-module of E decomposes into that of B.
Analytic decomposition applies to both equivariant and non-equivariant cases.
Abstract
We prove that the equivariant big quantum cohomology QH^*_T(E) of the total space of a toric bundle E \to B converges provided that the big quantum cohomology QH^*(B) converges. The proof is based on Brown's mirror theorem for toric bundles. It has been observed by Coates, Givental and Tseng that the quantum connection of E splits into copies of that of B. Under the assumption that QH^*(B) is convergent, we construct a decomposition of the quantum D-module of E into a direct sum of that of B, which is analytic with respect to parameters of QH^*_T(E). In particular, we obtain an analytic decomposition for the equivariant/non-equivariant big quantum cohomology of E.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models
