Gauge theory and mirror symmetry

Abstract

Outlined in this paper is a description of \emph{equivariance} in the world of 2-dimensional extended topological quantum field theories, under a topological action of compactLie groups. In physics language, I am gauging the theories --- coupling them to a principal bundle on the surface world-sheet. I describe the data needed to gauge the theory, as well as the computation of the gauged theory, the result of integrating over all bundles. The relevant theories are A-models, such as arise from the Gromov-Witten theory of a symplectic manifold with Hamiltonian group action, and the mathematical description starts with a group action on the generating category (the Fukaya category, in this example) which is factored through the topology of the group. Their mirror description involves holomorphic symplectic manifolds and Lagrangians related to the Langlands dual group. An application recovers the complex mirrors of flag varieties proposed by Rietsch.

References in corpus (5)

• The Vacuum Structure and Spectrum of N=2 Supersymmetric SU(N) Gauge Theory
• Triangulated categories of singularities and D-branes in Landau-Ginzburg models
• Integrable systems and supersymmetric gauge theory
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Cited by in corpus (16)

• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• Perversely categorified Lagrangian correspondences
• Symmetries of categorical representations and the quantum Ngô action
• Coulomb Branches of Star-Shaped Quivers
• BV quantization of the Rozansky-Witten model
• Generating the Fukaya categories of Hamiltonian G-manifolds
• Continuous and coherent actions on wrapped Fukaya categories
• Smooth constructions of homotopy-coherent actions
• Holomorphic curves and continuation maps in Liouville bundles
• Geometry of canonical bases and mirror symmetry
• Kostant-Toda lattices and the universal centralizer
• Equivariant localization in factorization homology and applications in mathematical physics I: Foundations
• Equivariant localization in factorization homology and applications in mathematical physics II: Gauge theory applications
• Twin Lagrangian fibrations in mirror symmetry
• Homological Mirror Symmetry for the universal centralizers I: The adjoint group case