59 research outputs found
Gauge theory and mirror symmetry
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
Relative quantum field theory
We highlight the general notion of a relative quantum field theory, which
occurs in several contexts. One is in gauge theory based on a compact Lie
algebra, rather than a compact Lie group. This is relevant to the maximal
superconformal theory in six dimensions.Comment: 19 pages, 4 figures; v2 small changes for publication; v3 small final
changes for publicatio
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Topological dualities in the Ising model
We relate two classical dualities in low-dimensional quantum field theory:
Kramers-Wannier duality of the Ising and related lattice models in
dimensions, with electromagnetic duality for finite gauge theories in
dimensions. The relation is mediated by the notion of boundary field theory:
Ising models are boundary theories for pure gauge theory in one dimension
higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t
Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects
the multiplicity of topological boundary states. In the process we describe
lattice theories as (extended) topological field theories with boundaries and
domain walls. This allows us to generalize the duality to non-abelian groups;
finite, semi-simple Hopf algebras; and, in a different direction, to finite
homotopy theories in arbitrary dimension
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