Misc,
Analytic Continuation Of Chern-Simons Theory
(Aug 28, 2010)
Abstract
The title of this article refers to analytic continuation of
three-dimensional Chern-Simons gauge theory away from integer values of the
usual coupling parameter k, to explore questions such as the volume conjecture,
or analytic continuation of three-dimensional quantum gravity (to the extent
that it can be described by gauge theory) from Lorentzian to Euclidean
signature. Such analytic continuation can be carried out by rotating the
integration cycle of the Feynman path integral. Morse theory or
Picard-Lefschetz theory gives a natural framework for describing the
appropriate integration cycles. An important part of the analysis involves flow
equations that turn out to have a surprising four-dimensional symmetry. After
developing a general framework, we describe some specific examples (involving
the trefoil and figure-eight knots in S^3). We also find that the space of
possible integration cycles for Chern-Simons theory can be interpreted as the
"physical Hilbert space" of a twisted version of N=4 super Yang-Mills theory in
four dimensions.
