Sergei Gukov, Quantum Geometry of hyperbolic 3-manifolds

The lectures will focus on three-dimensional Chern-Simons theory
with complex gauge group SL(2,C), which has many interesting
connections with knot theory, representation theory,
3-dimensional quantum gravity, and hyperbolic geometry.

It is very well known that the Chern-Simons-Witten invariant
of a knot K, colored by a N-dimensional representation of SU(2),
is related to the value of the N-colored Jones polynomial,
J_N (K,q), where q is a root of unity. Using the ideas from
Chern-Simons theory, I will explain that, when q is not
a root of unity, the colored Jones polynomial encodes quantum
invariants associated with flat SL(2,C) connections on the knot
complement.

This approach allows to explain a number of curious facts and
to predict some new and rather surprising relations between
the A-polynomial, the colored Jones polynomial, and new invariants
of hyperbolic 3-manifolds. In particular, these relations generalize
the volume conjecture and the Melvin-Morton-Rozansky conjecture,
and suggest an intriguing connection between the colored Jones
polynomial and the arithmetic of hyperbolic 3-manifolds.

                      * * *

The main reference will be:

  S.Gukov, "Three-Dimensional Quantum Gravity, Chern-Simons Theory,
  and the A-Polynomial,"
  http://www.arxiv.org/abs/hep-th/0306165

Other useful references:

  E.Witten, "Quantum Field Theory And The Jones Polynomial,"
  Commun. Math. Phys.  121, 351 (1989)

  W. Thurston, "The Geometry and Topology of Three-Manifolds,"
  http://www.msri.org/publications/books/gt3m/

  D.Cooper, M.Culler, H.Gillet, D.D.Long, P.B.Shalen,
  "Plane curves associated to character varieties of 3-manifolds,"
   Invent. Math. 118 (1994) 47

  H.Murakami, J.Murakami, M.Okamoto, T.Takata, Y.Yokota, "Kashaev's
  conjecture and the Chern-Simons invariants of knots and links,"
  http://www.arxiv.org/abs/math.GT/0203119