Abstracts for Maths/Physics
seminars
Talks will be held in 153 Sloan, unless otherwise noted.
The Formalism of
twisted loop groups in Gromov-Witten theory
Alexander Givental
On Friday, I plan to talk mostly about genus 0 GW-theory:
the axiomatic structure, its loop group invariance, classification
of semisimple theories, the quantum Lefschetz
theorem, a recent toric reduction theorem, and how to get
from them all genus 0 mirror formulas for free.
On Monday, I intend to talk about higher genus theory:
the loop group invariance conjecture, the quantum Riemann-Roch
theorem,
a quantization construction of tau-functions of semisimple
theories,
relationships with integrable hierarchies and vanishing cycles, and
applications to Witten's W_n-gravity conjecture.
Renormalization and Galois symmetries
Matilde Marcolli
I will show how the data of the counterterms that
govern the structure of divergences in the perturbative
renormalization of quantum field theories can be expressed
in terms of certain classes of differential systems with
irregular singularities. These form a Tannakian category
of flat equisingular vector bundles, which is universal
with respect to different physical theories. The corresponding
Galois group is a universal group of symmetries acting on
the coupling constants of physical theories, which contains
the renormalization group as a canonical one-parameter subgroup.
As an affine group scheme it is isomorphic to the motivic
Galois group of a category of mixed Tate motives.
(based on joint work with Connes)
Topological Field Theory and
Representation Theory
David Ben-Zvi (October 30, 2007 1-3PM)
I will discuss some interactions between gauge theory and
representation theory coming from joint work with David Nadler. In
particular I will discuss a partially defined three-dimensional
topological gauge theory, called the character theory, associated
to any
complex semisimple Lie group, and which captures a great deal of
the
corresponding representation theory. Our main result gives an
equivalence
of character theories for Langlands dual groups, which may be
considered
as a three-dimensional aspect of the electric-magnetic duality of
four-dimensional supersymmetric gauge theories.
The mathematical constructions center around the notion of derived
loop
spaces, which are objects of the new world of derived algebraic
geometry,
a fusing of homotopy theory and classical algebraic geometry.
In particular I will describe work with Nadler and John Francis
in which we associate (partial) three-dimensional TFTs
to objects of derived algebraic geometry. Time permitting, I will
mention
how a version of "orientifolding" allows us to
gain new insight into the representation theory of
real semisimple Lie groups through topological field theory.