April 30
Lisa Jeffrey, McGill U
APPLICATIONS OF QUANTUM FIELD THEORY TO GEOMETRY
Abstract:Over the past twenty years there has been a resurgence of
interaction between theoretical physics (particularly quantum field
theory) and pure mathematics (notably topology and geometry).
Physicists' ideas have led to a number of discoveries in pure mathematics.
Two complementary approaches are applied in quantum field theory.
In the first approach (quantization), one associates a Hilbert space to a
distinguished manifold -- the phase space -- associated to the physical system,
and the behavior of the system is computed by studying linear transformations
of the Hilbert space. In the second approach (functional integration), the
physical system is specified by a space of fields (which is roughly
the space of functions on some manifold) equipped with an action
functional (a real or complex valued function on the space of fields):
one averages over all possible paths the physical system could take,
weighted in such a way that the dominant contribution comes from the
extrema of the action functional, which are the paths satisfying the
classical equations of motion.
I shall describe some examples of quantum field theories that
have arisen in the context of geometrical problems associated to
manifolds, and two recent developments in geometry that are strongly
related to the two approaches in quantum field theory:
(i) the properties of the quantizing Hilbert space when the
phase space is acted on by a symmetry group;
(ii) certain formulas for integrals over finite dimensional
manifolds which are computed as sums of extrema of an action
functional on the manifold (the stationary phase approximation,
which in certain circumstances yields an exact formula).
[Note: Lisa Jeffrey will also give a more technical Special
Colloquium lecture on Monday, April 29, 4pm as follows:
FLAT CONNECTIONS ON RIEMANN SURFACES AND
TWO DIMENSIONAL YANG-MILLS THEORY
Abstract: If G = U(n), the space M of flat G-connections on a
Riemann surface S has been the subject of intense interest for
the past thirty years in algebraic and symplectic geometry as well as
through its applications to topology. It appears in two additional
guises: in topology as the space of representations of the fundamental
group of the Riemann surface into G, and in algebraic geometry as
the moduli space of holomorphic vector bundles of rank n (and degree
0) on S. Of associated interest are the spaces M(n,d) which
appear in algebraic geometry as the moduli spaces of holomorphic
vector bundles of rank n, degree d and fixed determinant: if n
and d are coprime these spaces are smooth Kaehler manifolds.
In a fundamental 1982 paper studying the Morse theory of the
Yang-Mills functional, Atiyah and Bott found formulas for the Betti
numbers of the spaces M(n,d), so that the characterization of the
cohomology as a vector space is complete: its ring structure (or
equivalently the value of the intersection numbers in the cohomology
ring) has however remained obscure. In 1992, Witten used physical
methods to find formulas for these intersection numbers: his work
involved a two dimensional quantum field theory for which the
Lagrangian was the Yang-Mills functional. In recent joint work with
F. Kirwan (announced in Elec. Res. Notices AMS) we have proved
Witten's formulas using methods from symplectic geometry (notably the
technique of localization in equivariant cohomology). In this lecture I
will survey the background leading up to the proof of Witten's
formulas: I will also describe how the formulas for intersection
numbers lead straightforwardly to a proof of the Verlinde formula.]
