String Theory Seminar
Thursday, September 11, 2003, 4:00 pm, 332 Phillips Hall, Mathematics Colloquium
at UNC
Katrin Wendland (University of Warwick),
Decoding the Geometry of Conformal Field Theories
Geometric structures can be encoded by several competing means
in mathematics. In differential geometry,
for instance, and very loosely speaking,
the Riemannian metric tensor of a
manifold is the most fundamental object to
study, whereas in algebraic geometry
the commutative algebra of specific functions
on a variety gives the basic structure.
In Connes' noncommutative geometry, these
ideas are combined. Here, geometry
is encoded in terms of spectral triples, that is
function spaces together with the
action of specific multiplicative and differential
operators on them.
Since a similar structure arises from conformal field theory, Connes'
noncommutative geometry
provides a useful tool and guideline in order to extract
geometric data from conformal field
theories. This deciphering of geometry from
conformal field theory is the topic
of our talk. However, we do not presume knowledge
of conformal field theory or noncommutative
geometry. We argu that degeneration
phenomena in geometry and thereby
also in conformal field theory play an
important role for the deciphering,
and we present a non-standard degenerating
example along with its geometric
interpretation.