Fall, 2002
- Speaker Yi-Zhi Huang, Rutgers University
- Title Differential equations, duality and modular invariance
- Time/place Friday, 9/20/2002 3:00 pm
in Hill 705
- Abstract I will explain a result obtained
recently on genus-one conformal field theories. Let V be a vertex
operator algebra satisfying the C_2-cofiniteness condition
and certain finite reductive properties. Then the q-traces of products
of geometrically-modified intertwining operators are
shown to satisfy systems of differential equations which can be chosen
to be regular at any given possible singular point.
Genus-one correlation functions are constructed as the analytic
extensions of these q-traces. We prove duality properties
for these genus-one correlation functions, including commutativity and
associativity. Using the associativity property and
the modular invariance for one-point functions, we establish the
modular invariance for genus-one correlation functions. I
will start with the definition of conformal field theory and will
explain briefly the notion of vertex operator algebra.
- Speaker Matthias Gaberdiel, Kings College and IAS
- Title Conformal field theory and vertex operator algebras
- Time/place Friday, 9/27/2002 3:00 pm
in Hill 705
- Abstract I plan to give an
informal introduction into conformal field theory and its relation to
vertex operator algebras. Towards the end I shall also
discuss the so-called C_2 condition of Zhu and some of its
implications.
- Speaker Earl Taft, Rutgers University
- Title Is there a one-sided quantum group?
- Time/place Friday, 10/18/2002 3:00 pm
in Hill 705
- Abstract In the 1980's, J.A. Green, W.D.Nichols and
EJT constructed a left Hopf algebra, i.e., a bialgebra with a linear
map S satisfying the left antipode condition, but not the
right one. It has a freeness feature that places it outside the realm
of quantum groups. Recently, S. Rodriguez-Romo and
EJT tried to construct a left Hopf algebra in the world of quantum
groups. We did not yet succeed, but the effort led to
some new quantum groups, modeled partially on quantum GL(2), with the
peculiar property that they remain
noncommutative when q=1 [ Letters in Mathematical Physics 61 (2002),
41-50.] We are trying to modify our procedures,
with the hope of finding a left quantum group. If such a thing exists,
it might reflect some lack of symmetry of interest to
physicists.
- Speaker Benjamin Doyon, Physics Department, Rutgers University
- Title Twisted vertex operator algebra modules and
Bernoulli polynomials
- Time/place Friday, 11/1/2002 3:00 pm
in Hill 705
- Abstract In
the construction of twisted modules for vertex operator algebras, one
can define twisted vertex operators by normal-ordered
products of more ``basic'' twisted vertex operators. One also needs to
introduce a certain subtle formal operator in the
construction; this gives, in particular, a correction term for the
action of the zero mode of the Virasoro algebra on a twisted
module. The generalization of this term to the case of a central
extension of an algebra of differential operators of higher
order is a very non-trivial problem from this point of view.
We start from the twisted Jacobi identity and derive various
``commutativity'' and ``associativity'' relations. They allow
us to define twisted vertex operators from more basic ones without
explicit reference to an extra formal operator, and to
calculate correction terms more conceptually. Bernoulli polynomials
appear when we use ``cylindrical coordinates'', where,
as shown by J. Lepowsky, the algebra simplifies drastically.
This is joint work with J. Lepowsky and A. Milas.
- Speaker Takashi Kimura, Boston University and IAS
- Title Integrable systems and topology
- Time/place Friday, 11/15/2002 3:00 pm
in Hill 705
- Abstract
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