Computation of an asymptotics related to Nahm's conjecture

Speaker: Dr. Masha Vlasenko (MPIM, Bonn)

Time: 3:00PM
Date: Wed 3rd November 2010

Location: Mathematical Sciences Seminar Room

We consider certain functions F_{A,B,C} in the upper half-plane given as a q-series depending on parameters (A,B,C) where A is a positive definite rxr-matrix with rational entries, B is a rational r-vector and C is a rational number. Several years ago Werner Nahm has formulated a conjectural criterion for a matrix A to be such that F_{A,B,C} is modular for some B and C. Don Zagier proved that there are exactly 7 ''modular'' triples (A,B,C) for r=1, and these triples satisfy Nahm's criterion. For r=2 we have already found a counterexample, hence the criterion should be corrected. We search for modular F_{A,B,C} using the following idea. As Zagier observed, if
F_{A,B,C} were modular its asymptotics when q->1 would have special form and this gives infinite number of polynomial equations on the entries of A and B. However these equations look rather complicated.
In the talk we show how one can produce them and discuss the task of finding their common zeros.

(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)