University of Georgia
Mathematics Department Colloquium
Professor Rolf Schimmrigk
Kennesaw State University
November 21, 2002
Arithmetic algebraic geometry in string theory
Several recent results indicate that methods from arithmetic algebraic
geometry should be useful in understanding the string theoretic nature
of Calabi-Yau manifolds. The main focus of the present talk is on a
particular problem that points in this direction. Black holes in
string theory compactifications on Calabi-Yau manifolds a priori might
be expected to have moduli dependent features. For example the
entropy of a black hole, measured essentially by the surface area of
the black hole horizon, might be expected to depend on the complex
structure moduli of the Calabi-Yau variety. This would be inconsistent
with known properties of black holes. Supersymmetric black holes
appear to evade this inconsistency by having moduli fields that flow
to fixed points in moduli space that depend only on the charges of the
black hole. Moore observed in the case of simple compactifications
with elliptic factors that these fixed points in moduli space lead to
curves with complex multiplication. This talk describes how the
concept of complex multiplication of black hole attractor varieties
can be generalized to higher dimensional Calabi-Yau manifolds with
finite fundamental group.