國立清華大學 數學系

Abstract:
Consider a smooth even dimensional compact manifold without boundary M and a Z2-graded vector bundle E=(E+)+(E-) over M. Let D be a Dirac-type operator acting on the sections of E, odd with respect to the grading of E. Let D+, D- be the restrictions of D to the space of sections of E+, E-. The index of D is the integer dim(Ker D+)-dim(Ker D-) and the Atiyah-Singer index formula computes this global analytic number in terms of characteristic classes associated with M and E. Assume now that a compact Lie group acts on M and on E in such a way that D commutes with the action of G. Then the kernel of D is a finite dimensional representation of G and the equivariant index of D computed at g in G is Tr(g on Ker D+)-Tr(g on Ker D-). In this lecture I will give a gentle introduction to the Atiyah-Segal-Singer formula, computing this number in terms of characteristic classes of the fixed point submanifolds and of the normal bundle to them.