A RIEMANN-ROCH FORMULA FOR SINGULAR REDUCTIONS BY CIRCLE ACTIONS

Abstract

We compute a Hirzebruch-Riemann-Roch type formula for the invariant Riemann-Roch number of a quantizable Hamiltonian

-manifold

( M , ω , J ) $(M,\omega ,{ \mathcal J})$ left parenthesis upper M comma omega comma script upper J right parenthesis

, allowing

0 $0$ 0

to be a singular value of the moment map

J : M → R ${ \mathcal J}:M\to {\mathbb R}$ script upper J colon upper M right arrow double struck upper R

. Our formula represents an instance of the Guillemin-Sternberg principle, which states that quantization should commute with reduction. The conceptual novelty of our result is that the involved reduced system only depends on the symplectic data of M . To establish this, we derive a complete singular stationary phase expansion of the Witten integral without appealing to any kind of desingularization. As a consequence, our formula expresses the invariant Riemann-Roch number purely in terms of symplectic invariants of the singular symplectic quotient. In particular, it involves a new explicit symplectic invariant of the singularities.