A Riemann-Roch Formula for Singular Reductions By circle actions

We compute a Hirzebruch-Riemann-Roch type formula for the invariant Riemann-Roch number of a quantizable Hamiltonian S 1 S¹ upper S Superscript 1 -manifold ( M , ω , J ) (M,ω , 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 J:M→ 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.