Year: 2026
Type: article
Abstract: We compute a Hirzebruch-Riemann-Roch type formula for the invariant Riemann-Roch number of a quantizable Hamiltonian S 1 $S^1$ upper S Superscript 1 -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.
Authors: Benjamin Delarue, Pablo Ramacher, Louis Ioos
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W4321471604