Symplectic Surgery and the Spin-C Dirac operator

Details Symplectic Surgery and the Spin-C Dirac operator Symplectic Surgery and the Spin-C Dirac operator

1995-04-26

arxiv.org/abs/dg-ga/9504002v4

Mathematics

Differential Geometry

30 pages, AMS-LaTeX. To appear in Advances in Mathematics. Revised version: Minor errors corrected, proofs simplified

Scientific paper

Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian $G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of $E$. In the present paper, we study gluing properties of the equivariant index under "symplectic cutting" operations. Our main application is a proof of the Guillemin-Sternberg conjecture, which says that if $E=L$ is a quantizing line bundle and $0$ a regular value of $J$, the multiplicity of the trivial representation in the equivariant index is equal to the Riemann-Roch number of the symplectic quotient. This generalizes previous results for the case that $G=T$ is abelian.

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