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Symplectic Surgery and the Spinc–Dirac Operator

✍ Scribed by Eckhard Meinrenken

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
512 KB
Volume
134
Category
Article
ISSN
0001-8708

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✦ Synopsis

Let G be a compact connected Lie group, and (M, |) a compact Hamiltonian G-space, with moment map J : M Ä g*. Under the assumption that these data are pre-quantizable, one can construct an associated Spin c Dirac operator % C , whose equivariant index yields a virtual representation of G. We prove a conjecture of Guillemin and Sternberg that if 0 is a regular value of J, the multiplicity N(0) of the trivial representation in the index space ind( % C ), is equal to the index of the Spin c Dirac operator for the symplectic quotient M 0 =J &1 (0)ÂG. This generalizes previous results for the case that G=T is abelian, i.e., a torus.

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