TY - JOUR

T1 - Equivariant Index and the Moment Map for Completely Integrable Torus Actions

AU - Grossberg, Michael D.

AU - Karshon, Yael

N1 - Funding Information:
* The second author is partially supported by NSF Grant 9404404-DMS.
Funding Information:
This work evolved from earlier work of the authors and of Tolman [G, KT]; see the end of Section 3. Earlier versions of this paper appeared as notes that the authors distributed in a sequence of talks at Tel-Aviv University in August 1992, and in the second author’s thesis [K, Chaps. 2, 3]. The second author was supported by the Weizmann Institute of Science in the summer of 1992, by the Alfred P. Sloan dissertation fellowship in the academic year 1992 93, and partially by NSF Grant 9404404-DMS in 1994.

PY - 1998/2/10

Y1 - 1998/2/10

N2 - Consider a completely integrable torus action on a compact Spincmanifold. The equivariant index of the Dirac operator is a virtual representation of the torus and is determined by the multiplicities of the weights which occur it in. We prove that these multiplicities are equal to values of the density function for the Duistermaat-Heckman measure, once this is defined appropriately. (The two-form that we take is half of the curvature of the line bundle which is associated to the Spincstructure. It is closed and invariant but not necessarily symplectic.) We deduce that these multiplicities are equal to the topological degree of the "descended moment map"Φ:M/T→t*, which in nice cases can be described as sums of certain winding numbers.

AB - Consider a completely integrable torus action on a compact Spincmanifold. The equivariant index of the Dirac operator is a virtual representation of the torus and is determined by the multiplicities of the weights which occur it in. We prove that these multiplicities are equal to values of the density function for the Duistermaat-Heckman measure, once this is defined appropriately. (The two-form that we take is half of the curvature of the line bundle which is associated to the Spincstructure. It is closed and invariant but not necessarily symplectic.) We deduce that these multiplicities are equal to the topological degree of the "descended moment map"Φ:M/T→t*, which in nice cases can be described as sums of certain winding numbers.

UR - http://www.scopus.com/inward/record.url?scp=0002632604&partnerID=8YFLogxK

U2 - 10.1006/aima.1997.1686

DO - 10.1006/aima.1997.1686

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AN - SCOPUS:0002632604

SN - 0001-8708

VL - 133

SP - 185

EP - 223

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 2

ER -