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String structures and the index of the Dirac-Ramond operator on orbifolds. (English) Zbl 0661.58038
A Dirac operator is defined on a manifold M possessing a spin structure, which happens if the second Stiefel-Whitney class \(w_ 2(M)\) vanishes; the index of this operator is then given by the Atiyah-Singer theorem.
This paper considers the corresponding issues in string theory. The Dirac-Ramond operator is defined as an operator on the loop space of M, when M possesses a “string structure”. The authors discuss the link between existence of such structures and a class \(\lambda \in H^ 4(M,{\mathbb{Z}})\), defined as one half the first Pontryagin class of the tangent bundle of M. For non simply connected M, they suggest that the vanishing of \(\lambda\) should be replaced by a stronger condition.
Then they investigate to what extent the character valued index formula for the Dirac-Ramond operator detects the \(\lambda\)-class. The index formula is also extended to orbifolds, quotients of arbitrary manifolds by discrete symmetry groups.
Reviewer: F.Rouvière

MSC:
58Z05 Applications of global analysis to the sciences
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
58J20 Index theory and related fixed-point theorems on manifolds
81T60 Supersymmetric field theories in quantum mechanics
53C80 Applications of global differential geometry to the sciences
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