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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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