Abstract
We describe a family of differential operators parametrized by the transversal vector potentials of a Riemannian foliation relative to the Clifford algebra of the foliation. This family is non-elliptic but in certain ways behaves like a standard Dirac family in the absolute case as a result of its elliptic-like regularity properties. The analytic and topological indices of this family are defined as elements of K-theory in the parameter space. We indicate how the cohomology of the parameter space is described via suitable maps to Fredholm operators. We outline the proof of a theorem of Vafa-Witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument. A determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation. With respect to a generalized coupled Dirac-Yang-Mills system, we indicate how chiral anomalies are located relative to the foliation.
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Communicated by A. Jaffe
Work supported in part by a grant from the National Science Foundation
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Glazebrook, J.F., Kamber, F.W. Transversal Dirac families in Riemannian foliations. Commun.Math. Phys. 140, 217–240 (1991). https://doi.org/10.1007/BF02099498
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DOI: https://doi.org/10.1007/BF02099498