Spinors on manifolds with boundary: APS index theorems with torsion

Abstract

Index theorems for the Dirac operator allow one to study spinors on manifolds with boundary and torsion. We analyse the modifications of the boundary Chern-Simons correction and APS η invariant in the presence of torsion. The bulk contribution must also be modified and is computed using a supersymmetric quantum mechanics representation. Here we find agreement with existing results which employed heat kernel and Pauli-Villars techniques. Nonetheless, this computation also provides a stringent check of the Feynman rules of de Boer et al. for the computation of quantum mechanical path integrals. Our results can be verified via a duality relation between manifolds admitting a Killing-Yano tensor and manifolds with torsion. As an explicit example, we compute the indices of Taub-NUT and its dual constructed using this method and find agreement for any finite radius to the boundary. We also suggest a resolution to the problematic appearance of the Nieh-Yan invariant multiplied by the regulator (mass) 2 in computations of the chiral gravitational anomaly coupled to torsion.

Physical aspects of the space–time torsion

“…In this paper, however, the explicit result has not been achieved because of the cumbersome way of calculation. Let us indicate only some of the subsequent calculations [116,142,135,193,52,51,65] and other papers devoted to the closely related issues like index theorems, topological structures and Wess-Zumino [189] conditions [143,46,128,192,151]. We shall not go into details of these works but only present the most important and simple expression for the anomaly and give a brief review of other results.…”

“…The same action has been rediscovered in [155], where it was also checked using the index theorems. In [151]) the action of a massive particle has been obtained through the squaring of the Dirac operator. This action does not possess supersymmetry, and does not have explicit link to the supersymmetric action of [157,155].…”

“…In [84] the action of a spinning particle on the background of torsion and electromagnetic field has been derived in the framework of the Berezin-Marinov path integral approach. This action possesses local supersymmetry in the standard approximation of weak torsion field and includes the previous actions of [157,155,151]) as a limiting cases.…”

An index theorem for non-standard Dirac operators

“…Let us again stress that for the study of the index, where one is interested in the spectrum of states, one must consider the quantization of the spinning particle model. This problem is taken up in detail in [14] One may expect that the equality of the indices of the standard and non-standard Dirac operators now translates to the equality of the Witten index for the standard and non-standard supersymmetry. However, this translation involves some subtleties, which we briefly discuss here.…”

“…For the case we are interested in, namely the chiral gravitational anomaly on manifolds with boundary, Atiyah, Patodi and Singer [16] have shown how to impose boundary conditions for spinors which ensure a well-posed index problem (case (I) in fact). Essentially their non-local boundary conditions stipulate the spectrum of the bulk modes in terms of the discrete spectrum of the compact boundary manifold (see [14,16,17,18,19] for further details). Similarly, a quantity for the non-standard supersymmetry can be obtained by defining the regularized expression index f = lim β→0 Tr (−1) F e −βK .…”