A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold

Abstract

We give a rigorous construction of the path integral in $${\mathcal {N}}=1/2$$ N = 1 / 2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler–Jones–Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Güneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah–Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaumé, Atiyah, Bismut and Witten.

A Short Proof of the Localization Formula for the Loop Space Chern Character of Spin Manifolds

“…This formula and its close relation to the Atiyah-Singer index theorem has sparked interest for more than 30 years (see [Ati85,AG85] and the introduction of [GL19] for further references). The path integral formula (0.1) has been finally given a rigorous interpretation for a certain class of differential forms θ in [HL17,GL19]; see also [Lud20]. Essentially, this is the class of iterated integrals, first considered by Chen [Che73] and later extended by Getzler, Jones and Petrack [GJP91] in order to contain the Bismut-Chern character forms first introduced by Bismut [Bis85].…”

“…Pulling back the integration functional I of [HL17] with the iterated integral map, one obtains a coclosed functional on the cyclic chain complex of Ω(M), which we denote by Ch D ; namely, it has then been observed in [GL19] that this functional can be viewed as a non-commutative Chern character associated to a Fredholm module over Ω(M) determined by the Dirac operator D on M, with a combinatorial formula similar to the JLO cocycle [JLO88].…”

“…The most remarkable feature of the loop space path integral I and its combinatorial counterpart Ch D is that they satisfy a localization formula of Duistermaat-Heckmann type, as though the loop space LM was a finite-dimensional manifold; see [HL17,Thm. 3.19] and [GL19,Thm.…”