Abstract
We give a rigorous construction of the path integral in {mathcal {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.
Highlights
Chern character for Fredholm modules over differential graded algebras [25]. Applying this construction to the dg algebra (X ) of differential forms on a compact spin manifold, this is a functional ChD on the cyclic chain complex of (X ), which is given by a formula closely resembling that of the JLO-cocycle [30] for Connes’ non-commutative Chern character [14,15], but with further correction terms coming from the fact that the action of (X ) by Clifford multiplication on the Hilbert space of L2-sections of the spinor bundle is not multiplicative; see formula (3.13) below
Over the ordinary smooth loop space locally convex topology on (LX), where S is the energy functional and ω is the canonical two-form on LX (see (2.7) below), exponentiated in the exterior algebra
The task of giving a rigorous construction of this supersymmetric path integral has received a lot of attention ever since Atiyah [3] and Bismut [8–10] used it formally as a tool to give a “proof” of the Atiyah–Singer index theorem for twisted Dirac oprators
Summary
Chern character for Fredholm modules over differential graded algebras [25]. Applying this construction to the dg algebra (X ) of differential forms on a compact spin manifold, this is a functional ChD on the cyclic chain complex of (X ), which is given by a formula closely resembling that of the JLO-cocycle [30] for Connes’ non-commutative Chern character [14,15], but with further correction terms coming from the fact that the action of (X ) by Clifford multiplication on the Hilbert space of L2-sections of the spinor bundle is not multiplicative; see formula (3.13) below. It turns out that the Chern character vanishes on the kernel of the iterated integral map and can be pushed forward to a functional ρ!ChD on int(LX ).By the properties of the Chern character, it is equivariantly closed (this is the supersymmetry property) and satisfies a localization formula; see (3.16) below. In order to make sense of the formal expression on the right hand side of (1.1), in a first step, one has to make sense of the top degree part, or Berezinian of the differential form e−ω ∧ θ for any iterated integral θ This is clearly problematic by infinite-dimensionality of the loop space. After recalling some basic facts regarding loop space differential forms and spin geometry, we define the bar complex, which is the home for both the iterated integral map and the Chern character ChD, which are introduced next.
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