Spectral asymmetry, supersymmetry and the equivariant Riemann-Roch defect

TL;DR

This paper explores the relationship between equivariant Riemann-Roch defects and supersymmetry, introducing a new partitioning of the $L^{2}$-spinor space that simplifies the computation of invariants.

Contribution

It presents a novel threefold partitioning of the $L^{2}$-spinor space and defines new equivariant invariants that are easier to compute, linking algebraic, analytic, and physical perspectives.

Findings

Identifies supersymmetric cancellations called lifted supersymmetry on the cone.

Defines complex equivariant $\xi_T$ and $\xi_T$ invariants equivalent to traditional ones.

Connects algebraic and analytic descriptions of Riemann-Roch defects and supersymmetry.

Abstract

We investigate the relationship between two interpretations of equivariant Riemann-Roch defects of complex spaces with conic singularities; as (i) equivariant $\eta_{T}$ and $\xi_{T}$ invariants, and as (ii) supertraces over local cohomology groups. This leads to a novel threefold partitioning of the $L^{2}$-spinor space on the link and a corresponding splitting of $\xi_{T}$. Two partitions correspond to cohomological contributions coming from $\bar\partial$-Neumann and $\bar\partial$-Dirichlet operators on the cone, while the third partition makes no contribution to the equivariant index defect, which we show is due to supersymmetric cancellations on the cone that we call lifted supersymmetry. We use this to define complex equivariant $\xi_T$ and $\eta_T$ invariants, which are equivalent to the usual invariants but are easier to compute. We highlight connections to related algebraic and analytic descriptions of Riemann-Roch defects in the literature, both at the level of numbers and their categorifications, and explore connections to existing notions of supersymmetric cancellations in physics and mathematics.