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Spectral asymmetry, supersymmetry and the equivariant Riemann-Roch defect

Gayana Jayasinghe, Alex R. Taylor, Xinran Yu

Abstract

We investigate the relationship between two interpretations of equivariant Riemann-Roch defects of complex spaces with conic singularities; as (i) equivariant $η_{T}$ and $ξ_{T}$ 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 $ξ_{T}$. 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 $ξ_T$ and $η_T$ 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.

Spectral asymmetry, supersymmetry and the equivariant Riemann-Roch defect

Abstract

We investigate the relationship between two interpretations of equivariant Riemann-Roch defects of complex spaces with conic singularities; as (i) equivariant and invariants, and as (ii) supertraces over local cohomology groups. This leads to a novel threefold partitioning of the -spinor space on the link and a corresponding splitting of . Two partitions correspond to cohomological contributions coming from -Neumann and -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 and 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.

Paper Structure

This paper contains 17 sections, 7 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Wedge Geometry and Dirac Operators
  3. Wedge geometry and Dirac operators
  4. Twisted Dolbeault complexes and domains
  5. Rolled-up extension and generalized boundary conditions
  6. Main results
  7. Review of $\eta$ and $\xi$ invariants
  8. Complex equivariant $\xi$ function
  9. Spectral decomposition theorem
  10. Equivariant index defect
  11. Examples and special cases
  12. $\eta$ invariants for the spin Dirac operator on round spheres
  13. Orbifolds
  14. Context and conclusions
  15. Supersymmetric counting and different categorifications
  16. ...and 2 more sections

Key Result

Theorem A

Assume the spectral Witt condition for the induced Dirac operator on $Z$. Then:

Theorems & Definitions (23)

  • Theorem A: Spectral decomposition and vanishing of the third sector
  • Theorem B
  • Remark 2.1: anti-commutation
  • Example 2.2
  • Example 2.3: Dolbeault complex on a complex cone
  • Remark 3.1
  • Lemma 3.2: Boundary conditions and the $\beta$-splitting
  • Lemma 3.3
  • Theorem 3.4
  • Example 3.5
  • ...and 13 more