Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differential Models for Anderson Dual to Twisted $\mathrm{Spin}^c$-Bordism and Twisted Anomaly Map

Fei Han, Yuanchu Li

TL;DR

The paper develops differential refinements of twisted $ ext{Spin}^c$-bordism and its Anderson dual, constructing two parallel, equivalent models: a differential extension built from twisted de Rham complexes and currents, and a gerbe-theoretic framework using bundle gerbe modules to encode twists and Dirac-operator data. It then defines a differential twisted anomaly map from $igl( ext{K}^{0}(X, au^{-1})igr)$ to the twisted Anderson dual, realized via the twisted Chern character and reduced eta-invariants in the presence of a differential twist $ au$. The work extends Yamashita–Yonekura’s de Rham models to the twisted setting and proves the equivalence of the differential extensions, including a differential multiplication and pushforward, along with an explicit geometric realization via gerbes. The resulting twisted anomaly map provides a geometric bridge between differential twisted $K$-theory and invertible field theories, aligning with the Stolz–Teichner program and enabling index-theoretic computations for twisted 1|1-dimensional supersymmetric theories.

Abstract

We construct differential models for twisted $\mathrm{Spin}^c$-bordism and for its Anderson dual, and employ the latter to define a twisted anomaly map whose source is the differential twisted $K$-theory. Our differential model for the twisted Anderson dual follows the formalism developed in [YY23]. To connect these constructions with the geometric framework of the Atiyah-Singer index theory, we further present a gerbe-theoretic formulation of our models in terms of bundle gerbes and gerbe modules [Mur96] [BCMMS02]. Within this geometric setting, we define the twisted anomaly map \[ \widehatΦ_{\widehat{\mathcal{G}}}\colon \widehat{K}^{0}(X,\widehat{\mathcal{G}}^{-1}) \longrightarrow \bigl(\widehat{IΩ^{\mathrm{Spin}^c}_{\mathrm{dR}}}\bigr)^{n}(X,\widehat{\mathcal{G}}), \] whose construction naturally involves the reduced eta-invariant of Dirac operators acting on Clifford modules determined by the twisted data. Conceptually, this map is expected to encode the anomalies of twisted $1|1$-dimensional supersymmetric field theories, in accordance with the perspectives developed in [ST11] and [FH21].

Differential Models for Anderson Dual to Twisted $\mathrm{Spin}^c$-Bordism and Twisted Anomaly Map

TL;DR

The paper develops differential refinements of twisted -bordism and its Anderson dual, constructing two parallel, equivalent models: a differential extension built from twisted de Rham complexes and currents, and a gerbe-theoretic framework using bundle gerbe modules to encode twists and Dirac-operator data. It then defines a differential twisted anomaly map from to the twisted Anderson dual, realized via the twisted Chern character and reduced eta-invariants in the presence of a differential twist . The work extends Yamashita–Yonekura’s de Rham models to the twisted setting and proves the equivalence of the differential extensions, including a differential multiplication and pushforward, along with an explicit geometric realization via gerbes. The resulting twisted anomaly map provides a geometric bridge between differential twisted -theory and invertible field theories, aligning with the Stolz–Teichner program and enabling index-theoretic computations for twisted 1|1-dimensional supersymmetric theories.

Abstract

We construct differential models for twisted -bordism and for its Anderson dual, and employ the latter to define a twisted anomaly map whose source is the differential twisted -theory. Our differential model for the twisted Anderson dual follows the formalism developed in [YY23]. To connect these constructions with the geometric framework of the Atiyah-Singer index theory, we further present a gerbe-theoretic formulation of our models in terms of bundle gerbes and gerbe modules [Mur96] [BCMMS02]. Within this geometric setting, we define the twisted anomaly map whose construction naturally involves the reduced eta-invariant of Dirac operators acting on Clifford modules determined by the twisted data. Conceptually, this map is expected to encode the anomalies of twisted -dimensional supersymmetric field theories, in accordance with the perspectives developed in [ST11] and [FH21].

Paper Structure

This paper contains 30 sections, 18 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Brief review of twisted (co)homology and differential extensions
  5. Bundle of spectra and twisted (co)homology
  6. Differential extensions to twisted (co)homology theories
  7. Differential twisted $\mathrm{Spin}^c$-bordism and Anderson dual
  8. Review of twisted $\mathrm{Spin}^c$-bordism
  9. Differential twisted $\mathrm{Spin}^c$-structures
  10. Differential twisted $\mathrm{Spin}^c$-bordism
  11. Twisted de Rham complex with coefficients and its dual
  12. Twisted Chern-Weil construction and differential extension
  13. The model for differential extension of twisted $\mathrm{Spin}^c$-bordism
  14. Anderson dual to twisted $\mathrm{Spin}^c$-bordism and its differential model
  15. Review on Anderson dual and Yamashita-Yonekura's differential model
  16. ...and 15 more sections

Key Result

Proposition 2.1

For the twisted theories represented by the bundle of spectra $P_{\tau}(k)\to X$, there are natural spectral sequences where $L_q(X,P_{\tau}(k))$ is the local system with fiber $\pi_q(k)$ and monodromy induced by the principal $K(\mathbb{Z},2)$-bundle $P\to X$ and the $K(\mathbb{Z},2)$-action on $\pi_q(k)$.

Theorems & Definitions (59)

  • Conjecture 1.1: freed2021reflection
  • Conjecture 1.2: Segal-Stolz-Teichner ST11
  • Definition 1.1: Wang07
  • Definition 1.2: Definition \ref{['def: difftwiSpincStru']}
  • Definition 1.3: Definition \ref{['def: diffSpincBG']}
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1: Wang07
  • Definition 3.2
  • ...and 49 more