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The role of p_1-structures in 3-dimensional Chern-Simons theories

Daniel S. Freed, Constantin Teleman

Abstract

Our recent paper~\cite{FST} with Claudia Scheimbauer uses the cobordism hypothesis to construct fully local Chern-Simons theories. Here we expose some physics motivations: Yang-Mills plus Chern-Simons in the bosonic case and the free Majorana-Weyl spinor field in the fermionic case. We also give expositions of tangential structures and invertible field theories, in particular the 'gravitational Chern-Simons theory' used by Witten to obtain topological field theories from the physical theories.

The role of p_1-structures in 3-dimensional Chern-Simons theories

Abstract

Our recent paper~\cite{FST} with Claudia Scheimbauer uses the cobordism hypothesis to construct fully local Chern-Simons theories. Here we expose some physics motivations: Yang-Mills plus Chern-Simons in the bosonic case and the free Majorana-Weyl spinor field in the fermionic case. We also give expositions of tangential structures and invertible field theories, in particular the 'gravitational Chern-Simons theory' used by Witten to obtain topological field theories from the physical theories.
Paper Structure (23 sections, 4 theorems, 65 equations)

This paper contains 23 sections, 4 theorems, 65 equations.

Table of Contents

  1. Yang--Mills $+$ Chern--Simons
  2. The physics origin of Chern--Simons theory
  3. The Spin variation
  4. A formula for the central charge
  5. Tangential structures
  6. Framings
  7. Orientations and Spin structures
  8. $(w_1,\,p_1)$-structures
  9. $(w_1,\,w_2, \,p_1)$-structures
  10. Maps of tangential structures
  11. Invertible field theories
  12. Some invertible topological field theories
  13. Framed theories
  14. Spin theories
  15. $(w_1,\,p_1)$-theories
  16. ...and 8 more sections

Key Result

Lemma 1

Let $\pi_n\colon\mathscr{X}_n\to B\!\mathop{\mathrm{O}}\nolimits_n$ be induced from $\pi\colon\mathscr{X}\to B\!\mathop{\mathrm{O}}\nolimits$, as in eq:1, and suppose $M$ is an $n$-manifold. Then there is a homeomorphism between the space of $\pi_n$-structures on $M$ and the space of $\pi$-structure

Theorems & Definitions (38)

  • Definition 1
  • Remark 1
  • Remark 2
  • Conjecture 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 2
  • Definition 3
  • Lemma 1
  • ...and 28 more