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Supersymmetric field theories and generalized cohomology

Stephan Stolz, Peter Teichner

Abstract

This survey discusses our results and conjectures concerning supersymmetric field theories and their relationship to cohomology theories. A careful definition of supersymmetric Euclidean field theories is given, refining Segal's axioms for conformal field theories. We state and give an outline of the proof of various results relating field theories to cohomology theories.

Supersymmetric field theories and generalized cohomology

Abstract

This survey discusses our results and conjectures concerning supersymmetric field theories and their relationship to cohomology theories. A careful definition of supersymmetric Euclidean field theories is given, refining Segal's axioms for conformal field theories. We state and give an outline of the proof of various results relating field theories to cohomology theories.

Paper Structure

This paper contains 34 sections, 22 theorems, 114 equations, 4 figures.

Table of Contents

  1. Results and conjectures
  2. Field theories
  3. Field theories over a manifold
  4. Supersymmetric field theories
  5. Concordance classes of field theories
  6. Field theories of non-zero degree
  7. $2|1\text{- }\operatorname{EFT}$'s and topological modular forms
  8. Gauged field theories and equivariant cohomology
  9. Comparison with our 2004 survey
  10. Summary of Contents
  11. Geometric field theories
  12. Segal's definition of a conformal field theory
  13. Internal categories
  14. The internal Riemannian bordism category
  15. The internal category of vector spaces
  16. ...and 19 more sections

Key Result

Theorem 1.4

The groupoid $1 \text{- } \operatorname{TFT}(X)$ of $1$-dimensional topological field theories over a manifold $X$ is equivalent to groupoid of finite-dimensional vector bundles over $X$ with connections.

Figures (4)

  • Figure 1: An object $(Y,Y^c,Y^\pm)$ of ${2\text{- {\sf RBord}}}_0$
  • Figure 2: A Riemannian bordism (object of ${2\text{- {\sf RBord}}}_1$)
  • Figure 3: The object $K_\ell$ of $2\text{- {\sf EBord}}_0$
  • Figure 4: The object $C_{\ell,\tau}\in 2\text{- {\sf EBord}}_1(K_\ell,K_\ell)$

Theorems & Definitions (85)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: DST
  • Theorem 1.5: HKST
  • Theorem 1.6: HKST
  • Definition 1.7
  • Corollary 1.8: HKST
  • Theorem 1.9: HKST
  • Theorem 1.10: ST3,ST6
  • ...and 75 more