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String Connections and Chern-Simons Theory

Konrad Waldorf

TL;DR

<3-5 sentence high-level summary> We develop a finite-dimensional smooth reformulation of string structures on spin bundles as trivializations of the Chern-Simons 2-gerbe, CS_P. String connections are defined as compatible connections on these trivializations, tying into Chern-Simons theory and degree-three differential cohomology. The main results show that geometric string structures form a 2-category, with isomorphism classes parameterized by differential cohomology $oldsymbol{reve H}^3(M,Z)$; every string structure admits a string connection and the space of such connections is an affine space. The framework clarifies the relation to Stolz–Teichner and provides a finite-dimensional, smooth model for extended CS theory within bundle 2-gerbe theory.

Abstract

We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string connections: it enables us to prove that every string structure admits a string connection and that the possible choices form an affine space. Further we discover a new relation between string connections, 3-forms on the base manifold, and degree three differential cohomology. We also discuss in detail the relation between our formulation of string connections and the original version of Stolz and Teichner.

String Connections and Chern-Simons Theory

TL;DR

<3-5 sentence high-level summary> We develop a finite-dimensional smooth reformulation of string structures on spin bundles as trivializations of the Chern-Simons 2-gerbe, CS_P. String connections are defined as compatible connections on these trivializations, tying into Chern-Simons theory and degree-three differential cohomology. The main results show that geometric string structures form a 2-category, with isomorphism classes parameterized by differential cohomology ; every string structure admits a string connection and the space of such connections is an affine space. The framework clarifies the relation to Stolz–Teichner and provides a finite-dimensional, smooth model for extended CS theory within bundle 2-gerbe theory.

Abstract

We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string connections: it enables us to prove that every string structure admits a string connection and that the possible choices form an affine space. Further we discover a new relation between string connections, 3-forms on the base manifold, and degree three differential cohomology. We also discuss in detail the relation between our formulation of string connections and the original version of Stolz and Teichner.

Paper Structure

This paper contains 26 sections, 35 theorems, 97 equations, 4 figures.

Table of Contents

  1. Acknowledgements.
  2. Summary of the Results
  3. String Structures as Trivializations
  4. String Connections as Connections on Trivializations
  5. Results on String Connections
  6. Bundle 2-Gerbes and their Trivializations
  7. The Chern-Simons Bundle 2-Gerbe
  8. Trivializations and String Structures
  9. Connections on Bundle 2-Gerbes
  10. Connections on Chern-Simons 2-Gerbes
  11. Connections and Geometric String Structures
  12. Existence and Classification of compatible Connections on Trivializations
  13. Trivializations of Chern-Simons Theory
  14. Chern-Simons Theory as an Extended 3d TFT
  15. Models for $n$-Bundles with Connection
  16. ...and 11 more sections

Key Result

Theorem 2

Let $\pi:P \!\xymatrix@C=0.5cm{\ar[r] &} M$ be a principal ${\mathrm{Spin}}(n)$-bundle over $M$.

Figures (4)

  • Figure 1: The pentagon axiom for a bundle gerbe product $\mu$. It is an equality between transformations over $Y^{[5]}$.
  • Figure 2: The compatibility condition between the associator $\mu$ of a bundle 2-gerbe and the transformation $\sigma$ of a trivialization. It is an equality of transformations over $Y^{[4]}$.
  • Figure 4: Sections of $n$-bundles with connection.
  • Figure 5: The compatibility between the transformations $\sigma$ and $\sigma'$ of two trivializations $\mathbb{T}$ and $\mathbb{T}'$ and the transformation $\beta$ of a 1-morphism $\mathbb{B}=(\mathcal{B},\beta)$ between $\mathbb{T}$ and $\mathbb{T}'$. It is an equality of transformations over $Y^{[3]}$.

Theorems & Definitions (55)

  • Definition 1: redden1
  • Theorem 2: stolz1
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Corollary 6
  • Theorem 1
  • Definition 2
  • Theorem 3
  • Corollary 1
  • ...and 45 more