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Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group

Tudor Dimofte, Sergei Gukov, Jonatan Lenells, Don Zagier

Abstract

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of "arithmetic TQFT" and conjecture (with supporting numerical evidence) that SL(2,C) Chern-Simons theory is an example of such a theory.

Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group

Abstract

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of "arithmetic TQFT" and conjecture (with supporting numerical evidence) that SL(2,C) Chern-Simons theory is an example of such a theory.

Paper Structure

This paper contains 18 sections, 210 equations, 1 table.

Table of Contents

  1. Introduction and summary
  2. Perturbation theory around a non-trivial flat connection
  3. Feynman diagrams and arithmetic TQFT
  4. Quantization of ${{\cal M}}_{{\rm flat}} (G_{{\mathbb{C}}}, \Sigma)$
  5. Analytic continuation
  6. A hierarchy of differential equations
  7. Classical and quantum symmetries
  8. Brane quantization
  9. A state integral model for perturbative $SL(2,{\mathbb{C}})$ Chern-Simons theory
  10. Ideal triangulations of hyperbolic 3-manifolds
  11. Hikami's invariant
  12. Quantum dilogarithm
  13. A state integral model for $Z^{(\rho)} (M;\hbar)$
  14. Examples and computations
  15. Numerical verification of the arithmeticity conjecture
  16. ...and 3 more sections