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Perturbative Chern-Simons invariants from non-acyclic flat connections

Pavel Mnev, Konstantin Wernli

TL;DR

The paper extends perturbative Chern-Simons invariants to non-acyclic flat connections by constructing a Chern-Simons volume form on the smooth irreducible component $\mathcal{M}'$ of the flat-connection moduli space and integrating it to produce a metric-independent invariant for framed 3-manifolds. It develops a BV/formal-geometry framework with an extended effective action on zero-modes and a sum over Feynman graphs, yielding a global object on $\mathcal{M}'$ whose cohomology class is invariant under metric variations. The first nontrivial degree term reveals explicit graph-based contributions ($\Phi_1$ to $\Phi_6$) and a relation to Ray–Singer torsion, connecting to the WRT asymptotics in favorable cases. Overall, the work provides a higher-loop, non-acyclic generalization of perturbative Chern-Simons theory, establishing a modular and geometric approach to invariants on non-acyclic loci and suggesting avenues for further refinement and applications.

Abstract

We give a short review of our construction of a higher-loop perturbative invariant of framed 3-manifolds, generalizing the perturbative Chern-Simons invariant of Witten-Axelrod-Singer, associated to an acyclic flat connection, to an invariant given by the integral of a certain "Chern-Simons volume form" over a smooth closed component of the moduli space of flat connections.

Perturbative Chern-Simons invariants from non-acyclic flat connections

TL;DR

The paper extends perturbative Chern-Simons invariants to non-acyclic flat connections by constructing a Chern-Simons volume form on the smooth irreducible component of the flat-connection moduli space and integrating it to produce a metric-independent invariant for framed 3-manifolds. It develops a BV/formal-geometry framework with an extended effective action on zero-modes and a sum over Feynman graphs, yielding a global object on whose cohomology class is invariant under metric variations. The first nontrivial degree term reveals explicit graph-based contributions ( to ) and a relation to Ray–Singer torsion, connecting to the WRT asymptotics in favorable cases. Overall, the work provides a higher-loop, non-acyclic generalization of perturbative Chern-Simons theory, establishing a modular and geometric approach to invariants on non-acyclic loci and suggesting avenues for further refinement and applications.

Abstract

We give a short review of our construction of a higher-loop perturbative invariant of framed 3-manifolds, generalizing the perturbative Chern-Simons invariant of Witten-Axelrod-Singer, associated to an acyclic flat connection, to an invariant given by the integral of a certain "Chern-Simons volume form" over a smooth closed component of the moduli space of flat connections.

Paper Structure

This paper contains 16 sections, 3 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The construction of the invariant
  3. The lowest (nontrivial) degree invariant
  4. Outline of proof of Theorem \ref{['main thm']}
  5. Discussion
  6. Formal Geometry on the moduli space of flat connections
  7. Elements of formal geometry
  8. Formal exponential maps
  9. Grothendieck connection
  10. Formal differential forms
  11. Formal half-densities
  12. Including parameters
  13. Formal geometry on $\mathcal{M}'$.
  14. Sum-over-trees formal exponential map
  15. A formula for the Grothendieck connection of $\underline{\varphi}^g$
  16. ...and 1 more sections

Key Result

Theorem 2.1

There exists a universal formal power seriesCoefficients of $c$ are (up to normalization) the anomalies $\beta_l$ appearing in AS94. in $\epsilon$ with coefficients depending only on the Lie algebra $\mathfrak{g}$ (but not on $M$) such that, for any connected component $\mathcal{M}'_\alpha$ of $\mathcal{M}'$ which is a closed smooth manifold, the formal power series does not depend on the metric

Figures (2)

  • Figure 1: Six graphs contributing to (\ref{['Lambda_1']}). An edge/leaf with $n\in \{0,1,2\}$ circle vertices on it corresponds to the term of degree $n$ in $\zeta$ in $\widehat{K}$, $\widehat{i}$, or in $\Theta$ (for graphs $\Gamma_5,\Gamma_6$). Dashed half-edges incident to circle vertices are decorated by $\zeta$ and solid leaves are decorated by $\mathsf{a}^2$. Dashed edges show Wick contractions between $\zeta$ and $\mathsf{a}^2$ in (\ref{['Y']}).
  • Figure 2: Four graphs contributing to (\ref{['Xi']}). The black vertex is decorated by $\lambda_{\delta g}$.

Theorems & Definitions (8)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Lemma A.1: CSglob
  • Lemma A.2