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Globalization of perturbative Chern-Simons theory on the moduli space of flat connections in the BV formalism

Pavel Mnev, Konstantin Wernli

TL;DR

This work develops a BV-formalism framework for perturbative Chern-Simons theory around non-acyclic flat connections on a closed 3-manifold, organizing the analysis via the smooth irreducible moduli stratum $\mathcal M'$. It introduces desynchronized perturbative data and a Gauss-Manin/Grothendieck-connection formalism to achieve horizontality across families of flat connections and gauge-fixing choices. By extending the perturbative partition function to nonhomogeneous forms and implementing metric-renormalization, the authors construct a global volume form $Z^{\mathrm{glob}}$ on $\mathcal M'$, whose cohomology class is metric-independent and yields a framed 3-manifold invariant upon integration over components. The theory is supplemented by a detailed formal geometry of the moduli space, a perturbative-dQME framework, and a motivating $\\Sigma\times S^1$ example that aligns with Reshetikhin–Turaev invariants in the large level limit. Overall, the paper provides a rigorous route from local perturbative data to global, metric-insensitive invariants arising from the moduli space of flat connections in the BV formalism.

Abstract

We study the perturbative path integral of Chern-Simons theory (the effective BV action on zero-modes) in Lorenz gauge, expanded around a (possibly non-acyclic) flat connection, as a family over the smooth irreducible stratum $\mathcal{M}' \subset \mathcal{M}$ of the moduli space of flat connections. We prove that it is horizontal with respect to the Grothendieck connection up to a BV-exact term. From it, we construct a volume form on $\mathcal{M'}$ - the "global partition function" - whose cohomology class is independent of the metric, and so is a 3-manifold invariant. As an element of the construction, we construct an extension of the perturbative partition function to a nonhomogeneous form on the space of triples $(A,A',g)$ consisting of (1) a "kinetic" flat connection $A$ around which Chern-Simons action is expanded, (2) a "gauge-fixing" flat connection $A'$, (3) a metric $g$. This extension is horizontal with respect to an appropriate Gauss-Manin superconnection (which involves the BV operator as a degree zero component).

Globalization of perturbative Chern-Simons theory on the moduli space of flat connections in the BV formalism

TL;DR

This work develops a BV-formalism framework for perturbative Chern-Simons theory around non-acyclic flat connections on a closed 3-manifold, organizing the analysis via the smooth irreducible moduli stratum . It introduces desynchronized perturbative data and a Gauss-Manin/Grothendieck-connection formalism to achieve horizontality across families of flat connections and gauge-fixing choices. By extending the perturbative partition function to nonhomogeneous forms and implementing metric-renormalization, the authors construct a global volume form on , whose cohomology class is metric-independent and yields a framed 3-manifold invariant upon integration over components. The theory is supplemented by a detailed formal geometry of the moduli space, a perturbative-dQME framework, and a motivating example that aligns with Reshetikhin–Turaev invariants in the large level limit. Overall, the paper provides a rigorous route from local perturbative data to global, metric-insensitive invariants arising from the moduli space of flat connections in the BV formalism.

Abstract

We study the perturbative path integral of Chern-Simons theory (the effective BV action on zero-modes) in Lorenz gauge, expanded around a (possibly non-acyclic) flat connection, as a family over the smooth irreducible stratum of the moduli space of flat connections. We prove that it is horizontal with respect to the Grothendieck connection up to a BV-exact term. From it, we construct a volume form on - the "global partition function" - whose cohomology class is independent of the metric, and so is a 3-manifold invariant. As an element of the construction, we construct an extension of the perturbative partition function to a nonhomogeneous form on the space of triples consisting of (1) a "kinetic" flat connection around which Chern-Simons action is expanded, (2) a "gauge-fixing" flat connection , (3) a metric . This extension is horizontal with respect to an appropriate Gauss-Manin superconnection (which involves the BV operator as a degree zero component).
Paper Structure (70 sections, 63 theorems, 330 equations, 15 figures)

This paper contains 70 sections, 63 theorems, 330 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Perturbative Chern-Simons partition function at a non-acyclic flat connection
  3. Path integral heuristics
  4. Mathematical definition of the perturbative partition function
  5. Aside: BV pushforward perspective
  6. $Z$ as a family over the moduli space of flat connections.
  7. The global partition function
  8. Dependence of the perturbative partition function on the flat connection: horizontality of $Z$, recovering $Z$ from $Z^\mathrm{glob}$
  9. The sum-over-trees formal exponential map on $\mathcal{M}'$.
  10. Main result 1
  11. "Desynchronized" Chern-Simons partition function -- main result 2
  12. Metric (in)dependence of the global partition function -- main result 3
  13. Extension of the partition function to a nonhomogeneous form in $A,A',g$ -- main result 4
  14. Motivating example: $\Sigma \times S^1$
  15. Comparison to literature and historical remarks
  16. ...and 55 more sections

Key Result

Theorem 1.3

This is Proposition prop ulZ finite horizontality, Corollary cor ulZ nabla^G horizontality and Corollary cor: Tphi^* Zglob=Z+Delta(...) put together.

Figures (15)

  • Figure 1: Desynchronized partition function $Z$ is a section of the bundle of formal half-densities on $H_{A}[1]$ over a neighborhood of the diagonal in $\mathrm{FC}'\times \mathrm{FC}'$. Under harmonic shifts of $A$ the section is $\widetilde{\nabla}^G$-horizontal, shifts of $A'$ change $Z$ by a BV-exact term.
  • Figure 2: Trees with labeling defining $\lambda_T$
  • Figure 3: Trees with the labeling defining $\mu_T$
  • Figure 4: Vertical and horizontal parallel transport to the diagonal on $\mathrm{FC}'\times \mathrm{FC}'$ (Proposition \ref{['prop 3.5']} (\ref{['prop 3.5 (c)']})).
  • Figure 5: A Feynman graph $\Gamma$ with $X$- and $Y$-subtrees and the resulting $\Gamma'$ and $\Gamma"$ graphs. $\mathsf{a}$-paths are shown in red; edges belonging to several $\mathsf{a}$ paths are thick red edges.
  • ...and 10 more figures

Theorems & Definitions (152)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1: Cattaneo2014
  • Theorem 2.3
  • Proposition 2.4
  • ...and 142 more