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On deformation quantization of the space of connections on a two manifold and Chern Simons Gauge Theory

Jonathan Weitsman

TL;DR

The paper develops an explicit deformation quantization of the space ${\mathcal A}(\Sigma)$ of connections by deriving a closed-form star product from three-dimensional Chern-Simons theory in axial gauge, tying the deformation to Wilson-loop observables and their Poisson structure. It shows that the Poisson bracket on Wilson loops reproduces Goldman’s bracket on the moduli space and provides a concrete star-product formula $W_C \star_h W_{C'}$ via $e^{\frac{h}{2}D}$ acting on translated curves, with detailed cases for $SU(2)$ and $GL(n)$ groups. The work connects deformation quantization to geometric quantization, TQFT, and skein theory, while highlighting gauge-invariance issues that prevent descent to moduli spaces and proposing avenues for resolving them, potentially informing Teichmüller theory and the volume conjecture. Overall, it offers explicit, computable tools linking knot invariants, moduli-space quantization, and the algebraic structure of observables in Chern-Simons theory.

Abstract

We use recent progress on Chern-Simons gauge theory in three dimensions [18] to give explicit, closed form formulas for the star product on some functions on the affine space ${\mathcal A}(Σ)$ of (smooth) connections on the trivialized principal $G$-bundle on a compact, oriented two manifold $Σ.$ These formulas give a close relation between knot invariants, such as the Kauffman bracket polynomial, and the Jones and HOMFLY polynomials, arising in Chern Simons gauge theory, and deformation quantization of ${\mathcal A}(Σ).$ This relation echoes the relation between the manifold invariants of Witten [20] and Reshetikhin-Turaev [16] and {\em geometric} quantization of this space (or its symplectic quotient by the action of the gauge group). In our case this relation arises from explicit algebraic formulas arising from the (mathematically well-defined) functional integrals of [18].

On deformation quantization of the space of connections on a two manifold and Chern Simons Gauge Theory

TL;DR

The paper develops an explicit deformation quantization of the space of connections by deriving a closed-form star product from three-dimensional Chern-Simons theory in axial gauge, tying the deformation to Wilson-loop observables and their Poisson structure. It shows that the Poisson bracket on Wilson loops reproduces Goldman’s bracket on the moduli space and provides a concrete star-product formula via acting on translated curves, with detailed cases for and groups. The work connects deformation quantization to geometric quantization, TQFT, and skein theory, while highlighting gauge-invariance issues that prevent descent to moduli spaces and proposing avenues for resolving them, potentially informing Teichmüller theory and the volume conjecture. Overall, it offers explicit, computable tools linking knot invariants, moduli-space quantization, and the algebraic structure of observables in Chern-Simons theory.

Abstract

We use recent progress on Chern-Simons gauge theory in three dimensions [18] to give explicit, closed form formulas for the star product on some functions on the affine space of (smooth) connections on the trivialized principal -bundle on a compact, oriented two manifold These formulas give a close relation between knot invariants, such as the Kauffman bracket polynomial, and the Jones and HOMFLY polynomials, arising in Chern Simons gauge theory, and deformation quantization of This relation echoes the relation between the manifold invariants of Witten [20] and Reshetikhin-Turaev [16] and {\em geometric} quantization of this space (or its symplectic quotient by the action of the gauge group). In our case this relation arises from explicit algebraic formulas arising from the (mathematically well-defined) functional integrals of [18].
Paper Structure (11 sections, 7 theorems, 81 equations, 2 figures)

This paper contains 11 sections, 7 theorems, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Poisson Bracket
  3. Chern Simons gauge theory in axial gauge on $\Sigma \times {\mathbb R}$
  4. Chern Simons gauge theory in three dimensions and the star product on the space of connections in two dimensions
  5. Explicit formulas for the star product
  6. Some Comments and Speculations: Relation to Geometric Quantization; Quantization of Moduli spaces
  7. Geometric Quantization, the star product on Kahler manifolds, and Topological Quantum Field Theory
  8. Gauge invariance, quantization of moduli spaces, and the Second Reidemeister move.
  9. Skein modules and quantized Teichmuller space; possible relation to the volume conjecture
  10. Some comments and speculations on a coproduct structure
  11. A geometric coproduct

Key Result

Theorem 1

For any two curves $C, C^\prime$ in $\Sigma$ intersecting transversally, for any $\epsilon > 0,$ and any $A \in {\mathcal{A}}(\Sigma),$ with similar formulas for the Poisson brackets of functions given by polynomials in Wilson loops.

Figures (2)

  • Figure 1: The composition of $T_f$ and $T_g$ acting on the Hilbert space associated to $\Sigma.$
  • Figure 2: The second Reidemeister move

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Definition 1.9
  • Theorem 3
  • Remark 2.2
  • Remark 2.14
  • Theorem 2.15: Nachbin nachbin
  • Remark 2.16: $SL(2,{\mathbb R})$ and $SL(2,{\mathbb C})$
  • Theorem 1
  • Theorem 2
  • ...and 3 more