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Ponzano-Regge model revisited II: Equivalence with Chern-Simons

Laurent Freidel, David Louapre

TL;DR

The paper builds a rigorous link between the gauge-fixed Ponzano-Regge model in 3D gravity and Chern-Simons theory by showing that the PR partition function can be recast as a Reshetikhin-Turaev evaluation of a chain-mail link built from the quantum double ${\mathcal{D}}(SU(2))$, despite the noncompactness of the underlying quantum group. It formalizes a gauge-fixing procedure that yields a well-defined volume of the flat connection moduli space and proves triangulation and gauge-independence via Pachner moves and Bianchi identities. It extends the correspondence to observables, including Wilson lines and spinning particles, by translating their insertions into chain-mail link evaluations; this provides a cohesive framework connecting spin-foam quantization with Chern-Simons quantization in 3D gravity without a cosmological constant. The results establish a robust, calculable bridge between discrete gravity amplitudes and topological quantum field theory techniques with potential implications for noncompact gauge groups and quantum geometry.

Abstract

We provide a mathematical definition of the gauge fixed Ponzano-Regge model showing that it gives a measure on the space of flat connections whose volume is well defined. We then show that the Ponzano-Regge model can be equivalently expressed as Reshetikhin-Turaev evaluation of a colored chain mail link based on D(SU(2)): a non compact quantum group being the Drinfeld double of SU(2) and a deformation of the Poincare algebra. This proves the equivalence between spin foam quantization and Chern-Simons quantization of three dimensional gravity without cosmological constant. We extend this correspondence to the computation of expectation value of physical observables and insertion of particles.

Ponzano-Regge model revisited II: Equivalence with Chern-Simons

TL;DR

The paper builds a rigorous link between the gauge-fixed Ponzano-Regge model in 3D gravity and Chern-Simons theory by showing that the PR partition function can be recast as a Reshetikhin-Turaev evaluation of a chain-mail link built from the quantum double , despite the noncompactness of the underlying quantum group. It formalizes a gauge-fixing procedure that yields a well-defined volume of the flat connection moduli space and proves triangulation and gauge-independence via Pachner moves and Bianchi identities. It extends the correspondence to observables, including Wilson lines and spinning particles, by translating their insertions into chain-mail link evaluations; this provides a cohesive framework connecting spin-foam quantization with Chern-Simons quantization in 3D gravity without a cosmological constant. The results establish a robust, calculable bridge between discrete gravity amplitudes and topological quantum field theory techniques with potential implications for noncompact gauge groups and quantum geometry.

Abstract

We provide a mathematical definition of the gauge fixed Ponzano-Regge model showing that it gives a measure on the space of flat connections whose volume is well defined. We then show that the Ponzano-Regge model can be equivalently expressed as Reshetikhin-Turaev evaluation of a colored chain mail link based on D(SU(2)): a non compact quantum group being the Drinfeld double of SU(2) and a deformation of the Poincare algebra. This proves the equivalence between spin foam quantization and Chern-Simons quantization of three dimensional gravity without cosmological constant. We extend this correspondence to the computation of expectation value of physical observables and insertion of particles.

Paper Structure

This paper contains 40 sections, 22 theorems, 149 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The gauge fixed Ponzano-Regge invariant
  3. Definitions and notations
  4. The invariance theorem
  5. State-sum expression of the Ponzano-Regge invariant
  6. Chain-mail invariant using ${\mathcal{D}}(\mathrm{SU}(2))$
  7. Construction of the chain-mail link $L_\Delta$
  8. Closed manifold
  9. Open manifold
  10. ${\mathcal{D}}(\mathrm{SU}(2))$ and its Reshetikhin-Turaev evaluation
  11. Definition of the chain-mail invariant for ${\mathcal{D}}(\mathrm{SU}(2))$
  12. Main theorem
  13. Properties of the chain-mail construction
  14. Sliding, killing and fusion properties
  15. Insertion of Wilson lines and Hilbert space
  16. ...and 25 more sections

Key Result

Lemma 1

and where the sum is over all connected components of $\partial { M}$if there is no such component the RHS of (diffg) is $0$ and $g_{\Sigma_i}>0$ denotes the genus of each connected component $\Sigma_i \subset \partial M$. If there is no such component the RHS of (diffg) should be $0$

Figures (9)

  • Figure 1: Dual complex around a tetrahedra and a boundary triangle.
  • Figure 2: Construction of chain-mail link around an edge and resulting picture for a tetrahedra.
  • Figure 3: Construction of the chain-mail link near the boundary.
  • Figure 4: The Chain mail around a marked vertex of $\daleth$
  • Figure 5: Piece of link coming from a vertex of $\Delta$
  • ...and 4 more figures

Theorems & Definitions (33)

  • Lemma 1
  • Definition 1: Gauge fixed Ponzano-Regge invariant
  • Theorem 1
  • Definition 2
  • Proposition 1: Ponzano-Regge invariant as a state sum
  • Definition 3: Diagonal endomorphism and reduced trace
  • Definition 4: Amplitude for a colored link
  • Definition 5: Chain-mail invariant
  • Theorem 2
  • Proposition 2
  • ...and 23 more