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Quantum deformation of two four-dimensional spin foam models

Winston J. Fairbairn, Catherine Meusburger

TL;DR

The paper constructs two fully q-deformed four-dimensional EPRL spin foam models, one Lorentzian (via the quantum Lorentz group) and one Euclidean (via tilting modules of $U_q^{(\mathrm{res})}(\mathfrak{su}(2))$ at root of unity). It defines quantum EPRL intertwiners using Haar measures on the quantum Lorentz group (Lorentzian) and a modular-category approach (Euclidean), proving convergence of intertwiners and 4-simplex amplitudes in both cases. The Lorentzian amplitude is shown to converge and to transform nontrivially under braiding, while the Euclidean amplitude is finite and decomposes into quantum 15j symbols, reflecting a finite fusion structure. The results suggest q-deformation acts as an infrared regulator for 4D gravity spin foams, with the deformation parameter $q$ plausibly linked to a positive cosmological constant through a cosmological length scale, and they open avenues for asymptotic and coherent-state analyses in the q-deformed setting.

Abstract

We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the EPRL model. The q-deformed models are based on the representation theory of two copies of U_q(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.

Quantum deformation of two four-dimensional spin foam models

TL;DR

The paper constructs two fully q-deformed four-dimensional EPRL spin foam models, one Lorentzian (via the quantum Lorentz group) and one Euclidean (via tilting modules of at root of unity). It defines quantum EPRL intertwiners using Haar measures on the quantum Lorentz group (Lorentzian) and a modular-category approach (Euclidean), proving convergence of intertwiners and 4-simplex amplitudes in both cases. The Lorentzian amplitude is shown to converge and to transform nontrivially under braiding, while the Euclidean amplitude is finite and decomposes into quantum 15j symbols, reflecting a finite fusion structure. The results suggest q-deformation acts as an infrared regulator for 4D gravity spin foams, with the deformation parameter plausibly linked to a positive cosmological constant through a cosmological length scale, and they open avenues for asymptotic and coherent-state analyses in the q-deformed setting.

Abstract

We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the EPRL model. The q-deformed models are based on the representation theory of two copies of U_q(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.

Paper Structure

This paper contains 34 sections, 6 theorems, 168 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Lorentzian model
  3. The quantum Lorentz group
  4. Hopf algebra structure
  5. The Hopf algebra $U_q(\mathfrak{su}(2))$.
  6. Representation theory of $U_q(\mathfrak{su}(2))$
  7. The Hopf algebra $F_q(\mathrm{SU}(2))$.
  8. The quantum Lorentz group $D(U_q(\mathfrak{su}(2)))$.
  9. Irreducible representations, duals and $R$-matrix
  10. Irreducible representations of the quantum Lorentz group
  11. Algebra of functions on the quantum Lorentz group.
  12. Universal $R$-matrix and braiding.
  13. The quantum EPRL intertwiner
  14. Quantum EPRL representations
  15. Quantum EPRL intertwiners
  16. ...and 19 more sections

Key Result

Theorem 2.3

Consider the $4$-tuple $K=(K_1,...,K_4)$ of irreducible representations of $U_q(\mathfrak{su}(2))$ associated with parameters in the label set $K_i\in \mathcal{L}$ and denote by $\alpha = (\alpha_1(K_1), ..., \alpha_4(K_4))$ the corresponding $4$-tuple of EPRL representations. Let $\lambda_{K,N}$ be This series converges absolutely and defines an element of $\mathrm{Hom}_{D(U_q(\mathfrak{su}(2)))}

Figures (4)

  • Figure 1: Diagram components: a) Clebsch-Gordan intertwiner $C_{K}^{IJ}: V_K\rightarrow V_I\otimes V_J$ for $U_q(\mathfrak{su}(2))$. b) Inclusion map $f^\alpha_K: V_K\rightarrow V_{\alpha(K)}$. c) Intertwiner $\phi_\alpha: V_\alpha\rightarrow V_\alpha^*$. d) Bilinear form $\beta_\alpha: V_\alpha\otimes V_\alpha\rightarrow \mathbb{C}$. e) Braiding $c_{\alpha_2,\alpha_1}: V_{\alpha_2}\otimes V_{\alpha_1}\rightarrow V_{\alpha_1}\otimes V_{\alpha_2}$.
  • Figure 2: $\quad$ a) EPRL tensor $\psi^\alpha\in\mathcal{L}( \mathbb{C},\otimes_{i=1}^4 V_{\alpha_i})\otimes F_q(SL(2,\mathbb{C})_\mathbb{R})$, b) Short notation for the EPRL tensor $\psi^\alpha$. c) EPRL intertwiner $\iota_\alpha:\otimes_{i=1}^4 V_{\alpha_i}\rightarrow\mathbb{C}$, d) Short notation for the EPRL intertwiner $\iota_\alpha$.
  • Figure 3: Diagram for the four-simplex amplitude $ev(\Gamma)$. The oriented lines carry EPRL representations. The solid black circles represent EPRL tensors, and the white circles denote the bilinear forms $\beta_\alpha$ on the representation spaces $V_\alpha$.
  • Figure 4: Diagram for the four-simplex amplitude $ev(\Gamma)$ with the labeling used in the proof of Theorem \ref{['ampconv']}.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 11 more