Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants

Robert Oeckl

TL;DR

This work introduces a unified, diagrammatic framework for lattice gauge theory where the gauge 'group' is replaced by a semisimple tensor category, unifying ordinary LGT with quantum groups and supersymmetric generalizations. The partition function is computed as a sum over category-morphism diagrams attached to a cellular decomposition, with the weak coupling limit recovering BF-type state sums and known topological invariants. The formalism extends to manifolds with boundaries, yielding boundary spin networks and, in the topological limit, TQFTs; it also accommodates Wilson loop and spin-network observables. The approach recovers classical results (Ponzano–Regge, Turaev–Viro, Crane–Yetter) in appropriate limits and provides a flexible, coordinate-free language for exploring quantum gravity spin foams and quantum gauge theories on the lattice.

Abstract

We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold. In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension <=4) and with supersymmetric gauge group (in any dimension). Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover. The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to TQFTs with or without embedded spin networks.

Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants

TL;DR

This work introduces a unified, diagrammatic framework for lattice gauge theory where the gauge 'group' is replaced by a semisimple tensor category, unifying ordinary LGT with quantum groups and supersymmetric generalizations. The partition function is computed as a sum over category-morphism diagrams attached to a cellular decomposition, with the weak coupling limit recovering BF-type state sums and known topological invariants. The formalism extends to manifolds with boundaries, yielding boundary spin networks and, in the topological limit, TQFTs; it also accommodates Wilson loop and spin-network observables. The approach recovers classical results (Ponzano–Regge, Turaev–Viro, Crane–Yetter) in appropriate limits and provides a flexible, coordinate-free language for exploring quantum gravity spin foams and quantum gauge theories on the lattice.

Abstract

We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold. In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension <=4) and with supersymmetric gauge group (in any dimension). Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover. The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to TQFTs with or without embedded spin networks.

Paper Structure

This paper contains 32 sections, 16 theorems, 70 equations, 32 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Categories and Diagrams
  3. Monoidal Categories with Structure
  4. Diagrams and Isotopy Invariance
  5. Semisimplicity
  6. Representation Theory
  7. Groups
  8. Representation Categories
  9. Representative Functions
  10. Integration and Semisimplicity
  11. Hopf Algebras and Quantum Groups
  12. Representation Categories
  13. "Representative Functions"
  14. Integration and Semisimplicity
  15. The Partition Function
  16. ...and 17 more sections

Key Result

Lemma 2.4

In a pivotal category the identities $\mathop{\mathrm{tr}}\nolimits_-(\Phi)=\mathop{\mathrm{tr}}\nolimits_+(\Phi^*)$ and $\mathop{\mathrm{tr}}\nolimits_+(\Phi)=\mathop{\mathrm{tr}}\nolimits_-(\Phi^*)$ hold for any morphism $\Phi$.

Figures (32)

  • Figure 2.1: Elementary tangle diagrams and their assigned morphisms.
  • Figure 2.2: Coupon.
  • Figure 2.3: Trace property defining a spherical category.
  • Figure 2.4: Additional ribbon diagrams and their assigned morphisms.
  • Figure 2.5: The crossing $\psi_{V,W}$ in the symmetric case.
  • ...and 27 more figures

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 28 more