Lattice Gauge Theories  

In General > s.a. non-commutative field theory; quantum gauge theory.
* History: First proposed by K Wilson in 1973 in order to explain quark confinement.
* Idea: Given a matter field φ in an N-dimensional representation of a gauge group G, one assigns a φ(x) to each lattice site and a U(x, x') in G to each (arbitrarily) oriented link; U corresponds to the holonomy exp( ie xx' Aa dxa).
* Action: The sum of a Wilson action Sg for U and a term Sm for the field φ

S:= βgplaquettes Ep + βmi,j φi Uij φj ,

where β:= 2N/g2; The energy for a plaquette (the smallest possible Wilson loop) along directions i, k is

EP:= 1 – (1/2N) tr (UP + UP) → (g2/N) ∑l a4 (Fikl)2      (continuum limit) ,

UP:= U1 U2 U3U4, and l labels the Lie algebra generators.
* Uses: Used to "show" confinement in non-abelian gauge theories, although there are difficulties coming from the fact that one encounters phase transitions going to the weak coupling limit (which corresponds to the continuum).

Specific Theories > s.a. graphs in physics [QED]; lattice QCD; QED and modified QED [Schwinger Model]; yang-mills gauge theory [chaos].
@ Electromagnetism: Aroca & Fort PLB(94), et al PLB(94) [Lagrangian loop formulation]; Orthuber qp/03; He & Teixeira PLA(05) [polyhedral complex, degrees of freedom], PLA(06) [geometric, Galerkin duality]; Di Bartolo et al MPLA(05) [Wilson loops]; Giuliani et al AP(12) [honeycomb lattice gauge theory for graphene]; > s.a. light [propagating on a lattice].
@ SU(2): Bloch et al NPB(04) [propagators and coupling]; Golterman & Shamir NPPS(05)hl/04 [chiral]; Gutbrod NPB(05) [gauge singularities].
@ SU(n): Bringoltz & Teper PLB(05)hl [bulk thermodynamic properties].
@ Standard model: Preparata & Xue PLB(91) [electroweak]; Creutz et al PLB(97); Zubkov PRD(10)-a1008 [electroweak, continuum limit].
@ Supersymmetric: Bietenholz MPLA(99) [Wess-Zumino]; Fujikawa NPB(02)ht [Leibniz rule]; Kaplan NPPS(03)hl/02, et al JHEP(03)hl/02; Itoh et al JHEP(03); Feo MPLA(04); Catterall et al PRP(09)-a0903; Joseph a1409-conf [with matter fields]; Catterall & Schaich JHEP(15)-a1505 [N = 4 supersymmetric Yang-Mills theory]; Bergner & Catterall a1603-IJMPA [rev]; > s.a. supersymmetric theories.
@ Other theories: Wadia pr(79)-a1212 [3D U(N) lattice gauge theory]; Bodwin PRD(96) [chiral gauge theories]; Kawamoto et al NPB(00)ht/99 [BF]; Larsson mp/02, Wise gq/05 [p-forms]; Wiese AdP(13)-a1305 [and ultracold quantum gases]; Wetterich NPB(13) [scalar lattice gauge theory]; Lipstein & Reid-Edwards JHEP(14)-a1404 [2-form gauge field, lattice gerbe theory].
> Other: see connection representation and loop quantum gravity.

References > s.a. lattice field theory; monopoles; topological defects; Tensor Networks; Wilson Loops.
@ Intros / reviews: Hasenfratz & Hasenfratz ARNPS(85); Montvay & Münster 94; Sharpe hl/98-conf; Münster & Walzl hl/00-ln; Wilson NPPS(05)hl/04 [history]; Oeckl 05; Rothe 12.
@ Loop representation and states: Brügmann PRD(91); Aroca et al PRD(96)ht [path integral]; Mathur PLB(06)hl/05, NPB(07).
@ Approches, formulations: Kogut & Susskind PRD(75) [Hamiltonian formulation]; Loll NPB(92) [variables, constraints]; Milton NPPS(97)hl/96 [alternative approach]; Berges et al PRD(07)hl/06 [real time – Lorentzian]; Zohar & Burrello PRD(15)-a1409 [quantum simulations]; Vairinhos & de Forcrand JHEP(14)-a1409 [with link variables integrated out]; Martinez et al a1605 [on a few-qubit quantum computer].
@ At finite T: Fodor & Katz PLB(02) [finite chemical potential]; Meyer NPB(08) [sum rules].
@ Continuum limit: Gross CMP(83) [3D U(1) theory]; McIntosh & Hollenberg PLB(02); Thiemann CQG(01)ht/00.
@ Simplicial lattice: Rajeev ht/04-conf [2+1 Yang-Mills]; Christiansen & Halvorsen JMP(12)-a1006 [gauge-invariant discrete action].
@ Random lattice: Christ et al NPB(82), NPB(82), NPB(82); Itzykson in(84), & Drouffe 89; Burda et al PRD(99)hl, NPPS(00)hl/99 [fermions].
@ Other lattices: Chodos PRD(78) [dynamical structure]; Andersen a1210 [Lorentz-covariant lattice graph].
@ Duality: Oeckl & Pfeiffer NPB(01)ht/00 [and spin-foam models]; Grosse & Schlesinger IJTP(01) [categorical methods]; Mathur & Sreeraj a1604 [SU(N) lattice theory and dual SU(N) spin model]; Riello a1706 [self-dual phase space].
@ Symmetry-preserving coarse-graining schemes: Tagliacozzo & Vidal PRB(11)-a1007; Bahr et al NJP(11)-a1011 [and linearized gravity].
@ Related topics: Boulatov CMP(97) [deformation]; Ma MPLA(00) [gluon propagator]; Burgio et al NPB(00) [\(\cal H\)phys]; Caselle IJMPA(00) [and AdS-cft]; Adams NPB(02)hl/01 [fermionic topological charge], NPB(02) [space of lattice fields]; de Forcrand & Jahn NPB(03) [SO(3) vs SU(2)]; Golterman & Shamir PRD(03)hl, NPPS(04)hl/03 [localization]; Silva & Oliveira NPB(04) [Gribov copies]; Stannigel et al PRL(14)-a1308 [generating constrained dynamics via the Zeno effect using engineered classical noise]; > s.a. entanglement entropy; Tomboulis-Yaffe Inequality.

Generalizations > s.a. spin networks [gauge networks].
* Tensor categories: The role of the gauge group is played by a tensor category, the admissible type (spherical, ribbon, symmetric) depending 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.
@ Tensor categories: Oeckl JGP(03)ht/01.


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