Gauge Field Theories |

**In General** > s.a. gauge
transformations and gauge choices;
history of physics; lorentz
group phenomenology; symmetry.

* __Motivation__: Make
a global symmetry into a local one (observers at different points can
choose independently); Masslessness of gauge particles related to
renormalizability (but see the Higgs mechanism); Can treat monopoles
without singularities in potentials; Geometric picture of fields obtained
using fiber bundle language.

* __History__: The
principle was introduced by Weyl; The fiber bundle picture appeared
in the late 1960s, but was accepted only around 1973.

* __Approaches__:
Modern mathematical formulations include ordinary differential geometry of
fiber bundles, compactified extra dimensions in Kaluza-Klein theories,
Grassmanian models, non-commutative geometry, and transitive Lie algebroids.

* __Idea__: In the
differential geometry approach, the basic objects are a semisimple (in
order for it to have a non-degenerate metric) Lie group *G*, with
Lie algebra *g*, and a principal *G*-bundle *P*
over spacetime; The variables are a *g*-valued connection 1-form
(i *e*) *A* on this principal fiber bundle (often used
interchangeably with a gauge potential, the pullback of the connection
1-form), and possibly coupled matter fields (cross-sections *φ* of
associated *G*-bundles); If (i *e*) *F* is the
curvature of the connection, and *D* its associated covariant
derivative, one field equation is the Bianchi identity,

*DF* := d*F* + [*A*,* F*] = 0 ;

Other field equations will depend on the form of the action chosen (careful,
*F* = d*A* + *A* ∧ *A* ≠ *DA* !).

@ __Texts and reviews__: Göckeler & Schücker 87;
Cheng & Li AJP(88)jul [RL];
Chan & Tsou 93;
Tsou ht/00-ln.

@ __Texts, and differential geometry__: Marathe & Martucci 92;
Naber 00, 11.

@ __Potentials generating the same fields__:
Majumdar & Sharatchandra PRD(01)ht/98.

**Line / Loop and Other Variables** > s.a. BF
theory; connection; Field
Line; holonomy; QCD;
quantum gauge theory; topological field theories.

@ __Wilson loops__:
Mandelstam AP(62);
Wu & Yang PRD(75),
PRD(76),
PRD(76);
Kozameh & Newman PRD(85) [differential holonomies and Yang-Mills equations];
Chan et al AP(86);
Gambini & Trias NPB(86);
Diakonov & Petrov PLB(89);
Bezerra & Letelier CQG(91)refs;
Rajeev & Turgut JMP(96)ht/95.

@ __Gauge-invariant__: Newman & Rovelli PRL(92) [lines of force];
Loll CQG(93)gq [inequalities on traces of holonomies];
Armand-Ugón et al PRD(94)ht/93 [loop variables];
Frittelli et al PRD(94) [Faraday lines];
Chechelashvili et al TMP(96)ht/95;
Ganor & Sonnenschein IJMPA(96)ht/95;
Haagensen ht/95,
et al NPB(96)ht/95;
Kijowski et al RPMP(87);
Zapata JMP(97)gq [graphs];
Faddeev & Niemi PRL(99)ht/98,
PLB(99)ht/98,
PLB(99)ht;
Blaschke et al ht/00 [topological invariants for QCD];
Orland PRD(04)ht;
Ferreira & Luchini a1109 [and global properties];
Wetterich a1710 [and flow equations].

@ __Fluxes__: Dzhunushaliev et al PLB(00) [flux tubes];
Freed et al AP(07)ht/06,
CMP(07) [non-commutativity];
> s.a. lattice gauge theory [flux and charge].

@ __Related topics__: Brambilla & Prosperi ht/94-conf [and potentials];
Watson PLB(94) [identities];
Gukov & Witten a0804 [surface operators];
Schroer FP(11)-a1012 [alternative setting, stringlike approach];
Ferreira & Luchini NPB(12) [integral formulation, in loop spaces];
Chung & Lu PRD(16)-a1609 [basis tensor fields];
> s.a. knots in physics;
Nicolai Map.

**Features, Techniques** > s.a. constrained
systems [including reduction];
fiber bundles.

* __Configuration space__:
The natural one is the moduli space of all gauge equivalence classes of
connections on a principal *G*-bundle over the spatial manifold Σ
(superspace) or connections over all such principal bundles over Σ (grand
superspace); > see connections.

* __Alice configurations__:
Fields in theories with disconnected groups such that the disconnectedness
has physical effects; > s.a. monopoles.

@ __With boundaries__: Śniatycki et al CMP(96);
Avramidi & Esposito CMP(99)ht/97,
gq/99-conf;
Ferrara & Frønsdal PLB(98)ht;
Díaz-Marín Sigma(15)-a1407 [*n*-dimensional abelian gauge fields, general-boundary formulation];
Geiller NPB(17)-a1703 [edge modes and corner ambiguities];
Gomes et al a1808 [unified geometric framework for boundary charges];
> s.a. quantum gauge theories.

@ __Measure__: Pickrell JGP(96);
Fleischhack mp/01,
mp/01;
> s.a. connection.

@ __Related topics__: Gomis et al PRP(95) [antibrackets];
Loll et al JGP(96) [complexification];
Lenz et al AP(00)ht [residual symmetries];
McInnes JPA(98) [Alice configurations];
Stoilov ht/05 [Lagrange multipliers];
Mišković & Pons JPA(06)ht/05 [dynamics and symmetries of perturbations];
Feng et al JHEP(07)ht [counting gauge invariants];
Kubyshin 89 [dimensional reduction];
Anderson CQG(08)-a0711 [new interpretation of variational principle];
Pommaret JModP(14)-a1310-talk [formal theory of systems of partial differential equations and Lie pseudogroups];
Berger et al a1806 [complete set of invariant tensors];
Balachandran & Reyes-Lega a1807 [role of the Gauss law].

> __Features, effects__:
see Gribov Effect; instantons;
monopoles; phase transitions;
Reference Frames [accelerated]; solutions.

> __Techniques, tools__:
see homology [chain complexes]; manifold types [gauge
orbit stratification]; Moduli Space; Seiberg-Witten Theory.

**Types of Theories and Related Concepts** > s.a. types of gauge theories.

* __Applications__:
They are very useful (especially the non-Abelian ones) in mathematics, to
get insights on 4D differential topology; In condensed-matter physics,
gauge fields provide the only means of describing the long-range
interactions of vortices or defects in terms of local fields, rendering
them accessible to standard field theoretic techniques.

@ __References__: Kleinert 89 [in condensed-matter physics];
García del Moral a1107 [new gauging mechanism];
Pivovarov PPN(13)-a1209-conf [inaction approach];
Margalli & Vergara PLA(15)-a1507 [hidden gauge symmetry in complex holomorphic systems].

> __Theories__: see
lattice gauge theory; non-commutative
gauge theories; yang-mills theories [including hamiltonian formulation].

> __Related concepts__:
see BRST transformations; charge;
energy-momentum tensor; noether symmetries;
particle models.

**Other References**

@ __General__: Moriyasu 83 [primer];
Healey 07 [conceptual];
Robinson et al a0810-ln [algebraic];
in Scheck 12;
Hamilton a1512-ln [intro, for mathematicians].

@ __Geometric picture, approaches__: Lubkin AP(63);
Hermann 70, 78;
Trautman RPMP(70),
CzJP(79);
Wu & Yang PRD(75);
Atiyah 79;
Daniel & Viallet RMP(80);
Eguchi et al PRP(80);
Balachandran et al 83,
update a1702;
Svetlichny ht/99-ln;
Aldrovandi & Barbosa IJTP(00)mp/01 [non-bundle structure],
IJTP(00)mp/01 [as optical medium];
Ferrantelli MSc(02)-a1002 [gauge-natural formulation, including suypersymmetries];
Harikumar et al PLB(03)ht/02 [topology];
Kubyshin mp/03-conf;
Robinson et al a0908-ln;
Alsid & Serna FP(15)-a1308,
Jordan et al a1404-ch [approaches];
Zharinov TMP(14) [algebraic and geometric methods];
Mielke 17;
> s.a. 2-spinors.

@ __Origin, gauge symmetry as emergent__: 't Hooft AIP(07)-a0707;
Donoghue et al proc(10)-a1007 [and violations];
Bjorken a1008-conf [vacuum condensate and QED];
Freund a1008 [extension of Verlinde's entropic gravity proposal];
Zaanen & Beekman AP(12)-a1108;
Kirillov et al PLB(12)-a1205;
Chkareuli PLB(13) [from spontaneously broken supersymmetry];
Levin & Wen PRB(05),
RMP(05)cm/04
[gauge bosons and fermions from "string-net condensation" in condensed-matter theory];
Canarutto IJGMP(14)-a1404-conf [from the geometry of Weyl spinors];
Arias et al PRB(15)-a1511 [elastic deformations in graphene];
Urrutia a1607-conf [from Nambu models];
Wetterich NPB(17)-a1608 [from decoupling];
Barceló et al JHEP(16)-a1608 [systematic study];
Sachdev RPP(18)-a1801.

@ __Meaning__: de Souza ht/98,
ht/99,
ht/99 [discrete];
Guttmann & Lyre phy/00 [physics vs math];
Gubarev et al PRL(01)hp/00 [of *A*^{2}];
Lyre PhSc(01)qp-conf [conceptual];
Healey PhSc(01)dec [reality of *A*];
Sánchez a0803;
Rovelli FP(14)-a1308 [why gauge?];
Weatherall a1411,
a1505-conf;
Afriat a1706;
Nguyen et al a1712
[the need for surplus structure].

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