Topological (Quantum) Field Theories  

In General > s.a. category [n-categories]; path-integral quantum field theory.
* Idea: (Quantum) field theories in which correlators depend only on the topology of the manifold.
* Motivation: They generate global smooth invariants for the manifold.
* Applications: Chern-Simons theories have found application in the description of some exotic strongly-correlated electron systems and the corresponding concept of topological quantum computing, and topological sigma models describe digital memcomputing machines DMMs for computing with instantons.
@ General references: Ivanenko & Sardanashvili MUPB(79); Witten CMP(88); Baulieu PLB(89); Horne NPB(89); Myers & Periwal PLB(89); in Atiyah 90; Rajeev PRD(90); Birmingham et al PRP(91); Wu CMP(91); Roca RNC(93); Anselmi CQG(97) [invariants]; Becchi et al PLB(97) [gauge dependence]; Vafa ht/00-conf; Jones BAMS(09) [development, and subfactor theory]; Boi IJGMP(09); Hellmann PhD-a1102 [and state sums on triangulated manifolds].
@ Books and reviews: Kaku 91; in Nash 91; Fré & Soriani 95; Labastida ht/95-conf; Labastida & Lozano ht/97-ln; Kaul ht/99; Schwarz ht/00; Labastida ht/01-talk; Kaul et al ht/05-en [Schwarz-type, including Chern-Simons and BF]; Labastida & Mariño 05; Ivancevic & Ivancevic a0810-ln; Qiu a1201-ln; Carqueville & Runkel a1705-ln.
@ Action: Labastida & Pernici PLB(88); Dayi NPB(90).
@ Observables: Labastida CMP(89); Ouvry & Thompson NPB(90).
@ Algebraic / geometric structure: Crane & Frenkel JMP(94); Crane & Yetter pr(94).
@ BRST approach: Birmingham et al NPB(89); Chen PRD(90).
@ Applications: Di Ventra et al a1609 [DMMs and computing with instantons].
@ Related topics: Atiyah IHES(89) [axioms]; Birmingham et al NPB(90) [renormalization]; Gegenberg & Kunstatter AP(94) [partition function]; Kauffman & Lomonaco SPIE(06)qp, qp/06 [q-deformed spin network approach], SPIE(07)-a0707-in [and quantum computation]; > s.a. stochastic quantization.

Specific Theories > s.a. chern-simons theory; path-integral approach; yang-mills theories.
@ Electromagnetism as a topological field theory: Rañada LMP(89); JPA(92).
@ General relativity / quantum gravity from topological field theory: Toon CQG(94)ht/93; Barrett JMP(95)gq; Mielke PRD(08), GRG(08) [BRST quantization]; Gielen JPCS(11)-a1109 [with linear constraints]; Morales et al EPJC(16)-a1602 [from 5D CS theory]; > s.a. 3D quantum gravity [lqg].
@ Topological quantum mechanics: Dunne et al PRD(90); Skagerstam & Stern IJMPA(90) [2+1 dimensions]; Rogers NPPS(00)ht.
@ Topological gauge theory: Ouvry et al PLB(89) [supersymmetric, quantization]; Brooks & Lue JMP(96) [monopoles]; Losev et al NPB(98) [Gromov-Witten paradigm]; Boldo et al IJMPA(03)ht, IJMPA(04)ht/03, NPPS(04)ht [observables]; Leal & Pineda MPLA(08) [abelian, and Milnor's link invariant]; Chen IJGMP(13)-a0803 [conceptual, historical]; Escalante & López-Osio IJPAM(12)-a1203 [Euler and second-Chern classes, Hamiltonian analysis].
@ Topological gauge theory, deformed: Kondo PRD(98)ht; García-Compeán & Paniagua GRG(05)ht/04 [non-commutative].
@ Topological (super)gravity: Chamseddine NPB(90); Koehler et al NPB(90); > s.a. Topological Gravity.
@ Gravity and topological matter: Gegenberg & Mann PRD(99)ht.
@ Homotopy quantum field theory: Brightwell & Turner m.QA/01, Brightwell et al IJMPA(03)m.AT/02; Turaev 10.
@ Other theories: Floreanini & Percacci MPLA(90) [pregeometry]; Gozzi & Reuter PLB(90) [classical mechanics as a topological field theory]; Birmingham et al IJMPA(90); Blau & Thompson AP(91), PLB(91) [forms]; Witten IJMPA(91) [cohomological]; Gamboa IJMPA(92); Dijkgraaf & Moore CMP(97) [balanced]; Husain & Jaimungal PRD(99)ht/98 [holographic]; Adams & Prodanov LMP(00) [Schwarz's, Z]; Malik JPA(01)ht/00 [2D]; Ferrari mp/01 [simple cubic model]; Koroteev & Zayakin ht/05-wd [example based on Morse theory]; Husain PRL(06) [harmonic oscillator duals and background-independence in quantum gravity]; Mathews a1201 [elementary, combinatorial theory]; > s.a. BF theory; m-theory.

Related Topics > s.a. 4D manifolds; dynamical systems; spin networks [invariants].
@ On manifolds with boundary: Husain & Jaimungal PRD(99)ht/98; Bel'kov et al a0907 [triangulated boundary]; Amoretti et al PRD(14)-a1410.
@ On lattices: Wheater PLB(89) [Ising-like], PLB(91) [gauge theories]; Bonzom & Smerlak LMP(10)-a1004 [degree of bubble divergences]; Bietenholz et al JHEP(10)-a1009 [actions].
@ (2+1)D theories and (3+1)D theories with defects: Dittrich a1701 [and and self-dual quantum geometries].
@ And knots / links: Horowitz & Srednicki CMP(90) [linking numbers]; Blanchet et al Top(95) [knot invariants]; Leal PRD(02); Lemes et al PLB(99) [linking observables]; Labastida ht/00-conf, ht/00-ln; Leal & Pineda MPLA(08)-a0705 [topological field theory of Milnor's link invariant]; Sleptsov MPLA(14) [generalization, superpolynomial invariants of knots]; > s.a. knot invariants.
@ Other topics: Eguchi MPLA(92) [and singularities]; Toon MPLA(94)ht/92 [particle content]; Archer JGP(95) [on PL manifolds]; Brooks & Lifschytz NPB(95) [Donaldson topological invariants and quantum gravity]; Mukku et al JPA(97) [order-chaos]; Freedman et al CMP(02)qp/00 [and quantum computing]; Rovelli & Speziale GRG(07) [expansion of field theories around a topological field theory]; Freed BAMS(13)-a1210 [extended topological quantum field theories and the cobordism hypothesis]; Carqueville a1607-proc [2D with defects, functorial and algebraic description, intro].
> Phenomenology: see cosmological-constant problem.

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