Topology in Physics  

In General
@ General references, reviews: Finklelstein IJTP(78) [field theory]; Balachandran FP(94)ht/93; Nash in(98)ht/97; Rong & Yue 99; Lantsman mp/01; Heller et al JMP(11)-a1007 [significance of non-Hausdorff spaces]; Eschrig 11; Asorey et al a1211 [fluctuating spacetime topology]; Bhattacharjee a1606-ln; Aidala et al a1708 [and experimental distinguishability].
@ Topological quantum numbers, invariants: Thouless 98; Kellendonk & Richard mp/06-conf [bulk vs boundary, and topological Levinson theorem]; > s.a. yang-mills gauge theory.
> Related topics: see Generic Property; Stability.
> Online resources: see Frederic Schuller 2015 lecture.

In Classical Theories > s.a. electromagnetism [knotted solutions]; phenomenology of magnetism; Susceptibility [topological]; theta sectors.
* Gauge theory: The three paradigms of topological objects are the Nielsen-Olesen vortex of the abelian Higgs model, the 't Hooft-Polyakov monopole of the non-abelian Higgs model, and the instanton of Yang-Mills theory.
* Condensed-matter-type systems: The best-known examples are the quantization of the magnetic flux that pierces a superconductor, which can only increase in units of the flux quantum h/2e, the Hall conductance of a 2D, low-temperature and high-magnetic-field electron gas, which is quantized in units of the conductance quantum \(e^2/h\), and more recently the magneto-optical response of a 3D topological insulator, quantized in units of the vacuum fine-structure constant, \(\alpha = e^2/\hbar c = 1/137\).
* Hydrodynamics: Topology appears in the notion of vortex, relevant both for classical and for quantum fluids.
@ Gauge theory: Monastyrsky 93 [and condensed matter]; Gross JMP(96) [cohomology and connections]; Lenz LNP(05)ht/04, Jackiw ht/05-en [rev]; Yang IJMPA(12); > s.a. types of yang-mills theories [on a circle].
@ Condensed matter: Monastyrsky 93 [and gauge theory]; Avdoshenko et al SRep(13)-a1301 [electronic structure of graphene spirals]; news nPhys(17)jul; Sergio & Pires 19.
@ Topological charges: Saaty ht/01; > s.a. field theory [topological currents].
@ Related topics: Kiehn mp/01 [topology-changing evolution]; Díaz & Leal JMP(08) [invariants from field theories]; Radu & Volkov PRP(08) [stationary vortex rings]; Seiberg JHEP(10)-a1005 [sum over topological sectors and supergravity]; Mouchet a1706 [in fluid dynamics, rev]; Candeloro et al a2104 [and precision of a finite thermometer]; > s.a. thermodynamic systems [Maxwell theory].

In Gravitation and Cosmology > s.a. Alexandrov Topology; geon; lorentzian and riemannian geometry [space of geometries]; spacetime topology.
* Spacetime topology: The spacetime manifold can be assigned different topologies; The most natural ones are the manifold topology, the Alexandroff topology (generated by gab and the I ±s), and Johan's strong topology for compact spacetimes (more stable under limits).
* Spatial topology: Any compact 3-topology can occur classically, since it can be given a metric such that R = −k, with k a positive constant, and with this metric it can be made to satisfy the constraints, with \(K_{ab} = \lambda\,g_{ab}\), for some constant λ.
@ References: Clarke GRG(71) [and general relativity]; Friedman & Mayer JMP(82) [angular momentum and charge]; Carcassi & Aidala PS(20)-a2006 [spacetime structure may be topological]; > s.a. topology at cosmological scales; topology change.

In Quantum Mechanics > s.a. path integrals [non-trivial configuration-space topology]; theta sectors.
@ General references: Sudarshan et al AIHP(88); Balachandran et al 91; Thouless 98 [topological quantum numbers]; Aharonov & Reznik PRL(00)qp/99 [local/non-local complementarity]; Suzuki a1107 [homotopy and path integrals]; Asorey et al a1211 [survey]; Neori & Goyal a1501 [fundamental groupoid approach].
@ Topological quantum phases: Buniy & Kephart PLA(08)ht/06 [second-order]; Thiang AHP(15)-a1406, IJGMP(15)-a1412 [homotopic versus isomorphic]; Witten RNC(16)-a1510-ln [topological phases of matter, and fractional quantum Hall effect]; Asorey nPhys(16)-a1607 [topological matter]; Aguilar et al a1903.
@ Topology and quantum states: Balachandran Pra(01)qp/00-conf; Dürr et al AHP(06)qp, JPA(07) [Bohmian mechanics]; Prudêncio & Cirilo-Lombardo IJGMP(13)-a1402 [entanglement and non-trivial topologies]; Pérez-Pardo et al IJGMP(15)-a1501 [boundary dynamics and topology change]; Qin et al NJP(18)-a1611 [topological invariants in strongly interacting quantum systems]; Arkinstall et al PRB(17) [lattice with topological states]; > s.a. entanglement; entanglement and spacetime; models of topology change.
@ Topology on the space of states: Bugajski PLA(94); Zhu & Ma PLA(10).
> Related topics: see aharonov-bohm effect; Aharonov-Casher Effect.

In Quantum Field Theory > s.a. CPT [violation mechanism]; QCD; QED [in non-trivial backgrounds]; qft in curved backgrounds.
@ General references: Monastyrsky 87; Schwartz 93, 94 [III, IV]; Bandyopadhyay 03.
@ Related topics: Blau IJMPA(89) [representation-independence]; Jackiw mp/05 [fractional quantum numbers, non-trivial phonons]; Baez & Stay LNP-a0903 [physics, topology, logic and computation]; Brunetti et al AHP(09)-a0812 [2D massive bosons]; Buchholz et al LMP(19)-a1808 [linking numbers]; Bessa & Rebouças a1910 [charged-particle motion in topologically non-trivial spaces]; Acquaviva et al a2012 [topologically inequivalent quantizations]; > s.a. charge.

In Other Theories > s.a. Kink; knot theory in physics; spacetime topology.
@ String theory: Balachandran et al NPB(87); Boi IJGMP(09).
@ Quantum topology: Isham CQG(89); Isham et al CQG(90); Isham in(91); Finkelstein & Hallidy IJTP(91) [and quantum logic]; Grib & Zapatrin IJTP(96)gq/95 [topology as an observable, and the space of topologies]; Schlesinger JMP(98); > s.a. quantum spacetime [relational topology].
@ Other quantum gravity: Patrascu JMP(16)-a1410 [and the black-hole information paradox]; > s.a. canonical quantum gravity.

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