Topologyin 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 Field 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*/2*e*, 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,
α = *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.

@ __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];
> 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 *g*_{ab} 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} = *λ*
*g*_{ab}, for some constant
*λ*.

@ __References__: Clarke GRG(71) [and general relativity];
Friedman & Mayer JMP(82) [angular momentum and charge];
> s.a. topology at cosmological scales.

**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].

@ __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; 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 a1808 [linking numbers];
> 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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