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 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 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 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]; > 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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send feedback and suggestions to bombelli at olemiss.edu – modified 21
aug
2017