Editing Global anomaly

{{short description|Quantum breakdown of large gauge transformations}}
{{more citations needed|date=May 2021}}

== Primary Examples ==

In [[theoretical physics]], a '''global anomaly''' is a type of [[anomaly (physics)|anomaly]]: in this particular case, it is a quantum effect that invalidates a [[large gauge transformation]] that would otherwise be preserved in the classical theory. This leads to an inconsistency in the theory because the space of configurations which is being integrated over in the functional integral involves both a configuration and the same configuration after a large gauge transformation has acted upon it and the sum of all such contributions is zero and the space of configurations cannot be split into connected components for which the integral is nonzero.

Alternatively, the existence of a global anomaly implies that the measure of [[Feynman]]'s [[functional integral]] cannot be defined globally.

The adjective "global" refers to the properties of a [[group (mathematics)|group]] that are detectable via large gauge or diffeomorphism transformations, but are not detectable locally via infinitesimal transformations. For example, all features of a discrete group (as opposed to a [[Lie group]]) are global in character.

A famous example is an SU(2) [[Yang–Mills theory]] in 4D with an odd number of [[chiral fermion]]s in the fundamental representation '''2''' or the isospin 1/2 of SU(2), transforming as [[doublet state|doublets]] under SU(2). This is known as [[Anomaly (physics)#Witten anomaly and Wang–Wen–Witten anomaly|the Witten SU(2) anomaly]].<ref name="An SU(2) Anomaly">{{cite journal | last=Witten | first=Edward | title=An SU(2) Anomaly | journal=Phys. Lett. B | volume=117 | issue=5 | date= November 1982 | doi=10.1016/0370-2693(82)90728-6 | page=324 | bibcode=1982PhLB..117..324W }}</ref>

Another new but much more subtle example is found in 2018, also for the SU(2) gauge theory in 4D, with an odd number of [[chiral fermion]]s in the representation '''4''' or the [[isospin]] 3/2 of SU(2). This is known as [[Anomaly (physics)#Witten anomaly and Wang–Wen–Witten anomaly|the new SU(2) anomaly]].<ref name="1810.00844">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | last3=Witten | first3=Edward | title=A New SU(2) Anomaly | journal=Journal of Mathematical Physics | volume=60 | issue=5 | date= May 2019 | issn= 1089-7658 | doi=10.1063/1.5082852 | page=052301 |arxiv=1810.00844| bibcode=2019JMP....60e2301W | s2cid=85543591 }}</ref> [[Anomaly (physics)#Witten anomaly and Wang–Wen–Witten anomaly|The new SU(2) anomaly]] has an important application to rule out the existence of any global anomaly for the SO(10) [[Grand Unified Theory|grand unified theory]]. This new anomaly is a mixed gauge-gravitational anomaly and a nonperturbative global anomaly.<ref name=1810.00844/><ref name="1809.11171">{{cite journal | last1=Wang | first1=Juven | last2=Wen | first2=Xiao-Gang | title=Nonperturbative definition of the standard models | journal=Physical Review Research  | volume=2 | issue=2 | date=1 June 2020 | issn=2469-9896 | doi=10.1103/PhysRevResearch.2.023356 | page=023356 |arxiv=1809.11171| bibcode= 2018arXiv180911171W| s2cid=53346597 }}</ref>

Many types of global anomalies must be canceled for a theory to be consistent. An example is [[modular invariance]], the requirement of anomaly cancellation for a part of a global [[gravitational anomaly]] 
<ref name="wittenglobal">{{Cite journal |author-link=Edward Witten |first=Edward |last=Witten |title=Global gravitational anomalies
 |journal=[[Communications in Mathematical Physics]] |volume=100 |issue=2 |pages=197–229 |year=1985 |mr=0804460  |doi = 10.1007/BF01212448|  url=https://link.springer.com/article/10.1007/BF01212448 }}</ref>
that deals with the [[large diffeomorphism]]s over two dimensional [[worldsheet]]s of [[genus]] 1 or more.

== Applications to Beyond the Standard Model Physics ==
 
In 2020, a concept known as "ultra unification"<ref>{{Cite journal |last=Wang |first=Juven |date=2021-03-31 |title=Unified model beyond grand unification |journal=Physical Review D |volume=103 |issue=10 |page=105024 |doi=10.1103/PhysRevD.103.105024 |arxiv=2012.15860|bibcode=2021PhRvD.103j5024W }}</ref> was introduced. It combines the [[Standard Model]] and [[grand unification]], particularly for the models with 15 Weyl fermions per generation, without the necessity of right-handed sterile neutrinos, by adding new gapped topological phase sectors or new gapless interacting conformal sectors consistent with the nonperturbative global anomaly cancellation and [[cobordism]] constraints 
<ref name="1604.06527">{{cite journal | last1= Freed | first1=Daniel S. | last2=Hopkins | first2=Michael J. | title=Reflection positivity and invertible topological phases
 | journal=Geometry & Topology | year=2021 | volume=25 | issue=3 | pages=1165–1330 | doi=10.2140/gt.2021.25.1165 |issn=1465-3060 |arxiv=1604.06527| bibcode= 2016arXiv160406527F| s2cid=119139835 }}</ref>
<ref name="1808.00009">{{cite journal | last1=García-Etxebarria | first1=Iñaki | last2=Montero | first2=Miguel | title=Dai-Freed anomalies in particle physics  | journal=JHEP | volume=2019 | issue=8 | date=August 2019 | page=3 | issn=1029-8479 | doi=10.1007/JHEP08(2019)003 |arxiv=1808.00009| bibcode=2019JHEP...08..003G | s2cid=73719463 }}</ref>
<ref name="1808.02881">{{cite arXiv | last1=Hsieh | first1=Chang-Tse   | title=Discrete gauge anomalies revisited
  |arxiv=1808.02881   }}</ref>
<ref>{{Cite journal |last1=Wan |first1=Zheyan |last2=Wang |first2=Juven |date=2020-07-09 |title=Beyond Standard Models and Grand Unifications: anomalies, topological terms, and dynamical constraints via cobordisms |url=https://doi.org/10.1007/JHEP07(2020)062 |journal=Journal of High Energy Physics |language=en |volume=2020 |issue=7 |pages=62 |doi=10.1007/JHEP07(2020)062 |issn=1029-8479|arxiv=1910.14668 |bibcode=2020JHEP...07..062W }}</ref> 
(especially from the mixed gauge-gravitational [[Anomaly (physics)|anomaly]], such as a '''Z'''/''16'''''Z''' class anomaly, associated with the [[baryon]] minus lepton number [[B − L|'''B'''−'''L''']] and the [[Weak hypercharge|electroweak hypercharge]] Y).<ref>{{cite journal |last1=Wilczek |first1=Frank |last2=Zee |first2=A. |year=1979 |title=Operator analysis of nucleon decay |journal=Physical Review Letters |volume=43 |issue=21 |pages=1571–1573 |bibcode=1979PhRvL..43.1571W |doi=10.1103/PhysRevLett.43.1571}}</ref>
 
Gapped topological phase sectors are constructed via the symmetry extension (in contrast to the symmetry breaking in the Standard Model's [[Higgs mechanism|Anderson-Higgs mechanism]]), whose low energy contains unitary Lorentz invariant Schwarz type <ref name="Schwarz">{{Citation | last1=Schwarz | first1=Albert | title= Topological quantum field theories| year=2000 | eprint=hep-th/0011260 }}</ref> 
[[topological quantum field theories]] (TQFTs such as [[Chern-Simons theory]]<ref name="wittenjonespolynomial">{{Cite journal |author-link=Edward Witten |first=Edward |last=Witten |title=Quantum Field Theory and the Jones Polynomial |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |year=1989 |mr=0990772 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 |url=http://projecteuclid.org/euclid.cmp/1104178138 }}</ref>), such as 4-dimensional noninvertible, 5-dimensional noninvertible, or 5-dimensional invertible entangled gapped phase TQFTs.

Alternatively, ultra unification suggests there could also be right-handed sterile neutrinos, gapless [[Unparticle physics|unparticle]] physics, or some combination of more general interacting [[Conformal field theory|conformal field theories (CFTs)]], to together cancel the mixed gauge-gravitational [[Anomaly (physics)|anomaly]]. This proposal can also be understood as coupling the Standard Model (as quantum field theory) to the Beyond the Standard Model sector (as TQFTs or CFTs being [[dark matter]]) via the discrete gauged [[B − L|'''B'''−'''L''']] topological force.

In a colloquium summary, ultra unification has two conceptual additions to the Standard Model. First, beyond-the-Standard-Model dark matter partly consists of [[topological order]] with low energy TQFT, while there are [[anyon]] statistics string excitations above the energy gap. Second, there exists the fifth force as a topological discrete gauge force of [[B − L|'''B'''−'''L''']] that mediates between the Standard Model particles, beyond-the-Standard-Model topological order dark matter, and gapped anyon string non-particle excitations.

In either TQFT or CFT scenarios, the implication is that a new high-energy physics frontier beyond the conventional 0-dimensional [[particle physics]] relies on new types of topological forces and matter. This includes gapped extended objects such as 1-dimensional line and 2-dimensional surface operators or conformal defects, whose open ends carry deconfined fractionalized particle or [[anyon]]ic string excitations.

Understanding and characterizing these gapped extended objects requires mathematical concepts such as [[cohomology]], [[cobordism]], or [[Category (mathematics)|category]] into particle physics. The topological phase sectors signify a departure from the conventional particle physics paradigm, indicating a frontier in beyond-the-Standard-Model physics.

==References==
{{reflist}}

[[Category:Anomalies (physics)]]


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