Editing Anomaly (physics) (section)

==Anomalies and cobordism==

In the modern description of anomalies classified by [[cobordism]] theory,<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> the [[Feynman diagram|Feynman-Dyson graphs]] only captures the perturbative local anomalies classified by integer '''Z''' classes  also known as the free part. There exists nonperturbative global anomalies classified by [[cyclic group|cyclic groups]] '''Z'''/''n'''''Z''' classes also known as the torsion part.

It is widely known and checked in the late 20th century that the [[standard model]] and chiral gauge theories are free from perturbative local anomalies (captured by [[Feynman diagram|Feynman diagrams]]). However, it is not entirely clear whether there are any nonperturbative global anomalies for the [[standard model]] and chiral gauge theories. 
Recent developments <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="1910.11277">{{cite journal | last1=Davighi | first1=Joe | last2=Gripaios | first2=Ben | last3=Lohitsiri | first3=Nakarin | title=Global anomalies in the Standard Model(s) and Beyond  | journal=JHEP | volume=2020 | issue=7 | date=July 2020 | page=232 | issn=1029-8479 | doi=10.1007/JHEP07(2020)232 |arxiv=1910.11277| bibcode=2020JHEP...07..232D | s2cid=204852053 }}</ref>
<ref name="1910.14668">{{cite journal | last1=Wan | first1=Zheyan | last2=Wang | first2=Juven | title=Beyond Standard Models and Grand Unifications: Anomalies, Topological Terms, and Dynamical Constraints via Cobordisms  | journal=JHEP | volume=2020 | issue=7 |  date=July 2020 | page=62 | issn=1029-8479 | doi=10.1007/JHEP07(2020)062 |arxiv=1910.14668| bibcode=2020JHEP...07..062W | s2cid=207800450 }}</ref>
based on the [[cobordism theory]] examine this problem, and several additional nontrivial global anomalies found can further constrain these gauge theories. There is also a formulation of both perturbative local and nonperturbative global description of anomaly inflow in terms of [[Michael Atiyah|Atiyah]], [[Vijay Kumar Patodi|Patodi]], and [[Isadore Singer|Singer]]
<ref name="APS">{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Patodi | first2=V. K. | last3=Singer | first3=I. M. | title=Spectral asymmetry and Riemannian geometry | doi=10.1112/blms/5.2.229  | mr=0331443  | year=1973 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=5 | issue=2 | pages=229–234| citeseerx=10.1.1.597.6432 }}</ref>
<ref name="APS1">{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Patodi | first2=V. K. | last3=Singer | first3=I. M. | title=Spectral asymmetry and Riemannian geometry. I | doi=10.1017/S0305004100049410 | mr=0397797  | year=1975 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=77 | issue=1 | pages=43–69| bibcode=1975MPCPS..77...43A | s2cid=17638224 }}</ref> [[eta invariant]] in one higher dimension. This [[eta invariant]] is a cobordism invariant whenever the perturbative local anomalies vanish.<ref name="1909.08775">{{cite arXiv| last1=Witten | first1=Edward | last2=Yonekura | first2=Kazuya | title=Anomaly Inflow and the eta-Invariant  | year=2019 | class=hep-th |eprint=1909.08775}}</ref>