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Anomaly (physics)
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===Higher anomalies involving higher global symmetries: Pure Yang–Mills gauge theory as an example=== The concept of global symmetries can be generalized to higher global symmetries,<ref name="1412.5148">{{cite journal | last1=Gaiotto | first1=Davide | last2=Kapustin | first2=Anton | last3=Seiberg | first3=Nathan | last4=Willett | first4=Brian | title=Generalized Global Symmetries | journal=JHEP | volume=2015 | issue=2 | date=February 2015 | page=172 | issn=1029-8479 | doi=10.1007/JHEP02(2015)172 |arxiv=1412.5148| bibcode=2015JHEP...02..172G | s2cid=37178277 }}</ref> such that the charged object for the ordinary 0-form symmetry is a particle, while the charged object for the n-form symmetry is an n-dimensional extended operator. It is found that the 4 dimensional pure Yang–Mills theory with only SU(2) gauge fields with a topological theta term <math>\theta=\pi,</math> can have a mixed higher 't Hooft anomaly between the 0-form time-reversal symmetry and 1-form '''Z'''<sub>''2''</sub> center symmetry.<ref name="1703.00501">{{cite journal | last1=Gaiotto | first1=Davide | last2=Kapustin | first2=Anton | last3=Komargodski | first3=Zohar | last4=Seiberg | first4=Nathan | title=Theta, Time Reversal, and Temperature | journal=JHEP | volume=2017 | issue=5 | date=May 2017 | page=91 | issn=1029-8479 | doi=10.1007/JHEP05(2017)091 |arxiv=1412.5148| bibcode=2017JHEP...05..091G | s2cid=119528151 }}</ref> The 't Hooft anomaly of 4 dimensional pure Yang–Mills theory can be precisely written as a 5 dimensional invertible topological field theory or mathematically a 5 dimensional bordism invariant, generalizing the anomaly inflow picture to this '''Z'''<sub>''2''</sub> class of global anomaly involving higher symmetries.<ref name="1904.00994">{{cite journal | last1=Wan | first1=Zheyan | last2=Wang | first2=Juven | last3=Zheng | first3=Yunqin | title=Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory | journal=Physical Review D | volume=100 | issue=8 | date=October 2019 | issn=2470-0029 | doi=10.1103/PhysRevD.100.085012 | page= 085012 |arxiv=1904.00994| bibcode=2019PhRvD.100h5012W | s2cid=201305547 }}</ref> In other words, we can regard the 4 dimensional pure Yang–Mills theory with a topological theta term <math>\theta=\pi</math> live as a boundary condition of a certain '''Z'''<sub>''2''</sub> class invertible topological field theory, in order to match their higher anomalies on the 4 dimensional boundary.<ref name="1904.00994">{{cite journal | last1=Wan | first1=Zheyan | last2=Wang | first2=Juven | last3=Zheng | first3=Yunqin | title=Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory | journal=Physical Review D | volume=100 | issue=8 | date=October 2019 | issn=2470-0029 | doi=10.1103/PhysRevD.100.085012 | page= 085012 |arxiv=1904.00994| bibcode=2019PhRvD.100h5012W | s2cid=201305547 }}</ref>
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