Editing Anomaly (physics) (section)

===Large gauge transformations===

Global anomalies in [[symmetries]] that approach the identity sufficiently quickly at [[infinity]] do, however, pose problems.  In known examples such symmetries correspond to disconnected components of gauge symmetries.  Such symmetries and possible anomalies occur, for example, in theories with chiral fermions or self-dual [[differential form]]s coupled to [[gravity]] in 4''k''&nbsp;+&nbsp;2 dimensions, and also in the [[#Witten anomaly and Wang–Wen–Witten anomaly|Witten anomaly]] in an ordinary 4-dimensional SU(2) gauge theory.

As these symmetries vanish at infinity, they cannot be constrained by boundary conditions and so must be summed over in the path integral.  The sum of the gauge orbit of a state is a sum of phases which form a subgroup of U(1).  As there is an anomaly, not all of these phases are the same, therefore it is not the identity subgroup.  The sum of the phases in every other subgroup of U(1) is equal to zero, and so all path integrals are equal to zero when there is such an anomaly and a theory does not exist.

An exception may occur when the space of configurations is itself disconnected, in which case one may have the freedom to choose to integrate over any
subset of the components.  If the disconnected gauge symmetries map the system between disconnected configurations, then there is in general a consistent truncation of a theory in which one integrates only over those connected components that are not related by large gauge transformations.  In this case the large gauge transformations do not act on the system and do not cause the path integral to vanish.

====Witten anomaly and Wang–Wen–Witten anomaly====

In SU(2) [[gauge theory]] in 4 dimensional [[Minkowski space]], a gauge transformation corresponds to a choice of an element of the [[special unitary group]] SU(2) at each point in spacetime.  The group of such gauge transformations is connected.

However, if we are only interested in the subgroup of gauge transformations that vanish at infinity, we may consider the 3-sphere at infinity to be a single point, as the gauge transformations vanish there anyway.  If the 3-sphere at infinity is identified with a point, our Minkowski space is identified with the 4-sphere.  Thus we see that the group of gauge transformations vanishing at infinity in Minkowski 4-space is [[isomorphic]] to the group of all gauge transformations on the 4-sphere.

This is the group which consists of a continuous choice of a gauge transformation in SU(2) for each point on the 4-sphere.  In other words, the gauge symmetries are in one-to-one correspondence with maps from the 4-sphere to the 3-sphere, which is the group manifold of SU(2).  The space of such maps is ''not'' connected, instead the connected components are classified by the fourth [[homotopy group]] of the 3-sphere which is the [[cyclic group]] of order two.  In particular, there are two connected components.  One contains the identity and is called the ''identity component'', the other is called the ''disconnected component''.

When a theory contains an odd number of flavors of chiral fermions, the actions of gauge symmetries in the identity component and the disconnected component of the gauge group on a physical state differ by a sign.  Thus when one sums over all physical configurations in the [[functional integration|path integral]], one finds that contributions come in pairs with opposite signs.  As a result, all path integrals vanish and a theory does not exist.

The above description of a global anomaly is for the SU(2) gauge theory coupled to an odd number of (iso-)spin-1/2 Weyl fermion in 4 spacetime dimensions. This is known as 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> In 2018, it is found by Wang, Wen and Witten that the SU(2) gauge theory coupled to an odd number of (iso-)spin-3/2 Weyl fermion in 4 spacetime dimensions has a further subtler non-perturbative global anomaly detectable on certain non-spin manifolds without [[spin structure]].<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> This new anomaly is called the new SU(2) anomaly. Both types of anomalies<ref name="An SU(2) Anomaly"/><ref name=1810.00844/> have analogs of (1) dynamical gauge anomalies for dynamical gauge theories and (2) the 't Hooft anomalies of global symmetries. In addition, both types of anomalies are mod 2 classes (in terms of classification, they are both finite groups '''Z'''<sub>''2''</sub>  of order 2 classes), and have analogs in 4 and 5 spacetime dimensions.<ref name=1810.00844/> More generally, for any natural integer N, it can be shown that an odd number of fermion multiplets in representations of (iso)-spin 2N+1/2 can have the SU(2) anomaly; an odd number of fermion multiplets in representations of (iso)-spin 4N+3/2 can have the new SU(2) anomaly.<ref name=1810.00844/> For fermions in the half-integer spin representation, it is shown that there are only these two types of SU(2) anomalies and the linear combinations of these two anomalies; these classify all global SU(2) anomalies.<ref name=1810.00844/> This new SU(2) anomaly also plays an important rule for confirming the consistency of [[SO(10)]] grand unified theory, with a Spin(10) gauge group and chiral fermions in the 16-dimensional spinor representations, defined on non-spin manifolds.<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>