nLab 't Hooft anomaly

Idea

As described in ‘t Hooft (1980), a global symmetry of a quantum field theory is said to have a ‘t Hooft anomaly if it is non-anomalous as a global symmetry but has a quantum anomaly if one attempts to turn it into a gauge symmetry.

Informally, a global symmetry described by the action of a group $G$ may be “coupled” to a gauge potential $A$ (a connection on a bundle). Denoting the partition function coupled to such a connection as $Z(A)$, the partition function of the gauged theory is, modulo normalization factors, $Z=\sum_A Z(A).$ If the symmetry has a ‘t Hooft anomaly, then performing a gauge transformation on all connections results in a nontrivial phase multiplying each partition function $Z_g=\sum_A \phi(A,g) Z(g\cdot A)$ where each phase $\phi(A,g)$ depends on the particular background field and gauge transformation, so that generally $Z\neq Z_g$. This is problematic since the partition function is supposed to be gauge invariant.

For $G$ a finite group, and when the $n$-dimensional spacetime $\Sigma$ is the boundary of an $(n+1)$-dimensional space $X$, this situation may be remedied by “coupling” the $n$-dimensional theory with symmetry $G$ to a $(n+1)$-dimensional topological quantum field theory (a Dijkgraaf-Witten theory classified by $H^{n+1}(G,U(1))$) on $X$ such that the phase contributions of a gauge transformation of both cancel each other. This is known as anomaly inflow (see e.g. Freed, Hopkins, Lurie, and Teleman (2009) and Freed (2014)).

Any generalized global symmetry is also thought to potentially exhibit ‘t Hooft anomalies described by a TQFT. In the literature, this is referred to as the Anomaly TFT, an invertible field theory (cf. Freed 2014), but little is known about what this TQFT is supposed to be for cases not equivalent to group-like cases (for which the TQFT is DW).

References

• Gerard 't Hooft, Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking In: G. ‘Hooft et al., Recent Developments in Gauge Theories, NATO Advanced Study Institutes Series 59, Springer (1980) [doi:10.1007/978-1-4684-7571-5_9]

• Daniel Freed, Michael Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups, in P. R. Kotiuga (ed.) A celebration of the mathematical legacy of Raoul Bott AMS (2010) [arXiv:0905.0731, ISBN:978-0-8218-4777-0]

• Daniel Freed. Anomalies and Invertible Field Theories, talk at StringMath2013 [arXiv:1404.7224]