't Hooft Anomaly

Definition

A ‘t Hooft anomaly is an obstruction to gauging a global symmetry: a phase acquired by the partition function $Z[A]$ under a background gauge transformation that no local counterterm built from the background $A$ can remove. Unlike a dynamical anomaly, the symmetry is preserved — but the theory cannot be coupled consistently to a background for it without a partner in one higher dimension.

Two properties make anomalies powerful:

• Anomaly matching (‘t Hooft): the anomaly is renormalization-group invariant, so the UV and IR must reproduce it. A trivially gapped symmetric vacuum is forbidden when the anomaly is nonzero — the IR must break the symmetry, be gapless, or carry a nontrivial topological sector.
• Anomaly inflow: the non-invariance is cancelled by a bulk symmetry-protected topological (SPT) phase in $d+1$ dimensions whose boundary is the anomalous theory. “An anomaly is the boundary of an SPT.”

Anomalies need not be exotic. A symmetry realized projectively on the Hilbert space is the 0+1d germ; the Lieb–Schultz–Mattis theorem is a lattice mixed anomaly between translation and internal symmetry, forbidding a unique gapped ground state for a half-integer-spin chain.

Role in Research

Anomalies enter the generalized-symmetry program as sharp, RG-stable constraints on phase diagrams. The centerpiece example in the course is Yang–Mills at $\theta =\pi$: a mixed anomaly between the ${\mathbb{Z}}_N$ 1-form center symmetry and time reversal (Gaiotto–Kapustin–Komargodski–Seiberg) forbids a trivially gapped, $T$-symmetric, confining vacuum. On the lattice the anomaly is a fractional instanton number in the presence of a 2-form background, built from the cup product $B\cup B$ (Pontryagin square).

• Mixed anomalies with higher-form symmetries directly constrain confinement scenarios in the group’s confinement-duality program.
• The modified Villain construction realizes these anomalies exactly on a finite lattice.

Relations

• higher-form-symmetries — anomalies of and mixed anomalies with $p$-form symmetries
• spt-phases — inflow: the anomalous theory lives on the boundary of a bulk SPT
• lattice-gauge-theory — the θ = π anomaly as fractional instanton number with a center background
• villain-action — exact lattice backgrounds that make the anomaly visible without a continuum limit
• quantum-hall-effect — the chiral quantum Hall edge is the textbook continuum example of inflow
• chern-simons-theory — the bulk term whose boundary non-invariance the edge cancels

Papers

‘t Hooft (1980, anomaly matching); Gaiotto–Kapustin–Komargodski–Seiberg, JHEP 05 (2017) 091 [arXiv:1703.00501]; Lieb–Schultz–Mattis (1961), Oshikawa (2000), Hastings (2004). See the course bibliography.

Notes

• The Pontryagin square $\mathcal{P}(B)$ is the honest lattice object; the non-commutativity of the cup product is what carries the anomaly, so it must not be “simplified away.”
• LSM being an anomaly is the clean statement that “an anomaly can live in an ordinary spin chain” — anomalies are not the exclusive property of chiral gauge theories.
• The quantum Hall edge is the textbook continuum example of inflow (Witten 2026, §3). On a film with boundary, $\exp (2\pi ik\mathrm{CS}(A))$ is not a number but a unit vector in a line fixed by the boundary data (Ramadas–Singer–Weitsman). A chiral edge theory whose partition function lives in the inverse line compensates it (Callan–Harvey). The edge is not unique for given $k$; the simplest is $k$ chiral fermions. The edge current’s anomaly coefficient, i.e. its Schwinger term, is the Hall conductance. See quantum-hall-effect and chern-simons-theory.