Anomalous Symmetries End at the Boundary

TL;DR

The paper shows that 't Hooft anomalies obstruct symmetric boundary conditions: any preserved subgroup $G_{\mathcal{B}}$ at a boundary must be anomaly-free, with perturbative and gravitational anomalies treated via descent and Ward identities and tied to anomaly inflow.Two complementary formalisms are developed: a descent/Wess-Zumino analysis of currents and an symmetry-defect (G-foam) framework that translates anomalies into group cohomology and generalized cohomology invariants, including AHSS data.In unitary theories, boundary obstructions are stringent, disallowing Lorentz-invariant boundaries that preserve anomalous symmetries in dimensions $D$ up to those considered, while non-unitary theories can exhibit more exotic boundary behavior and require caution in interpreting anomalies.The results extend to gauge-gravity anomalies and higher-form symmetries, showing that symmetric boundaries are only possible when the relevant invariants vanish or can be canceled by bulk-boundary inflow decorations, with implications for domain walls and emergent anomalies.

Abstract

A global symmetry of a quantum field theory is said to have an 't Hooft anomaly if it cannot be promoted to a local symmetry of a gauged theory. In this paper, we show that the anomaly is also an obstruction to defining symmetric boundary conditions. This applies to Lorentz symmetries with gravitational anomalies as well. For theories with perturbative anomalies, we demonstrate the obstruction by analyzing the Wess-Zumino consistency conditions and current Ward identities in the presence of a boundary. We then recast the problem in terms of symmetry defects and find the same conclusions for anomalies of discrete and orientation-reversing global symmetries, up to the conjecture that global gravitational anomalies, which may not be associated with any diffeomorphism symmetry, also forbid the existence of boundary conditions. This conjecture holds for known gravitational anomalies in $D \le 3$ which allows us to conclude the obstruction result for $D \le 4$.

Figures (4)

• Figure 1: The elementary recombination for $D=2$ (sometimes called an $F$-move or crossing relation), drawn here as relating the red foam at bottom to the red foam at top, is captured by a point-like singularity of a three dimensional foam which is Poincaré dual to a tetrahedron and for which the initial and final foams form the top and bottom boundaries.
• Figure 2: The argument for Theorem \ref{['propwog']} illustrated for $D = 2$. At top left we have the special bubble introduced near the boundary. From left to right, top to bottom, this bubble is absorbed by the boundary. In purple we have indicated where absorbing a point-like junction causes a recombination of the boundary defects, and produces compensating phases. The condition that the boundary correlation function is non-vanishing requires that these phases precisely cancel $e^{2\pi i\omega}$ of the original bubble. This requires $\omega$ be exact in group cohomology, hence that there is no anomaly.
• Figure 3: Here we have a spatial picture of a 2+1D fermionic system with a unitary symmetry $C^2 = (-1)^F$. At the white circle, two co-oriented $C$ defects (green) fuse to a fermion parity defect (dashed). This junction may trap a Majorana fermion. However, if it does, then by layering a $p+ip$ superconductor (so taking $\beta_3 = 1 \in \Omega^3_{Spin}$), since the fermion parity defect ends at the fusion junction, the $p+ip$ superconductor sees a vortex there (red star), which also traps a Majorana fermion Alicea_2012. This Majorana may be paired with the other one to create a featureless fusion junction. Thus, this decoration does not actually contribute an anomaly in this symmetry class. With more work, one can show that for this symmetry class and dimension, there are in fact no nontrivial anomalies (see Appendix C.4 of Garc_a_Etxebarria_2019).
• Figure 4: A bubble halfway absorbed into a symmetric boundary condition drawn for $D = 2$. The bulk $G_{\mathcal{B}}$-foam is drawn in green. Where it ends on the boundary (white circles) there is a possible gravitational anomaly, described by a function $\beta_1: G_{\mathcal{B}} \to \Omega^1$, which in a spin theory could be a fermionic operator, $\Omega^1_{spin} = \mathbb{Z}_2$. This gravitational anomaly means that the $G_{\mathcal{B}}$-foam with boundary is not isotopy invariant, and so we cannot apply the argument we gave in Section \ref{['sec:defect']} for Prop. \ref{['propwog']}. However, by introducing the subliminal $G_{\mathcal{B}}$-foam (orange) carrying the invertible phase corresponding to $\beta_1$ (for $\Omega^1_{spin}$ it is a fermionic world line) the gravitational anomaly is cured and the entire object is isotopy invariant.

Theorems & Definitions (10)

• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 1
• proof
• Theorem 3
• proof
• Conjecture 1
• Theorem 4
• proof