Anomalies of non-invertible symmetries in (3+1)d
Clay Córdova, Po-Shen Hsin, Carolyn Zhang
Abstract
Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d “inflow” symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the “Frobenius-Schur indicator” of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing \mathbb{Z}_{4} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>4</mml:mn> </mml:msub> </mml:math> ordinary symmetry host worldvolume theories with time-reversal symmetry \mathsf{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> </mml:math> obeying the algebra \mathsf{T}^{2}=C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> or \mathsf{T}^{2}=(-1)^{F}C, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>F</mml:mi> </mml:msup> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> with C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>C</mml:mi> </mml:math> a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.