Anomaly Obstructions to Symmetry Preserving Gapped Phases

TL;DR

This work shows that discrete and higher-form anomalies can obstruct the existence of symmetry-preserving gapped phases in quantum field theories. By formulating a mapping-torus obstruction via anomaly inflow, the authors derive universal constraints that exclude certain gapped TQFTs carrying given anomalies and apply them to 4d Yang-Mills at $\theta=\pi$ and adjoint QCD. The approach unifies LSM-type conclusions with higher-form symmetry analyses and provides concrete no-go results for long-distance IR behavior in broad classes of continuum theories. These obstructions offer a powerful tool for constraining IR dynamics and guiding expectations for the phase structure of gauge theories.

Abstract

Anomalies are renormalization group invariants that constrain the dynamics of quantum field theories. We show that certain anomalies for discrete global symmetries imply that the underlying theory either spontaneously breaks its generalized global symmetry or is gapless. We identify an obstruction, formulated in terms of the anomaly inflow action, that must vanish if a symmetry preserving gapped phase, i.e. a unitary topological quantum field theory, exits with the given anomaly. Our result is similar to the $2d$ Lieb-Schultz-Mattis theorem but applies more broadly to continuum theories in general spacetime dimension with various types of discrete symmetries including higher-form global symmetries. As a particular application, we use our result to prove that certain $4d$ non-abelian gauge theories at $θ=π$ cannot flow at long distances to a phase which simultaneously, preserves time-reversal symmetry, is confining, and is gapped. We also apply our obstruction to $4d$ adjoint QCD and constrain its dynamics.

Figures (9)

• Figure 1: The action of a $0$-form symmetry operator $U_g$ associated to the group element $g\in G^{(0)}$ on a local operator $\mathcal{O}$. When $U_g$ surrounds $\mathcal{O}$ it converts $\mathcal{O}$ to $g(\mathcal{O})$.
• Figure 2: The action of a $p$-form ($p\ge1$) symmetry operator $U_g$ associated to the group element $g\in G^{(p)}$ on a $p$-dimensional extended operator $\mathcal{O}$. When $U_g$ surrounds $L_p$, it acts by a phase $P(\mathcal{O},g)$ satisfying the group law $P(\mathcal{O},g_1)P(\mathcal{O},g_2) = P(\mathcal{O},g_1g_2)$. The extended operator $\mathcal{O}$ is charged under the symmetry group element $g$ when $P(\mathcal{O},g)\neq 1$.
• Figure 3: (a): The right hand side of \ref{['partvan']}. The symmetry defect $U_{g_1}$ corresponding to the background $A_1$ is localized at a point in $S^{p_1+1}$ and wraps $S^{p_2+1}$, while the defect $U_{g_2}$ corresponding to $A_2$ is localized at a point in $S^{p_2+1}$ and wraps $S^{p_1+1}$. (b): Assuming that $g_1$ and $g_2$ are not spontaneously broken, the operators $U_{g_1}$ and $U_{g_2}$ admit topological boundary conditions with which the correlation function does not change. (See section \ref{['bcsec']}, for details. In particular these "boundary conditions" need not satisfy Cardy-like constraints.) (c): Deforming the boundaries, the operators can be contracted to their intersection point. This contraction defines a local operator. (d): Finally, projecting onto a sector where this operator acts as the identity, the local operator is identified with a nonzero number. Therefore \ref{['partvan']} implies that the partition function on $S^{p_1+1}\times S^{p_2+1}$ vanishes, which contradicts with unitarity.
• Figure 4: The geometry defining the state $\ket{\phi_0}$ in the Hilbert space $\mathcal{H}$ on $S^{p+1}\times S^{d-p-2}_{U_g}$. The picture is on the special case of $p=0$ and $d=2$, or can be thought as depicting a 2-dimensional slice of a more general case.
• Figure 5: The geometry defining the state $\ket{0}$ in the Hilbert space $\widetilde{\mathcal{H}}$ on $S^{d-1}$ with $U_g$ inserted along $S^{d-p-2}$.

Theorems & Definitions (5)

• Lemma 1
• Corollary 1
• Proposition 1
• Remark
• Proposition 2