1-form symmetry versus large N QCD

TL;DR

The paper addresses whether large-$N$ QCD harbors an emergent $Z_N$ 1-form symmetry and how this relates to center symmetry. It systematically contrasts the 1-form symmetry structure in pure YM with the presence of fundamental quarks, identifying endability and quark-loop non-suppression as core obstructions, and tests the ideas in a tractable 2d scalar QCD model using a hopping expansion. The results show that large-$N$ QCD has a ${Z}_N$ center symmetry but no nontrivial $\,Z_N$ 1-form symmetry, since the necessary codimension-2 topological operators fail to be topological at large $N$ and quark loops remain non-suppressed in their correlation functions. This points to a need for generalized or non-invertible symmetry notions to explain confinement-inspired selection rules, beyond the standard 1-form symmetry paradigm.

Abstract

We show that large N QCD does not have an emergent $\mathbb{Z}_N$ 1-form symmetry. Our results suggest that a symmetry-based understanding of (approximate) confinement in QCD would require some further generalization of the notion of generalized global symmetries.

Figures (10)

• Figure 1: Wilson loops in large $N$ QCD with quarks of mass $m$ obey the same selection rules as in large $N$ pure YM theory. Some of these selection rules (the ones inside the blue circle) can be explained by a ${\mathbb Z}_N$ center symmetry. Some of them can be explained by a ${\mathbb Z}_N$$1$-form symmetry (beige circle), but only in the YM limit $m \to \infty$. There is no ${\mathbb Z}_N$$1$-form symmetry in large $N$ QCD with finite $m$, and so there are selection rules outside both the blue and beige circles. Moreover, as we review in Section \ref{['sec:conclusions']}, there are also many large $N$ selection rules that cannot be explained by a ${\mathbb Z}_N$$1$-form symmetry even when $m \to \infty$.
• Figure 2: Action of a ${\mathbb Z}_N$$1$-form symmetry generator $U_k(C)$ on a Wilson loop in representation $R$ in $d=3$.
• Figure 3: Different ways of shrinking a topological codimension-2 operator in the presence of an open fundamental Wilson line give different results. This gives rise to the so-called 'endability' constraints on the existence of topological operators with non-trivial actions on Wilson loops. Here we depict the situation in three spacetime dimensions, where the would-be topological line can be collapsed in two distinct ways.
• Figure 4: To prove a finite-volume selection rule for a non-contractible line operator $P$ charged under an invertible $1$-form symmetry, one can consider a correlation function of $P$ and a generator $U_k$ of the $1$-form symmetry with its inverse $U_{-k}$. In the figure we assume the spacetime manifold is $M_{d-2} \times S^1_L \times S^1_{\beta}$, $P$ wraps around $S^1_{\beta}$ localized at $\vec{x} \in M_{d-2}$ and a point $z \in S^1_L$, while the symmetry generators wrap around $M_{d-2}$ and are point-like on $S^1_L \times S^1_{\beta}$. We have suppressed the dependence on $M_{d-2}$ in the figure to reduce clutter.
• Figure 5: To prove a selection rule for contractible closed line operators $W_R$ charged under a $1$-form symmetry, one can consider a correlation function of $W_R$ and a generator $U_{\alpha}$ of the $1$-form symmetry. Then one uses the action of the symmetry generator along with the principle of cluster decomposition. For visual clarity, the figure assumes that $M_{d-2}$ is point-like in the plane of the curve $C$. To justify the invocations of cluster decomposition indicated on the right-hand side of the figure, the spacetime area must be large compared to the area bounded by $C$, so that $U_{\alpha}$ can be moved far from $C$ once it is outside of $C$. Similarly, to justify the use of cluster decomposition on the left-hand side, the area bounded by $C$ must also be large. If $\langle U_{\alpha}\rangle \neq 0, f(\alpha,R) \neq 1$, we get the selection rule that large contractible closed line operators $W_R(C)$ must vanish in any renormalization scheme. This selection rule only holds when $C$ bounds a diverging area, in infinite spacetime volume, with an appropriate ordering of limits. As explained in the text, the selection rule is satisfied by an area-law behavior for the contractible line operator, but not by a perimeter-law behavior.