Symmetries and Strings of Adjoint QCD${}_2$ (2008.07567v2)

Abstract: We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group $SU(N)$. The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits $\sim 2^{2N}$ non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all $N$ these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, $m\ll g_\mathrm{YM}$, the theory confines and the non-invertible symmetries are softly broken. We use them to compute analytically the $k$-string tension for $N\leq 5$. Our results suggest that the $k$-string tension, $T_k$, is $T_k\sim |m| \sin(\pi k /N)$ for all $N$. We also consider the dynamics of adjoint QCD deformed by symmetric quartic fermion interactions. These operators are not generated by the RG flow due to the non-invertible symmetries, thus violating the ordinary notion of naturalness. We conjecture partial confinement for the deformed theory by these four-fermion interactions, and prove it for $SU(N\leq5)$ gauge theory. Comparing the topological phases at zero and large mass, we find that a massless particle ought to appear on the string for some intermediate nonzero mass, consistent with an emergent supersymmetry at nonzero mass. We also study the possible infrared phases of adjoint QCD allowed by the non-invertible symmetries, which we are able to do exhaustively for small values of $N$. The paper contains detailed reviews of ideas from fusion category theory that are essential for the results we prove.

An Analysis of Symmetries and Strings in Adjoint QCD$_2$

The paper presents an in-depth examination of the symmetries in two-dimensional adjoint QCD (quantum chromodynamics) with gauge group $\SU(N)$, focusing on the unique characteristics that manifest due to the presence of non-invertible symmetries. Unlike the usual approach that only characterizes the dynamics through ordinary symmetries and anomalies, this work highlights the emergence of $\sim 2^{2N}$ non-invertible symmetries that impose strict constraints on the infrared phases and massive excitations of the theory.

Key Points and Results

• Non-Invertible Symmetries: The existence of a large number of non-invertible symmetries in massless adjoint QCD$_2$ is demonstrated through a mathematical and physical framework, suggesting that these symmetries are closely tied to the topological nature of the theory's line operators.
• Deconfinement of Fundamental Quark: For all $N$, these symmetries imply the deconfinement of fundamental quarks, countering initial intuitions based solely on conventional symmetry analysis. This is specifically interesting as it challenges previous conclusions drawn from anomaly considerations alone, especially for $N>2$.
• Mass Deformations and Confinement: Introducing a small mass to the adjoint quark results in the confinement scenario, breaking the non-invertible symmetries softly. This transition is quantitatively analyzed, especially with respect to the $k$-string tension, which provides insights into confinement mechanisms traditionally explored in higher-dimensional QCD.
• Quartic Fermion Interactions: The paper also explores the effects of deformations by quartic interactions involving adjoint quarks. Some of these operators do not emerge from the renormalization group (RG) flow due to the presence of non-invertible symmetries, violating the notion of naturalness.
• Emergent Supersymmetry: Theoretical conjectures are presented on the appearance of massless particles on strings at intermediate nonzero quark mass, suggesting an emergent supersymmetry, a feature consistent with the hints from certain anomalies and symmetry arguments.

Implications

Theoretical Insights

The treatment of adjoint QCD$_2$ as presented opens several avenues for exploring nontrivial dynamics in lower-dimensional theories as models for real-world QCD. The mathematical tools and interpretations of fusion categories applied here are vital to understanding complex quantum systems where typical symmetry-based approaches are insufficient. This work extends our understanding of how non-invertible lines influence confinement and deconfinement transitions, offering a fresh perspective on symmetry breaking and the structure of quantum field theories.

Practical and Computational Developments

Exact computations of quantities such as the $k$-string tensions provide valuable benchmarks for future analytical and numerical studies. These results may inspire new methods for probing non-Abelian gauge theory models, leading potentially to new algorithmic strategies particularly beneficial in lattice QCD simulations.

Speculations on Future Developments in AI

The formal development and manipulation of fusion categories in the context of theoretical physics may find analogs in AI, particularly in areas focusing on categorical learning and the integration of complex, non-linear data. The mathematical methods employed in this research might inspire computational models that better adapt learned symmetries to improve decision-making processes, particularly in AI systems designed to handle vast amounts of structured data.

In summary, the paper presents a comprehensive exploration of adjoint QCD$_2$ with crucial implications for symmetry theory, improved understanding of QCD-like models, and novel computational methods in analytical quantum field theory. The elucidation of non-invertible symmetries broadens the horizon for future research in both theoretical physics and potential interdisciplinary innovations.