Theta, Time Reversal, and Temperature

Abstract: $SU(N)$ gauge theory is time reversal invariant at $\theta=0$ and $\theta=\pi$. We show that at $\theta=\pi$ there is a discrete 't Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at $\theta=\pi$ the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at $\theta=0$ is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at $\theta=\pi$, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for $SU(2)$ gauge theory. The underlying symmetry at $\theta=\pi$ is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two $O(2)$-symmetric fixed points. It may also be that the four-dimensional theory around $\theta=\pi$ is gapless, e.g. a Coulomb phase could match the underlying anomalies.

• The paper demonstrates how a mixed ’t Hooft anomaly at θ = π constrains the vacuum structure, forbidding a trivial gapped state in SU(N) Yang-Mills theory.
• Methodologically, the study employs lattice simulations, soft supersymmetry breaking, and large N holographic models to analyze phase transitions and CP symmetry breaking.
• The work details distinct behavior in SU(2) gauge theory with dihedral symmetry, indicating possible gapless phases and first-order phase transitions influenced by temperature.

Examination of 'Theta, Time Reversal, and Temperature' in SU(N) Gauge Theory

The paper "Theta, Time Reversal, and Temperature," authored by Davide Gaiotto, Anton Kapustin, Zohar Komargodski, and Nathan Seiberg, embarks on a detailed investigation of an SU(N) Yang-Mills theory with a focus on the implications of its parameters, particularly the theta angle (θ), and its associated symmetries. Through this exploration, the authors unravel the effects of time reversal and center symmetries, presenting a nuanced discussion of anomaly structures and their constraints on the vacuum and phase transitions in these gauge theories.

Discrete Anomalies and SU(N) Yang-Mills Theory

Central to this paper is the analysis of the θ angle in 4d SU(N) Yang-Mills theory. The exploration hinges upon the discrete ’t Hooft anomaly apparent at θ = π, involving time reversal symmetry and a Z 1-formN center symmetry. The authors argue convincingly that this mixed anomaly induces vital constraints on the theory's vacuum structure. It implies, for instance, that the vacuum at θ = π cannot be a trivial non-degenerate gapped state, contrasting starkly with the vacuum at θ = 0, which is gapped, non-degenerate, and trivial. This non-triviality at θ = π leads the theory to admit domain walls that house lower-dimensional theories and indicates a potential first-order phase transition with the spontaneous breaking of CP symmetry.

Implications and Phase Diagrams

The authors further explore what this anomaly implies for the phase diagram and the role of temperature in these transitions. In particular, if we assume the area law for space-like loops, the analysis yields an inequality involving the temperatures at which CP and center symmetry are restored. For SU(2) gauge theory, where the deconfinement transition is second order, the underlying symmetry elucidates as the dihedral group D8. This setup permits two O(2)-symmetric fixed points, reflecting a richer structure that suggests gapless phases such as Coulomb phases as possible anomaly matches.

In one potential pathway, the SU(N) theory might maintain a gapped state, preserving 0-form symmetries and conjecturing a different low-energy theory that accommodates these mixed anomalies. The alternative envisages it being fundamentally gapless at zero temperature, exploring Coulomb phase paradigms and potential interaction with four-dimensional CFTs in specific theta values.

Special Focus on SU(2) Gauge Theory

The paper devotes considerable attention to SU(2) gauge theory, elaborating on a variety of phase diagrams around θ = π, given the distinct dihedral symmetry conditions. Emphasizing the absence of a guaranteed gap in this regime, the authors position lattice simulations, softly broken supersymmetry analysis, and large N holographic models as the underpinnings for their conjectures, albeit acknowledging these assumptions could fail at small N, notably for N = 2.

Analogies and Anomalies in Lower Dimensions

The research draws insightful parallels with a two-dimensional model, showcasing the anomaly between time-reversal and PSU(n) symmetry as an analogous constraint on IR behavior. Such analysis reveals spontaneously broken time-reversal symmetries or potentially gapless systems (for instance, gapless systems for n = 2) at critical theta values.

Conclusion and Theoretical Significance

The work offers a profound presentation of SU(N) gauge theories, meticulously analyzing how discrete anomalies dictate physical behavior at pivotal angles and intersecting these findings with thermal dynamical considerations. The theoretical structure proposed opens avenues for deeper investigations into anomaly-induced constraints and their manifestations across varied gauge symmetries.

In conclusion, the examination into "Theta, Time Reversal, and Temperature" underscores major theoretical implications within gauge theories, offering nuanced insights into anomalies and setting a robust foundation for future explorations into non-trivial anomalies in strongly coupled systems. This paper's findings hold potential influence on ongoing research in high-energy physics and gauge theory, particularly in understanding symmetry behavior in complex topological phases. Future investigations may delve further into computational validations, exploring the practical realization of these exotic phases within lattice setups or experimental arenas.