Generalized Global Symmetries (1412.5148v2)

Abstract: A $q$-form global symmetry is a global symmetry for which the charged operators are of space-time dimension $q$; e.g. Wilson lines, surface defects, etc., and the charged excitations have $q$ spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries ($q$=0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a subgroup). They can also have 't Hooft anomalies, which prevent us from gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.

• The paper introduces generalized q-form symmetries that extend conventional global symmetry concepts in quantum field theory.
• It rigorously analyzes selection rules, background field couplings, and 't Hooft anomalies to constrain physical processes.
• The study links these symmetries to applications in gauge theories and topological phases, highlighting implications for symmetry breaking and SPT phases.

Overview of "Generalized Global Symmetries"

The paper "Generalized Global Symmetries" by Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett explores the paper of higher-dimensional extensions of conventional global symmetries in quantum field theories. Defined as q-form global symmetries, these extensions are characterized by charged operators of space-time dimension q, such as Wilson lines and surface defects, and corresponding charged excitations of q spatial dimensions, including strings and membranes. This comprehensive analysis broadens the classical understanding of global symmetries, introducing novel theoretical frameworks, selection rules, and anomalies.

Theoretical Foundations

The authors detail a framework where generalized global symmetries, like their 0-form counterparts, induce Ward identities and selection rules for quantum amplitudes. They can be coupled to classical background fields and gauged, albeit often exhibiting 't Hooft anomalies that constrain these processes. These anomalies may result in anomaly matching conditions and influence exotic phases like Symmetry Protected Topological (SPT) phases.

A notable result discussed is the ambiguity in presenting higher-form gauge symmetries, where differing presentations can represent the same physical system. In contrast, higher-form global symmetries are unambiguous, hence offering a robust organizational structure to the spectrum of charged operators.

Applications in Gauge Theories and Topological Phases

The implications of these generalized symmetries are analyzed within the context of various gauge theories. The discussion spans from lattice gauge theories to string theory and supergravity, underpinning their prevalence and importance across different physics domains. The paper highlights that many known phenomena, including duality mappings in quantum field theories, can be reinterpreted through the lens of higher-form global symmetries.

In particular, the work examines the role of generalized global symmetries in the context of discrete gauge theories were non-trivial cohomological aspects become significant. The paper extends to the implications for domain walls and boundary conditions, most notably in the theories with varied and non-trivial global one-form symmetries.

Symmetries, Anomalies, and Spontaneous Symmetry Breaking

A crucial contribution of the paper is the formalism related to the spontaneous breaking of these symmetries. The work draws connections between symmetry breaking and the emergence of Goldstone bosons, here represented by massless photons in gauge theories. The authors navigate through the Coleman-Mermin-Wagner theorem's generalization, providing insights into when such breaking is achievable given the dimensional constraints of the theory and symmetry type.

Moreover, the analysis extends to understanding how these generalized symmetries underpin and potentially protect certain topological phases of matter—specifically, the SPT phases. The paper argues for a revised view of symmetry protection, stipulating that these generalized symmetries could safeguard specific topological orders and phase boundaries.

Future Directions and Speculations

The authors speculate that harnessing the concept of generalized global symmetries could catalyze new advances in understanding quantum field theories' dualities and phase structures. The potential to uncover novel SPT phases and elucidate the nature of topological insulators is suggested as a lucrative endeavor in both theoretical and condensed matter physics. Moreover, extending these ideas to higher-dimensional theories and clarifying their implications for non-Lagrangian field theories could reveal deeper insights into the unification of quantum theory with gravity.

Ultimately, "Generalized Global Symmetries" offers far-reaching implications for the enrichment of quantum field theory, promising new ways to unify various physical phenomena under a coherent theoretical framework. The advanced discussion plays a pivotal role in refining our understanding of both existing and beyond-conventional paradigms in physics.