Higher symmetry and gapped phases of gauge theories (1309.4721v2)

Abstract: We study topological field theory describing gapped phases of gauge theories where the gauge symmetry is partially Higgsed and partially confined. The TQFT can be formulated both in the continuum and on the lattice and generalizes Dijkgraaf-Witten theory by replacing a finite group by a finite 2-group. The basic field in this TQFT is a 2-connection on a principal 2-bundle. We classify topological actions for such theories as well as loop and surface observables. When the topological action is trivial, the TQFT is related to a Dijkgraaf-Witten theory by electric-magnetic duality, but in general it is distinct. We propose the existence of new phases of matter protected by higher symmetry.

• The paper generalizes Dijkgraaf-Witten theories by replacing finite groups with finite 2-groups to capture novel gapped phases and duality structures.
• It classifies topological actions in 2, 3, and 4 dimensions, revealing unique interactions of 1-form and 2-form gauge fields.
• The research introduces symmetry 2-group protected phases that offer deeper insights into gauge confinement and duality in quantum field theories.

Higher Symmetry and Gapped Phases of Gauge Theories

The paper "Higher Symmetry and Gapped Phases of Gauge Theories," authored by Anton Kapustin and Ryan Thorngren, explores the generalization of topological quantum field theories (TQFTs) that describe gapped phases of gauge theories with higher symmetry. The paper presents these TQFTs as extensions of the well-known Dijkgraaf-Witten (DW) theories, replacing a finite group with a finite 2-group. This framework introduces more complexity and depth into the description of gapped phases, revealing novel phases of matter that conventional DW theories fail to describe.

The main focus is on TQFTs involving 1-form and 2-form gauge fields, alongside 0-form and 1-form gauge symmetries. The authors conceptualize these theories using the apparatus of 2-groups, denoted as a quadruple $G = (G,H,t,\alpha)$, which encapsulates the interactions between group $G$, group $H$, a homomorphism $t$, and an action $\alpha$. The research demonstrates that massive phases of gauge theories, where the microscopic gauge group is partially confined and partially Higgsed, can be effectively analyzed using this TQFT formulation.

Key Findings and Contributions

1. Generalization of DW Theories: The paper extends DW theories through the introduction of 2-groups, capturing a broader spectrum of TQFTs that include both electric and magnetic gauge groups.
2. Classification of Topological Actions: For dimensions 2, 3, and 4, the paper classifies topological actions of these models. Each dimension presents unique features, such as dualizability of certain forms, and the emergence of nontrivial actions at higher dimensions.
3. Symmetry-Protected Phases: The authors propose the existence of phases protected by a symmetry 2-group, advocating for more complex structures protecting gapped phases compared to traditional symmetry groups.
4. Quantum Field Theory Dualities: The paper discusses dualities in TQFTs, explaining how dualization can be performed under certain conditions, broadening the understanding of gauge symmetries.

Implications and Future Directions

This research has significant theoretical implications, offering a richer description of TQFTs that could lead to new insights into the nature of gauge theories. The use of 2-groups and higher categorical structures provides a novel perspective on symmetry and confinement in quantum field theories. Practically, these findings may have applications in condensed matter physics, particularly in the paper of topological phases and quantum computation where higher symmetries and dualities play pivotal roles.

Moving forward, further exploration into the algebraic and geometric structures of higher categories will likely expand the utility of this approach. Moreover, applying the framework in higher-dimensional contexts could reveal deeper connections between various quantum field theories and string theory. Researchers in theoretical and mathematical physics should find this approach insightful in addressing complex problems related to symmetry, duality, and topological phases.