[Paper Review] Jena lectures on generalized global symmetries: principles and applications

Nabil Iqbal|arXiv (Cornell University)|Jul 30, 2024

Algebraic and Geometric Analysis4 citations

TL;DR

This paper presents an accessible introduction to generalized global symmetries—specifically higher-form and non-invertible symmetries—demonstrating their presence in familiar quantum field theories like QED, Yang-Mills, and the 2D Ising model. It shows how these symmetries, framed as topological operators, unify concepts like confinement, anomalies, and hydrodynamics, with key results including the identification of 1-form symmetries in pure gauge theories and the effective field theory description of magnetohydrodynamics via global symmetry principles.

ABSTRACT

This is an elementary set of lectures on generalized global symmetries originally given at the Jena TPI School on QFT and Holography, designed to be accessible to the reader with a basic knowledge of quantum field theory. Topics covered include an introduction to higher-form symmetries with selected applications: Abelian and non-Abelian gauge theories in the continuum and on the lattice, statistical mechanical systems, the Adler-Bell-Jackiw anomaly from the point of view of non-invertible symmetry, and magnetohydrodynamics. Mathematical formalism is kept to a minimum and an emphasis is placed on understanding the global symmetry structure present in well-known models.

Motivation & Objective

• To introduce generalized global symmetries—higher-form and non-invertible symmetries—as fundamental structures in quantum field theory, accessible to readers with basic QFT knowledge.
• To demonstrate that these symmetries are not abstract constructs but underlie well-known physical systems, such as free electrodynamics, SU(N) gauge theories, and the 2D Ising model.
• To show how the global symmetry structure, particularly via topological operators, provides deep insights into phase transitions, confinement, and anomalies.
• To connect generalized symmetries to effective field theories, including magnetohydrodynamics and finite-temperature hydrodynamics, via symmetry-based formalism.
• To advocate for the integration of generalized symmetry concepts into standard quantum field theory curricula, emphasizing their conceptual clarity and broad applicability.

Proposed method

• The paper uses a minimal mathematical formalism, focusing on physical intuition and concrete examples to introduce higher-form symmetries as topological operators.
• It applies the concept of symmetry as a topological operator to analyze phases of matter, including confinement in SU(N) gauge theories via 1-form symmetry.
• The Adler-Bell-Jackiw anomaly is reinterpreted through the lens of non-invertible symmetries, particularly for the U(1)_A axial symmetry.
• A systematic effective field theory approach is used to describe finite-temperature hydrodynamics, including magnetohydrodynamics, based on 1-form symmetry and charge defect operators.
• The formalism derives the hydrodynamic dispersion relation ω = -i(r/χ)k², with χ as the 1-form charge susceptibility, using only global symmetry and effective field theory principles.
• Lattice realizations of QED are analyzed to show that the EFT description remains valid even at strong coupling, where e ~ 1.

Figure 1: Particle world-lines can’t end: no matter at what time-slice we evaluate $Q(t)$ on we get the same number.

Experimental results

Research questions

• RQ1 How do higher-form symmetries—specifically 1-form symmetries—emerge in standard quantum field theories like QED and Yang-Mills theory?
• RQ2 In what way do generalized global symmetries, particularly non-invertible ones, provide a deeper understanding of anomalies such as the ABJ anomaly?
• RQ3 How can the dynamics of magnetohydrodynamics be systematically derived from global symmetry principles, without relying on gauge field redundancies?
• RQ4 What is the role of topological operators in characterizing phases of matter beyond the Landau paradigm, especially in confined and topologically ordered phases?
• RQ5 Can effective field theories based on generalized global symmetries describe strongly correlated systems, such as those on lattices with order-1 coupling?

Key findings

• The 1-form symmetry in pure SU(N) gauge theory is identified as a key mechanism for confinement, with the symmetry being spontaneously broken in the confining phase.
• The axial anomaly of the U(1)_A symmetry is reinterpreted as a non-invertible symmetry anomaly, providing a new perspective on chiral anomalies in QFT.
• The hydrodynamic dispersion relation ω = -i(r/χ)k² for magnetohydrodynamics is derived from a global symmetry-based effective field theory, with χ being the 1-form charge susceptibility.
• The effective field theory approach successfully explains unexplained results in real-time lattice simulations of QED at strong coupling, where e ~ 1.
• The 2D compact boson model exhibits a non-genuine local operator that realizes a 1-form symmetry, illustrating the subtlety of symmetry realization in statistical systems.
• The paper demonstrates that generalized global symmetries are already present in elementary models like free electrodynamics and the 2D Ising model, suggesting their foundational role in QFT.

Figure 2: Integrating divergenceless current $\star j$ on a codimension-1 manifold ${{\mathcal{M}}}_{d-1}$ .