Jena lectures on generalized global symmetries: principles and applications (2407.20815v2)

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.

• The paper introduces generalized global symmetries by extending conventional QFT symmetries to include higher-form symmetries governing extended objects.
• It applies these principles to gauge theories and condensed matter systems, offering fresh insights into phenomena like confinement and phase transitions.
• The work examines non-invertible symmetries and anomalies, paving the way for novel applications in magnetohydrodynamics and quantum gravity.

Overview of "Jena lectures on generalized global symmetries: principles and applications"

The paper "Jena lectures on generalized global symmetries: principles and applications" by Nabil Iqbal is a comprehensive set of lecture notes aiming to introduce the concept of generalized global symmetries in quantum field theory (QFT) and its several applications. This text, initially prepared for a lecture series on QFT and holography, explores higher-form symmetries and their implications for various physical systems, providing a consolidated introduction designed to be accessible even to those with a limited background in QFT.

Introduction to Generalized Global Symmetries

The paper elucidates how traditional symmetries, familiar in many physics courses, form a subset within a broader framework of generalized global symmetries. These symmetries have gained attention in recent years for their potential to simplify the understanding of complex gauge theories and provide a unifying language across different physical phenomena. Importantly, generalized global symmetries extend beyond the standard 0-form global symmetries to include higher-form symmetries, which associate symmetries with extended objects rather than just point particles.

Key Concepts and Applications

1. Higher-Form Symmetries: These symmetries relate to p-forms, which naturally correspond to the conservation of higher-dimensional objects, like strings or branes, rather than just point particles. For instance, a 1-form symmetry relates to strings or Wilson lines, and its conservation is depicted through a two-index antisymmetric current. This concept is crucial in understanding the behavior of Abelian gauge theories, both in continuum and lattice formulations.
2. Applications to Gauge Theories: The paper discusses how generalized global symmetries apply to Abelian and non-Abelian gauge theories important for numerous applications. For example, in pure Yang-Mills theories, the center symmetries of the gauge groups emerge as higher-form symmetries, providing insights into topics such as confinement and the phases of gauge theories.
3. Non-Invertible Symmetries and Anomalies: Iqbal addresses non-invertible symmetries' role in understanding the Adler-Bell-Jackiw anomaly. These symmetries arise when the fusion of symmetry defects does not follow a group composition law, creating a complex symmetry structure that extends conventional symmetry discussions in QFT.
4. Statistical and Condensed Matter Systems: The paper also draws connections to statistical mechanics, explaining how generalized global symmetries can be employed to comprehensively describe phase transitions and critical phenomena in systems like the 2d Ising model.
5. Magnetohydrodynamics and Real-World Applications: Beyond theoretical exercises, these symmetries find utility in magnetohydrodynamics, relevant to both astrophysical phenomena and modern condensed matter systems. By framing magnetohydrodynamics in terms of higher-form symmetries, the text provides a new lens through which to view the macroscopic behavior of charged fluids and plasmas.

Theoretical and Practical Implications

The exploration of generalized global symmetries offers profound theoretical insights, potentially reshaping the understanding of symmetries' role in QFT. Practically, it enhances the ability to model complex systems across various domains of physics, possibly leading to new developments in fields ranging from condensed matter physics to cosmology.

Speculation on Future Developments

Going forward, the continued investigation into higher-form and non-invertible symmetries could pave the way for novel insights into string theory and quantum gravity, with potential ramifications for the formulation of unified physical theories. Additionally, understanding these symmetries might lead to advanced computational techniques applicable to simulate strongly interacting systems where conventional methods fall short.

Overall, Iqbal's notes serve as a critical resource, not only summarizing the foundational knowledge required to navigate the landscape of generalized global symmetries but also pointing towards future research trajectories that hold promise for significant theoretical and applied advancements.