Higher Gauge Theory (2401.05275v1)

Abstract: This is an invited survey article on higher gauge theory for the Encyclopedia of Mathematical Physics, 2nd edition. In particular, we provide a lightning introduction to higher structures and to the construction of the kinematical data of higher gauge theories, i.e. connections on higher principal bundles. We also summarise the most important applications and dynamical principles that have appeared in the literature, and we close with comments on related areas.

• The paper introduces higher gauge theory by extending conventional gauge frameworks with categorified structures and higher principal bundles.
• It develops a robust kinematical framework using BRST complexes to overcome fake-flatness, enabling non-Abelian generalizations of gauge symmetries.
• It investigates applications such as higher Chern–Simons theories and AKSZ sigma models, highlighting their significance in quantum gravity and unified field theories.

Higher Gauge Theory: A Survey

The paper "Higher Gauge Theory" presents a comprehensive survey of the theoretical framework, mathematical constructions, and potential applications of higher gauge theory. Authored by leading researchers, the survey focuses on higher structures in mathematical physics, particularly highlighting the development of higher gauge theories and their kinematical and dynamical principles.

At the heart of gauge theory lies the concept of principal bundles and their connections, which facilitate the parallel transport of data across manifolds. In higher gauge theory, this idea is extended via categorification, where traditional notions such as groups and spaces are replaced by higher groups and higher groupoids. This allows for the exploration of parallel transport along higher-dimensional objects like surfaces and volumes, rather than just paths.

Kinematical Framework

Within the kinematical framework of higher gauge theory, a central role is played by higher principal bundles and their connections. These bundles are associated with higher groups, where morphisms of varying dimensions enact the transformations that describe gauge symmetries in these extended contexts. The local and infinitesimal approach to these structures is elegantly captured by the Becchi--Rouet--Stora--Tyutin (BRST) complex, which incorporates the infinitesimal transformations and their algebraic properties, providing a robust mathematical ground for field theories.

A significant challenge in higher gauge theory is overcoming the limitations imposed by the fake-flatness condition. This status, which frequently emerges in attempts to generalize conventional gauge theories with non-Abelian symmetries, restricts the curvature and gauge freedoms of a theory, often reducing it to an Abelian form locally. The introduction of adjustment mechanisms allows for the development of non-Abelian structures by modifying the curvature definitions and enabling compatibility with non-trivial symmetry transformations.

Dynamical Aspects and Applications

The survey explores various dynamical scenarios where higher gauge theories demonstrate their utility. Prominent among these are higher Chern--Simons theories, which generalize classical Chern--Simons theories to incorporate forms of higher degree, providing insights into new types of topological invariants. Similarly, Alexandrov--Kontsevich--Schwarz--Zaboronsky (AKSZ) sigma models are explored as a context for realizing gauge symmetries in infinite-dimensional settings using $N$Q-manifolds and BV formalism.

Applications in supersymmetric field theories, particularly in the context of supergravity and superconformal field theories, illustrate the theoretical richness provided by higher gauge structures. The paper highlights the integration of these ideas in constructing tensor hierarchies, a mechanism prevalent in supergravity models for mediating interactions between multiple fields of varying forms.

Implications and Future Directions

The research on higher gauge theory posits profound implications for theoretical physics, particularly in developing consistent theories of quantum gravity and unified field theories. By expanding the horizon of gauge symmetries to encompass higher geometrical dimensions and structures, the paper paves the way for advancing the fundamental understanding of spacetime and field interactions.

Furthermore, the inclusion of non-local and extended objects, such as forms and gerbes, enriches the landscape of mathematical physics, offering potential solutions to longstanding challenges like anomaly cancellation and dualities in string theory. The speculative trajectory includes the extension of these principles into manifest higher symmetry operations, specifically addressing higher-form symmetries and potential for non-invertible symmetries within a refined mathematical formalism.

The continuing evolution of mathematical tools such as higher topos theory and derived categorical structures promises further connections between higher gauge theories and other aspects of modern theoretical physics, enabling novel developments in quantum field theory and beyond.

In summary, the survey presented in "Higher Gauge Theory" offers an invaluable resource, charting the advancement of higher gauge structures and illuminating their central role in the future of theoretical and mathematical physics. As further developments emerge, these theories are likely to occupy an increasingly pivotal position in efforts to unify and extend the principles underlying physical laws, providing deeper insights into the universe's fabric and its symmetries.