Foundations of General-Relativistic Gauge Field Theory

Scientific Program

The topic of this conference school is General-Relativistic Gauge Field Theory: its mathematical foundations, its first principle conceptual and technical development, state-of-the-art applications, and model building. The event also features contributed talks.

Organizing Committee: Lucrezia Ravera (Organizer & Diversity Coordinator; Politecnico di Torino, DISAT, Italy), Jordan François (Scientific Advisor; University of Graz, Austria), Vincenzo Antonelli (Organizer; Politecnico di Torino, DISMA, Italy)

The Scientific Program includes the following modules:
- [Module DG, 4.5h], 3 lectures, by Jordan François (University of Graz, Austria);
- [Module SR, 2.5h], 2 lectures, by Lucrezia Ravera (Politecnico di Torino, Italy);
- [Module AS, 4.5h], 3 lectures, by Marc Geiller (ENS de Lyon, CNRS, France);
- [Module FA, 4.5h], 3 lectures, by Philipp Berghofer (University of Graz, Austria).

Lectures DG

Three lectures on Bundle Differential Geometry (DG) of Gauge Field Theory, to provide the mathematical and geometric tools underpinning classical gauge theories: e.g., Ehresmann connection and Yang-Mills Gauge Field Theory, Cartan connection and gauge gravity, twisted connections on field space and anomalies, differential geometry of field space and gauge-fixing.

Some References: J. François, "Differential geometry of gauge theory: an introduction", PoS Modave 2020 (2021) 002; J. François, L. Ravera, "Cartan geometry, supergravity, and group manifold approach", Archivum Math. 60 (2024) 4, 243-281; M. J. D. Hamilton, J. A. de Azcarraga, J. M. Izquierdo, "Lie groups, Lie algebras, cohomology and some applications in physics", Cambridge Monographs on Mathematical Physics (1995), CUP; R. W. Sharpe, "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program", Springer (1997); A. Cap, J. Slovák, "Parabolic Geometries I: Background and General Theory", Mathematical Surveys and Monographs, Volume 154 (2009), AMS.

Topics of Module DG:

• Principal bundles
• Gauge group
• Ehresmann and Cartan connections
• Cartan geometry
• Yang-Mills and gravitational gauge theories
• Bundle geometry of field space
• Gauge-fixing
• Cocyclic forms and anomalies
• Field-dependent gauge transformations
• Basic cohomology of field space

Lectures SR

Two lectures on Symmetry Reduction (SR) schemes – in particular, the Dressing Field Method (DFM) and its relational interpretation – in General-Relativistic Gauge Field Theory, including state-of-the-art model building and applications (among which, e.g., General Relativity, Supersymmetric Field Theories, etc.).

Some References: J. François, L. Ravera, "Geometric relational framework for general-relativistic gauge field theories", Fortsch. Phys. 2024 (2024) 2400149; J. François, L. Ravera, "Spacetime boundaries do not break diffeomorphism and gauge symmetries", Phys. Rev. D 112 (2025) 12, 12; J. François, L. Ravera, "Dressing fields for supersymmetry: the cases of the Rarita-Schwinger and gravitino fields", JHEP 07 (2024) 041; J. François, L. Ravera, "Unconventional supersymmetry via the dressing field method", Phys. Rev. D 111 (2025) 12, 12; J. François, L. Ravera, "Relational Supersymmetry via the Dressing Field Method and Matter-Interaction Supergeometric Framework", Annalen der Physik 537, no. 9 (2025): 537, e00121.

Topics of Module SR:

• Symmetry Reduction via the DFM and its relational interpretation
• Field-theoretic application to Maxwell theory, U(1) case plus complex scalar and Abelian Higgs model
• DFM in GR physics, gravity plus scalar fields (scalar coordinatization)
• "Boundary problem" and diffeo-dressing
• DFM in supersymmetric field theory
• Matter-interaction supergeometric framework and the DFM

Lectures AS

Three lectures on Asymptotic Symmetries (AS), covariant phase space formalism, and applications to electromagnetism and General Relativity.

Some References: S. Speziale, "GGI lectures on boundary and asymptotic symmetries", [arXiv:2512.16810 [hep-th]]; G. Compère, A. Fiorucci, "Advanced Lectures on General Relativity", [arXiv:1801.07064 [hep-th]]; M. Bañados, I. A. Reyes, "A short review on Noether’s theorems, gauge symmetries and boundary terms", Int. J. Mod. Phys. D 25 (2016) 10, 1630021, [arXiv:1601.03616 [hep-th]].

Topics of Module AS:

• First Noether theorem in mechanics and field theory
• Second Noether theorem in gauge field theory
• Covariant phase space formalism
• Applications to electromagnetism and general relativity
• Asymptotic symmetries

Lectures FA

Three lectures on Foundational Aspects (FA) of General-Relativistic Gauge Field Theory, to allow a clearer understanding of the current state of general-relativistic physics and gauge theories, starting from first principles and conceptual analysis.

Some References: P. Berghofer, J. François, S. Friederich, H. Gomes, G. Hetzroni, A. Maas, R. Sondenheimer, "Gauge Symmetries, Symmetry Breaking, and Gauge-Invariant Approaches", Elements in the Foundations of Contemporary Physics, Cambridge University Press (2023); P. Berghofer, J. François, "Dressing vs. fixing: On how to extract and interpret gauge-invariant content", Foundations of Physics, 54(6):72, 2024; P. Berghofer, J. François, L. Ravera, "What Price Fiber Bundle Substantivalism? On How to Avoid Holes in Fibers", [arXiv:2505.12876 [physics.hist-ph]].

Topics of Module FA:

• Philosophy of physics more generally, with a focus on the discussion of the nature of time and space in classical physics (Leibniz-Clarke debate)
• The Hole argument
• Gauge symmetries and gauge-invariant approaches

Contributed Talks

List of contributed talks, each lasting 30 minutes (questions included).

Title: Soft charges in 3D Maxwell theory at null infinity and horizons
Speaker: Olivera Mišković (Pontificia Universidad Católica de Valparaíso, Chile)
Abstract: Asymptotic symmetries at null boundaries often signal the presence of soft degrees of freedom and associated conserved charges. Here, these symmetries are studied in three-dimensional electromagnetism in two complementary contexts: null infinity of flat spacetime and the null boundary provided by the BTZ horizon. Both analyses lead to two sets of soft charges: the usual electric charge and an additional charge tied to a shift asymptotic symmetry. Together, these charges close into an infinite-dimensional Kac–Moody–type charge algebra with a nontrivial central extension.

Title: A Journey through Higher-Derivative Models in Contemporary Field Theory
Speaker: Ronaldo Thibes (Universidade Estadual do Sudoeste da Bahia, Brazil)
Abstract: With a initial motivation of treating point charge divergences in electrodynamics, higher-derivative (HD) models have come a long way in quantum in field theory passing through important challenges concerning their unitarity evolution and existence of propagating ghost fields. In modern physics, the relevance of HD models relies not only on abstract mathematical curiosities, but actually brings new possibilities for approaching open problems. In fact, since the 1950 well-known Pais-Uhlenbeck groundbreaking paper, models containing derivatives of order higher than two have abounded in the literature. In this talk, we briefly review and contextualize important HD models including Bopp-Podolsky, Lee-Wick, Pais-Uhlenbeck and HD Klein-Gordon generalizations while discussing their roles within recent new physics proposals. We present a consistently parametrized family of higher-order generalizations for the Klein-Gordon equation with their corresponding HD actions, illustrate the BRST quantization of the Pais-Uhlenbeck oscillator and discuss the null-plane dynamics and constraint structure of the Bopp-Podolsky model. We show that through the introduction of higher-derivatives, it is possible to generate massive modes for the fields whitout breaking gauge invariance. Different alternative interpretations are formulated concerning connections to the Pauli-Vilars regularization scheme, existence of new massive physical fields or disposal of consistent building blocks for constructions of larger relevant effective theories. We end with some remarks on the consistent reduction of order for higher-derivative models as recently presented in the literature.

Title: Classical gauge theories and gravity
Speaker: Sebastian Brezina (Comenius University in Bratislava, Slovakia)
Abstract: It belongs to folklore that electromagnetic, electroweak, and strong interactions are understood as gauge theories. In contrast, the situation regarding gravitational interaction is often considered more complex. Since the 1960s, this complexity has led to extensive research on the question: Can gravity be understood as a gauge theory?
In this talk, we explore a framework for describing classical gauge theories developed by Andrzej Trautman in the 1970s. This framework utilizes the theory of Ehresmann connections on principal G-bundles. Within the framework, gravity can be understood as a classical gauge theory, where the metric tensor field is treated as a classical Higgs field. One significant advantage of this framework is that it enables the description of gravity and gauge theories of internal symmetries using the same mathematical formalism. This unification leads to a deeper understanding of the crucial geometrical differences between these two distinct classes of classical gauge theories, which have non-trivial physical implications.

Title: Weyl-Invariant Equations from Gauge PDEs with Asymptotic Boundaries
Speaker: Mikhail Markov (University of Mons, Belgium)
Abstract: I plan to discuss the applications of the gauge PDE approach to the study of the boundary structure of gauge fields on the conformal boundary of asymptotically AdS spaces.

The main result is the construction of an efficient calculus for the gauge PDE induced on the boundary, which allows one to systematically derive Weyl-invariant equations induced on the boundary. The so-called obstruction equations (e.g. Conformal Gravity in dimension d=4), higher conformal Yang–Mills equations, and GJMS operators are derived systematically, as the constraints on the leading boundary value of, respectively, the metric, YM field, and the critical scalar field. In particular,
the higher conformal Yang–Mills equation in dimension d=8, obtained within this framework appears to be new. The Weyl-invariant equations on the subleading boundary data for these fields are also derived.

The approach is very general and can be considered as an extension of theFefferman-Graham construction that is applicable to generic gauge fields and explicitly takes into account both the leading and the subleading sector.

Title: The Katz, Bičák and Lynden-Bell (KBL) regularization and its applications
Speaker: Nelson Merino (Universidad Arturo Prat, Iquique, Chile)
Abstract: The Katz boundary term provides a well-defined variational principle under Dirichlet boundary conditions and, when combined with a subtraction of the action evaluated on a background, known as the KBL regularization procedure, yields finite Noether charges and a finite on-shell action. This boundary term is constructed from the dynamical metric and the difference between the Christoffel symbols associated with the dynamical manifold and a reference background. So far, this method has been tested only for specific solutions. In this work (soon to be submitted for publication), using the Fefferman-Graham gauge, we show that the finiteness of conserved charges can be proven for families of asymptotically locally Anti–de Sitter spacetimes in general relativity. The finiteness of the charges can be established in arbitrary dimensions; however, the prescription for defining the background in this framework distinguishes between even and odd spacetime dimensions. Other possible applications and potential relations with the covariant phase space method will also be discussed.

Title: From generalized principal connections to generalized Yang-Mills theories
Speaker: Hartwig Winterroth (University of Torino, Italy)
Abstract: In the framework of the fiber bundle approach to covariant Lagrangian field theories we investigate the notions of generalized principal bundle and generalized principal connection as introduced by Castrillón López and Rodríguez Abella (2023), aiming at the development of an instance of generalized mathematical gauge theories. We provide a characterization of Lie group fiber bundle connections and generalized principal connections in order to obtain the local coordinate representation of all such structures. In particular, studying the curvature of generalized principal connections, we specialize the Bianchi identities obtaining a generalized version of the classical homogeneous field equations. As an application, we prove also that vector bundles are an example of generalized principal bundles, that a generalized principal connection on a vector bundle is an affine connection and that the generalized homogeneous field equations can be rephrased, in this case, in terms of basic soldering forms and torsion tensors. Finally, resorting to gauge theories and variational calculus on fiber bundles, we propose a first approach to (an instance of) generalized Yang-Mills theories. We accordingly prove that the corresponding variational field equations (i.e. Euler-Lagrange equations) generalize the classical Yang-Mills equations. While generalized Yang-Mills theories need more developments to give a full answer, we expect that the Einstein equations can be formulated ultimately as an example of generalized Yang-Mills equations. This is a joint work with Lorenzo Fatibene.

Title: Gauge-Invariant Renormalisation Group Flows
Speaker: Paul Philip Sprenger (Institute for Theoretical Physics, Heidelberg University, Germany)
Abstract: In this talk I give a brief introduction to the functional renormalisation group, a non-perturbative framework widely used to study gauge field theories. While this approach has been successfully applied to Yang–Mills theory and quantum gravity, standard implementations generally modify gauge invariance. This modification obscures the physical interpretation of results. I present a novel formalism of renormalisation group flows that systematically resolves the modification of gauge redundancies. I apply this to quantum gravity and highlight the consequences of recovering a gauge invariant renormalisation group flow.

Title: Gravity is gauge? A unification perspective
Speaker: Priidik Gallagher (University of Tartu, Estonia)
Abstract: The question of the extent to which gravity is a gauge theory can prove to be surprisingly complicated, considering constraint classes, internal and external symmetries, spacetime tangent structure, and connection versus metric variables. Furthermore, unifying gravity with the rest of the interactions provides even more conundrum, but also an arena for theory development. The premise for classical gravity unification in field theory is reviewed, with emphasis in form of a recently proposed Khronon Lorentz gauge theory of gravity, and with motivations in Cartan geometry, topological field theory, and pre(geo)metric ideas. Unification in particle physics has a well-motivated list of requirements, and rigorous limitations, e.g. by the Coleman-Mandula theorem among others. There are a multitude of ways to bypass this, but it will be argued that it seems also an inherently spacetime-geometric obstruction. Special attention will be brought to how to distinguish gravity and internal gauge theory, and how to approach a gravity-unified phase for theory development.

Title: Holographic thermal propagator from modularity
Speaker: Katarina Trailović (Institute Jožef Stefan, Slovenia)
Abstract: We revisit the low-temperature thermal propagator of a holographic conformal field theory in four spacetime dimensions by exploiting its connection to the Nekrasov–Shatashvili (NS) limit of the Omega-deformed N = 2 supersymmetric SU(2) Yang–Mills theory with N_f = 4 hypermultiplets. In the regime of vanishing energy, the low-temperature expansion corresponds to a large adjoint vacuum expectation value expansion. In this limit, we show that a second expansion in instanton numbers organizes into quasi-modular forms, which can be resummed into closed-form expressions in terms of Eisenstein series. The resulting thermal propagator series in positive powers of small temperatures exhibits clear signs of being asymptotic. Our method—combining modular properties, q-recursion techniques, and the NS prepotential—provides a systematic and computationally efficient framework for analyzing retarded Green's functions of holographic black branes in the low-temperature limit.

Title: The autoparallel equations with non-metricity as Finsler geodesics
Speaker: Lehel Csillag (Transilvania University of Brasov, Romania)
Abstract: Autoparallel curves in metric-affine geometries are generally non-variational and do not generally coincide with the Euler-Lagrange equations of any Lagrangian. For symmetric connections with vectorial nonmetricity, we show that the autoparallel equations can be realized as geodesics of a suitably chosen Finsler metric, reducing the problem of variationality to Finsler metrizability. By formulating this as a first-order partial differential equation, we obtain necessary and sufficient conditions and classify all (alpha, beta)-metrics whose geodesics coincide with these autoparallels. For generalized (alpha, beta)-metrics, necessary conditions are obtained.