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 oral contributions by young researchers.
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 (18 hours of lectures in total):
- [Module DG, 4.5h], 3 lectures, by Jordan François (University of Graz, Austria);
- [Module SR, 4.5h], 3 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).
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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, "Mathematical Gauge Theory : With Applications to the Standard Model of Particle Physics", Universitext (2017), Springer; 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
- Variational principle
- Basic cohomology of field space
- Physical degrees of freedom
- Introduction to the Dressing Field Method
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Lectures SR
Three 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, "On the Meaning of Local Symmetries: Epistemic-Ontological Dialectics", Found. Phys. 55 (2025) no.3, 38; J. François, L. Ravera, "Geometric relational framework for general-relativistic gauge field theories", Fortsch. Phys. 2024 (2024) 2400149; J. Attard, J. François, S. Lazzarini, "Weyl gravity and Cartan geometry", Phys. Rev. D 93, 085032 (2016); J. François, L. Ravera, "Reassessing the foundations of Metric-Affine Gravity", Eur. Phys. J. C 85, 902 (2025); 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; J. François, L. Ravera, "Relational bundle geometric formulation of non-relativistic quantum mechanics", Fortschr. Phys. 2025 (2025): e70040.
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, Abelian Higgs model (Spontaneous Symmetry Breaking and the DFM)
- Application of the DFM to gravity plus scalar fields
- "Boundary problem" and diffeo-dressing
- Scalar coordinatization in general-relativistic physics
- Conformal gravity and dressing
- The case of Metric-Affine Gravity
- Gauge-invariance in supersymmetric field theory
- Matter-interaction supergeometric framework and the DFM
- Symmetry reduction in quantum theory
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Lectures AS
Three lectures on Asymptotic Symmetries (AS) in General Relativity and applications to gravitational (and electromagnetic) radiation and gravitational waves physics.
Some References: ...
Topics of Module AS:
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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
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Contributed Talks
T.B.A.
