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Summer School Schedule

Lecture series

Lorenzo Fatibene (University of Torino, Italy)

• Noether symmetries
• Natural and gauge-natural bundles
• Natural/gauge-natural symmetries
• Superpotentials in relativistic/gauge-natural theories

Salvatore Capozziello ( Universita di Napoli "Federico II", Italy)

• Metric-affine theories
• Why extending/modifying General Relativity?
• Geometric objects
• Tetrads and spin
• Cartan structure equations
• Spin connection
• Lorentz group and algebra
• The Geometric Trinity of Gravity: Lagrangians, field equations, and solutions. Extending the Geometric Trinity of Gravity
• Summary
• Open problems

Zbynek Urban (VSB-TU Ostrava, Czech Republic)

• Jets and contact elements as underlying structures, ideal of contact forms
• Variational sequences, Lepage forms, and the inverse problem of the calculus of variations
• Local and global aspects of variational principles
• Noether-type symmetry theoremss
• Applications in geometric mechanics and field theory

Workshop Lectures

• Jan Brajerčík: (University of Prešov, Slovakia)

  SO(3)-invariant Finsler structures
  

  In our contribution, we deal with parameter invariant variational functionals on manifold of 1-velocities T1Q of a smooth manifold Q, which are also SO(3)-invariant. We explicitly describe Lepageforms on T1Q serving as an integrand in variational functionals. Corresponding conservation law equations are discussed and illustrated by an example.

• Marián Fecko (Comenius University in Bratislava, Slovakia)

  2D vector calculus and forms on 4D space-time
  

  The presentation provides a brief introduction to two-dimensional vector calculus together with its application to work with differential forms on usual four-dimensional space-time. The construction closely mimics the parametrization of differential forms in E3 by scalars and three- dimensional vectors. Four basic two-dimensional differential operators appear. As an illustration we apply it to corresponding decomposition of Maxwell’s equations and to dimensional reduction of the latter.

• Manuel Hohmann (University of Tartu, Estonia)

  Geometric generalization of the Vainberg-Tonti Lagrangian
  

  One of the most fundamental principles used in theoretical physics is the stationary action principle. It states that the classical solutions of a physical system are the stationary solutions of an action functional, defined on its phase space. This leads to the Euler-Lagrange equations by applying variational calculus, in a straightforward calculation. A more difficult question is the so-called inverse problem of variational calculus: given a set of differential equations, do these equations arise from variation of an action, and how can one find the action? Both questions can be answered by the method of variational completion, and calculating the so-called Vainberg-Tonti Lagrangian. However, this classical method is dependent on the choice of coordinates, and requires certain conditions on the domain of definition of the system. In my talk, I make use of the proper geometric framework for the Lagrange formalism and the inverse problem in terms of jet bundles, which is known as the variational bicomplex. In this context, the differential equations to be completed are known as source forms, and the question whether these are variational is answered by the Helmholtz expressions, which vanish in the affirmative case. Using this framework, I will show how the classical Vainberg-Tonti procedure can be generalized in a coordinate-independent fashion.

• Damianos Iosifidis (Scuola Superiore Meridionale, Italy)

  Belinfante-Rosenfeld Symmetrization from Metric-Affine Conservation Laws: The Cases of QED and QCD
  

  We derive the Belinfante–Rosenfeld symmetrization procedure from the metric-affine conservation law by means of an affine lift of flat-spacetime field theories. Following minimal coupling to an independent affine connection, variation of the action with respect to the connection defines the hypermomentum current. Taking the flat-spacetime limit of the resulting conservation law, we recover the Belinfante–Rosenfeld relation, with the divergence of the hypermomentum current reproducing the Belinfante improvement term. We thoroughly study the form of the couplings with this property and their physical significance. The construction is illustrated for Quantum Electrodynamics and Quantum Chromodynamics.

• Martin Krššák (Comenius University in Bratislava, Slovakia)

  Teleparallel regularization of the gravitational action
  

  Classical solutions of the field equations with a finite Euclidean action play an important role in quantum gravity, where they determine the dominant contributions to the gravitational path integral and can be used to derive black hole thermodynamics. In the teleparallel formulation of general relativity, finite-action solutions can be obtained without introducing Gibbons–Hawking-like boundary terms, but the result depends on a non-dynamical spin connection that effectively encodes the regularization. We study different choices of the teleparallel spin connection and various methods for evaluating the gravitational action. We show that consistent results require properly accounting for the role of singularities, while the ambiguities associated with the choice of spin connection reveal underlying symmetries of the teleparallel action.

• Marcella Palese (University of Torino, Italy)

  Conservation laws and globality for the variation of local Noether strong currents
  

  The interplay of inverse variational problems with invariance properties is exploited. We focus on symmetries of globally defined locally variational field equations and find conditions for the variation of local Noether strong currents to be conserved and variationally equivalent to a system of global and conserved currents.

• Aneta Wojnar (University of Wroclaw, Poland)

  Wave Phenomena Across Scales as Tests of Gravity: The FuSe CA24101 Initiative
  

  This presentation introduces the FuSe Action and its main scientific challenges to a broad audience working on the mathematical formulation of fundamental interactions. A central focus of the talk is the role of thermodynamics in understanding physical phenomena. In particular, the presentation highlights the need for a consistent formulation of thermodynamics in systems governed by long-range interactions, especially gravitation, where standard approaches face significant limitations. The presentation will also outline the importance of the FuSe Action in fostering collaboration across disciplines and engaging researchers working on the foundations of physical theories, with the aim of advancing both theoretical understanding and practical applications.

Poster session

Participants are welcome to present a poster on their current work related to the school topics, and publish it on our website:

Sebastian Brezina: Teleparallel gravity through Noether’s second theorem
Lorenzo Fatibene, Pietro Isaia: A variational framework for linearized theories
Renata Ferent: Navigation problems in Finsler geometry
Paul Hafemann: A real Clifford algebra approach to Plebanski gravity
Roald Heinrich Ivask: Spacetime energy in general parallel gravity
Martin Kováč: Hamiltonian formulation of teleparallel gravity with the Nieh–Yan term
Paul Martin Kull: Bianchi cosmologies with four-dimensional groups of motion in new general relativity
Miroslav Kureš: Weil contact elements and contact correspondences
Peter Mészáros: Interaction of fluids described through relative motion and application to Bianchi type-I spacetime
Marcella Palese: Conservation laws and globality for the variation of local Noether strong currents
Marcella Palese, Ekkehart Winterroth: Topological obstructions in Lagrangian field theories
Stylianos Papadopoulos: Gravitational instantons in Spin(4) gauge formulation of space-time with Cartan-khronon
Mateo Paulišić: Galilean electrodynamics and duality in the galilean limit
Margaréta Verebesová: Connected component structure for automorphism groups of Weil algebras and its applications
Hartwig Winterroth: Towards General Relativity as a generalized Yang-Mills theory

Social Program

A tour of the historic center of Brașov will be organized on Monday, August 17, in the afternoon. An excursion to the surroundings of Brașov, followed by the conference dinner, will take place on Wednesday, August 19. A short trip to Tâmpa Mountain (by cable car or on foot), offering breathtaking panoramic views of Brașov, will be organized on Friday, August 21, after the official closing of the conference. Participants arriving one day earlier are welcome to join an informal mountain hike on Sunday, August 16.