2nd International Workshop on Geometric Field Theory
July 6-8, Warsaw
July 6-8, Warsaw
Title: Instantons from projective plane
Abstract: I will show how combining some 19th century projective geometry with integrable harmonic maps equations leads to a construction of a new instanton, ultimately answering (negatively!) a question of S. T. Yau.
Title: Relative derivations, classification problems and PDEs Slides
Abstract: I will explain a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations. The concept of relative algebroid also unifies the theories of algebroids and (formal) PDEs. This talk is based on joint work with my former PhD student Wilmer Smilde (Cornell).
Title: Multicontact brackets and multisymplectization Slides
Abstract: While contact geometry provides robust tools for mathematical physics, extending these structures to multicontact manifolds opens powerful new avenues for modeling dissipation. In this talk, I will introduce a novel graded bracket of forms on multicontact manifolds. We will explore how this bracket extends well-known notions in contact geometry by satisfying a graded Jacobi identity, alongside two distinct versions of the Leibniz rule, including a weak formulation.
Moving from pure geometry to physical application, the presentation will develop the multisymplectization of multicontact structures. This connects our new brackets to multisymplectic geometry, allowing us to derive field equations in an abstract context. I will then demonstrate how the Jacobi bracket provides a natural framework to study the evolution of physical observables and directly address dissipation phenomena. Finally, we will apply these abstract geometric to classical dissipative field theories.
Title: Generalized JT gravity from Chern-Simons reduction
Abstract: II will describe the dimensional reduction of AdS3/Z2 gravity formulated as an so(2, 2) Chern–Simons theory on a three-manifold with toroidal boundary. The resulting theory is a two-dimensional BF-like model on a disk together with a one-dimensional boundary dynamics on its S1 boundary. Under suitable boundary conditions the universal one-dimensional action reproduces the standard Drinfel’d–Sokolov restriction to coadjoint orbits of the Virasoro algebra, among them the Schwarzian dynamics associated with Jakiw-Teitelboim gravity. I will also discuss how the so(2, 2) algebra of the 3D Chern–Simons model naturally leads to current-dressed Kac–Moody extensions of the boundary dynamics.
Title: Beyond Riemann: Finsler Spacetime and the Geometry of Cosmic Acceleration Slides
Abstract: Finsler geometry extends the Riemannian framework by adopting a fully general notion of arc length. In this talk, I will outline key physical motivations for modeling spacetime within a Finslerian framework, emphasizing its reach beyond standard Riemannian geometry. Then, I will highlight the interplay between Finsler gravity and metric-affine gravity, and the mutual insights they provide. At last, I will introduce a Finsler gravitational model that, under cosmological symmetry, naturally produces exponential expansion—without invoking a cosmological constant or additional fields.
Title: Reconstruction in Lie-Poisson reduction for field theories Slides
Abstract: Any reduction procedure naturally poses a reconstruction problem: given a solution of the reduced system, can one recover a corresponding solution of the original unreduced problem? While reconstruction is always possible in mechanics, in Euler–Poincaré and Lagrange–Poincaré reduction for field theories, the obstruction to reconstruction is characterised by the curvature of a connection constructed from the reduced solution. Far less is known, however, in the Poisson–covariant formulation of Hamiltonian field theories. In this talk, we present explicit reconstruction conditions for Lie–Poisson systems and extend these results to the case where the symmetry group is a subgroup of the structure group of the configuration space.
Title: Double groupoids and composites
Abstract: We will present an application of double groupoids to the characterization of uniformity and homogeneity of composite materials.
Title: Generalising Floer’s proof of the Arnold conjecture from symplectic to multisymplectic geometry Slides
Abstract:In symplectic geometry, Floer theory is the fundamental framework for establishing the existence of time-periodic solutions in Hamiltonian systems. For closed symplectic manifolds, the Arnold conjecture predicts a lower bound on the number of such solutions in terms of topological invariants of the underlying manifold. A key insight is that the L^2-gradient flow lines of the symplectic action functional correspond to pseudo-holomorphic curves, allowing the application of elliptic PDE techniques. Focusing on the two-dimensional setting, we introduce a novel multisymplectic framework that enables the extension of elliptic methods from symplectic geometry. Within this framework, we formulate a generalisation of the Arnold conjecture and show that the L^2-gradient flow lines of the associated multisymplectic action functional are pseudo-Fueter curves defined with respect to a compatible almost hyperkähler structure.
Title: To be or not to be... multisymplectic Slides
Abstract: There is a vast literature about the role of multisymplectic geometry in the Hamiltonian formulation of classical field theory. It is considered to be the field-theoretic counterpart of the symplectic structure of a phase space in mechanics. But is it really the case? In my talk, I shall examine different approaches to Hamiltonian field theory in order to try to answer this question.
Title: Black Hole Thermodynamics from Multisymplectic
Abstract: We analyze the Noether currents and surface charges associated with spacetime diffeomorphisms in General Relativity using the multisymplectic geometry of jet bundles and the variational bicomplex.
Title: Variational principles in physics (how to use them and how not to use them) Slides
Abstract: The history of variational principles in physics will be discussed. The role of boundary conditions will be thoroughly analyzed. In particular, the fundamental difference between mathematical "optimization problems" and the physical approach to field dynamics based on the variational principles of Maupertuis and Hamilton will be strongly emphasized.
Title: The Arnold-Liuoville Theorem for field theories
Abstract: A Hamiltonian system on a $(2n)$-dimensional symplectic manifold $(M, \omega, H)$, is said to be Liouville integrable if the Hamiltonian can be written in terms of an independent family of conserved quantities $(f_{i})_{i=1}^n: M \to \mathbb{R}^n$ that commute with respect to the Poisson bracket. Then, under some regularity conditions, the level sets of $(f_{i})_{i=1}^n$ are a foliation of $(M, \omega)$ by Lagrangian submanifolds diffeomorphic to Abelian Lie groups $(\mathbb{T^k}\times\mathbb{R}^{n-k})$. The Hamiltonian dynamics is tangent to this foliation. Moreover, one can find the so-called angle--action coordinates, which are Darboux coordinates for $\omega$. In these special coordinates, the equations of motion are linear with constant coefficients in each of the submanifolds. This is the Arnold--Liouville theorem (in the non-compact case). We will discuss its generalization to Field Theories in terms of conserved currents on the multisymplectic framework. Here, we obtain a foliation by Abelian lie groups of the fiber at each point of spacetieme. We are also able to simplify the Hamilton-de Donder-Weyl equations if the Hamiltonian is preserved by the Hamiltonian vector fields of the currents.
Title: Generalised differentiable structure Slides
Abstract:In this talk, we present a method for extending the notion of a differentiable manifold to arbitrary subsets of Rn. Using tools from groupoid theory and the characteristic distribution, we show how these sets can be decomposed into integrable foliations arising from singular distributions. These tools could enable us to endow a set with a differentiable structure, in the absence of any regularity properties. In various fields of science, particularly in Hamiltonian and Lagrangian mechanics, it is common to want to work with sets that do not satisfy the often overly restrictive conditions of a differentiable manifold. This raises the need to develop a more flexible geometric framework that allows for the treatment of such singular structures.
Title: Lie Theorem for superposition rules for PDEs revisited Slides
Abstract:A superposition rule is a time-independent map through which the general solution of a non-autonomous system of partial differential equations in normal form can be expressed in terms of a finite family of generic particular solutions and some constants related to the initial conditions. The celebrated Lie's Theorem (1893) states that such a system admits a superposition rule if and only if it is a Lie system; that is, a time-dependent vector field taking values in a finite-dimensional Lie algebra. While Lie's original proof contained some technical issues, these were corrected by the geometric approach of Cariñena, Grabowski and Marmo via foliation theory.
In this talk, we reformulate the construction by Cariñena, Grabowski and Marmo in terms of germs, yielding a natural one-to-one correspondence between superposition rules and certain regular foliations. We further show that the space of initial conditions can be replaced by other manifolds of the same dimension, which may be regarded (locally) as leaf spaces of the associated regular foliations. If time permits, we will also sketch this correspondence in terms of Lie groupoids.
Title: Multisymplectic structures as G-structures Slides
Abstract:Multisymplectic geometry provides a geometric framework to express the equations of motion of classical field theories, in analogy with how symplectic geometry is used in classical mechanics. A multisymplectic form encodes key structures of a given theory, including symmetries, conserved quantities, and the Poisson bracket.
In the study of these structures, the use of adapted (or Darboux) coordinates is of central importance. However, unlike in symplectic geometry, the closedness of a multisymplectic form is not sufficient to guarantee the existence of flat coordinates. Indeed, there are examples of closed forms of constant linear type that are not flat, such as those arising in $G_2$-structures.
These issues can be understood in a unified way through the theory of G-structures. In this talk, I will introduce the basic notions of G-structures and structure tensors. I will then explain how this framework can be used to construct forms whose integrability depends strictly on conditions of order $k$, for arbitrary $k$. Finally, if time permits, I will discuss some possible directions and ideas related to integrability.
Title: Manifolds with boundaries, field theory and multisymplectic formalism Slides
Abstract:In this talk, I will present a new formalism for studying the field theories on manifolds with boundary. Based on the ideas of relative cohomology [Margalef-Bentabol & Villaseñor, 2021], I will extend the definition of multisymplectic structures to manifolds with boundary. We will see how this structure reproduces the field equations of variational principles with boundary. I will explain how to generalize the observables, the graded Poisson brackets, and the conserved charges to manifolds with boundary. Moreover, I will present the Lagrangian formalism, deriving the Poincaré-Cartan form and the Euler-Lagrange equations. Finally, I will illustrate the formalism with several examples.
Title: Evolution operators for describing linearly singular systems Slides
Abstract: Linearly singular systems provide a unified geometric framework to describe general first-order singular differential equations on a manifold, naturally capturing dynamics that cannot be written in standard normal form due to a linear operator multiplying the derivatives. In this poster, we review the formal geometric theory of these systems and focus on a relevant particular case: their description in terms of a vector field along a map.
This formulation generalizes the so-called $K$ evolution operator utilized in classical and contact mechanics. We will demonstrate how these evolution operators provide an alternative description of the dynamics and how this framework is useful for relating constraint functions arising from different, but related, implicit differential equations.
Title: The multisymplectic structure of the Universal Covariant Phase Space Slides
Abstract:In this talk I will introduce the spaces in which the order r momenta of a field theory live, the affine duals of the order r jet bundles. It is already known that one can take the projective limit of the sequence of jet bundles of all orders to get the infinite dimensional jet bundle. The passage to this projective limit means, among other things, that the differential forms on the infinite jet bundle naturally acquire a bigrading, and thus split into horizontal and vertical forms. Here I will explore what happens when one considers the now direct limit of the system of all finite order phases spaces, and how the multisymplectic structures that exist for finite orders are preserved in the limit. Furthermore, I will show how Hamiltonian field theories can naturally be formulated in this context.
Title: From Local Observables to Multilocal Observables via Configuration Spaces Slides
Abstract: This talk describes a construction of multilocal algebraic structures from local geometric data. Given a vector bundle over a manifold, one can build equivariant bundles over powers of the manifold, interpreted as objects over configuration spaces with diagonals included. These bundles carry natural algebraic operations reflecting both pointwise multiplication and the combination of independent configurations.
I will explain how this framework leads to a free commutative algebra of multilocal observables generated by local fields. A skew-symmetric pairing on the local generators then extends uniquely to a Poisson bracket satisfying appropriate Leibniz rules. The construction provides a geometric and algebraic way to pass from local field-theoretic data to a Poisson algebra of observables.
The emphasis will be on the role of configuration spaces, diagonals, and equivariance under symmetric groups. The categorical language of two compatible tensor products will be used mainly as an organizing principle.
Title: Skinner-Rusk for action-dependent field theories Slides
Abstract:In this work, we present the application of the well known Skinner-Rusk formalism to the multicontact setting, a powerful framework providing geometrical structure suitable for the treatment of action-dependent classical field theories. The Skinner-Rusk formalism applied in the multicontact context, allows us to provide a combined version of both Lagrangian and Hamiltonian formalisms particularly suitable for the study and description of singular theories.