Research

The content of a classical field theory is usually determined by a fiber bundle $\pi \colon E \rightarrow \mathcal{M}$ and a (first-order) Lagrangian density $\mathcal{L} \colon J^1 \pi \rightarrow \bigwedge^n \mathrm{T}^\ast \mathcal{M}$, where $n = \dim \mathcal{M}$. The main objective of the multisymplectic framework is to describe, from a geometric perspective, the information we may gather from the field within the manifold $J^1 \pi$. In particular, we can show that there is a canonically determined closed differential form $\Omega_\mathcal{L} \in \Omega^{n+1}(J^1 \pi)$ which encodes the equations of motion. This is the multisymplectic form. Just as a symplectic form induces a Poisson algebra on functions, the multisymplectic form induces a graded bracket $$ \{\cdot, \cdot \} \colon \Omega^{a}_H(J^1 \pi) \otimes \Omega^{b}_H(J^1 \pi) \rightarrow \Omega^{(a + b) - (n -1)}_H(J^1 \pi)\,.$$ on a subfamily of differential forms called Hamiltonian forms. These forms have been widely studied in the literature in relation to symmetries, conserved charges, the Peierls bracket, and other topics.

My research focuses on studying the properties of the graded algebra induced by a multisymplectic form, together with the geometry encoding it. In particular, we are using higher and graded Poisson and Dirac structures as the geometries encoding such brackets. These notions are relatively new (although they were not introduced by us). In \cite{di2025j.math.phys}, the first article of my PhD, we summarized all the notions appearing in the literature and, more importantly, proved that specifying such a structure is equivalent to specifying a bracket of forms satisfying the properties of the bracket above. This, to us at least, justifies the study of these structures as the structures encoding the graded brackets of the multisymplectic framework.

More recently, in \cite{li2026int.j.geom.methodsmod.phys}, we have given a way of describing the equations of motion in such a manifold as $$\psi^\ast(\mathrm{d} \alpha) = \left( \{\alpha, \mathcal{H}\} + \mathrm{d} \alpha\right) \circ \psi \,,$$ for some section $\psi \colon \mathcal{M} \rightarrow P$, in a similar way to the Poisson-manifold case. Here, $P \rightarrow \mathcal{M}$ is a particular fiber bundle endowed with a graded Poisson or graded Dirac structure (in the case above, $J^1 \pi \rightarrow \mathcal{M}$). By considering this question, we showed that the equation above only makes sense for certain degrees of forms, but that there is a particular subclass of Hamiltonian forms, which we call special Hamiltonian forms and denote by $\widetilde \Omega^a_H(P) \subset \Omega^a_H(P)$ within the full graded algebra. These forms satisfy the following properties:

• The formula above is still well defined (and hence the exterior differential of each form is completely determined).
• They define a subalgebra with respect to the Poisson bracket.

This subalgebra appears to be important. In particular, in the cited article we compute it for the specific case of Yang--Mills theories.

Detailed explanation coming soon.

Detailed explanation coming soon.