From symplectic to multisymplectic topology

Over the past four decades, symplectic geometry has established itself as a central mathematical area of research. At its core lies Hamiltonian mechanics, a reformulation of Newton’s equations that builds on the concept of energy of physical systems. Instead of studying solutions of second order ordinary differential equations, after applying so-called Legendre transforms, dynamical problems can be geometrically described as flows on even-dimensional phase spaces. This geometric viewpoint has proven extraordinarily powerful, not least because it allows the use of sophisticated tools from partial differential equations (PDEs) and topology to address fundamental questions about dynamics, such as the existence of periodic motions.

Oliver Fabert and his PhD student Ronen Brilleslijper are working to extend these successful ideas far beyond their traditional setting. Their research focuses on multisymplectic geometry, a higher-dimensional generalization of symplectic geometry that naturally arises in Hamiltonian field theory. Whereas classical mechanics describes systems evolving along a one-dimensional time line, many physical and biological phenomena—ranging from soap films over chemical reactions to the spread of diseases—are governed by fields defined over higher-dimensional domains. In these cases, Newton’s equation is replaced by nonlinear elliptic equations such as the nonlinear Laplace equation, which, among other phenomena, describes minimal surfaces and steady states of reaction–diffusion systems.

In symplectic geometry, one of the most influential tools is Floer theory, developed in the late 1980s. Floer theory provides a systematic method for proving the existence of time-periodic solutions in Hamiltonian systems. Its key insight is that the gradient flow lines of an action functional can be interpreted as (pseudo-)holomorphic curves, defined using a compatible complex structure. These are maps satisfying a Cauchy-Riemann type equation, which opens the door to powerful elliptic PDE methods.

Multisymplectic geometry aims to play a similar foundational role for Hamiltonian field theory, but extending Floer-theoretic ideas to this setting is highly nontrivial: Even if the field equations are elliptic themselves, in the standard De Donder-Weyl approach the Hamiltonian field equations are no longer elliptic due to the presence of an infinite-dimensional kernel. As a consequence of the resulting compactness problems, standard techniques from symplectic geometry do not generalize properly from the symplectic to the multisymplectic setting.
In their recent work, Fabert and Brilleslijper introduce a novel multisymplectic framework tailored to overcome these obstacles, focusing in particular on the case of two-dimensional domains. By incorporating extra complex structures, they construct a novel multisymplectic framework. Remarkably, the gradient lines of the resulting action functional now turn out to be pseudo-Fueter curves, a higher-dimensional analogue of pseudo-holomorphic curves, defined using a compatible almost quaternionic structure. 

Since the underlying Fueter equation from almost quaternionic geometry is elliptic in the same way as the Cauchy-Riemann equation from almost complex geometry, their work provides a fascinating pathway for importing elliptic PDE methods—so successful in symplectic geometry—into the realm of Hamiltonian field theory. Beyond this central result, the authors establish foundational geometric properties of their framework, including a multisymplectic version of the Darboux theorem.

Fabert and Brilleslijper wrote up their findings in their recent papers "Generalizing symplectic topology from 1 to 2 dimensions" and “Floer sections in multisymplectic geometry”. The latter one has just been accepted for publication in the International Journal of Geometric Methods in Mathematical Physics (IJGMMP).