Generalizing symplectic topology from 1 to 2 dimensions and beyond

Classical physics essentially divides into classical mechanics and classical field theory. The ODEs from classical mechanics may be described using symplectic geometry. From the 1980s on, symplectic geometry flourished following the discovery of so-called elliptic methods, like Gromov’s pseudo-holomorphic curve theory and Floer theory. These techniques led to powerful analytic tools that revealed deep connections between dynamics, geometry, and topology. On the other hand, classical field theories describe systems whose states depend on several space and/or time variables and therefore are governed by PDEs. A natural question is whether a generalization of elliptic methods still exists in this framework. One approach is to reinterpret the PDEs as infinite-dimensional symplectic systems ([FL23b]), but this requires singling out one variable as “time”, thereby breaking the symmetry between space–time coordinates. Finite-dimensional alternatives, such as multisymplectic and polysymplectic geometry, preserve this symmetry. However, the associated field equations are typically not elliptic, which obstructs the use of the analytic methods that have been so successful in symplectic geometry. This thesis introduces a framework for studying field theories which is both symmetric in the coordinates on the domain and provides elliptic equations. By restricting to complex manifolds, we are able to regularize the equations from poly- and multisymplectic geometry. The pseudo-holomorphic curves from symplectic topology turn into pseudo-Fueter curves and correspondingly Floer maps may be defined. Our main result is Theorem 4.2.2, which provides a lower bound on the number of periodic solutions to the field equations, similar to the Arnold conjecture in symplectic geometry. The proof covers the Fredholm and compactness theories for the moduli space of Floer maps and establishes C0-bounds. Other notable results from the thesis are the extension of the compactness results to more general target manifolds (Theorem 5.1.2), and the Darboux-type theorems for both the poly- and multisymplectic cases (Corollary 3.2.3 and theorem 6.5.3).