Using Lagrangian Multiforms on Integrable Lattices to Investigate Correspondence of Manifolds

Manifolds are used everywhere within math and physics. As a strong example, within string theory, it is arguable that by better understanding compactification of manifolds, the ability to constrain the theory would be vastly increased. Also, within mathematics, by better understanding the equivalence of manifolds in high dimensions (n>4), more correspondences would be establishable. Despite the clear application, direct progress is slow, as manifolds, in generality, are difficult to relate concretely past n>4 dimensions. However, there are still many existing mathematical tools that establish correspondces through invariants such as the Euler characteristic and the cohomology group.

In the first part of this work, connections to geometric reduction are introduced. By utilizing such introduction on the geometry of integrable discrete lattices, an incredible generation of structure appears. Using this structure, this work explores how, generally, two manifolds may be related geometrically and topologically. A specific example - periodic reduction of the linearized integrable KdV lattice equation first introduced by Frank Nijohff is re-introduced, and then discussed topologically. By exploring the KdV equation, it is shown that the reduction has a geometric correspondence. The geometric correspondence, coming precisely from lagrangian multiforms, through the extended Euler-Lagrange equations (Nijhoff), shows concretely that it's possible to relate lattice equations to their underlying geometric structure.

In the second part, the KdV reduction is generalized to higher dimensions, demonstrating the power of the integrability criterion. Upon exploring this reduction, findings in homotopy theory and differential geometry are compared to findings from the generalization, with the relation explored. As a final example, geometric compactification of (3+1) dimensional GR to (2+1) dimensional GR coupled with E&M from Masato Shimono is re-introduced and considered. By applying the theory introduced in the work, it is found to be consitent with the findings of Kaluza-Klein compactification, thus hinting at an application past integrable systems. Further work within lagrangian multiforms is dicussed with, with an aside on compactification.