Gauge theory and symplectic structures

par PARTHA SARATHI GHOSH (Institut de Mathématiques de Jussieu - Paris Rive Gauche)

vendredi 16 oct. 2026, 11:15 → 12:15 Europe/Paris

112 (ICJ)

Motivated by the problem of finding obstructions to the existence of symplectic structures in dimensions higher than four, I present an elliptic system of equations on almost Hermitian manifolds that reduces to the Seiberg-Witten equations with Taubes' perturbation in dimension four. To define the equations, one needs to choose a spin-c structure on the manifold. We prove that the system has index zero for the canonical spin-c structure. Moreover, on almost Kähler (symplectic) manifolds, we construct a canonical solution and prove its transversality with large values of the perturbation parameter, while in dimension six we obtain uniqueness of the canonical solution under a certain assumption. We also show that on Kähler manifolds the equations reduce, under a natural ansatz, to the vortex equations. These results provide evidence toward a possible higher dimensional gauge theory capable of producing invariants of almost complex structures and ultimately, obstructions to compatible symplectic forms.