I am mainly interested in geometric analysis and its applications to gauge theory and mathematical physics. More concretely, I usually deal with elliptic PDE's arising from physical gauge theories.
Currently I am working on 4 main projects:
The construction and the moduli space of BPS monopoles with arbitrary symmetry breaking for the classical (compact) Lie groups. This is a joint work with Benoit Charbonneau.
The geometric-analytic properties of Majorana spinors on closed spinc manifolds.
The constraction of instantons on the Euclidean Schwarzschild manifold. This is a joint work with Gonçalo Oliveira.
The geometry of the moduli spaces of gauge non-linear sigma models. This is a joint work with Nuno Romão.
Preprints of these projects are coming "soon"!
I am also trying to learn about Higgs bundles, gauge theory on G2 and Spin(7) manifolds, and supersymmetry. If you know a good project in these fields (or in any other fields really) that involves (elliptic) geometric PDE's, shoot me an email!
Ginzburg–Landau fields are the solutions of the Ginzburg–Landau equations which depend on two positive parameters, α and β. We give conditions on α and β for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in ℝ2, spheres, tori, etc.) with de Gennes–Neumann boundary conditions. We also prove that, for each such manifold and all positive α and β, Ginzburg–Landau fields exist for only a finite set of energy values and the Ginzburg–Landau free energy is a Palais–Smale function on the space of gauge equivalence classes.
Ákos Nagy:The Berry connection of the Ginzburg–Landau vortices, Communications in Mathematical Physics, 350(1), 105-128 (2017)
The Berry connection of the Ginzburg–Landau vortices
We analyze 2-dimensional Ginzburg–Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit.
Gábor Etesi and Ákos Nagy:S-duality in Abelian gauge theory revisited, Journal of Geometry and Physics 61, 693-707 (2011)
Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub–NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a non-trivial curvature correction to the modular weights of the partition function. But S-duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter non-trivial difficulties stemming from original non-compact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case.
Benoit Charbonneau and Ákos Nagy:Monopoles with non-maximal symmetry breaking,
Ákos Nagy:Concentrating Majorana spinors on Riemannian manifolds.