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 coming from physics and/or gauge theories.
Currently I am interested in/working on the following projects:
The L2-geometry of moduli spaces in gauged non-linear sigma models (also referred to as Hamiltonian Gromov–Witten theory).
This is a joint work with Nuno Romão.
Construction of SU(N) monopoles with arbitrary symmetry breaking via the Nahm transform.
This is a joint work with Benoit Charbonneau.
Construction of SU(N) instantons on asymptotically locally flat (ALF) black hole metrics via solving Kazdan–Warner type equations.
This is a joint work with Gonçalo Oliveira and a continuation of a previous project.
Preprints of these projects are coming "soon".
I am also learning about Higgs bundles, gauge theory on G2 and Spin(7) manifolds, and supersymmetry.
Abstract. 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
Abstract. 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)
Abstract. 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.
Ákos Nagy and Gonçalo Oliveira:From vortices to instantons on the Euclidean Schwarzschild manifold, submitted (2017)
From vortices to instantons on the Euclidean Schwarzschild manifold
Abstract. The first irreducible solution of the SU(2) self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on ℝ4 were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on ℝ4, the case of the ES manifold has resisted many efforts for the past 40 years.
By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES manifold, we obtain: a complete description of a connected component of the moduli space of unit energy SU(2) instantons; new examples of instantons with non-integer energy and non-trivial holonomy at infinity; a complete classification of finite energy, spherically symmetric, SU(2) instantons.
As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin.