We formulate and solve a $G$-equivariant analogue of the Seiberg-Witten equations on any irreducible Hermitian symmetric space $G/K$. We use a canonical $K$-invariant form to define a twisted Hodge operator generalising self-duality to higher dimensions. The resulting equivariant Seiberg-Witten equations arise, for a compact $G/K$, as conditions for the absolute minima of an energy functional. Their linear part has a representation theoretic interpretation: the same Weyl group data control the highest-weight realisation of the positive Dirac kernel in the compact case and its discrete-series realisation on the non-compact dual. This space of harmonic spinors collapses almost entirely by the quadratic conditions of the monopole equation. In the compact case, the equivariant moduli set, when non-empty, is a single gauge class, identifying with the $G$-invariant part of Bradlow's trichotomy for vortices, while in the non-compact case, every irreducible solution sits outside the discrete series.
Motivated by the problem of finding obstructions to the existence of symplectic structures in dimensions higher than four, we introduce 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 nee...
We study the notion of generalized Kahler Bismut--Hermitian--Einstein (GKBHE) manifolds, namely generalized K\"ahler manifolds for which both Bismut Ricci forms vanish, and we develop a theory for such structures on compact threefolds. We organize the analysis according to the horizontal rank, an invariant taking value...
We develop a non-Abelian Hodge-theoretic refinement of asymptotic limits in type-$A$ class-$\mathcal S$ theories, enriching the geometric description of the degenerating UV curve with gauge-theoretic data encoded by the associated Hitchin system. A degeneration of the UV curve produces a long plumbing tube, which we eq...
In this article, we give a global construction of the Virasoro conformal blocks on Teichm\"uller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which enc...
Guillaume Baverez, C. Guillarmou, Antti Kupiainen et al.· 1 citation
We construct non-flat non-Hermitian Yang--Mills connections with zero Einstein constant on smoothly trivial rank-two bundles over compact K\"ahler surfaces, with positive harmonic metrics and stable induced and adjoint holomorphic bundles. Thus vanishing Chern classes do not force flatness even under stability on both...
We solve a well-known question of Polterovich from 2002 regarding the rigidity of Hamiltonian diffeomorphisms in Hofer's metric in dimension two: every non-trivial Hamiltonian diffeomorphism of a surface of genus at least one has all positive iterations separated from the identity in Hofer's metric. In higher dimension...
Habib Alizadeh, E. Shelukhin· 0 citations
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