Hitchin-Kobayashi correspondence, quivers, and vortices

A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kaelher, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin--Kobayashi correspondence for twisted quiver bundles over a compact Kaehler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian--Einstein equations.Comment: 28 pages; larger introduction, added references for the introduction, added a short comment in Section 1, typos corrected, accepted in Comm. Math. Phy

Coupled equations for Kähler metrics and Yang-Mills connections

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kahler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kahler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K-energy and geodesic stability. We finish by giving some examples of solutions.Comment: 61 pages; v2: introduction partially rewritten; minor corrections and improvements in presentation, especially in Section 4; added references; v3: To appear in Geom. Topol. Minor corrections and improvements, following comments by referee

Noncommutative Poisson vertex algebras and Courant-Dorfman algebras

We introduce the notion of double Courant-Dorfman algebra and prove that it satisfies the so-called Kontsevich-Rosenberg principle, that is, a double Courant-Dorfman algebra induces Roytenberg's Courant-Dorfman algebras on the affine schemes parametrizing finite-dimensional representations of a noncommutative algebra. The main example is given by the direct sum of double derivations and noncommutative differential 1-forms, possibly twisted by a closed Karoubi-de Rham 3-form. To show that this basic example satisfies the required axioms, we first prove a variant of the Cartan identity $[L_X,L_Y]=L_{[X,Y]}$ for double derivations and Van den Bergh's double Schouten-Nijenhuis bracket. This new identity, together with noncommutative versions of the other Cartan identities already proved by Crawley-Boevey-Etingof-Ginzburg and Van den Bergh, establish the differential calculus on noncommutative differential forms and double derivations and should be of independent interest. Motivated by applications in the theory of noncommutative Hamiltonian PDEs, we also prove a one-to-one correspondence between double Courant-Dorfman algebras and double Poisson vertex algebras, introduced by De Sole-Kac-Valeri, that are freely generated in degrees 0 and 1.Comment: v3: New author added. Major revision. Comments are welcome

Gravitating vortices, cosmic strings, and the Kähler--Yang--Mills equations

In this paper we construct new solutions of the K ahler{Yang{Mills equations, by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, that we call gravitating vortex equations, describe abelian vortices on the Riemann surface with back reaction of the metric. As a particular case of these gravitating vortices on the Riemann sphere we nd solutions of the Einstein{Bogomol'nyi equations, which physically correspond to Nielsen{ Olesen cosmic strings in the Bogomol'nyi phase. We use this to provide a Geometric Invariant Theory interpretation of an existence result by Y. Yang for the Einstein{Bogomol'nyi equations, applying a criterion due to G. Sz ekelyhidi.Partially supported by the Spanish MINECO under the ICMAT Severo Ochoa grant No. SEV-2011- 0087, and under grant No. MTM2013-43963-P. The work of the second author has been partially supported by the Nigel Hitchin Laboratory under the ICMAT Severo Ochoa grant. The research leading to these results has received funding from the European Union's Horizon 2020 Programme (H2020-MSCA-IF-2014) under grant agreement No. 655162, and by the European Commission Marie Curie IRSES MODULI Programme PIRSES-GA-2013-612534.Peer reviewe

(0,2) Mirror Symmetry on homogeneous Hopf surfaces

In this work we find the first examples of (0,2) mirror symmetry on compact non-K\"ahler complex manifolds. For this we follow Borisov's approach to mirror symmetry using vertex algebras and the chiral de Rham complex. Our examples of (0,2) mirrors are given by pairs of Hopf surfaces endowed with a Bismut-flat pluriclosed metric. Requiring that the geometry is homogeneous, we reduce the problem to the study of Killing spinors on a quadratic Lie algebra and the construction of associated $N=2$ superconformal structures on the superaffine vertex algebra, combined with topological T-duality.Comment: 55 pages. Minor changes and corrections. References update