arXiv subjects

Mario Garcia-Fernandez

Publications and source records attributed to Mario Garcia-Fernandez.

36 records · Page 2Linked to original sources

arXiv2020

On the Kähler-Yang-Mills-Higgs equations

In this paper we introduce a set of equations on a principal bundle over a compact complex manifold coupling a connection on the principal bundle, a section of an associated bundle with Kähler fibre, and a Kähler structure on the base. These equations are a generalization of the Kähler-Yang-Mills equations introduced by the authors. They also generalize the constant scalar curvature for a Kähler metric studied by Donaldson and others, as well as the Yang-Mills-Higgs equations studied by Mundet i Riera. We provide a moment map interpretation of the equations, construct some first examples, and study obstructions to the existence of solutions.

math.DG↗

arXiv2019

T-dual solutions of the Hull-Strominger system on non-Kähler threefolds

We construct new examples of solutions of the Hull-Strominger system on non-Kähler torus bundles over K3 surfaces, with the property that the connection $\nabla$ on the tangent bundle is Hermite-Yang-Mills. With this ansatz for the connection $\nabla$, we show that the existence of solutions reduces to known results about moduli spaces of slope-stable sheaves on a K3 surface, combined with elementary analytical methods. We apply our construction to find the first examples of T-dual solutions of the Hull-Strominger system on compact non-Kähler manifolds with different topology.

math.DG↗

arXiv2019

Ricci flow, Killing spinors, and T-duality in generalized geometry

We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Hull-Strominger system. Our main result investigates a method to produce new solutions of the Ricci flow and the Killing spinor equations. For this, we consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrate that solutions of the equations are exchanged under this symmetry. As applications, we give a mathematical explanation of the dilaton shift in string theory and prove that the Hull-Strominger system is preserved by T-duality.

math.DG↗

arXiv2018

Gravitating vortices and the Einstein--Bogomol'nyi equations

In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations. As a particular case of the gravitating vortex equations on $\mathbb{P}^1$, we find the Einstein--Bogomol'nyi equations, previously studied in the theory of cosmic strings in physics. We prove two main results in this paper. Our first main result gives a converse to an existence theorem of Y. Yang for the Einstein--Bogomol'nyi equations, establishing in this way a correspondence with Geometric Invariant Theory for these equations. In particular, we prove a conjecture by Y. Yang about the non-existence of cosmic strings on $\mathbb{P}^1$ superimposed at a single point. Our second main result is an existence and uniqueness result for the gravitating vortex equations in genus greater than one.

math.DG↗

arXiv2018

Lectures on the Strominger system

These notes give an introduction to the Strominger system of partial differential equations, and are based on lectures given in September 2015 at the GEOQUANT School, held at the Institute of Mathematical Sciences (ICMAT) in Madrid. We describe the links with the theory of balanced metrics in hermitian geometry, the Hermite-Yang-Mills equations, and its origins in physics, that we illustrate with many examples. We also cover some recent developments in the moduli problem and the interrelation of the Strominger system with generalized geometry, via the cohomological notion of string class.

math.DG↗

arXiv2017

Stability data, irregular connections and tropical curves

We study a class of meromorphic connections $\nabla(Z)$ on $\mathbb{P}^1$, parametrised by the central charge $Z$ of a stability condition, with values in a Lie algebra of formal vector fields on a torus. Their definition is motivated by the work of Gaiotto, Moore and Neitzke on wall-crossing and three-dimensional field theories. Our main results concern two limits of the families $\nabla(Z)$ as we rescale the central charge $Z \mapsto RZ$. In the $R \to 0$ "conformal limit" we recover a version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for enumerative invariants), although with a different construction yielding new explicit formulae. In the $R \to \infty$ "large complex structure" limit the connections $\nabla(Z)$ make contact with the Gross-Pandharipande-Siebert approach to wall-crossing based on tropical geometry. Their flat sections display tropical behaviour, and also encode certain tropical/relative Gromov-Witten invariants.

math.AG↗

arXiv2016

Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry

We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable Killing spinor equations on a Courant algebroid. As an application, we propose a unifying framework for metrics with holonomy $\SU(3)$ and solutions of the Strominger system.

math.DG↗

arXiv2016

Moduli of $G_2$ structures and the Strominger system in dimension 7

We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. The resulting system of partial differential equations can be regarded as an analogue of the Strominger system in $7$-dimensions. We initiate the study of the moduli space of solutions and show that it is finite dimensional using elliptic operator theory. We also relate the associated geometric structures to generalized geometry.

math.DG↗

arXiv2016

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

In this paper we construct new solutions of the Kahler-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 find 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. Szekelyhidi.

math.DG↗

arXiv2016

Quantization of Hitchin's equations for Higgs Bundles I

We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit towards the solution of the Hitchin equation. We relate the existence of balanced metrics to the Gieseker stability of the Higgs bundle.

math.DG↗

arXiv2014

Balanced metrics on twisted Higgs Bundles

A twisted Higgs bundle on a Kähler manifold $X$ is a pair $(E,ϕ)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $ϕ\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were first considered by Hitchin when $X$ is a curve and $M$ is the tangent bundle of $X$, and also by Simpson for higher dimensional base. The Hitchin-Kobayashi correspondence for such pairs states that $(E,ϕ)$ is polystable if and only if $E$ admits a hermitian metric solving the Hitchin equation. This correspondence is a powerful tool to decide whether there exists a solution of the equation, but it provides little information as to the actual solution. In this paper we study a quantization of this problem that is expressed in terms of finite dimensional data and balanced metrics that give approximate solutions to the Hitchin equation. Motivation for this study comes from work of Donagi--Wijnholt on arXiv:1104.2610 concerning balanced metrics for the Vafa-Witten equations.

math.DG↗

arXiv2013

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.

math.DG↗

arXiv2013

Torsion-free generalized connections and Heterotic Supergravity

This work revisits the notions of connection and curvature in generalized geometry, with emphasis on torsion-free generalized connections on a transitive Courant algebroid, compatible with a generalized metric. Non-exact Courant algebroids have been considered recently by R. Rubio in the context of $B_n$-generalised geometry and arise naturally from the theory of generalized reduction of Burzstyn, Cavalcanti and Gualtieri. As an application, we provide a mathematical derivation of the equations of motion of heterotic supergravity, inspired by the work of Coimbra, Strickland-Constable and Waldram.

math.DG↗

arXiv2013

Deformations of complex structures and the coupled Kähler-Yang-Mills equations

In this work we define a deformation theory for the Coupled Kähler-Yang-Mills equations in arXiv:1102.0991, generalizing work of Székelyhidi on constant scalar curvature Kähler metrics. We use the theory to find new solutions of the equations via deformation of the complex structure of a polarised manifold endowed with a holomorphic vector bundle. We also study the deformations of the recent examples of Keller and Tønnesen-Friedman.

math.DG↗

arXiv2012

Solutions of the Strominger System via Stable Bundles on Calabi-Yau Threefolds

We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3) holonomy then the equations of motion derived from the heterotic string effective action are also satisfied by the solutions we obtain.

math.DG↗

arXiv2011

Limits of balanced metrics on vector bundles and polarised manifolds

We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter α, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of α, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einstein metric on E and a constant scalar curvature Kähler metric in c_1(L). For special values of α, limits of balanced metrics are solutions of a system of coupled equations relating a Hermitian-Einstein metric on E and a Kähler metric in c_1(L). For this, we compute the top two terms of the density of states expansion of the Bergman kernel of E \otimes L^k.

math.DG↗

arXiv2011

Coupled equations for Kähler metrics and Yang-Mills connections (Thesis)

We study equations on a principal bundle over a compact complex manifold coupling connections on the bundle with Kähler structures in the base. These equations generalize the conditions of constant scalar curvature for a Kähler 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 character and the Mabuchi K-energy. We explain their relationship to the algebro-geometric moduli problem for pairs consisting of a polarized variety and a holomorphic vector bundle.

math.DG↗

arXiv2010

Heterotic Non-Kahler Geometries via Polystable Bundles on Calabi-Yau Threefolds

In arXiv:1008.1018 it is shown that a given stable vector bundle $V$ on a Calabi-Yau threefold $X$ which satisfies $c_2(X)=c_2(V)$ can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux compactifications via non-Kahler deformation of Calabi-Yau geometries with polystable bundles. As an application, we obtain examples of non-Kahler deformations of some three generation GUT models.

hep-th↗