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U. Bruzzo

Publications and source records attributed to U. Bruzzo.

15 recordsLinked to original sources
arXiv2020

Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free $\mathcal O_X$-module $\mathcal A$ which has a Lie algebroid structure. Here $X$ is either a complex manifold or a regular scheme over an algebraically closed field $k$. One spectral sequence can be associated with $\mathcal A$ by choosing a global section $V$ of $\mathcal A$, and considering a Koszul complex with a differential given by inner product by $V$. This spectral sequence is shown to degenerate at the second page by using Deligne's degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $\mathcal D_E$ of a holomolorphic vector bundle $E$ on a complex manifold. If $V$ is a differential operator on $E$ with scalar symbol, i.e, a global section of $\mathcal D_E$, we associate with the pair $(E,V)$ a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($E=0$) case

math.KT↗
arXiv2018

D-branes, surface operators, and ADHM quiver representations

A supersymmetric quantum mechanical model is constructed for BPS states bound to surface operators in five dimensional SU(r) gauge theories using D-brane engineering. This model represents the effective action of a certain D2-brane configuration, and is naturally obtained by dimensional reduction of a quiver $(0,2)$ gauged linear sigma model. In a special stability chamber, the resulting moduli space of quiver representations is shown to be virtually smooth and isomorphic to a moduli space of framed quotients on the projective plane. A precise conjecture relating a K-theoretic partition function of this moduli space to refined open string invariants of toric lagrangian branes is formulated for conifold and local P^1 x P^1 geometries.

hep-th↗
arXiv2008

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and A a Lie algebroid on M equipped with a compatible infinitesimal G-action. Out of these data we construct an equivariant Lie algebroid cohomology and prove for compact G a related localization formula. As an application we prove a Bott-type formula.

math.DG↗
arXiv2005

Normal bundles to Laufer rational curves in local Calabi-Yau threefolds

We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.

math-ph↗
arXiv2005

Semistability vs. nefness for (Higgs) vector bundles

According to Miyaoka, a vector bundle E on a smooth projective curve is semistable if and only if a certain numerical class in the projectivized bundle PE is nef. We establish a similar criterion for the semistability of Higgs bundles: namely, such a bundle is semistable if and only if for every integer s between 0 and the rank of E, a suitable numerical class in the scheme parametrizing the rank s locally-free Higgs quotients of E is nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the above mentioned classes is equivalent to the semistability of the Higgs bundle E together with the vanishing of the discriminant of E.

math.AG↗
arXiv2004

Relatively stable bundles over elliptic fibrations

We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah's and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.

math.AG↗
arXiv2001

A Fourier transform for sheaves on Lagrangian families of real tori

We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian submanifolds of $X$ and holomorphic vector bundles with compatible unitary connections supported on complex submanifolds of the relative Jacobian of $X$ (suitable conditions being verified on both sides).

math.DG↗
arXiv2000

Complex Lagrangian embeddings of moduli spaces of vector bundles

By means of a Fourier-Mukai transform we embed moduli spaces of stable bundles on an algebraic curve C as isotropic subvarieties of moduli spaces of mu-stable bundles on the Jacobian variety J(C). When g(C)=2 this provides new examples of special Lagrangian submanifolds.

math.AG↗
arXiv1998

Categorial mirror symmetry for K3 surfaces

We study the structure of a modified Fukaya category ${\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of $\fF(X)$ is equivalent to a subcategory of the derived category ${\bold D}(\hat X)$ of coherent sheaves on the mirror K3 surface $\hat X$.

math-ph↗
arXiv1998

Mirror symmetry on K3 surfaces as a hyperkaehler rotation

We show that under the hypotheses of Strominger, Yau and Zaslow's paper, a mirror partner of a K3 surface $X$ with a fibration in special Lagrangian tori can be obtained by rotating the complex structure of $X$ within its hyperkähler family of complex structures. The same hypotheses force the B-field to vanish.

math-ph↗
arXiv1996

A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle $F$ is stable and has the same Euler characteristic as $F$.

alg-geom↗
arXiv1994

Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.

hep-th↗
arXiv1994

Transformation de Fourier-Mukai sur les Surfaces Hyperkählériennes

Given two compact hyperkähler surfaces $X$ and $Y$ and a holomorphic vector bundle $Q$ on $X\times Y$, which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on $X$ to vector bundles on $Y$. If $X$ and $Y$ are dual complex tori, this transform maps instantons on $X$ to instantons on $Y$. After a quick review of these results, we define a Fourier-Mukai transform in the case when $X$ is a K3 surface, and study the behaviour of instantons on $X$ under this transform. Hard copies may be mailed on request

dg-ga↗