We classify nonabelian extensions of Lie algebroids in the holomorphic or algebraic category, and introduce and study a spectral sequence that one can attach to any such extension and generalizes the Hochschild-Serre spectral sequence associated to an ideal in a Lie algebra. We compute the differentials of the spectral sequence up to $d_2$

}, keywords = {Lie algebroids, nonabelian extensions, spectral sequences}, author = {Ugo Bruzzo and Igor Mencattini and Pietro Tortella and Vladimir Rubtsov} } @article {2013, title = {Symplectic instanton bundles on P3 and {\textquoteright}t Hooft instantons}, year = {2013}, note = {This preprint has been published with the title "Moduli of symplectic instanton vector bundles of higher rank on projective space P-3 " in CENTRAL EUROPEAN JOURNAL OF MATHEMATICS, Volume: 10, issue 4, Augst 2012, pages 1232-1245.}, institution = {arXiv:1312.5554 [math.AG]}, abstract = {We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $I^*_{n,r}$ of tame symplectic instantons is irreducible and has the expected dimension, equal to $4n(r+1)-r(2r+1)$. The proof is inherently based on a relation between the spaces $I^*_{n,r}$ and the moduli spaces of {\textquoteright}t Hooft instantons.}, url = {http://urania.sissa.it/xmlui/handle/1963/34486}, author = {Ugo Bruzzo and Dimitri Markushevich and Alexander Tikhomirov} } @article {2012, title = {On localization in holomorphic equivariant cohomology}, journal = {Central European Journal of Mathematics, Volume 10, Issue 4, August 2012, Pages 1442-1454}, number = {arXiv:0910.2019;}, year = {2012}, publisher = {Springer}, abstract = {We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu{\textquoteright}s localization formulas.}, keywords = {Lie algebroids}, doi = {10.2478/s11533-012-0054-2}, url = {http://hdl.handle.net/1963/6584}, author = {Ugo Bruzzo and Vladimir Rubtsov} } @article {2011, title = {Moduli of symplectic instanton vector bundles of higher rank on projective space $\\mathbb{P}^3$}, journal = {Central European Journal of Mathematics 10, nr. 4 (2012) 1232}, number = {arXiv:1109.2292v1;}, year = {2012}, note = {14 pages}, publisher = {SISSA}, abstract = {Symplectic instanton vector bundles on the projective space $\\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\\mathbb{P}^3$ with $r\\ge2$ and second Chern class $n\\ge r,\\ n\\equiv r({\\rm mod}2)$. We give an explicit construction of an irreducible component $I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$ has the expected dimension $4n(r+1)-r(2r+1)$.}, doi = {10.2478/s11533-012-0062-2}, url = {http://hdl.handle.net/1963/4656}, author = {Ugo Bruzzo and Dimitri Markushevich and Alexander Tikhomirov} } @article {2011, title = {D-branes, surface operators, and ADHM quiver representations}, number = {SISSA;01/2011/FM}, year = {2011}, note = {45 pages, v2: minor corrections}, institution = {SISSA}, abstract = {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 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.}, url = {http://hdl.handle.net/1963/4133}, author = {Ugo Bruzzo and Duiliu-Emanuel Diaconescu and M. Yardim and G. Pan and Yi Zhang and Chuang Wu-yen} } @article {2011, title = {Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle}, journal = {Differential Geometry and its Applications 29 (2011) 147-153}, number = {SISSA;05/2010/FM}, year = {2011}, publisher = {Elsevier}, doi = {10.1016/j.difgeo.2011.02.001}, url = {http://hdl.handle.net/1963/3830}, author = {Indranil Biswas and Ugo Bruzzo} } @article {2011, title = {Moduli of framed sheaves on projective surfaces}, journal = {Doc. Math. 16 (2011) 399-410}, number = {arXiv:0906.1436;}, year = {2011}, publisher = {Documenta Mathematica}, abstract = {We show that there exists a fine moduli space for torsion-free sheaves on a\\r\\nprojective surface, which have a \\\"good framing\\\" on a big and nef divisor. This\\r\\nmoduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of\\r\\nHuybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space, and discuss some examples on rational surfaces.}, url = {http://hdl.handle.net/1963/5126}, author = {Ugo Bruzzo and Dimitri Markushevich} } @article {2011, title = {Poincar{\'e} polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces}, journal = {Communications in Mathematical Physics 304 (2011) 395-409}, volume = {304}, number = {SISSA;56/2009/FM}, year = {2011}, month = {06/2011}, pages = {395-409}, publisher = {Springer}, abstract = {We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.

}, doi = {10.1007/s00220-011-1231-z}, url = {http://hdl.handle.net/1963/3738}, author = {Ugo Bruzzo and Rubik Poghossian and Alessandro Tanzini} } @article {2011, title = {Q-factorial Laurent rings}, number = {arXiv:1108.4116v1;}, year = {2011}, note = {5 pages}, institution = {SISSA}, abstract = {Dolgachev proved that, for any field k, the ring naturally associated to a\\r\\ngeneric Laurent polynomial in d variables, $d \\\\geq 4$, is factorial. We prove a\\r\\nsufficient condition for the ring associated to a very general complex Laurent\\r\\npolynomial in d=3 variables to be Q-factorial.}, url = {http://hdl.handle.net/1963/4183}, author = {Ugo Bruzzo and Antonella Grassi} } @article {2011, title = {Semistable and numerically effective principal (Higgs) bundles}, journal = {Advances in Mathematics 226 (2011) 3655-3676}, number = {SISSA;27/2009/FM}, year = {2011}, publisher = {Elsevier}, abstract = {We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.\\r\\n\\r\\nIn a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian{\textendash}Yang{\textendash}Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.}, doi = {10.1016/j.aim.2010.10.026}, url = {http://hdl.handle.net/1963/3638}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2010, title = {Cohomology of Skew-holomorphic lie algebroids}, number = {SISSA;15/2010/FM}, year = {2010}, abstract = {We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.}, url = {http://hdl.handle.net/1963/3853}, author = {Ugo Bruzzo and Vladimir Rubtsov} } @article {2010, title = {Gauge theory: from physics to geometry}, journal = {Rend. Istit. Mat. Univ. Trieste 42 (2010) 103-128}, number = {SISSA;80/2010/FM}, year = {2010}, publisher = {Istituto di matematica. Universita\\\' di Trieste}, abstract = {Maxwell theory may be regarded as a prototype of gauge theory and generalized to nonabelian gauge theory. We briey sketch the history of the gauge theories, from Maxwell to Yang-Mills theory, and the identification of gauge fields with connections on fibre bundles. We introduce the notion of instanton and consider the moduli spaces of such objects. Finally, we discuss some modern techniques for studying the topology of these moduli spaces.}, url = {http://hdl.handle.net/1963/4105}, author = {Ugo Bruzzo} } @article {2010, title = {Picard group of hypersurfaces in toric varieties}, number = {SISSA;78/2010/FM}, year = {2010}, abstract = {We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.}, url = {http://hdl.handle.net/1963/4103}, author = {Ugo Bruzzo and Antonella Grassi} } @article {2010, title = {On semistable principal bundles over complex projective manifolds, II}, journal = {Geom. Dedicata 146 (2010) 27-41}, number = {SISSA;85/2008/FM}, year = {2010}, abstract = {Let (X, \\\\omega) be a compact connected Kaehler manifold of complex dimension d and E_G a holomorphic principal G-bundle on X, where G is a connected reductive linear algebraic group defined over C. Let Z (G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P of G and a holomorphic reduction of the structure group of E_G to P (say, E_P) such that the bundle obtained by extending the structure group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat connection; (2) The adjoint vector bundle ad(E_G) is numerically flat; (3) The principal G-bundle E_G is pseudostable, and the degree of the charateristic class c_2(ad(E_G) is zero.}, doi = {10.1007/s10711-009-9424-8}, url = {http://hdl.handle.net/1963/3404}, author = {Indranil Biswas and Ugo Bruzzo} } @article {2010, title = {Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces}, number = {SISSA;59/2010/FM}, year = {2010}, abstract = {We construct a compactification $M^{\\\\mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\\\\gamma \\\\colon M^s \\\\to M^{\\\\mu ss}$, where $M^s$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{\\\\mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin-Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.}, url = {http://hdl.handle.net/1963/4049}, author = {Ugo Bruzzo and Dimitri Markushevich and Alexander Tikhomirov} } @article {2009, title = {Equivariant cohomology and localization for Lie algebroids}, journal = {Funct. Anal. Appl. 43 (2009) 18-29}, number = {SISSA;40/2005/FM}, year = {2009}, abstract = {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.}, isbn = {978-981-270-377-4}, doi = {10.1007/s10688-009-0003-4}, url = {http://hdl.handle.net/1963/1724}, author = {Ugo Bruzzo and Lucio Cirio and Paolo Rossi and Vladimir Rubtsov} } @article {2009, title = {Holomorphic equivariant cohomology of Atiyah algebroids and localization}, number = {SISSA;65/2009/FM}, year = {2009}, url = {http://hdl.handle.net/1963/3774}, author = {Ugo Bruzzo and Vladimir Rubtsov} } @article {2008, title = {Instanton counting on Hirzebruch surfaces}, number = {SISSA;55/2008/FM}, year = {2008}, abstract = {We perform a study of the moduli space of framed torsion free sheaves on Hirzebruch surfaces by using localization techniques. After discussing general properties of this moduli space, we classify its fixed points under the appropriate toric action and compute its Poincare\\\' polynomial. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.}, url = {http://hdl.handle.net/1963/2852}, author = {Ugo Bruzzo and Rubik Poghossian and Alessandro Tanzini} } @article {2008, title = {On semistable principal bundles over a complex projective manifold}, journal = {Int. Math. Res. Not. vol. 2008, article ID rnn035}, number = {arXiv.org;0803.4042v1}, year = {2008}, publisher = {Oxford University Press}, abstract = {Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \\\\chi of P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by \\\\chi is numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.}, doi = {10.1093/imrn/rnn035}, url = {http://hdl.handle.net/1963/3418}, author = {Indranil Biswas and Ugo Bruzzo} } @article {2007, title = {Metrics on semistable and numerically effective Higgs bundles}, journal = {J. Reine Angew. Math. 612 (2007) 59-79}, number = {SISSA;17/2006/FM}, year = {2007}, abstract = {We consider fibre metrics on Higgs vector bundles on compact K\\\\\\\"ahler manifolds, providing notions of numerical effectiveness and numerical flatness in terms of such metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, providing semistability criteria for Higgs bundles on projective manifolds of any dimension.}, doi = {10.1515/CRELLE.2007.084}, url = {http://hdl.handle.net/1963/1840}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2007, title = {Numerically flat Higgs vector bundles}, number = {SISSA;39/2005/FM}, year = {2007}, abstract = {After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.}, doi = {10.1142/S0219199707002526}, url = {http://hdl.handle.net/1963/1757}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2007, title = {Semistable principal Higgs bundles}, number = {SISSA;89/2007/MP}, year = {2007}, url = {http://hdl.handle.net/1963/2533}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2006, title = {Normal bundles to Laufer rational curves in local Calabi-Yau threefolds}, number = {SISSA;88/2005/FM}, year = {2006}, abstract = {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.}, doi = {10.1007/s11005-006-0057-7}, url = {http://hdl.handle.net/1963/1785}, author = {Ugo Bruzzo and Antonio Ricco} } @article {2006, title = {Semistability vs. nefness for (Higgs) vector bundles}, journal = {Differential Geom. Appl. 24 (2006) 403-416}, number = {arXiv.org;math/0310040v3}, year = {2006}, abstract = {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.}, doi = {10.1016/j.difgeo.2005.12.007}, url = {http://hdl.handle.net/1963/2237}, author = {Ugo Bruzzo and Daniel Hernandez Ruiperez} } @article {2004, title = {Superlocalization formulas and supersymmetric Yang-Mills theories}, journal = {Nucl. Phys. B 678 (2004) 638-655}, number = {arXiv.org;math-ph/0310036v1}, year = {2004}, publisher = {Elsevier}, abstract = {By using supermanifold techniques we prove a generalization of the localization formula in equivariant cohomology which is suitable for studying supersymmetric Yang-Mills theories in terms of ADHM data. With these techniques one can compute the reduced partition functions of topological super Yang-Mills theory with 4, 8 or 16 supercharges. More generally, the superlocalization formula can be applied to any topological field theory in any number of dimensions.}, doi = {10.1016/j.nuclphysb.2003.11.033}, url = {http://hdl.handle.net/1963/2886}, author = {Ugo Bruzzo and Francesco Fucito} } @article {2003, title = {Multi-instanton calculus and equivariant cohomology}, journal = {J.High Energy Phys. 2003,no.5,054,24 pp.}, number = {SISSA;75/2002/FM}, year = {2003}, publisher = {SISSA Library}, doi = {10.1088/1126-6708/2003/05/054}, url = {http://hdl.handle.net/1963/1645}, author = {Ugo Bruzzo and Jose F. Morales and Francesco Fucito and Alessandro Tanzini} } @article {2002, title = {Relatively stable bundles over elliptic fibrations}, journal = {Math. Nachr. 238 (2002) 23-36}, number = {arXiv.org;math/0109123v2}, year = {2002}, publisher = {Wiley}, abstract = {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.}, url = {http://hdl.handle.net/1963/3132}, author = {Claudio Bartocci and Ugo Bruzzo and Daniel Hernandez Ruiperez and Jose M. Munoz Porras} } @article {2001, title = {Complex Lagrangian embeddings of moduli spaces of vector bundles}, journal = {Differential Geom. Appl. 14 (2001) 151-156}, number = {arXiv.org;math/0010249v1}, year = {2001}, publisher = {Elsevier}, abstract = {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.}, doi = {10.1016/S0926-2245(00)00040-1}, url = {http://hdl.handle.net/1963/2885}, author = {Ugo Bruzzo and Fabio Pioli} } @article {2001, title = {A Fourier transform for sheaves on real tori. I. The equivalence Sky(T)~ Loc (T)}, journal = {J. Geom. Phys. 39 (2001), no. 2, 174--182}, number = {SISSA;68/00/FM}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0393-0440(01)00009-2}, url = {http://hdl.handle.net/1963/1526}, author = {Ugo Bruzzo and Giovanni Marelli and Fabio Pioli} } @article {2001, title = {On the Multi-Instanton Measure for Super Yang-Mills Theories}, journal = {Nuclear Phys. B 611 (2001), no. 1-3, 205--226.}, number = {SISSA;74/00/FM}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0550-3213(01)00349-2}, url = {http://hdl.handle.net/1963/1531}, author = {Ugo Bruzzo and Francesco Fucito and Alessandro Tanzini and Gabriele Travaglini} } @article {1999, title = {Categorial mirror symmetry for K3 surfaces}, journal = {Comm. Math. Phys. 206 (1999) 265-272}, number = {SISSA;100/98/FM/GEO}, year = {1999}, publisher = {Springer}, abstract = {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 $\\\\hat X$.}, doi = {10.1007/s002200050705}, url = {http://hdl.handle.net/1963/2887}, author = {Claudio Bartocci and Ugo Bruzzo and Guido Sanguinetti} } @article {1998, title = {Mirror Symmetry on K3 Surfaces as a Hyper-K{\"a}hler Rotation}, journal = {Lett. Math. Phys. 45 (1998) 295-301}, number = {arXiv.org;physics/9802044v2}, year = {1998}, publisher = {Springer}, abstract = {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 hyperk\\\\\\\"ahler family of complex structures. The same hypotheses force the B-field to vanish.}, doi = {10.1023/A:1007446916202}, url = {http://hdl.handle.net/1963/2888}, author = {Ugo Bruzzo and Guido Sanguinetti} } @article {10572, title = {Hilbert schemes of points on some K3 surfaces and Gieseker stable boundles}, journal = {MATH PROC CAMBRIDGE 120: 255-261 Part 2}, number = {SISSA;34/95/GEO-FM}, year = {1994}, publisher = {SISSA Library}, abstract = {By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\\\\Hilb^n(X)$ can be identified, for all $n\\\\geq 1$, with moduli spaces of Gieseker stable vector bundles on $X$ of rank $1+2n$. We also introduce a new Fourier-Mukai type transform for such surfaces.

}, url = {http://hdl.handle.net/1963/937}, author = {Ugo Bruzzo and Antony Maciocia} } @article {1990, title = {Chern-Simons forms on principal superfiber bundles}, journal = {J.Math.Phys.31:45,1990}, number = {SISSA;109/87/FM}, year = {1990}, publisher = {SISSA Library}, abstract = {A graded Weil homomorphism is defined for principal superfiber bundles and the related transgression (or Chern-Simons) forms are introduced. As an example of the application of these concepts, a {\textquoteleft}{\textquoteleft}superextension\\\'\\\' of the Dirac monopole is discussed.}, doi = {10.1063/1.528826}, url = {http://hdl.handle.net/1963/590}, author = {Giovanni Landi and Claudio Bartocci and Ugo Bruzzo} }