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 $d_2$

U1 - 7293 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - RPRT T1 - Symplectic instanton bundles on P3 and 't Hooft instantons Y1 - 2013 A1 - Ugo Bruzzo A1 - Dimitri Markushevich A1 - Alexander Tikhomirov AB - 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 't Hooft instantons. PB - arXiv:1312.5554 [math.AG] UR - http://urania.sissa.it/xmlui/handle/1963/34486 N1 - 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. U1 - 34675 U2 - Mathematics U4 - 1 U5 - MAT/03 ER - TY - JOUR T1 - On localization in holomorphic equivariant cohomology JF - Central European Journal of Mathematics, Volume 10, Issue 4, August 2012, Pages 1442-1454 Y1 - 2012 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov KW - Lie algebroids AB - 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's localization formulas. PB - Springer UR - http://hdl.handle.net/1963/6584 U1 - 6543 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Moduli of symplectic instanton vector bundles of higher rank on projective space $\\mathbbP^3$ JF - Central European Journal of Mathematics 10, nr. 4 (2012) 1232 Y1 - 2012 A1 - Ugo Bruzzo A1 - Dimitri Markushevich A1 - Alexander Tikhomirov AB - 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 $4n(r+1)-r(2r+1)$. PB - SISSA UR - http://hdl.handle.net/1963/4656 N1 - 14 pages U1 - 4406 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - RPRT T1 - D-branes, surface operators, and ADHM quiver representations Y1 - 2011 A1 - Ugo Bruzzo A1 - Duiliu-Emanuel Diaconescu A1 - M. Yardim A1 - G. Pan A1 - Yi Zhang A1 - Chuang Wu-yen AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/4133 N1 - 45 pages, v2: minor corrections U1 - 3873 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - JOUR T1 - Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle JF - Differential Geometry and its Applications 29 (2011) 147-153 Y1 - 2011 A1 - Indranil Biswas A1 - Ugo Bruzzo PB - Elsevier UR - http://hdl.handle.net/1963/3830 U1 - 497 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Moduli of framed sheaves on projective surfaces JF - Doc. Math. 16 (2011) 399-410 Y1 - 2011 A1 - Ugo Bruzzo A1 - Dimitri Markushevich AB - 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. PB - Documenta Mathematica UR - http://hdl.handle.net/1963/5126 U1 - 4942 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - JOUR T1 - Poincaré polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces JF - Communications in Mathematical Physics 304 (2011) 395-409 Y1 - 2011 A1 - Ugo Bruzzo A1 - Rubik Poghossian A1 - Alessandro Tanzini AB -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.

PB - Springer VL - 304 UR - http://hdl.handle.net/1963/3738 IS - 2 U1 - 579 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Q-factorial Laurent rings Y1 - 2011 A1 - Ugo Bruzzo A1 - Antonella Grassi AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/4183 N1 - 5 pages U1 - 3907 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - JOUR T1 - Semistable and numerically effective principal (Higgs) bundles JF - Advances in Mathematics 226 (2011) 3655-3676 Y1 - 2011 A1 - Ugo Bruzzo A1 - Beatriz Grana-Otero AB - 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–Yang–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. PB - Elsevier UR - http://hdl.handle.net/1963/3638 U1 - 666 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Cohomology of Skew-holomorphic lie algebroids Y1 - 2010 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov AB - 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. UR - http://hdl.handle.net/1963/3853 U1 - 856 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Gauge theory: from physics to geometry JF - Rend. Istit. Mat. Univ. Trieste 42 (2010) 103-128 Y1 - 2010 A1 - Ugo Bruzzo AB - 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. PB - Istituto di matematica. Universita\\\' di Trieste UR - http://hdl.handle.net/1963/4105 U1 - 299 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Picard group of hypersurfaces in toric varieties Y1 - 2010 A1 - Ugo Bruzzo A1 - Antonella Grassi AB - 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. UR - http://hdl.handle.net/1963/4103 U1 - 301 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On semistable principal bundles over complex projective manifolds, II JF - Geom. Dedicata 146 (2010) 27-41 Y1 - 2010 A1 - Indranil Biswas A1 - Ugo Bruzzo AB - 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. UR - http://hdl.handle.net/1963/3404 U1 - 928 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces Y1 - 2010 A1 - Ugo Bruzzo A1 - Dimitri Markushevich A1 - Alexander Tikhomirov AB - 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. UR - http://hdl.handle.net/1963/4049 U1 - 353 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Equivariant cohomology and localization for Lie algebroids JF - Funct. Anal. Appl. 43 (2009) 18-29 Y1 - 2009 A1 - Ugo Bruzzo A1 - Lucio Cirio A1 - Paolo Rossi A1 - Vladimir Rubtsov AB - 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. SN - 978-981-270-377-4 UR - http://hdl.handle.net/1963/1724 U1 - 2427 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Holomorphic equivariant cohomology of Atiyah algebroids and localization Y1 - 2009 A1 - Ugo Bruzzo A1 - Vladimir Rubtsov UR - http://hdl.handle.net/1963/3774 U1 - 551 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Instanton counting on Hirzebruch surfaces Y1 - 2008 A1 - Ugo Bruzzo A1 - Rubik Poghossian A1 - Alessandro Tanzini AB - 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. UR - http://hdl.handle.net/1963/2852 U1 - 1848 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On semistable principal bundles over a complex projective manifold JF - Int. Math. Res. Not. vol. 2008, article ID rnn035 Y1 - 2008 A1 - Indranil Biswas A1 - Ugo Bruzzo AB - 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. PB - Oxford University Press UR - http://hdl.handle.net/1963/3418 U1 - 917 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Metrics on semistable and numerically effective Higgs bundles JF - J. Reine Angew. Math. 612 (2007) 59-79 Y1 - 2007 A1 - Ugo Bruzzo A1 - Beatriz Grana-Otero AB - 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. UR - http://hdl.handle.net/1963/1840 U1 - 2376 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Numerically flat Higgs vector bundles Y1 - 2007 A1 - Ugo Bruzzo A1 - Beatriz Grana-Otero AB - 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. JF - Commun. Contemp. Math. 9 (2007) 437-446 UR - http://hdl.handle.net/1963/1757 U1 - 2787 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Semistable principal Higgs bundles Y1 - 2007 A1 - Ugo Bruzzo A1 - Beatriz Grana-Otero UR - http://hdl.handle.net/1963/2533 U1 - 1585 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Normal bundles to Laufer rational curves in local Calabi-Yau threefolds Y1 - 2006 A1 - Ugo Bruzzo A1 - Antonio Ricco AB - 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. JF - Lett. Math. Phys. 76 (2006) 57-63 UR - http://hdl.handle.net/1963/1785 U1 - 2759 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Semistability vs. nefness for (Higgs) vector bundles JF - Differential Geom. Appl. 24 (2006) 403-416 Y1 - 2006 A1 - Ugo Bruzzo A1 - Daniel Hernandez Ruiperez AB - 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. UR - http://hdl.handle.net/1963/2237 U1 - 2007 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Superlocalization formulas and supersymmetric Yang-Mills theories JF - Nucl. Phys. B 678 (2004) 638-655 Y1 - 2004 A1 - Ugo Bruzzo A1 - Francesco Fucito AB - 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. PB - Elsevier UR - http://hdl.handle.net/1963/2886 U1 - 1814 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Multi-instanton calculus and equivariant cohomology JF - J.High Energy Phys. 2003,no.5,054,24 pp. Y1 - 2003 A1 - Ugo Bruzzo A1 - Jose F. Morales A1 - Francesco Fucito A1 - Alessandro Tanzini PB - SISSA Library UR - http://hdl.handle.net/1963/1645 U1 - 2473 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Relatively stable bundles over elliptic fibrations JF - Math. Nachr. 238 (2002) 23-36 Y1 - 2002 A1 - Claudio Bartocci A1 - Ugo Bruzzo A1 - Daniel Hernandez Ruiperez A1 - Jose M. Munoz Porras AB - 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. PB - Wiley UR - http://hdl.handle.net/1963/3132 U1 - 1201 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Complex Lagrangian embeddings of moduli spaces of vector bundles JF - Differential Geom. Appl. 14 (2001) 151-156 Y1 - 2001 A1 - Ugo Bruzzo A1 - Fabio Pioli AB - 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. PB - Elsevier UR - http://hdl.handle.net/1963/2885 U1 - 1815 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - A Fourier transform for sheaves on real tori. I. The equivalence Sky(T)~ Loc (T) JF - J. Geom. Phys. 39 (2001), no. 2, 174--182 Y1 - 2001 A1 - Ugo Bruzzo A1 - Giovanni Marelli A1 - Fabio Pioli PB - SISSA Library UR - http://hdl.handle.net/1963/1526 U1 - 2637 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On the Multi-Instanton Measure for Super Yang-Mills Theories JF - Nuclear Phys. B 611 (2001), no. 1-3, 205--226. Y1 - 2001 A1 - Ugo Bruzzo A1 - Francesco Fucito A1 - Alessandro Tanzini A1 - Gabriele Travaglini PB - SISSA Library UR - http://hdl.handle.net/1963/1531 U1 - 2632 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Categorial mirror symmetry for K3 surfaces JF - Comm. Math. Phys. 206 (1999) 265-272 Y1 - 1999 A1 - Claudio Bartocci A1 - Ugo Bruzzo A1 - Guido Sanguinetti AB - 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 $\\\\hat X$. PB - Springer UR - http://hdl.handle.net/1963/2887 U1 - 1813 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Mirror Symmetry on K3 Surfaces as a Hyper-Kähler Rotation JF - Lett. Math. Phys. 45 (1998) 295-301 Y1 - 1998 A1 - Ugo Bruzzo A1 - Guido Sanguinetti AB - 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. PB - Springer UR - http://hdl.handle.net/1963/2888 U1 - 1812 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Hilbert schemes of points on some K3 surfaces and Gieseker stable boundles JF - MATH PROC CAMBRIDGE 120: 255-261 Part 2 Y1 - 1994 A1 - Ugo Bruzzo A1 - Antony Maciocia AB -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.

PB - SISSA Library UR - http://hdl.handle.net/1963/937 U1 - 3517 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Chern-Simons forms on principal superfiber bundles JF - J.Math.Phys.31:45,1990 Y1 - 1990 A1 - Giovanni Landi A1 - Claudio Bartocci A1 - Ugo Bruzzo AB - 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 ``superextension\\\'\\\' of the Dirac monopole is discussed. PB - SISSA Library UR - http://hdl.handle.net/1963/590 U1 - 3314 U2 - Mathematics U3 - Mathematical Physics ER -