@article {han2020gauge, title = {On the gauge group of Galois objects}, year = {2020}, month = {03/2020}, abstract = {We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the classical gauge groupoid of a principal bundle. When the base algebra is in the centre of the total space algebra, the gauge group of the noncommutative principal bundle is isomorphic to the group of bisections of the bialgebroid. In particular we consider Galois objects (non-trivial noncommutative bundles over a point in a sense) for which the bialgebroid is a Hopf algebra. For these we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include Galois objects of group Hopf algebras and of Taft algebras.}, url = {https://arxiv.org/abs/2002.06097}, author = {Xiao Han and Giovanni Landi} } @article {doi:10.1142/S0129055X18500204, title = {Principal fibrations over noncommutative spheres}, journal = {Reviews in Mathematical Physics}, volume = {30}, number = {10}, year = {2018}, pages = {1850020}, abstract = {We present examples of noncommutative four-spheres that are base spaces of $SU(2)$-principal bundles with noncommutative seven-spheres as total spaces. The noncommutative coordinate algebras of the four-spheres are generated by the entries of a projection which is invariant under the action of $SU(2)$. We give conditions for the components of the Connes{\textendash}Chern character of the projection to vanish but the second (the top) one. The latter is then a non-zero Hochschild cycle that plays the role of the volume form for the noncommutative four-spheres.}, doi = {10.1142/S0129055X18500204}, url = {https://arxiv.org/abs/1804.07032}, author = {Michel Dubois-Violette and Xiao Han and Giovanni Landi} } @article {arici2016gysin, title = {The Gysin sequence for quantum lens spaces}, journal = {Journal of Noncommutative Geometry}, volume = {9}, number = {4}, year = {2016}, pages = {1077{\textendash}1111}, abstract = {

We define quantum lens spaces as {\textquoteleft}direct sums of line bundles{\textquoteright} and exhibit them as {\textquoteleft}total spaces{\textquoteright} of certain principal bundles over quantum projective spaces. For each of these quantum lens spaces we construct an analogue of the classical Gysin sequence in K-theory. We use the sequence to compute the K-theory of the quantum lens spaces, in particular to give explicit geometric representatives of their K-theory classes. These representatives are interpreted as {\textquoteleft}line bundles{\textquoteright} over quantum lens spaces and generically define {\textquoteleft}torsion classes{\textquoteright}. We work out explicit examples of these classes.

}, doi = {10.4171/JNCG/216}, author = {Francesca Arici and Simon Brain and Giovanni Landi} } @inbook {Arici2016, title = {Pimsner Algebras and Circle Bundles}, booktitle = {Noncommutative Analysis, Operator Theory and Applications}, year = {2016}, pages = {1{\textendash}25}, publisher = {Springer International Publishing}, organization = {Springer International Publishing}, address = {Cham}, abstract = {

We report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate several results with examples of quantum weighted projective and lens spaces and θ-deformations.

}, isbn = {978-3-319-29116-1}, doi = {10.1007/978-3-319-29116-1_1}, url = {https://doi.org/10.1007/978-3-319-29116-1_1}, author = {Francesca Arici and Francesco D{\textquoteright}Andrea and Giovanni Landi}, editor = {Alpay, Daniel and Cipriani, Fabio and Colombo, Fabrizio and Guido, Daniele and Sabadini, Irene and Sauvageot, Jean-Luc} } @article {arici2016pimsner, title = {Pimsner algebras and Gysin sequences from principal circle actions}, journal = {Journal of Noncommutative Geometry}, volume = {10}, year = {2016}, pages = {29{\textendash}64}, issn = {1661-6952}, doi = {10.4171/jncg/228}, url = {http://hdl.handle.net/2066/162951}, author = {Francesca Arici and Jens Kaad and Giovanni Landi} } @article {2012, title = {Moduli spaces of noncommutative instantons: gauging away noncommutative parameters}, journal = {Quarterly Journal of Mathematics (2012) 63 (1): 41-86}, number = {SISSA;62/2009/FM}, year = {2012}, publisher = {Oxford University Press}, abstract = {Using the theory of noncommutative geometry in a braided monoidal category, we improve upon a previous construction of noncommutative families of instantons of arbitrary charge on the deformed sphere S^4_\\\\theta. We formulate a notion of noncommutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as showing how to remove gauge parameters using a noncommutative quotient construction. Although the parameter spaces are a priori noncommutative, we show that one may always recover a classical parameter space by making an appropriate choice of gauge transformation.}, doi = {10.1093/qmath/haq036}, url = {http://hdl.handle.net/1963/3777}, author = {Simon Brain and Giovanni Landi} } @article {2009, title = {Families of Monads and Instantons from a Noncommutative ADHM Construction}, number = {SISSA;59/2008/FM}, year = {2009}, abstract = {We give a \\\\theta-deformed version of the ADHM construction of SU(2) instantons with arbitrary topological charge on the sphere S^4. Classically the instanton gauge fields are constructed from suitable monad data; we show that in the deformed case the set of monads is itself a noncommutative space. We use these monads to construct noncommutative {\textquoteleft}families\\\' of SU(2) instantons on the deformed sphere S^4_\\\\theta. We also compute the topological charge of each of the families. Finally we discuss what it means for such families to be gauge equivalent.}, url = {http://hdl.handle.net/1963/3478}, author = {Simon Brain and Giovanni Landi} } @article {2009, title = {Gauged Laplacians on quantum Hopf bundles}, journal = {Comm. Math. Phys. 287 (2009) 179-209}, number = {arXiv.org;0801.3376v2}, year = {2009}, publisher = {Springer}, abstract = {We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe {\textquoteleft}excitations moving on the quantum sphere\\\' in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect.}, doi = {10.1007/s00220-008-0672-5}, url = {http://hdl.handle.net/1963/3540}, author = {Giovanni Landi and Cesare Reina and Alessandro Zampini} } @article {2008, title = {The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere}, journal = {Comm. Math. Phys. 279 (2008) 77-116}, number = {SISSA;95/2007/MP}, year = {2008}, abstract = {Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the quantum Euclidean 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one of the spin structure of the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an {\textquoteleft}instanton\\\' projection. A real structure which satisfies all required properties modulo a suitable ideal of {\textquoteleft}infinitesimals\\\' is also introduced.}, doi = {10.1007/s00220-008-0420-x}, url = {http://hdl.handle.net/1963/2567}, author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski and Giovanni Landi} } @article {2008, title = {Noncommutative families of instantons}, journal = {Int. Math. Res. Not. vol. 2008, Article ID rnn038}, number = {arXiv.org;0710.0721v2}, year = {2008}, publisher = {Oxford University Press}, abstract = {We construct $\\\\theta$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_\\\\theta$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_\\\\theta(2,H)$ by $Sp_\\\\theta(2)$.}, doi = {10.1093/imrn/rnn038}, url = {http://hdl.handle.net/1963/3417}, author = {Giovanni Landi and Chiara Pagani and Cesare Reina and Walter van Suijlekom} } @article {2008, title = {The Noncommutative Geometry of the Quantum Projective Plane}, journal = {Rev. Math. Phys. 20 (2008) 979-1006}, number = {SISSA;100/2007/MP}, year = {2008}, abstract = {We study the spectral geometry of the quantum projective plane CP^2_q. In particular, we construct a Dirac operator which gives a 0^+ summable triple, equivariant under U_q(su(3)).}, doi = {10.1142/S0129055X08003493}, url = {http://hdl.handle.net/1963/2548}, author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski and Giovanni Landi} } @article {2007, title = {Dirac operators on all Podles quantum spheres}, journal = {J. Noncomm. Geom. 1 (2007) 213-239}, number = {arXiv.org;math/0606480v2}, year = {2007}, abstract = {We construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.}, doi = {10.4171/JNCG/5}, url = {http://hdl.handle.net/1963/2177}, author = {Francesco D{\textquoteright}Andrea and Ludwik Dabrowski and Giovanni Landi and Elmar Wagner} } @article {2006, title = {A Hopf bundle over a quantum four-sphere from the symplectic group}, journal = {Commun. Math. Phys. 263 (2006) 65-88}, number = {arXiv.org;math/0407342v2}, year = {2006}, abstract = {We construct a quantum version of the SU(2) Hopf bundle $S^7 \\\\to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU_q(2)$ on $S^7_q$ such that the algebra $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$.}, doi = {10.1007/s00220-005-1494-3}, url = {http://hdl.handle.net/1963/2179}, author = {Giovanni Landi and Chiara Pagani and Cesare Reina} } @article {2005, title = {The Dirac operator on SU_q(2)}, journal = {Commun. Math. Phys. 259 (2005) 729-759}, number = {arXiv:math/0411609;}, year = {2005}, note = {v2: minor changes}, publisher = {Springer}, abstract = {We construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.}, doi = {10.1007/s00220-005-1383-9}, url = {http://hdl.handle.net/1963/4425}, author = {Ludwik Dabrowski and Giovanni Landi and Andrzej Sitarz and Walter van Suijlekom and Joseph C. Varilly} } @article {2005, title = {The local index formula for SUq(2)}, journal = {K-Theory 35 (2005) 375-394}, number = {SISSA;01/2005/FM}, year = {2005}, abstract = {We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.}, doi = {10.1007/s10977-005-3116-4}, url = {http://hdl.handle.net/1963/1713}, author = {Walter van Suijlekom and Ludwik Dabrowski and Giovanni Landi and Andrzej Sitarz and Joseph C. Varilly} } @article {2005, title = {Principal fibrations from noncommutative spheres}, journal = {Comm. Math. Phys. 260 (2005) 203-225}, number = {SISSA;68/2004/FM}, year = {2005}, abstract = {We construct noncommutative principal fibrations S_\\\\theta^7 \\\\to S_\\\\theta^4 which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. The algebra inclusion $A(S_\\\\theta^4) \\\\into A(S_\\\\theta^7)$ is an example of a not trivial quantum principal bundle.}, doi = {10.1007/s00220-005-1377-7}, url = {http://hdl.handle.net/1963/2284}, author = {Giovanni Landi and Walter van Suijlekom} } @article {2005, title = {The spectral geometry of the equatorial Podles sphere}, journal = {C. R. Math. 340 (2005) 819-822}, number = {SISSA;71/2004/FM}, year = {2005}, abstract = {We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the equatorial Podles sphere we construct suq2-equivariant Dirac operator and real structure which satisfy these modified properties.}, doi = {10.1016/j.crma.2005.04.003}, url = {http://hdl.handle.net/1963/2275}, author = {Ludwik Dabrowski and Giovanni Landi and Mario Paschke and Andrzej Sitarz} } @article {2004, title = {Fredholm modules for quantum euclidean spheres}, journal = {J. Geom. Phys. 49 (2004) 272-293}, number = {SISSA;66/2002/FM}, year = {2004}, publisher = {SISSA Library}, abstract = {The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\\\\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres $S_q^{N-1}$. We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i. e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra $A(S^{N-1}_q)$.}, doi = {10.1016/S0393-0440(03)00092-5}, url = {http://hdl.handle.net/1963/1636}, author = {Eli Hawkins and Giovanni Landi} } @article {2003, title = {Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces}, journal = {Mod. Phys. Lett. A 18 (2003) 2371-2379}, number = {arXiv.org;math/0309143v1}, year = {2003}, publisher = {World Scientific}, abstract = {We study sigma-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space $CP^{q-1}$.}, doi = {10.1142/S0217732303012593}, url = {http://hdl.handle.net/1963/3215}, author = {Ludwik Dabrowski and Thomas Krajewski and Giovanni Landi} } @article {2002, title = {Instanton algebras and quantum 4-spheres}, journal = {Differential Geom. Appl. 16 (2002) 277-284}, number = {arXiv.org;math/0101177v2}, year = {2002}, publisher = {Elsevier}, abstract = {We study some generalized instanton algebras which are required to describe {\textquoteleft}instantonic complex rank 2 bundles\\\'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IC$, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form.}, doi = {10.1016/S0926-2245(02)00066-9}, url = {http://hdl.handle.net/1963/3134}, author = {Ludwik Dabrowski and Giovanni Landi} } @article {2001, title = {Instantons on the Quantum 4-Spheres S^4_q}, journal = {Comm. Math. Phys. 221 (2001) 161-168}, number = {arXiv.org;math/0012103v2}, year = {2001}, publisher = {Springer}, abstract = {We introduce noncommutative algebras $A_q$ of quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IR$, defined via a suspension of the quantum group $SU_q(2)$, and a quantum instanton bundle described by a selfadjoint idempotent $e\\\\in \\\\Mat_4(A_q)$, $e^2=e=e^*$. Contrary to what happens for the classical case or for the noncommutative instanton constructed in Connes-Landi, the first Chern-Connes class $ch_1(e)$ does not vanish thus signaling a dimension drop. The second Chern-Connes class $ch_2(e)$ does not vanish as well and the couple $(ch_1(e), ch_2(e))$ defines a cycle in the $(b,B)$ bicomplex of cyclic homology.}, doi = {10.1007/PL00005572}, url = {http://hdl.handle.net/1963/3135}, author = {Ludwik Dabrowski and Giovanni Landi and Tetsuya Masuda} } @article {2000, title = {Some Properties of Non-linear sigma-Models in Noncommutative Geometry}, journal = {Int. J. Mod. Phys. B 14 (2000) 2367-2382}, number = {SISSA;158/99/FM}, year = {2000}, publisher = {SISSA Library}, abstract = {We introduce non-linear $\\\\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some characteristic features of the corresponding $\\\\sigma$-models. In particular we construct a $\\\\sigma$-model instanton with topological charge equal to 1. We also define and investigate some properties of a noncommutative analogue of the Wess-Zumino-Witten model.}, doi = {10.1142/S0217979200001898}, url = {http://hdl.handle.net/1963/1373}, author = {Ludwik Dabrowski and Thomas Krajewski and Giovanni Landi} } @article {1990, title = {Algebraic differential calculus for gauge theories}, journal = {Nuclear Phys. B. Proc. Suppl. 18A (1990), 171}, number = {SISSA;135/89/FM}, year = {1990}, publisher = {SISSA Library}, doi = {10.1016/0920-5632(90)90649-F}, url = {http://hdl.handle.net/1963/891}, author = {Giovanni Landi and Giuseppe Marmo} } @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} } @article {1988, title = {Algebraic reduction of the \\\'t Hooft-Polyakov monopole to the Dirac monopole.}, journal = {Phys. Lett. B 201 (1988), no. 1, 101-104.}, number = {SISSA;97/87/FM}, year = {1988}, publisher = {SISSA Library}, doi = {10.1016/0370-2693(88)90088-3}, url = {http://hdl.handle.net/1963/578}, author = {Giovanni Landi and Giuseppe Marmo} } @mastersthesis {1988, title = {An Algebraic Setting for Gauge Theories}, year = {1988}, school = {SISSA}, url = {http://hdl.handle.net/1963/5828}, author = {Giovanni Landi} } @article {1988, title = {Einstein algebras and the algebraic Kaluza-Klein monopole.}, journal = {Phys. Lett. B 210 (1988), no. 1-2, 68--72.}, number = {SISSA;7/88/FM}, year = {1988}, publisher = {SISSA Library}, doi = {10.1016/0370-2693(88)90351-6}, url = {http://hdl.handle.net/1963/603}, author = {Giovanni Landi and Giuseppe Marmo} } @article {1987, title = {Extensions of Lie superalgebras and supersymmetric Abelian gauge fields.}, journal = {Phys. Lett. B 193 (1987), no. 1, 61-66.}, number = {SISSA;26/87/FM}, year = {1987}, publisher = {SISSA Library}, doi = {10.1016/0370-2693(87)90456-4}, url = {http://hdl.handle.net/1963/507}, author = {Giovanni Landi and Giuseppe Marmo} } @article {1987, title = {Graded Chern-Simons terms}, journal = {Phys. Lett. B 192 (1987), no. 1-2, 81-88.}, number = {SISSA;27/87/FM}, year = {1987}, publisher = {SISSA Library}, doi = {10.1016/0370-2693(87)91146-4}, url = {http://hdl.handle.net/1963/508}, author = {Giovanni Landi and Giuseppe Marmo} } @article {1987, title = {Lie algebra extensions and abelian monopoles.}, journal = {Phys. Lett. B 195 (1987), no. 3, 429-434}, number = {SISSA;25/87/FM}, year = {1987}, publisher = {SISSA Library}, doi = {10.1016/0370-2693(87)90043-8}, url = {http://hdl.handle.net/1963/506}, author = {Giovanni Landi and Giuseppe Marmo} } @article {1986, title = {The natural spinor connection on $S\\\\sb 8$ is a gauge field}, journal = {Lett. Math. Phys. 11 (1986), no. 2, 171-175}, number = {SISSA;61/85/MP}, year = {1986}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/448}, author = {Giovanni Landi} } @article {1985, title = {Flat connections for Lax hierarchies on coadjoint orbits}, journal = {Phys. Lett. A 108 (1985), no. 7, 311-314}, number = {SISSA;73/85/MP}, year = {1985}, publisher = {SISSA Library}, doi = {10.1016/0375-9601(85)90102-1}, url = {http://hdl.handle.net/1963/460}, author = {Giovanni Landi and Sergio De Filippo} } @article {1985, title = {Maximal acceleration and Sakharov{\textquoteright}s limiting temperature}, journal = {Lett. Nuovo Cim. 42 (1985) 70-72}, number = {SISSA;69/84/EP}, year = {1985}, publisher = {Societ{\`a} Italiana di Fisica}, abstract = {

It is shown that Sakharov{\textquoteright}s maximal temperature, derived by him from astrophysical considerations, is a straightforward consequence of the maximal acceleration introduced by us in previous works.

}, doi = {10.1007/BF02748306}, url = {http://hdl.handle.net/1963/372}, author = {Eduardo R. Caianiello and Giovanni Landi} }