arXiv subjects

Giovanni Landi

Publications and source records attributed to Giovanni Landi.

At least 55 records · Page 3Linked to original sources

arXiv2013

Algebraic deformations of toric varieties I. General constructions

We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus. We develop a sheaf theory using techniques from noncommutative algebraic geometry. The cases of projective varieties are studied in detail, and several explicit examples are worked out, including new noncommutative deformations of Grassmann and flag varieties. Our constructions set up the basic ingredients for thorough study of instantons on noncommutative toric varieties, which will be the subject of the sequel to this paper.

math.QA↗

arXiv2012

Instantons and vortices on noncommutative toric varieties

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantised twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

hep-th↗

arXiv2012

Calculi, Hodge operators and Laplacians on a quantum Hopf fibration

We describe Laplacian operators on the quantum group SUq (2) equipped with the four dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space S2q with the restricted left covariant three dimensional differential calculus. This is done by giving a family of Hodge dualities on both the exterior algebras of SUq (2) and S2q . We also study gauged Laplacian operators acting on sections of line bundles over the quantum sphere.

math.QA↗

arXiv2012

Algebraic deformations of toric varieties II. Noncommutative instantons

We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parametrizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build instanton gauge bundles with canonical connections on the noncommutative four-sphere that satisfy appropriate anti-selfduality equations. We construct projective moduli spaces for the torsion free sheaves and demonstrate that they are smooth. We define equivariant partition functions of these moduli spaces, finding that they coincide with the usual instanton partition functions for supersymmetric gauge theories on C^2.

hep-th↗

arXiv2012

Differential and Twistor Geometry of the Quantum Hopf Fibration

We study a quantum version of the SU(2) Hopf fibration $S^7 \to S^4$ and its associated twistor geometry. Our quantum sphere $S^7_q$ arises as the unit sphere inside a q-deformed quaternion space $\mathbb{H}^2_q$. The resulting four-sphere $S^4_q$ is a quantum analogue of the quaternionic projective space $\mathbb{HP}^1$. The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space $\mathbb{CP}^3_q$ and use it to study a system of anti-self-duality equations on $S^4_q$, for which we find an `instanton' solution coming from the natural projection defining the tautological bundle over $S^4_q$.

math.QA↗

arXiv2012

Moduli Spaces of Instantons on Toric Noncommutative Manifolds

We study analytic aspects of U(n) gauge theory over a toric noncommutative manifold $M_θ$. We analyse moduli spaces of solutions to the self-dual Yang-Mills equations on U(2) vector bundles over four-manifolds $M_θ$, showing that each such moduli space is either empty or a smooth Hausdorff manifold whose dimension we explicitly compute. In the special case of the four-sphere $S^4_θ$ we find that the moduli space of U(2) instantons with fixed second Chern number $k$ is a smooth manifold of dimension $8k-3$.

math-ph↗

arXiv2012

Geometry of Quantum Projective Spaces

In recent years, several quantizations of real manifolds have been studied, in particular from the point of view of Connes' noncommutative geometry. Less is known for complex noncommutative spaces. In this paper, we review some recent results about the geometry of complex quantum projective spaces.

math.QA↗

arXiv2012

Topological aspects of generalized Harper operators

A generalized version of the TKNN-equations computing Hall conductances for generalized Dirac-like Harper operators is derived. Geometrically these equations relate Chern numbers of suitable (dual) bundles naturally associated to spectral projections of the operators.

math-ph↗

arXiv2011

Dimensional reduction over the quantum sphere and non-abelian q-vortices

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kaehler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU_q(2), and also of invariant gauge connections on these bundles. The reduction of Yang-Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-Kaehler quotient construction.

hep-th↗

arXiv2011

Generalized TKNN-equations

We derive generalized TKNN-equations via bundle representations of the noncommutative torus with rational deformation parameter, the bundle coming from spectral projections in the torus algebra. These equations relate Chern numbers of dual bundles which we interpret as Hall conductances for Dirac-like Hamiltonians describing magnetic Bloch electrons in a strong magnetic field. We also present their generalizations for irrational values of the deformation parameter.

math-ph↗

arXiv2010

Moduli spaces of noncommutative instantons: gauging away noncommutative parameters

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_θ. 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.

math.QA↗

arXiv2010

The 3D Spin Geometry of the Quantum Two-Sphere

We study a three-dimensional differential calculus on the standard Podles quantum two-sphere S^2_q, coming from the Woronowicz 4D+ differential calculus on the quantum group SU_q(2). We use a frame bundle approach to give an explicit description of the space of forms on S^2_q and its associated spin geometry in terms of a natural spectral triple over S^2_q. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.

math.QA↗

arXiv2010

Families of Monads and Instantons from a Noncommutative ADHM Construction

We give a θ-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 `families' of SU(2) instantons on the deformed sphere S^4_θ. We also compute the topological charge of each of the families. Finally we discuss what it means for such families to be gauge equivalent.

math.QA↗

arXiv2010

Lie prealgebras

We introduce a generalization of Lie algebras within the theory of nonhomogeneous quadratic algebras and point out its relevance in the theory of quantum groups. In particular the relation between the differential calculus on quantum group and the Koszul duality due to Positselski is made apparent.

math.QA↗

arXiv2010

Holomorphic structures on the quantum projective line

We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted positivity for twisted Hochschild and cyclic cocycles and exhibit an explicit twisted positive Hochschild cocycle for the complex structure on the sphere.

math.QA↗

arXiv2010

Anti-selfdual Connections on the Quantum Projective Plane: Monopoles

We present several results on the geometry of the quantum projective plane CP2q. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding 'monopoles' and 'instanton' charges; complete diagonalization of gauged Laplacians on these line bundles; quantum invariants via equivariant K-theory and q-indices.

math.QA↗

arXiv2010

Geometry of the quantum projective plane

We review some of the geometry of the quantum projective plane with emphasis on the construction of a differential calculus and of the Dirac operator (of a spin^c-structure). We also report on anti-self-dual connections on line bundles, the spectrum of associated (gauged) Laplacian operators, and on classical and quantum characteristic classes.

math.QA↗

arXiv2010

Bounded and unbounded Fredholm modules for quantum projective spaces

We construct explicit generators of the K-theory and K-homology of the coordinate algebra of `functions' on quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.

math.QA↗