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.
We construct $θ$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_θ$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_θ(2,H)$ by $Sp_θ(2)$.
We present an account of the ADHM construction of instantons on Euclidean space-time $\mathbb{R}^4$ from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parameterised by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal-Groenewold plane $\mathbb{R}^4_\hbar$ and the Connes-Landi plane …
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.
We construct a five-parameter family of gauge-nonequivalent SU(2) instantons on a noncommutative four sphere $S_θ^4$ and of topological charge equal to -1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on $S_θ^4$. They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the ``tangent space'' to the modul…
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 $8k-3$.
It is shown that the quantum instanton bundle introduced in Commun. Math. Phys. 226, 419-432 (2002) has a bijective canonical map and is, therefore, a coalgebra Galois extension.
Through techniques afforded by $C^*$-algebras and Hilbert modules, we study the topology of spaces which parametrize families of instanton gauge fields on noncommutative Euclidean four-spheres $S^4_σ$. By deforming the ADHM construction of instantons on the classical sphere $S^4$, we obtain families of instantons on the quantum sphere which are naturally parametrized by noncommutative topological spaces. Using the internal gauge theory of $S^4_σ$ determined by the inner automorphisms of its function algebra, we find that one may always recover a classical parameter space by making a suitable c…
We present an introduction to the use of noncommutative geometry for gauge theories with emphasis on a construction of instantons for a class of four dimensional toric noncommutative manifolds. These instantons are solutions of self-duality equations and are critical points of an action functional. We explain the crucial role of twisted symmetries as well as methods from noncommutative index theorems.
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 inst…
We construct (anti)instanton solutions of a would-be q-deformed su(2) Yang-Mills theory on the quantum Euclidean space R_q^4 [the SO_q(4)-covariant noncommutative space] by reinterpreting the function algebra on the latter as a q-quaternion bialgebra. Since the (anti)selfduality equations are covariant under the quantum group of deformed rotations, translations and scale change, by applying the latter we can generate new solutions from the one centered at the origin and with unit size. We also construct multi-instanton solutions. As they depend on noncommuting parameters playing the roles of `…
These notes aim at a pedagogical introduction to recent work on deformation of spaces and deformation of vector bundles over them, which are relevant both in mathematics and in physics, notably monopole and instanton bundles. We first decribe toric noncommutative manifolds (also known as isospectral deformations) and give a detailed introduction to gauge theories on a toric four-sphere. This includes a Yang-Mills action functional with associated equations of motion and self-duality equations. We construct a particular class of instanton solutions on a SU(2) bundle with a suitable use of twist…
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 insta…
I carefully study noncommutative version of ADHM construction of instantons, which was proposed by Nekrasov and Schwarz. Noncommutative ${\bf R}^4$ is described as algebra of operators acting in Fock space. In ADHM construction of instantons, one looks for zero-modes of Dirac-like operator. The feature peculiar to noncommutative case is that these zero-modes project out some states in Fock space. The mechanism of these projections is clarified when the gauge group is U(1). I also construct some zero-modes when the gauge group is U(N) and demonstrate that the projections also occur, and the mec…
Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of the 4-dimensional real affine space. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative projective plane, certain complexes of sheaves on a noncommutative 3-dimensional projective space, and the modified ADHM data. The modified ADHM construction itself is interpre…
We describe the statistical mechanics of a melting crystal in three dimensions and its relation to a diverse range of models arising in combinatorics, algebraic geometry, integrable systems, low-dimensional gauge theories, topological string theory and quantum gravity. Its partition function can be computed by enumerating the contributions from noncommutative instantons to a six-dimensional cohomological gauge theory, which yields a dynamical realization of the crystal as a discretization of spacetime at the Planck scale. We describe analogous relations between a melting crystal model in two d…
This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifolds are the instanton moduli spaces of the noncommutative vector bundles over noncommutative 4-manifolds associated with highest weights of the compact Lie groups. Moreover their dimensions are described explicitly in terms of given principal bundles and highest weights. As a cor…
We report on some recent work on deformation of spaces, notably deformation of spheres, describing two classes of examples. The first class of examples consists of noncommutative manifolds associated with the so called $θ$-deformations which were introduced out of a simple analysis in terms of cycles in the $(b,B)$-complex of cyclic homology. These examples have non-trivial global features and can be endowed with a structure of noncommutative manifolds, in terms of a spectral triple $(\ca, \ch, D)$. In particular, noncommutative spheres $S^{N}_θ$ are isospectral deformations of usual spherical…
Studied are moduli spaces of self dual or anti-self dual connections on noncommutative 4-manifolds, especially deformation quantization of compact spin Riemannian 4-manifolds and their isometry groups have 2-torus subgroup. Then such moduli spaces of irreducible modules associated with highestweights of compact connected semisimple Lie groups are smooth manifolds with dimension determined by their weights. This is a generalization of Atiyah-Hitchin-Singer's classical result as well as Landi-Suijlekom's noncommutative 4-sphere case.
We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of Yang-Mills theory defined on a projective module for arbitrary noncommutativity parameter θwhich is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of θ. The construction of noncommutative …
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 $CP^{q-1}$.
We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle S^7 --> S^4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson--Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU_q(2); it determines a new deformation of the 4-sphere Sigma^4_q as the algebra of coinvariants in S_q^7. We show that the quantum vector bundle associated to the fundamental corepresentation of …
Projections play crucial roles in the ADHM construction on noncommutative $\R^4$. In this article a framework for the description of equivalence relations between projections is proposed. We treat the equivalence of projections as ``gauge equivalence'' on noncommutative space. We find an interesting application of this framework to the study of U(2) instanton on noncommutative $\R^4$: A zero winding number configuration with a hole at the origin is ``gauge equivalent'' to the noncommutative analog of the BPST instanton. Thus the ``gauge transformation'' in this case can be understood as a nonc…
We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of $\Rb^n$. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres $S^4_θ$. The noncommutative algebras $\Ac=C^{\ify} (S^{4}_θ)$ of functions on NC-spheres are solutions to the vanishing, $ {\rm ch}_j (e) = 0, j < 2 $, of the Chern character in the cyclic homology of $\Ac$ of an idempotent $e \in M_4 (\Ac), e^2 = e, e = e^*$. The universal noncommu…
Noncommutative Donaldson-Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative N=2 gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated to the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson-Thomas invariants in the Coulomb branch in terms of gauge…
We construct explicit deformation quantizations of the noncompact complex surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons. We then show that the quantum instanton moduli space can be viewed as the étale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.
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 …
We briefly report on our recent results regarding the introduction of a notion of a q-quaternion and the construction of instanton solutions of a would-be deformed su(2) Yang-Mills theory on the corresponding SO_q(4)-covariant quantum space. As the solutions depend on some noncommuting parameters, this indicates that the moduli space of a complete theory will be a noncommutative manifold.
We have recently introduced the notion of a q-quaternion bialgebra and shown its strict link with the SO_q(4)-covariant quantum Euclidean space R_q^4. Adopting the available differential geometric tools on the latter and the quaternion language we have formulated and found solutions of the (anti)selfduality equation [instantons and multi-instantons] of a would-be deformed su(2) Yang-Mills theory on this quantum space. The solutions depend on some noncommuting parameters, indicating that the moduli space of a complete theory should be a noncommutative manifold. We summarize these results and ad…
We describe basic concepts of noncommutative geometry and a general construction extending the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. Basic tools of the theory, K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic index theorems, Hopf algebra symmetry are reviewed. They cover the global aspects of noncommutative spaces, such as the transformation $θ\to 1/θ$ for the NC torus $\Tb_θ^2$, unseen in perturbative expansions in $θ$ such as star or Moyal products. We discuss the foundational pro…
A noncommutative algebra of the complex $q$-twistors and their differentials is considered on the basis of the quantum $GL_q (4)\times SL_q (2)$ group. Real and pseudoreal $q$-twistors are discussed too. We consider the quantum-group self-duality equation in the framework of the local gauge algebra of differential forms on $q$-twistor spaces. Quantum deformations of the general multi-instanton solutions are constructed. The corresponding noncommutative algebras of moduli are introduced. The general $q$-instanton connection is a function of the $q$-twistors and the $q$-moduli .
We survey some of the AGT relations between N=2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces. We treat the cases of gauge theories on both flat space and ALE spaces in some detail, and with emphasis on the implications arising from embedding them into supersymmetric theories in six dimensions. Along the way we construct new toric noncommutative ALE spaces using the general theory of complex algebraic deformations of toric varieties, and indicate how t…
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 …
We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four dimensional sphere with fixed volume. The equation involves an id…
It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the end $q$-deformed instanton models are introduced for every integral index.
We investigate the quantum twistor bundle constructed as a $U(1)$-quotient of the quantum instanton bundle of Bonechi, Ciccoli and Tarlini. It is an example of a locally trivial noncommutative bundle fulfilling conditions of the framework recently proposed by Brzeziński and Szymański. In particular, we give a detailed description of the corresponding $C^*$-algebra of 'continuous functions' on its noncommutative total space. Furthermore, we analyse a different construction of a quantum instanton bundle due to Landi, Pagani and Reina, find a basis of its polynomial algebra and discover an intrig…
