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 corollary, it is shown that a smooth 4-manifold is the instanton moduli space of the canonocal noncommutative vector bundle over the noncommutative comples projective 2-space associated with the highest weight 1 of the 1-torus.
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 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.
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 approp…
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 $θ$-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 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…
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.
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…
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…
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 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 $8k-3$.
In this paper we study the moduli space of standard holomorphic structures on a noncommutative complex two torus. It will be shown that the moduli space is naturally identified with the moduli space of stable bundles on an elliptic curve. We also propose that the mirror reflection of the noncommutative complex torus is the mirror reflection of the elliptic curve together with a linear foliation. From this we identify the moduli space of super cycles on the mirror reflection with the moduli space of standard holomorphic bundles on a noncommutative complex torus.
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 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 …
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 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 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…
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…
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 (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 `…
In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Witten's conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.
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…
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincaré pairing) in terms of representation theory of some vertex operator algebras ("$\mathscr W$-algebras").
