Noncommutative Instantons Research Articles

Notes on the dynamics of noncommutative U(2) and commutative SU(3) instantons

We examine the dynamics of noncommutative instantons of instanton number 2 and commutative instantons of instanton number 3 in 5D super Yang-Mills theory. We begin by detailing the construction of the $1/4$-Bogamolyni-Prasad-Somerfeldt instanton solutions, their moduli space, and the moduli space potential using an explicit parametrization of the moduli space coordinates in terms of the biquaternions. We then go on to numerically analyze the dynamics on the moduli spaces we have constructed, discussing some of the numerical issues which arose, and describing the numerical algorithm we developed to solve them.

Quantizations of local surfaces and rebel instantons

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.

BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem

Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ (noncommutative) instantons on origami spacetimes. There, $X$ is modelled on a union of (up to six) coordinate complex planes ${\bf C}^{2}$ intersecting in $Z$ modelled on ${\bf C}^{4}$. The instantons are shared by the collection of four dimensional gauge theories sewn along two dimensional defect surfaces and defect points. We also define several quiver versions ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$ of ${\bf M}_{k}({\vec n})$, motivated by the considerations of sewn gauge theories on orbifolds ${\bf C}^{4}/{\Gamma}$. The geometry of the spaces ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$, more specifically the compactness of the set of torus-fixed points, for various tori, underlies the non-perturbative Dyson-Schwinger identities recently found to be satisfied by the correlation functions of $qq$-characters viewed as local gauge invariant operators in the ${\mathcal{N}}=2$ quiver gauge theories. The cohomological and K-theoretic operations defined using ${\bf M}_{k}({\vec n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects.

Moduli space dynamics of noncommutative U(2) instantons

We consider the low energy dynamics of charge two instantons on noncommutative ℝ NC 2 × ℝ NC 2 in U(2) 5-dimensional super-Yang-Mills, using the Manton approximation for slow-moving instantons to calculate the moduli space metric. By employing the ADHM construction, we are able to understand some aspects of the geometry and topology of the system. We also consider the effect of adding a potential to the moduli space, giving scattering results for noncommutative dyonic instantons.

Exact construction of noncommutative instantons

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and give some exact instanton solutions for various noncommutative settings. We also present a new formula which is crucial to show an origin of the instanton number for U(1) and to prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data.

Noncommutative Instantons Revisited

We discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction of U(N) instantons in noncommutative (NC) space and prove the one-to-one correspondence between moduli spaces of the noncommutative instantons and the ADHM data, together with an origin of the instanton number for U(1). We also give a derivation of the ADHM construction from the viewpoint of the Nahm transformation of instantons on four-torus.

Remarks on theN=1SU(M+p)×SU(p)quiver gauge theory with flavor

We study supersymmetric vacua of the $\mathcal{N}=1$ cascading $SU(M+p)\ifmmode\times\else\texttimes\fi{}SU(p)$ gauge theory with flavor---the theory on $p$ D3-branes and $M$ wrapped D5-branes at the tip of the conifold, and ${N}_{f}$ flavor D7-branes wrapping a holomorphic four-cycle inside the conifold. The Coulomb branch of the moduli space is inherited from the pure gauge theory without flavor and was thoroughly studied in the past. Besides, there is a Higgs branch where some D3- and/or D5-branes dissolve in the D7-branes forming the worldvolume gauge instantons. We study the Higgs branch both from the field theory and the bulk point of view. On the classical level the moduli space is closely related to the one of the $\mathcal{N}=2$ ${\mathbb{C}}^{2}/{\mathbb{Z}}_{2}$ orbifold theory, in particular, certain vacua of the $\mathcal{N}=1$ theory are related to noncommutative instantons on the resolved ${\mathbb{C}}^{2}/{\mathbb{Z}}_{2}$. On the quantum level the Higgs branch acquires corrections due to renormalization of the K\"ahler potential and nonperturbative effects in field theory. In the bulk this is encoded in the classical D7-brane geometry. We compute the vacuum expectation value (VEV)s of the protected operators and the field theory RG flow and find an agreement with the parallel computations in the bulk.

Noncommutative deformation of spinor zero mode and Atiyah-Drinfeld-Hitchin-Manin construction

A method to construct noncommutative instantons as deformations from commutative instantons was provided by Maeda and Sako [J. Geom. Phys. 58, 1784 (2008)]10.1016/j.geomphys.2008.08.006. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative \documentclass[12pt]{minimal}\begin{document}${\mathbb R}^4$\end{document}R4, and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Green's function associated with instantons on noncommutative \documentclass[12pt]{minimal}\begin{document}${\mathbb R}^4$\end{document}R4, as a smooth deformation of the commutative case. The feature of the zero modes of the Dirac operator and the Green's function derives noncommutative ADHM (Atiyah-Drinfeld-Hitchin-Manin) equations which coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one correspondence between the instantons on noncommutative \documentclass[12pt]{minimal}\begin{document}${\mathbb R}^4$\end{document}R4 and ADHM data. An example of a noncommutative instanton and a spinor zero mode are also given.

Index pairings for pullbacks of C*-algebras

In this overview, we study how to reduce the index pairing for a fibre-product C*-algebra to the index pairing for the C*-algebra over which the fibre product is taken. As an example we analyze the case of suspensions and apply it to noncommutative instanton bundles of arbitrary charges over the suspension of quantum deformations of the 3-sphere.

Noncommutative vortices and instantons from generalized Bose operators

Generalized Bose operators correspond to reducible representations of the harmonic oscillator algebra. We demonstrate their relevance in the construction of topologically non-trivial solutions in noncommutative gauge theories, focusing our attention to flux tubes, vortices, and instantons. Our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the generalized Bose operator. When used in conjunction with the noncommutative ADHM construction, we find that these new instantons are in general not unitarily equivalent to the ones currently known in literature.

Instantons, quivers and noncommutative Donaldson–Thomas theory

We construct noncommutative Donaldson–Thomas invariants associated with abelian orbifold singularities by analyzing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson–Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.

Gauge theories from D7-branes over vanishing 4-cycles

We study quiver gauge theories on D7-branes wrapped over vanishing holomorphic 4-cycles. We investigate how to incorporate O7-planes and/or flavor D7-branes, which are necessary to cancel anomalies. These theories are chiral, preserve four supercharges and exhibit very rich infrared dynamics. Geometric transitions and duality in the presence of O-planes are analyzed. We study the Higgs branch of these quiver theories, showing the emergence of fuzzy internal dimensions. This branch is related to noncommutative instantons on the divisor wrapped by the seven-branes. Our results have a natural application to the recently introduced F(uzz) limit of F-theory.

On the symplectic structure of instanton moduli spaces

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU ( 2 ) instantons on both commutative and non-commutative R 4 . We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero–Moser spaces to the instanton moduli spaces.

Recent Developments in Instantons in Noncommutative ℝ4

We review recent developments in noncommutative deformations of instantons in ℝ4. In the operator formalism, we study how to make noncommutative instantons by using the ADHM method, and we review the relation between topological charges and noncommutativity. In the ADHM methods, there exist instantons whose commutative limits are singular. We review smooth noncommutative deformations of instantons, spinor zero‐modes, the Green′s functions, and the ADHM constructions from commutative ones that have no singularities. It is found that the instanton charges of these noncommutative instanton solutions coincide with the instanton charges of commutative instantons before noncommutative deformation. These smooth deformations are the latest developments in noncommutative gauge theories, and we can extend the procedure to other types of solitons. As an example, vortex deformations are studied.

Examples of gauged Laplacians on noncommutative spaces

We present two (classes of) examples of gauged Laplacian operators. The first one is a model of spin-Hall effect on a noncommutative four-sphere Sϑ4 with isospin degrees of freedom, coming from a noncommutative instanton, and invariant under the quantum group SOϑ(5). The second one, a Hall effect on a quantum 2-dimensional sphere Sq2, describes ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole with symmetry coming from the quantum group SUq(2). For both models, ample symmetries provide a complete diagonalization.

Cohomological gauge theory, quiver matrix models and Donaldson–Thomas theory

We study the relation between Donaldson–Thomas theory of Calabi–Yau threefolds and a six-dimensional topological Yang–Mills theory. Our main example is the topological U ( N ) gauge theory on flat space in its Coulomb branch. To evaluate its partition function we use equivariant localization techniques on its noncommutative deformation. As a result the gauge theory localizes on noncommutative instantons which can be classified in terms of N-coloured three-dimensional Young diagrams. We give to these noncommutative instantons a geometrical description in terms of certain stable framed coherent sheaves on projective space by using a higher-dimensional generalization of the ADHM formalism. From this formalism we construct a topological matrix quantum mechanics which computes an index of BPS states and provides an alternative approach to the six-dimensional gauge theory.

Noncommutative deformation of instantons

We study instanton solutions on noncommutative Euclidean 4-space which are deformations of instanton solutions on commutative Euclidean 4-space. We show that the instanton numbers of these noncommutative instanton solutions coincide with the commutative solutions and conjecture that the instanton number in R 4 is preserved for general noncommutative deformations. We also study noncommutative deformation of instanton solutions on a T 4 with twisted boundary conditions.

Fayet-Iliopoulos terms in AdS/CFT with flavour

We construct the gravity dual of a field theory with flavour in which there are Fayet-Iliopoulos (FI) terms present. For this purpose we turn on a constant Kalb-Ramond B field in four internal space directions of AdS_5 x S^5 together with a D7 brane probe wrapping AdS_5 x S^3. The B field induces noncommutativity on the four internal directions of the D7 brane probe perpendicular to the AdS boundary. We argue on general grounds that the moduli space of the Higgs part of mixed Coulomb-Higgs states in the dual field theory at the boundary is described by the ADHM equations for noncommutative instantons on the four internal directions of the D7 brane probe. In particular, the global symmetries match. The FI term arises as the holographic dual of an anti-selfdual B field. We discuss possible applications for this construction.

Noncommutative instantons come in families

A construction of instantons in the context of noncommutative geometry, in particular SU(2) instantons on a noncommutative 4 sphere, has been recently reported. Firstly, a noncommutative principal fibration A(S4θ)↬ A(S7θ) which ‘quantizes’ the classical SU(2)-Hopf fibration over S4, has been constructed in [11] on the toric noncommutative four-sphere S4θ The generators of A4θ are the entries of a projection p which describes the basic instanton on A4θ. That is, p gives a projective module of finite type p[A(S4θ)]4 and a connection ▽ = p ° D on it which has a self-dual curvature and charge 1, in some appropriate sense; this is the basic instanton. In [12] infinitesimal instantons — ‘the tangent space to the moduli space’ -were constructed using infinitesimal conformal transformations, that is elements in a quantized enveloping algebra Uθ(so(5, 1)). In [10] we looked at a global construction and obtain generic charge 1 instantons by ‘quantizing’ the action of the Lie groups SL(2, ℌ) and SO(2) on the basic instanton which enter the classical construction [1]. We review all this here.

Noncommutative Instantons from Twisted Conformal Symmetries

We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a noncommutative four sphere \({S_{\theta}^4}\) 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 \({S_{\theta}^4}\) . 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 moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to five, a number that survives deformation.