Narrow Results

We present an account of the ADHM construction of instantons on Euclidean space-time ℝ⁴ 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, parametrized by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal–Groenewold plane and the Connes–Landi plane .

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 quantized 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.

We prove that the space of mathematical instantons with second Chern class 5 over ℙ₃ is smooth and irreducible. Unified and simple proofs for the same statements in case of second Chern class ≤ 4 are contained.

A framed symplectic sheaf on a smooth projective surface $X$ is a torsion-free sheaf $E$ together with a trivialization on a divisor $D\subseteq X$ and a morphism ${\mathrm{\Lambda }}^{2}E\to {{𝒪}}_X$ satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for $X={\mathbb{P}}_{ℂ}^2$. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map onto the space of framed symplectic ideal instantons.

The principle of indirect elimination states that an algorithm for solving discretized differential equations can be used to identify its own bad-converging modes. When the number of bad-converging modes of the algorithm is not too large, the modes thus identified can be used to strongly improve the convergence. The method presented here is applicable to any linear algorithm like relaxation or multigrid. An example from theoretical physics, the Dirac equation in the presence of almost-zero-modes arising from instantons, is studied. Using the principle, bad-converging modes are removed efficiently. It is sketched how the method can be used for a Conjugate Gradient algorithm. Applied locally, the principle is one of the main ingredients of the Iteratively Smoothing Unigrid algorithm.

The properties of the new analytic running coupling are investigated at the higher loop levels. The expression for this invariant charge, independent of the normalization point, is obtained by invoking the asymptotic freedom condition. It is shown that at any loop level the relevant β-function has the universal behaviors at small and large values of the invariant charge. Due to this feature the new analytic running coupling possesses the universal asymptotics in both the ultraviolet and infrared regions irrespective of the loop level. The consistency of the model considered with the general definition of the QCD invariant charge is shown.

The n-instanton contribution to the Seiberg–Witten prepotential of N = 2 supersymmetric d = 4 Yang–Mills theory is represented as the integral of the exponential of an equivariantly exact form. Integrating out an overall scale and a U(1) angle the integral is rewritten as (4n - 3)-fold product of a closed two-form. This two-form is, formally, a representative of the Euler class of the instanton moduli space viewed as a principal U(1) bundle, because its pullback under bundle projection is the exterior derivative of an angular one-form.

We describe how to reduce the fuzzy four-sphere algebra to a set of four independent raising and lowering oscillator operators. In terms of them we derive the projector valued operators for the fuzzy four-sphere, which are the global definition of k-instanton connections over this noncommutative base manifold.

We study σ-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 .

There exists a recursive algorithm for constructing BPST-type multi-instantons on commutative ℝ⁴. When deformed noncommutatively, however, it becomes difficult to write down non-singular instanton configurations with topological charge greater than one in explicit form. We circumvent this difficulty by allowing for the translational instanton moduli to become noncommutative as well. Such a scenario is natural in the self-dual Yang–Mills hierarchy of integrable equations where the moduli of solutions are seen as extended spacetime coordinates associated with higher flows. By judicious adjustment of the moduli-noncommutativity we achieve the ADHM construction of generalized 't Hooft multi-instanton solutions with everywhere self-dual field strengths on noncommutative ℝ⁴.

We calculate some nonperturbative (D-instanton) quantum corrections to the moduli space metric of several (n>1) identical matter hypermultiplets for the type-IIA superstrings compactified on a Calabi–Yau threefold, near conifold singularities. We find a nontrivial deformation of the (real) 4n-dimensional hypermultiplet moduli space metric due to the infinite number of D-instantons, under the assumption of n tri-holomorphic commuting isometries of the metric, in the hyper-Kähler limit (i.e. in the absence of gravitational corrections).

Using ϕ-mapping topological current theory and the expansion theory of the δ-function, we found a more exact expression of conventional instanton and multi-instanton. We established a novel approach to instanton. It is found that the instantons arise from the symmetric phase of the Higgs field ϕ=0, the fine topological structure of the instanton number is also given.

We show that vortices of Yang–Mills–Higgs model in R² space can be regarded as instantons of Yang–Mills model in R² × Z₂ space. For this, we construct the noncommutative Z₂ space by explicitly fixing the Z₂ coordinates and then show, by using the Z₂ coordinates, that BPS equation for the vortices can be considered as a self-dual equation. We also propose the possibility to rewrite the BPS equations for vortices as ADHM equations through the use of self-dual equation.

The ADHM construction of instantons on four-sphere is extended to hyperbolic four-space of constant negative curvature.

We discuss an alternative for baryon-violating six quarks transition in the context of low scale string theory. In particular, with M${}_S$ = 10–10³ TeV, such a transition can be mediated by two color-triplets through a quartic coupling with down-quarks, generated by exotic instantons, in a calculable and controllable way. We show how flavor-changing neutral currents (FCNCs) limits on color-triplet mass are well compatible with $n-\bar{n}$ oscillation ones. If an $n-\bar{n}$ transition was found, this would be an indirect hint for our model. This would strongly motivate searches for direct channels in proton–proton colliders. In fact, our model can be directly tested in an experimentally challenging 100–1000 TeV proton–proton collider, searching for our desired color-triplet states and an evidence for exotic instantons resonances, in addition to stringy Regge resonances, anomalous Z${}^{\prime }$-bosons and gauged megaxion. In particular, our scenario can be related to the 750 GeV diphoton hint identifying it with the gauged megaxion dual to the B-field. On the other hand, this scenario is compatible with TeV-ish color triplets visible at large hadron collider (LHC) and with 1–10 TeV string scale, i.e. stringy resonances at LHC.

Recent studies have emphasized the important role that a shape deformability of scalar-field models pertaining to the same class with the standard ${\varphi }^4$ field, can play in controlling the production of a specific type of breathing bound states so-called oscillons. In the context of cosmology, the built-in mechanism of oscillons suggests that they can affect the standard picture of scalar ultra-light dark matter. In this paper, kink scatterings are investigated in a parametrized model of bistable system admitting the classical ${\varphi }^4$ field as an asymptotic limit, with focus on the formation of long-lived low-amplitude almost harmonic oscillations of the scalar field around a vacuum. The parametrized model is characterized by a double-well potential with a shape-deformation parameter that changes only the steepness of the potential walls, and hence the flatness of the hump of the potential barrier, leaving unaffected the two degenerate minima and the barrier height. It is found that the variation of the deformability parameter promotes several additional vibrational modes in the kink-phonon scattering potential, leading to suppression of the two-bounce windows in kink–antikink scatterings and the production of oscillons. Numerical results suggest that the anharmonicity of the potential barrier, characterized by a flat barrier hump, is the main determinant factor for the production of oscillons in double-well systems.

In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N = 2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory.

This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology.

In the case of an invertible coordinate commutator matrix θ_ij, we derive a general instanton solution of the noncommutative gauge theories on d = 2n planes given in terms of n oscillators.

A procedure, allowing us to calculate the coefficients of the SW prepotential in the framework of the instanton calculus is presented. As a demonstration explicit calculations for 2, 3 and 4-instanton contributions are carried out.

We introduce a matrix model for noncommutative gravity, based on the gauge group U(2)⊗U(2). The vierbein is encoded in a matrix Y_μ, having values in the coset space U(4)/(U(2)⊗U(2)), while the spin connection is encoded in a matrix X_μ, having values in U(2)⊗U(2). We show how to recover the Einstein equations from the θ→0 limit of the matrix model equations of motion. We stress the necessity of a metric tensor, which is a covariant representation of the gauge group in order to set up a consistent second order formalism. We finally define noncommutative gravitational instantons as generated by U(2)⊗U(2) valued quasi-unitary operators acting on the background of the matrix model. Some of these solutions have naturally self-dual or anti-self-dual spin connections.