Instantons and confinement

The Euclidean vacuum functional of Yang-Mills theory is saturated by infinitely large two-dimensional sheets of calculable universal thickness. The gauge fields inside the sheets are characterised by a distribution of instantons held together by a smooth solution of the field equations. The sheets cannot grow in the transverse directions. Within each sheet the orientation in group space of the average field is correlated to the spatial orientation of the area, while different sheets are totally uncorrelated. We exhibit a class of fluctuations above this background which confine quarks. No assumption about the infra-red behaviour of the Gell-Mann-Low function is used; all quantities are calculable and no free parameters exist beyond the group structure constants.     

Instantons and hypercontact structures

Any (anti)-instanton yields a set of three contact forms on S⁷. The basic (anti)-instanton yields a hypercontact structure. Also, we indicate sufficient conditions for the components of an SU (2) connection to be contact forms. Finally, we prove, under a mild hypothesis, that the three contact forms of any hypercontact structure define the same contact structure.     

On Yang–Mills Instantons over Multi-Centered Gravitational Instantons

 In this paper we explicitly calculate the analogue of the 't Hooft SU(2) Yang–Mills instantons on Gibbons–Hawking multi-centered gravitational instantons, which come in two parallel families: the multi-Eguchi–Hanson, or A _k ALE gravitational instantons and the multi-Taub–NUT spaces, or A _k ALF gravitational instantons. We calculate their energy and find the reducible ones. Following Kronheimer we also exploit the U(1) invariance of our solutions and study the corresponding explicit singular SU(2) magnetic monopole solutions of the Bogomolny equations on flat ℝ³. Received: 16 September 2002 / Accepted: 22 October 2002 Published online: 21 February 2003 Communicated by A. Connes     

Instantons and algebraic geometry

Minimum action solutions for SU(2) Yang-Mills fields in Euclidean 4-space correspond, via the Penrose twistor transform, to algebraic bundles on the complex projective 3-space. These bundles in turn correspond to algebraic curves. The implication of these results for the Yang-Mills fields is described. In particular all solutions are rational and can be constructed from a series of AnsätzeA _l forl1.     

Instantons and Asymmetric Vacua

It is shown that with asymmetric classical vacua the quantum mechanical instanton approach to the energy splitting of degenerate states applies even though the degenerate state in one well is not the quantum mechanical ground state of that well. The instanton approach leads to the correct leading exponential behavior of the energy splitting E, but the prefactor is much more difficult to compute due to the asymmetric nature of the fluctuation potential V(_c(t)), which is in turn a direct consequence of the asymmetry of the two classical minima between which the instanton interpolates.     

Noncommutative Instantons Revisited

We find a new gauge in which U(1) noncommutative instantons are explicitly non-singular on noncommutative R ⁴. We also present a pedagogical introduction into noncommutative gauge theories.     

Instantons on Gravitons

Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-valued functions of a single variable, that are in turn organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions. For a judiciously chosen bow the Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spaces of instantons on ALF spaces.     

Noncommutative Instantons and Twistor Transform

Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of ℝ⁴. 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 ℙ², certain complexes of sheaves on a noncommutative ℙ³, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative ℙ² has a natural hyperk?hler metric and is isomorphic as a hyperk?hler manifold to the moduli space of framed torsion free sheaves on the commutative ℙ². The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative ℝ⁴ than the one considered by Nekrasov and Schwarz (a q-deformed ℝ⁴). Received: 3 May 2000 / Accepted: 3 April 2001     

Singular Instantons and Extra Dimensions

Hawking and Turok (HT) have recently proposedthat an open universe can be created from nothing. Theinstanton describing this process is singular, andtherefore its validity has been subject to question. In particular, Vilenkin has shown that aninstanton with the same singular structure as Hawkingand Turok's would lead to the unsuppressed decay of flatspace. However, Vilenkin's solution can be seen as the dimensional reduction of a five-dimensionalnonsingular instanton. In this context the unsuppressedinstability of flat space can be traded formetastability with a low decay rate, provided that the size of the extra dimension is large comparedwith the Planck scale. Implications for the HT model arediscussed.     

ABCD of Instantons

We solve = 2 supersymmetric Yang-Mills theories for an arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do this without resolving singularities of the compactified instanton moduli spaces.To Freeman Dyson, with admirationOn leave of absence from ITEP, Moscow, Russia     

Instantons in cutoff theories

It is shown that the inhomogeneous saddle points of scale invariant theories make the semiclassical expansion sensitive to the choice of non-renormalizable operators. In particular, the instanton fugacity and the beta function of the two-dimensional non-linear sigmamodel depend on apparently non-renormalizable operators. This represents a non-perturbative breakdown of that concept of universality which is based on low dimensional operators.     

Construction of Doubly-Periodic Instantons

We construct finite-energy instanton connections over ℝ⁴ which are periodic in two directions via an analogue of the Nahm transform for certain singular solutions of Hitchin's equations defined over a 2-torus. Received: 20 September 1999 / Accepted: 15 April 2000