Instanton Research Articles

Explicit construction of Hermitian Yang-Mills instantons on coset manifolds

In four dimensions, ’t Hooft symbols offer a compact and powerful framework for describing the self-dual structures fundamental to instanton physics. Extending this to six dimensions, the six-dimensional ’t Hooft symbols can be constructed using the isomorphism between the Lorentz group Spin(6) and the unitary group SU(4). We demonstrate that the six-dimensional self-dual structures governed by the Hermitian Yang-Mills equations can be elegantly organized using these generalized ’t Hooft symbols. We also present a systematic method for constructing Hermitian Yang-Mills instantons from spin connections on six-dimensional manifolds using the generalized ’t Hooft symbols. We provide a thorough analysis of the topological invariants such as instanton and Euler numbers.

Modes of the Sakai-Sugimoto soliton

Abstract The instanton in the Sakai-Sugimoto model corresponds to the Skyrmion on the holographic boundary - which is asymptotically flat - and is fundamentally different from the flat Minkowski space Yang-Mills instanton. We use the Atiyah-Patodi-Singer index theorem and a series of transformations to show that there are 6k zeromodes - or moduli - in the limit of infinite 't Hooft coupling of the Sakai-Sugimoto model. The implications for the low-energy baryons - the Skyrmions - on the holographic boundary, is a scale separation between 2k "heavy" massive modes and 6k-9 "light" massive modes for k>1; the 9 global transformations that correspond to translations, rotations and isorotations remain as zeromodes. For k=1 there are 2 "heavy" modes and 6 zeromodes due to degeneracy between rotations and isorotations.

Yang-Mills instantons as the endpoint of black hole evaporation

Yang-Mills instantons as the endpoint of black hole evaporation

An S3 × S3 bundle over S4 and new supergravity solutions

The metric of S7 can be written as an SU(2)-instanton bundle over S4. It is also possible to write it differently as an anti-instanton bundle. We use this observation to construct an instanton–anti-instanton, SU(2)×SU(2), bundle over S4. We show that this 10d manifold admits two Einstein metrics. We then rewrite the metric to isolate two U(1) directions of the fibre and dimensionally reduce along them to get an eight dimensional metric describing an S2×S2 bundle over S4. This metric allows a harmonic 4-form which we use to derive new supergravity solutions in eleven and ten dimensions as AdS3×M8 and AdS2×M8, respectively.

On definite lattices bounded by a homology 3–sphere and Yang–Mills instanton Floer theory

On definite lattices bounded by a homology 3–sphere and Yang–Mills instanton Floer theory

On rank 3 instanton bundles on P3$\mathbb {P}^3$

AbstractWe investigate rank 3 instanton vector bundles on of charge and its correspondence with rational curves of degree . For , we present a correspondence between stable rank 3 instanton bundles and stable rank 2 reflexive linear sheaves of Chern classes and we use this correspondence to compute the dimension of the family of stable rank 3 instanton bundles of charge 2. Finally, we use the results above to prove that the moduli space of rank 3 instanton bundles on of charge 2 coincides with the moduli space of rank 3 stable locally free sheaves on of Chern classes . This moduli space is irreducible, has dimension 16 and its generic point corresponds to a generalized't Hooft instanton bundle.

How to build a black hole out of instantons

An often fruitful route to study quantum gravity is the determination and study of quantum mechanical models — that is, models with finite degrees of freedom — that capture the dynamics of a black hole’s microstates. An example of such a model is the superconformal quantum mechanics of Yang-Mills instantons, which has a proposed gravitational dual description as M-theory on a background of the form X7 × S4. This model arises in the strongly-coupled limit of the BFSS matrix model with additional fundamental hypermultiplets, offering a route towards useful numerical simulation. We construct a six-parameter black hole solution in this theory, which is generically non-supersymmetric and non-extremal, and is shown to arise in an “ultra-spinning” limit of the recently-found six-parameter AdS7 solution. We compute its thermodynamic properties, and show that in the supersymmetric limit the entropy and on-shell action match precisely the expected results as computed from the superconformal index of the quantum mechanics, to leading order in the supergravity regime. The low-lying spectrum thus provides access to the dynamics of near-extremal black holes, whose spectra are expected to receive strong quantum corrections.

Generalization of instanton-induced inflation and dynamical compactification

It was shown that Yang-Mills instantons on an internal space can trigger the expansion of our four-dimensional universe as well as the dynamical compactification of the internal space. We generalize the instanton-induced inflation and dynamical compactification to general Einstein manifolds with positive curvature and also to the FLRW metric with spatial curvature. We explicitly construct Yang-Mills instantons on all Einstein manifolds under consideration and find that the homogeneous and isotropic universe is allowed only if the internal space is homogeneous. We then consider the FLRW metric with spatial curvature as a solution of the eight-dimensional Einstein-Yang-Mills theory. We find that open universe (k = −1) admits bouncing solutions unlike the other cases (k = 0, +1).

Bridgeland stability of minimal instanton bundles on Fano threefolds

We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of $X$ is at least 3, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.

Rational Skyrmions

A new method is introduced to construct approximations to Skyrmions that are explicit rational functions of the spatial Cartesian coordinates. The scheme uses ADHM data of a Yang–Mills instanton to produce a Skyrmion with a baryon number that is equal to the instanton number. The formula for the Skyrmion involves only the evaluation of the ADHM data, in contrast to the Atiyah–Manton construction that requires the solution of a differential equation that can only be solved explicitly in the case of a spherically symmetric Skyrmion. Examples with baryon numbers one and two are studied in detail. The energy of the rational Skyrmion with baryon number one is lower than that of the Atiyah–Manton Skyrmion, which is already within one percent of the energy of the true numerically computed Skyrmion. A family of baryon number two Skyrmions is presented, which includes an axially symmetric Skyrmion that smoothly transforms to a pair of well-separated single Skyrmions as the parameter is varied.

Calorons and Constituent Monopoles

We study anti-self-dual Yang–Mills instantons on {mathbb {R}}^{3}times S^{1}, also known as calorons, and their behaviour under collapse of the circle factor. In this limit, we make explicit the decomposition of calorons in terms of constituent pieces which are essentially charge 1 monopoles. We give a gluing construction of calorons in terms of the constituents and use it to compute the dimension of the moduli space. The construction works uniformly for structure group an arbitrary compact semi-simple Lie group.

Tangent, cotangent, normal and conormal bundles are almost never instanton bundles

In this very short note we give an elementary characteristic free proof of the result claimed in the title (see Theorem 1.2 for a more precise formulation), generalizing a recent result proved by Benedetti et al. 2023 (see reference list) for Ulrich bundles over the complex field. Moreover, we also give a similar result about the twists of the cotangent bundle and make some comments about the possibility to obtain an analogous result for twists of the tangent bundle.

Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.

(0, 4) Projective superspaces. Part I. Interacting linear sigma models

We describe the projective superspace approach to supersymmetric models with off-shell (0, 4) supersymmetry in two dimensions. In addition to the usual superspace coordinates, projective superspace has extra bosonic variables — one doublet for each SU(2) in the R-symmetry SU(2) × SU(2) which are interpreted as homogeneous coordinates on CP1 × CP1. The superfields are analytic in the CP1 coordinates and this analyticity plays an important role in our description. For instance, it leads to stringent constraints on the interactions one can write down for a given superfield content of the model. As an example, we describe in projective superspace Witten’s ADHM sigma model — a linear sigma model with non-derivative interactions whose target is R4 with a Yang-Mills instanton solution. The hyperkähler nature of target space and the twistor description of instantons by Ward, and Atiyah, Hitchin, Drinfeld and Manin are natural outputs of our construction.

Dynamical Compactification with Matter

In this work, we study cosmological solutions of the 8-dimensional Einstein Yang-Mills theory coupled to a perfect-fluid matter. A Yang-Mills instanton of extra dimensions causes a 4-dimensional expanding universe with dynamical compactification of the extra dimensions. To construct physically reliable situations, we impose the null energy condition on the matter. This energy condition is affected by the extra dimensions. Then, we consider cosmological constant to grasp the structure of the solution space. Even in this simple case, we find several interesting solutions, such as bouncing universes and oscillatory solutions, eventually arriving at a de Sitter universe with stabilized compact dimensions. In addition, we consider a class of matters whose energy density depends on the volume of the extra dimensions. This case shows another set of bouncing universes. Also, a real scalar with potential is taken into account. The scalar field model admits de Sitter solutions due to the choice of potential, and we demonstrate how potentials can be constructed using flow equations. Thus, what we discuss in this work is based on the 8-dimensional Einstein frame, which corresponds to the 4-dimensional Jordan frame by dimensional reduction. Consequently, the results are derived in the 4-dimensional Jordan frame, not in the 4-dimensional Einstein frame.

Black hole entropy from quantum mechanics

We provide evidence for a holographic duality between superconformal quantum mechanics on the moduli space of Yang-Mills instantons and M-theory in certain asymptotically AdS7 × S4 backgrounds with a plane-wave boundary metric. We show that the gravitational background admits a supersymmetric black hole solution whose entropy is precisely reproduced by the superconformal index of the dual quantum mechanics.

Yang–Mills Instantons in the Dual-Superconductor Vacuum Can Become Confining

As known, the realistic, exponential, fall-off of the rate of production of light mesons in the chromo-electric field of a quark–antiquark string, as a function of the meson mass, can be obtained from the Schwinger-formula Gaussian fall-off within a phenomenological approach which assumes a certain distribution of the string tension. This approach gets a clear meaning in the London limit of the dual superconductor, where the logarithmic increase of the chromo-electric field towards the core of the string leads precisely to the change of the Gaussian fall-off to the exponential one, thus allowing for an identification of the phenomenological distribution of the string tension. In this paper, we demonstrate that, for this distribution of the string tension, the distribution of large-size Yang–Mills instantons, which are interacting with the confining monopole background, becomes O(1/ρ3), where ρ is the size of an instanton. Since such a distribution of large-size instantons is known to yield confinement, we conclude that, in the London limit of the dual-superconductor vacuum, instantons can form a confining medium, and we evaluate their contribution to the total string tension.

Instanton sheaves on projective schemes

A h–instanton sheaf on a closed subscheme X of some projective space endowed with an ample and globally generated line bundle OX(h) is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with h–instanton sheaves relating them to Ulrich sheaves. Moreover, we study h–instanton sheaves on smooth curves and surfaces, cyclic n–folds, Fano 3–folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to h–instanton bundles on varieties satisfying some mild extra technical conditions.

Modified Lévy Laplacian on manifold and Yang–Mills instantons

In this paper, an infinite-dimensional Laplacian defined as the Cesáro mean of the second-order directional derivatives on manifold is considered. This Laplacian is parametrized by the choice of a curve in the group of orthogonal rotations. It is shown that under certain conditions on the curve, this operator is related to instantons on a four-dimensional manifold.

ADHM skyrmions

We propose, via the Atiyah–Manton approximation, a framework for studying skyrmions on using Atiyah–Drinfeld–Hitchin–Manin (ADHM) data for Yang–Mills instantons on . We provide a dictionary between important concepts in the Skyrme model and analogous ideas for ADHM data, and describe an efficient process for obtaining approximate Skyrme fields directly from ADHM data. We show that the approximation successfully describes all known skyrmions with charge , with energies reproduced within 2% of the true minimisers. We also develop factorisation methods for studying clusters of instantons and skyrmions, generalising early work by Christ–Stanton–Weinberg, and describe some relatively large families of explicit ADHM data. These tools provide a unified framework for describing coalesced highly-symmetric configurations as well as skyrmion clusters, both of which are needed to study nuclear systems in the Skyrme model.