Instanton Research Articles

A holographic description of theta-dependent Yang-Mills theory at finite temperature * * Supported by the research startup funds of Dalian Maritime University (02502608) and the Fundamental Research Funds for the Central Universities (017192608)

Theta-dependent gauge theories can be studied using holographic duality through string theory in certain spacetimes. By this correspondence we consider a stack of N0 dynamical D0-branes as D-instantons in the background sourced by Nc coincident non-extreme black D4-branes. According to the gauge-gravity duality, this D0-D4 brane system corresponds to Yang-Mills theory with a theta angle at finite temperature. We solve the IIA supergravity action by taking account into a sufficiently small backreaction of the Dinstantons and obtain an analytical solution for our D0-D4-brane configuration. Subsequently, the dual theory in the large Nc limit can be holographically investigated with the gravity solution. In the dual field theory, we find that the coupling constant exhibits asymptotic freedom, as is expected in QCD. The contribution of the theta-dependence to the free energy gets suppressed at high temperatures, which is basically consistent with the calculation using the Yang-Mills instanton. The topological susceptibility in the large Nc limit vanishes, and this behavior remarkably agrees with the implications from the simulation results at finite temperature. Moreover, we finally find a geometrical interpretation of the theta-dependence in this holographic system.

Instanton bundles on the flag variety F(0,1,2)

Instanton bundles on $\mathbb{P}^3$ have been at the core of the research in Algebraic Geometry during the last thirty years. Motivated by the recent extension of their definition to other Fano threefolds of Picard number one, we develop the theory of instanton bundles on the complete flag variety $F:=F(0,1,2)$ of point-lines on $\mathbb{P}^2$. After giving for them two different monadic presentations, we use it to show that the moduli space $MI_F(k)$ of instanton bundles of charge $k$ is a geometric GIT quotient and the open subspace $MI^s_F(k)\subset MI_F(k)$ of stable instanton bundles has a generically smooth component of dim $8k-3$. Finally we study their locus of jumping conics.

Irreducibility of the moduli space of orthogonal instanton bundles on $$\mathbb {P}^n$$

In order to obtain existence criteria for orthogonal instanton bundles on $$\mathbb {P}^n$$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on $$\mathbb {P}^n$$ and prove that every orthogonal instanton bundle with no global sections on $$\mathbb {P}^n$$ and charge $$c\ge 2$$ has rank $$r \le (n-1)c$$. We also prove that when the rank r of the bundles reaches the upper bound, $$\mathcal {M}_{\mathbb {P}^n}^{\mathcal {O}}(c,r)$$, the coarse moduli space of orthogonal instanton bundles with no global sections on $$\mathbb {P}^n$$, with charge $$c\ge 2$$ and rank r, is affine, smooth, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in $$\mathcal {M}_{\mathbb {P}^n}^{\mathcal {O}}(c,r)$$, whenever is non-empty.

Yang–Mills instanton sheaves with higher topological charges

We explicitly construct [Formula: see text] (or [Formula: see text] complex) Yang–Mills (YM) (weakly) three and four instanton sheaves on [Formula: see text]. These results extend the previous construction of YM (weakly) instanton sheaves with topological charge two [S. H. Lai, J. C. Lee and I. H. Tsai, Yang–Mills instanton sheaves, Ann. Phys. 377 (2017) 446, arXiv:1603.02860 ].

On definite lattices bounded by integer surgeries along knots with slice genus at most 2

We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the ( 2 , 5 ) (2,5) -torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most 2. The proofs use input from Yang–Mills instanton gauge theory, Heegaard Floer correction terms, and the topology of singular complex plane curves.

Sasakian quiver gauge theories and instantons on cones over round and squashed seven-spheres

We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing G-equivariance on the homogeneous space G/H=SU(4)/SU(3) endowed with its Sasaki-Einstein structure, and G/H=Sp(2)/Sp(1) as a 3-Sasakian manifold. In both cases we describe the equivariance conditions and the resulting quivers. We further study the moduli spaces of instantons on the metric cones over these spaces by using the known description for Hermitian Yang-Mills instantons on Calabi-Yau cones. It is shown that the moduli space of instantons on the hyper-Kähler cone can be described as the intersection of three Hermitian Yang-Mills moduli spaces. We also study moduli spaces of translationally invariant instantons on the metric cone R8/Zk over S7/Zk.

Sheaf lines of Yang–Mills instanton sheaves

We calculate a sheaf line in [Formula: see text] which is the real line supporting sheaf points on [Formula: see text] of [Formula: see text] Yang–Mills (YM) instanton (or [Formula: see text] complex YM instanton) sheaves for some given ADHM data we obtained previously. We found that this sheaf line is indeed a special jumping line over [Formula: see text] spacetime. In addition, we calculate the singularity structure of the connection [Formula: see text] and the field strength [Formula: see text] at the corresponding singular point on [Formula: see text] of this sheaf line. We found that the order of singularity at the singular point on [Formula: see text] associated with the sheaf line in [Formula: see text] is higher than those of other singular points associated with normal jumping lines. We conjecture that this is a general feature for sheaf lines among jumping lines.

Construction of Stable Rank 2 Bundles on ℙ3 Via Symplectic Bundles

In this article we study the Gieseker–Maruyama moduli spaces ℬ(e, n) of stable rank 2 algebraic vector bundles with Chern classes c1 = e ∈ {−1, 0} and c2 = n ≥ 1 on the projective space ℙ3. We construct the two new infinite series Σ0 and Σ1 of irreducible components of the spaces ℬ(e, n) for e = 0 and e = −1, respectively. General bundles of these components are obtained as cohomology sheaves of monads whose middle term is a rank 4 symplectic instanton bundle in case e = 0, respectively, twisted symplectic bundle in case e = −1. We show that the series Σ0 contains components for all big enough values of n (more precisely, at least for n ≥ 146). Σ0 yields the next example, after the series of instanton components, of an infinite series of components of ℬ(0, n) satisfying this property.

Expanding universe and dynamical compactification using Yang-Mills instantons

We consider an eight-dimensional Einstein-Yang-Mills theory to explore whether Yang-Mills instantons formed in extra dimensions can induce the dynamical instability of our four-dimensional spacetime. We show that the Yang-Mills instantons in extra dimensions can trigger the expansion of our universe in four-dimensional spacetime as well as the dynamical compactification of extra dimensions. We also discuss a possibility to realize a reheating mechanism via the quantum back-reaction from the contracting tiny internal space with a smeared instanton.

ADHM and the 4d quantum Hall effect

Yang-Mills instantons are solitonic particles in d = 4 + 1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions.It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function.

Yang–Mills instantons in Kähler spaces with one holomorphic isometry

We consider self-dual Yang–Mills instantons in 4-dimensional Kähler spaces with one holomorphic isometry and show that they satisfy a generalization of the Bogomol'nyi equation for magnetic monopoles on certain 3-dimensional metrics. We then search for solutions of this equation in 3-dimensional metrics foliated by 2-dimensional spheres, hyperboloids or planes in the case in which the gauge group coincides with the isometry group of the metric (SO(3), SO(1,2) and ISO(2), respectively). Using a generalized hedgehog ansatz the Bogomol'nyi equations reduce to a simple differential equation in the radial variable which admits a universal solution and, in some cases, a particular one, from which one finally recovers instanton solutions in the original Kähler space. We work out completely a few explicit examples for some Kähler spaces of interest.

New Divisors in the Boundary of the Instanton Moduli Space

Let ${\mathcal I}(n)$ denote the moduli space of rank $2$ instanton bundles of charge $n$ on ${\mathbb P}^3$. We know from several authors that ${\mathcal I}(n)$ is an irreducible, nonsingular and affine variety of dimension $8n-3$. Since every rank $2$ instanton bundle on ${\mathbb P}^3$ is stable, we may regard ${\mathcal I}(n)$ as an open subset of the projective Gieseker--Maruyama moduli scheme ${\mathcal M}(n)$ of rank $2$ semistable torsion free sheaves $F$ on ${\mathbb P}^3$ with Chern classes $c_1=c_3=0$ and $c_2=n$, and consider the closure $\overline{{\mathcal I}(n)}$ of ${\mathcal I}(n)$ in ${\mathcal M}(n)$. We construct some of the irreducible components of dimension $8n-4$ of the boundary $\partial{\mathcal I}(n):=\overline{{\mathcal I}(n)}\setminus{\mathcal I}(n)$. These components generically lie in the smooth locus of ${\mathcal M}(n)$ and consist of rank $2$ torsion free instanton sheaves with singularities along rational curves.

The Infinitesimal Moduli Space of Heterotic G2 Systems

Heterotic string compactifications on integrable G2 structure manifolds Y with instanton bundles {(V,A), (TY,tilde{theta})} yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative {mathcal{D}} and show that it is equivalent to a heterotic G2 system encoding the geometry of the heterotic string compactifications. This operator {mathcal{D}} acts on a bundle {mathcal{Q}=T^*Y oplus {rm End}(V) oplus {rm End}(TY)} and satisfies a nilpotency condition {check{{mathcal{D}}}^2=0} , for an appropriate projection of {mathcal D}. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group {check H^1_{check{{mathcal{D}}}}(mathcal{Q})}. We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the {alpha'} expansion.

The four-dimensional smooth Poincaré conjecture

We start with a discussion of the more familiar 3-dimensional Poincare conjecture whose solution constitutes a major achievement of modern geometric analysis. This motivates the precise formulation of the smooth 4 dimensional Poincare conjecture given in this article. Then we explain the difference between the topological and smooth structures on a manifold as well as the groundbreaking work of Donaldson in defining differential invariant for 4-dimensional manifolds using Yang-Mills instanton. At the end, we give some recent approaches in the attack of the conjecture

Hyperbolic monopoles from hyperbolic vortices

Yang–Mills–Higgs monopoles and vortices in hyperbolic space can be constructed from and invariant Yang–Mills instantons, respectively. We use this fact to describe a large class of hyperbolic monopoles directly in terms of hyperbolic vortices embedded into 3 dimensions, yielding a remarkably simple relation between their Higgs fields. The class of monopoles we obtain are fixed relative to a plane in hyperbolic space, in a way which will be made clear by a study of the monopole spectral curve. We will use the correspondence between vortices and monopoles to give new insight into the moduli space of hyperbolic monopoles. Finally, our technique allows an explicit construction of the fields of a hyperbolic monopole invariant under a action, which we compare to periodic monopoles in Euclidean space.

Vector bundles whose restriction to a linear section is Ulrich

An Ulrich sheaf on an n-dimensional projective variety \(X \subseteq \mathbb {P}^{N}\) is an initialized ACM sheaf which has the maximum possible number of global sections. Using a construction based on the representation theory of Roby–Clifford algebras, we prove that every normal ACM variety admits a reflexive sheaf whose restriction to a general 1-dimensional linear section is Ulrich; we call such sheaves \(\delta \)-Ulrich. In the case \(n=2,\) where \(\delta \)-Ulrich sheaves satisfy the property that their direct image under a general finite linear projection to \(\mathbb {P}^2\) is a semistable instanton bundle on \(\mathbb {P}^{2}\), we show that some high Veronese embedding of X admits a \(\delta \)-Ulrich sheaf with a global section.

Sasakian quiver gauge theory on the Aloff–Wallach space X1,1

We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form Md×X1,1 with d-dimensional Riemannian manifold Md and the Aloff–Wallach space X1,1=SU(3)/U(1) endowed with its Sasaki–Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff–Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly and then compare the results with the quiver gauge theory on Q3=SU(3)/(U(1)×U(1)), the leaf space underlying the Sasaki–Einstein manifold X1,1. Moreover, we study instanton solutions on the metric cone C(X1,1) by evaluating the Hermitian Yang–Mills equation. We briefly discuss some features of the moduli space thereof, following the main ideas of a treatment of Hermitian Yang–Mills instantons on cones over generic Sasaki–Einstein manifolds in the literature.

SL(2, C) Instanton Sheaves

We constructed SL(2, C) Yang-Mills instanton solutions which satisfy the complex ADHM equations and the monad construction by using the biquaternion method. And we also found that the SL(2, C) instanton solutions can be used to explicitly construct instanton sheaves on CP3 which similar to the case holomorphic vector bundles on CP3 of SU(2) ADHM construction. And the existence of these instanton sheaves is related to singularities on S4 which do not exist in SU(2) instanton case.

Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry

We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carri\'on and Fern\'andez and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H^1_{\check{d}_\theta}(Y,TY)$, and is given by the kernel of a map ${\cal\check F}$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map ${\cal\check F}$ is given in terms of the curvature of the bundle and maps $H^1_{\check{d}_\theta}(Y,TY)$ into $H^2_{\check{d}_A}(Y,\mathrm{End}(V))$, and moreover can be used to define a cohomology on an extension bundle of $TY$ by $\mathrm{End}(V)$. We comment further on the resemblance with the holomorphic Atiyah algebroid and connect the story to physics, in particular to heterotic compactifications on $(Y,V)$ when $\alpha'=0$.