Instanton Charge Research Articles

Out-of-time-order correlator as a detector of baryonic phase structure in holographic QCD with instanton

We study the out-of-time-order correlators (OTOC) of Skyrmion as baryon in the D0-D4/D8 model which is expected to be holographically dual to QCD with instantons as D0-branes or with a non-zero theta angle. Baryon states are identified to the excitations of the Skyrmion which are described by a holographic quantum mechanical system in this model. By employing the definition of OTOC in quantum mechanics, we derive the formulas and demonstrate explicitly the numerical calculations of the OTOC. Our calculation illustrates the quantum OTOC with imaginary Lyapunov coefficient indicates the possibly metastable baryonic status in the presence of the instanton while the classical OTOC can not, thus it reveals the instantonic or theta-dependent features of QCD are dominated basically by its quantum properties. Furthermore, the OTOC also implies the baryonic phase becomes really chaotic with real Lyapunov exponent if the instanton charge increases sufficiently which agrees with the unstable baryon spectrum presented in this model. In this sense, we believe the OTOC may be treated as a tool to detect the baryonic phase structure of QCD.

Contact instantons with Legendrian boundary condition: A priori estimates, asymptotic convergence and index formula

In this paper, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential [Formula: see text]-convergence result at a puncture under the uniform [Formula: see text] bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. Leaving the study of [Formula: see text]-estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [27], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [27]–[29] and [36] containing applications to contact topology and contact Hamiltonian dynamics.

Exact expressions for n-point maximal U(1)Y-violating integrated correlators in SU(N) mathcal{N} = 4 SYM

The exact expressions for integrated maximal U(1)Y violating (MUV) n-point correlators in SU(N) mathcal{N} = 4 supersymmetric Yang-Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of N and τ = θ/(2π) + 4πi/ {g}_{YM}^2 , and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights (w, −w) where w = n − 4. The correlators satisfy Laplace-difference equations that relate the SU(N+1), SU(N) and SU(N−1) expressions and generalise the equations previously found in the w = 0 case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight (w, −w). For any fixed value of N the perturbation expansion of this correlator is found to start at order ( {g}_{YM}^2 N)w. The contributions of Yang-Mills instantons of charge k > 0 are of the form qkf(gYM), where q = e2πiτ and f(gYM) = O( {g}_{YM}^{-2w} ) when {g}_{YM}^2 ≪ 1. Anti-instanton contributions have charge k < 0 and are of the form {overline{q}}^{left|kright|}hat{f}left({g}_{YM}right) , where hat{f}left({g}_{YM}right)=Oleft({g}_{YM}^{2w}right) when {g}_{YM}^2 ≪ 1. Properties of the large-N expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of n-point free-field MUV correlators with the integrands of (n − 4)-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rôle of SL(2, ℤ)-covariance in the construction.

Axial Anomaly in SU(N) Yang-Mills Matrix Models.

The SU(N) Yang-Mills matrix model admits self-dual and anti-self-dual instantons. When coupled to N_{f} flavors of massless quarks, the Euclidean Dirac equation in an instanton background has n_{+} positive and n_{-} negative chirality zero modes. The vacua of the gauge theory are N-dimensional representations of SU(2), and the (anti-) self-dual instantons tunnel between two commuting representations, the initial one composed of r_{0}^{(1)} irreps and the final one with r_{0}^{(2)} irreps. We show that the index (n_{+}-n_{-}) in such a background is equal to a new instanton charge T_{new}=±[r_{0}^{(2)}-r_{0}^{(1)}]. Thus T_{new}=(n_{+}-n_{-}) is the matrix model version of the Atiyah-Singer index theorem. Further, we show that the path integral measure is not invariant under a chiral rotation, and relate the noninvariance of the measure to the index of the Dirac operator. Axial symmetry is broken anomalously, with the residual symmetry being a finite group. For N_{f} fundamental fermions, this residual symmetry is Z_{2N_{f}}, whereas for adjoint quarks it is Z_{4N_{f}}.

Towards Lefschetz Thimbles in Sigma Models, I

We study two dimensional path integral Lefschetz thimbles, i.e. the possible path integration contours. Specifically, in the examples of the $O(N)$ and ${\bf CP}^{N-1}$ models, we find a large class of complex critical points of the sigma model actions which are relevant for the theory in finite volume at finite temperature, with various chemical potentials corresponding to the symmetries of the models. In this paper we discuss the case of the $O(2m)$ and the ${\bf CP}^{N-1}$ models in the sector of zero instanton charge, as well as some solutions of the $O(2m+1)$ model. The ${\bf CP}^{N-1}$-model for all instanton charges and a more general class of solutions of the $O(N)$-model with odd $N$ will be discussed in the forthcoming paper.

Symmetric calorons of higher charges and their large period limits

Periodic instantons, also called calorons, are the BPS solutions to the pure Yang–Mills theories on R3×S1. It is known that the calorons interconnect with the instantons and the BPS monopoles as the ratio of their size to the period of S1 varies. We give, in this paper, the action density configurations of the SU(2) calorons of higher instanton charges with several platonic symmetries through the numerical Nahm transform, after the construction of the analytic Nahm data. The calorons considered are 5-caloron with octahedral symmetry, 7-caloron with icosahedral symmetry, and 4-caloron interconnecting tetrahedral and octahedral symmetries. We also consider the large period, or the instanton, limits of the Nahm data, i.e., the ADHM limits, and observe the similar spatial distributions of the action densities with the calorons.

Resumming perturbative series in the presence of monopole bubbling effects

Monopole bubbling effect is screening of magnetic charges of singular Dirac monopoles by regular 't Hooft-Polyakov monopoles. We study properties of weak coupling perturbative series in the presence of monopole bubbling effects as well as instantons. For this purpose, we analyze supersymmetric 't Hooft loop in four dimensional $\mathcal{N}=2$ supersymmetric gauge theories with Lagrangians and non-positive beta functions. We show that the perturbative series of the 't Hooft loop is Borel summable along positive real axis for fixed instanton numbers and screened magnetic charges. It turns out that the exact result of the 't Hooft loop is the same as the sum of the Borel resummations over instanton numbers and effective magnetic charges. We also obtain the same result for supersymmetric dyonic loops.

Classical deformations of noncompact surfaces and their moduli of instantons

We describe semiuniversal deformation spaces for the noncompact surfaces Zk:=Tot(OP1(−k)) and prove that any nontrivial deformation Zk(τ) of Zk is affine.It is known that the moduli spaces of instantons of charge j on Zk are quasi-projective varieties of dimension 2j−k−2. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation Zk(τ) are empty.

Ultraviolet properties of the self-dual Yang-Mills theory

We compute the ultraviolet divergences in the self-dual Yang-Mills theory, both in the purely perturbative (zero instanton charge) and topologically non-trivial sectors. It is shown in particular that the instanton measure is precisely the same as the one-loop result in the standard Yang-Mills theory.

Phases of kinky holographic nuclear matter

Holographic QCD at finite baryon number density and zero temperature is studied within the five-dimensional Sakai-Sugimoto model. We introduce a new approximation that models a smeared crystal of solitonic baryons by assuming spatial homogeneity to obtain an effective kink theory in the holographic direction. The kink theory correctly reproduces a first order phase transition to lightly bound nuclear matter. As the density is further increased the kink splits into a pair of half-kink constituents, providing a concrete realization of the previously suggested dyonic salt phase, where the bulk soliton splits into constituents at high density. The kink model also captures the phenomenon of baryonic popcorn, in which a first order phase transition generates an additional soliton layer in the holographic direction. We find that this popcorn transition takes place at a density below the dyonic salt phase, making the latter energetically unfavourable. However, the kink model predicts only one pop, rather than the sequence of pops suggested by previous approximations. In the kink model the two layers produced by the single pop form the surface of a soliton bag that increases in size as the baryon chemical potential is increased. The interior of the bag is filled with abelian electric potential and the instanton charge density is localized on the surface of the bag. The soliton bag may provide a holographic description of a quarkyonic phase.

D-instantons in Klebanov-Witten model

We study D-instanton solutions in type IIB supergravity on $AdS_5\times T^{1,1}$, which has a dual ${\cal N}=1$ $SU(N)\times SU(N)$ super Yang-Mills theory. Apart from ordinary D(-1)-brane instantons, we discuss wrapped D1-branes over minimal 2-cycles and derive explicit solutions preserving half the supersymmetries. These solutions are identified with Yang-Mills instantons which are (anti)self-dual in both gauge group factors with instanton charge $(k,k')$. By examining the boundary behaviour of the solutions we discuss the coupling to the corresponding dual boundary operators, and identify their vacuum expectation values. We also discuss the boundary terms and compute the action for these solutions.

Semilocal fractional instantons

We find semi-local fractional instantons of codimension four in Abelian and non-Abelian gauge theories coupled with scalar fields and the corresponding ${\mathbb C}P^{N-1}$ and Grassmann sigma models at strong gauge coupling. They are 1/4 BPS states in supersymmetric theories with eight supercharges, carry fractional (half) instanton charges characterized by the fourth homotopy group $\pi_4 (G/H)$, and have divergent energy in infinite spaces. We construct exact solutions for the sigma models and numerical solutions for the gauge theories. Small instanton singularity in sigma models is resolved at finite gauge coupling (for the Abelian gauge theory). Instantons in Abelian and non-Abelian gauge theories have negative and positive instantons charges, respectively, which are related by the Seiberg-like duality that changes the sign of the instanton charge.

Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on R4 and with two-dimensional conformal field theory. We construct a stacky compactification of the minimal resolution Xk of the Ak−1 toric singularity C2/Zk, which is a projective toric orbifold Xk such that Xk∖Xk is a Zk-gerbe. We construct moduli spaces of torsion free sheaves on Xk which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasi-projective variety, compute its dimension, and classify its fixed points under the natural induced toric action. We use this construction to compute the partition functions and correlators of chiral BPS operators for N=2 quiver gauge theories on Xk with nontrivial holonomies at infinity. The partition functions are computed with and without couplings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas, which relate them to the corresponding Nekrasov partition functions on the affine toric open subsets of Xk. We compare our new partition functions with previous computations, explore their connections to the representation theory of affine Lie algebras, and find new constraints on fractional instanton charges in the coupling to fundamental matter. We show that the partition functions in the low energy limit are characterized by the Seiberg–Witten curves, and in some cases also by suitable blowup equations involving Riemann theta-functions on the Seiberg–Witten curve with characteristics related to the nontrivial holonomies.

The condensate from torus knots

We discuss recently formulated instanton-torus knot duality in $\Omega$-deformed 5D SQED on $\mathbb{R}^4 \times S^1$ focusing at the microscopic aspects of the condensate formation in the instanton ensemble. Using the chain of dualities and geometric transitions we embed the SQED with a surface defect into the $SU(2)$ SQCD with $N_f=4$ and identify the numbers $(n,m)$ of the torus $T_{n,m}$ knot as instanton charge and electric charge. The HOMFLY torus knot invariants in the fundamental representation provide entropic factor in the condensate of the massless flavor counting the degeneracy of the instanton--W-boson web with instanton and electric numbers $(n,m)$ but different spin and flavor content. Using the inverse geometrical transition we explain how our approach is related to the evaluation of the HOMFLY invariants in terms of Wilson loop in 3d CS theory. The reduction to 4D theory is briefly considered and some analogy with baryon vertex is conjectured.

Fractional instantons and bions in the principal chiral model on ℝ 2 × S 1 $$ {\mathrm{\mathbb{R}}}^2\times {S}^1 $$ with twisted boundary conditions

Bions are multiple fractional instanton configurations with zero instanton charge playing important roles in quantum field theories on a compactified space with a twisted boundary condition. We classify fractional instantons and bions in the $SU(N)$ principal chiral model on ${\mathbb R}^2 \times S^1$ with twisted boundary conditions. We find that fractional instantons are global vortices wrapping around $S^1$ with their $U(1)$ moduli twisted along $S^1$, that carry $1/N$ instanton (baryon) numbers for the ${\mathbb Z}_N$ symmetric twisted boundary condition and irrational instanton numbers for generic boundary condition. We work out neutral and charged bions for the $SU(3)$ case with the ${\mathbb Z}_3$ symmetric twisted boundary condition. We also find for generic boundary conditions that only the simplest neutral bions have zero instanton charges but instanton charges are not canceled out for charged bions. A correspondence between fractional instantons and bions in the $SU(N)$ principal chiral model and those in Yang-Mills theory is given through a non-Abelian Josephson junction.

Yang–Mills theory for semidirect products $$\mathrm{G}\ltimes \mathfrak {g}^*$$ G ⋉ g ∗ and its instantons

Yang-Mills theory with a symmetry algebra that is the semidirect product $\mathfrak{h}\ltimes\mathfrak{h}^*$ defined by the coadjoint action of a Lie algebra $\mathfrak{h}$ on its dual $\mathfrak{h}^*$ is studied. The gauge group is the semidirect product ${\rm G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$, a noncompact group given by the coadjoint action on $\mathfrak{h}^*$ of the Lie group ${\rm G}_{\mathfrak{h}}$ of $\mathfrak{h}$. For $\mathfrak{h}$ simple, a method to construct the self-antiself dual instantons of the theory and their gauge non\-equivalent deformations is presented. Every ${\rm G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$ instanton has an embedded ${\rm G}_{\mathfrak{h}}$ instanton with the same instanton charge, in terms of which the construction is realized. As an example,$\mathfrak{h}=\mathfrak{s}\mathfrak{u}(2)$ and instanton charge one is considered. The gauge group is in this case $SU(2)\ltimes{\bf R}^3$. Explicit expressions for the selfdual connection, the zero modes and the metric and complex structures of the moduli space are given.

Stringy instanton counting and topological strings

We study the stringy instanton partition function of four dimensional $$ \mathcal{N}=2 $$ U(N) supersymmetric gauge theory which was obtained by Bonelli et al. in 2013. In type IIB string theory on $$ {\mathrm{\mathbb{C}}}^2\times {T}^{*}{\mathrm{\mathbb{P}}}^1\times \mathrm{\mathbb{C}} $$ , the stringy U(N) instantons of charge k are described by k D1-branes wrapping around the $$ {\mathrm{\mathbb{P}}}^1 $$ bound to N D5-branes on $$ {\mathrm{\mathbb{C}}}^2\times {\mathrm{\mathbb{P}}}^1 $$ . The KK corrections induced by compactification of the $$ {\mathrm{\mathbb{P}}}^1 $$ give the stringy corrections. We find a relation between the stringy instanton partition function whose quantum stringy corrections have been removed and the K-theoretic instanton partition function, or by geometric engineering, the refined topological A-model partition function on a local toric Calabi-Yau threefold. We also study the quantum stringy corrections in the stringy instanton partition function which is not captured by the refined topological strings.

Fractional instantons and bions in the O(N) model with twisted boundary conditions

Recently, multiple fractional instanton configurations with zero instanton charge, called bions, have been revealed to play important roles in quantum field theories on compactified spacetime. In two dimensions, fractional instantons and bions have been extensively studied in the ${\mathbb C}P^{N-1}$ model and the Grassmann sigma model on ${\mathbb R}^1 \times S^1$ with the ${\mathbb Z}_N$ symmetric twisted boundary condition. Fractional instantons in these models are domain walls with a localized $U(1)$ modulus twisted half along their world volume. In this paper, we classify fractional instantons and bions in the $O(N)$ nonlinear sigma model on ${\mathbb R}^{N-2} \times S^1$ with more general twisted boundary conditions in which arbitrary number of fields change sign. We find that fractional instantons have more general composite structures, that is, a global vortex with an Ising spin (or a half-lump vortex), a half sine-Gordon kink on a domain wall, or a half lump on a "space-filling brane" in the $O(3)$ model (${\mathbb C}P^1$ model) on ${\mathbb R}^{1} \times S^1$, and a global monopole with an Ising spin (or a half-Skyrmion monopole), a half sine-Gordon kink on a global vortex, a half lump on a domain wall, or a half Skyrmion on a "space-filling brane" in the $O(4)$ model (principal chiral model or Skyrme model) on ${\mathbb R}^{2} \times S^1$. We also construct bion configurations in these models.

$${\mathcal{N} = 2}$$ N = 2 Quiver Gauge Theories on A-type ALE Spaces

We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in $\mathcal{N}=2$ gauge theories on ALE spaces based on quiver varieties and the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$, in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion free sheaves on a suitable orbifold compactification of $X_k$. We extend these computations to generic $\mathcal{N}=2$ superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg-Witten geometry.

Instanton bags, high density holographic QCD, and chiral symmetry restoration

We describe the simplest example of an instanton bag in Euclidean space. It consists of a monopole wall and a Kaluza-Klein monopole wall, lifted to one higher dimension, trapping the instanton charge in the middle. This object has finite instanton density in a three-dimensional volume. Baryon physics in holographic QCD models gets translated into a multi-instanton problem in the bulk, and a state with a high density baryonic charge consists of a non-diluted multi-instanton solution. The instanton bag is a good candidate for this high-density state. We compute its parameters via moduli stabilization. Chiral symmetry restoration is exhibited by this state, and it is a direct consequence of its non-diluted features.