The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge vortices in Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special lagrangian submanifold of the moduli space of instantons in non-commutative Yang-Mills theories. This submanifold is the fixed point set of a action on the instanton moduli space which rotates the instantons in a plane. To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory. We further describe a moduli space construction for semi-local vortices, lumps in the and grassmannian sigma-models, and vortices on the non-commutative plane. We argue that this relationship between vortices and instantons underlies many of the quantitative similarities between quantum field theories in two and four dimensions.

Vortices, instantons and branes

We review our recent work on solitons in the Higgs phase. We use U(NC) gauge theory with NF Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Moduli spaces of domain walls (kinks) and vortices, which are the only elementary solitons in the Higgs phase, are found in terms of the moduli matrix. Stable monopoles and instantons can exist in the Higgs phase if they are attached by vortices to form composite solitons. The moduli spaces of these composite solitons are also worked out in terms of the moduli matrix. Webs of walls can also be formed with characteristic difference between Abelian and non-Abelian gauge theories. Instanton–vortex systems, monopole–vortex–wall systems, and webs of walls in Abelian gauge theories are found to admit negative energy objects with the instanton charge (called intersectons), the monopole charge (called boojums) and the Hitchin charge, respectively. We characterize the total moduli space of these elementary as well as composite solitons. In particular the total moduli space of walls is given by the complex Grassmann manifold SU(NF)/[SU(NC) × SU(NF − NC) × U(1)] and is decomposed into various topological sectors corresponding to boundary condition specified by particular vacua. The moduli space of k vortices is also completely determined and is reformulated as the half ADHM construction. Effective Lagrangians are constructed on walls and vortices in a compact form. We also present several new results on interactions of various solitons, such as monopoles, vortices and walls. Review parts contain our works on domain walls (Isozumi Y et al 2004 Phys. Rev. Lett. 93 161601 (Preprint hep-th/0404198), Isozumi Y et al 2004 Phys. Rev. D 70 125014 (Preprint hep-th/0405194), Eto M et al 2005 Phys. Rev. D 71 125006 (Preprint hep-th/0412024), Eto M et al 2005 Phys. Rev. D 71 105009 (Preprint hep-th/0503033), Sakai N and Yang Y 2005 Comm. Math. Phys. (in press) (Preprint hep-th/0505136)), vortices (Eto M et al 2005 Phys. Rev. Lett. 96 161601 (Preprint hep-th/0511088), Eto M et al 2006 Phys. Rev. D 73 085008 (Preprint hep-th/0601181)), domain wall webs (Eto M et al 2005 Phys. Rev. D 72 085004 (Preprint hep-th/0506135), Eto M et al 2006 Phys. Lett. B 632 384 (Preprint hep-th/0508241), Eto M et al 2005 AIP Conf. Proc. 805 354 (Preprint hep-th/0509127)), monopole–vortex–wall systems (Isozumi Y et al 2005 Phys. Rev. D 71 065018 (Preprint hep-th/0405129), Sakai N and Tong D 2005 J. High Energy Phys. JHEP03(2005)019 (Preprint hep-th/0501207)), instanton–vortex systems (Eto M et al 2005 Phys. Rev. D 72 025011 (Preprint hep-th/0412048)), effective Lagrangian on walls and vortices (Eto M et al 2006 Phys. Rev. D (in press) (Preprint hep-th/0602289)), classification of BPS equations (Eto M et al 2005 Preprint hep-th/0506257) and Skyrmions (Eto M et al 2005 Phys. Rev. Lett. 95 252003 (Preprint hep-th/0508130)).

Solitons in the Higgs phase: the moduli matrix approach

at large distances. The AdS/CFT correspondence also provides insights into the inherently non-perturbative aspects of QCD such as the orbital and radial spectra of hadrons and the form of hadronic wavefunctions. In particular, we show that there is an exact correspondence between the fth-dimensional coordinate of anti{de Sitter (AdS) space z and a specic light-front impact variable which measures the separation of the quark and gluonic constituents within the hadron in ordinary space-time. This connection allows one to compute the analytic form of the frame-independent light-front wavefunctions of mesons and baryons, the fundamental entities which encode hadron properties and which allow the computation of decay constants, form factors and other exclusive scattering amplitudes. Relativistic light-front equations in ordinary space-time are found which reproduce the results obtained using the fth-dimensional theory. As specic examples we compute the pion coupling constant f , the pion charge radius r 2 and examine the propagation of the electromagnetic current in AdS space, which determines the space and time-like behavior of the pion form factor and the pole of the meson.

Light-Front Dynamics and AdS/QCD Correspondence: The Pion Form Factor in the Space- and Time-Like Regions

https://scholarworks.wm.edu/cgi/viewcontent.cgi?article=1956&context=aspubs

Light-front holographic QCD and emerging confinement

We consider aspects of dynamical baryons in a holographic dual of QCD that is formulated on the basis of a D4/D8-brane configuration. We construct a soliton solution carrying a unit baryon number and show that it is obtained as an instanton solution of four-dimensional Yang-Mills theory with fixed size. The Chern-Simons term on the flavor D8-branes plays a crucial role of protecting the instanton from collapsing to zero size. By quantizing the collective coordinates of the soliton, we derive the baryon spectra. Negative-parity baryons as well as baryons with higher spins and isospins can be obtained in a simple manner.

/pdf/baryons-from-instantons-in-holographic-qcd-4kwok5dgos.pdf

Baryons from instantons in holographic QCD

Baryons in AdS/QCD

We consider a general class of boundary terms of the open XYZ spin-1/2 chain compatible with integrability. We have obtained the general elliptic solution of a K-matrix obeying the boundary Yang-Baxter equation using the R-matrix of the eight-vertex model and derived the associated integrable spin-chain Hamiltonian.

https://iopscience.iop.org/article/10.1088/0305-4470/27/24/002/pdf

Integrable XYZ spin chain with boundaries

The S, T, U parameters in the $ SU(3)_C\times SU(3)_L\times U(1)$ model with right -handed neutrinos are calculated. Explicit expressions for the oblique and Z - Z' mixing contributions are obtained. We show that the bilepton oblique contributions to S and T parameters are bounded : $- 0.085 \stackrel{<}{\sim} S \stackrel{<}{\sim} 0.05$ and $- 0.001 \stackrel{<}{\sim} T \stackrel{<}{\sim} 0.08$. The Z - Z' mixing contribution is positive and above 10%, but it will increase fastly with the higher Z' mass. %can be negative. The consequent mass splitting of the bilepton is derived and to be 15%. The limit on the mass of the neutral bilepton in this model is obtained.

S,T,Uparameters in anSU(3)C×SU(3)L×U(1)model with right-handed neutrinos

We have evaluated by numerical simulation the average size ${R}_{K}$ of random polygons of fixed knot topology $K=\ensuremath{\emptyset}{,3}_{1}{,3}_{1}\ensuremath{\sharp}{4}_{1},$ and we have confirmed the scaling law ${R}_{K}^{2}\ensuremath{\sim}{N}^{2{\ensuremath{\nu}}_{K}}$ for the number N of polygonal nodes in a wide range; $N=100--2200.$ The best fit gives $2{\ensuremath{\nu}}_{K}\ensuremath{\simeq}1.11--1.16$ with good fitting curves in the whole range of N. The estimate of $2{\ensuremath{\nu}}_{K}$ is consistent with the exponent of self-avoiding polygons. In a limited range of N $(N\ensuremath{\gtrsim}600),$ however, we have another fit with $2{\ensuremath{\nu}}_{K}\ensuremath{\simeq}1.01--1.07,$ which is close to the exponent of random polygons.

Takeo Inami

Papers

Baryons in AdS/QCD

Integrable XYZ spin chain with boundaries

S,T,Uparameters in anSU(3)C×SU(3)L×U(1)model with right-handed neutrinos

Average size of random polygons with fixed knot topology

Low-energy dynamics of noncommutative CP1 solitons in 2+1 dimensions