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     

Multi-instanton check of the relation between the prepotential F and the modulus u in N = 2 SUSY Yang-Mills theory

By examining multi-instantons in N = 2 supersymmetric SU(2) gauge theory, we derive, on very general grounds, and to all orders in the instanton number, a relationship between the prepotential

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(Φ), and the coordinate on the quantum moduli space u = TrΦ². This relation was previously obtained by Matone in the context of the explicit Seiberg-Witten low-energy solution of the model. Our findings can be viewed as a multi-instanton check of the proposed exact results in supersymmetric gauge theory.     

Topological Quantization of Instantons in SU（2） Yang-Mills Theory

By decomposing SU（2） gauge potential in four-dimensional Euclidean SU（2） Yang-Mills theory in a new way, we find that the instanton number related to the isospin defects of a doublet order parameter can be topologically quantized by the Hopf index and Brouwer degree. It is also shown that the instanton number is just the sum of the topological charges of the isospin defects in the non-trivial sector of Yang-Mills theory.     

Transport theory for a two-flavor color superconductor

QCD with two light-quark flavors at high baryonic density is a color superconductor. The diquark condensate breaks the SU(3) gauge symmetry down to an SU(2) subgroup. We study thermal fluctuations of the superconductor for temperatures below the gap. These are described by a simple transport equation. In the collisionless limit and close to equilibrium, it gives rise to the "hard superconducting loop" effective theory for the SU(2) gauge fields. This theory describes Debye screening and Landau damping of the gauge fields in the presence of the diquark condensate. We explain how our effective theory follows to one-loop order from quantum field theory. Our approach provides a convenient starting point for the computation of transport coefficients of the two-flavor color superconductor.     

Non-abelian Gauge Symmetry for Fields in Phase Space: a Realization of the Seiberg-Witten Non-abelian Gauge Theory

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.

     

Higgs boson production in two photon processes

We discuss some properties of the (instanton—anti-instanton) solution of the SU(2) gauge theory with a massless Higgs doublet recently found by Lipatov, Bukhvostov and Malkov.     

On the topological equivalence of gauge and Higgs fields in the dyon sectors

We prove by topological methods methods that in the context of SU(2) gauge theory the integers labelling the homotopy classes of Yang-Mills and Higgs fields are equal for periodic instanton and dyon configurations. A similar statement is true for the group SO(3) and, in the periodic instanton case, whenever a simply connected group is broken down to an abelian subgroup. We briefly discuss how the result goes over to the canonically quantized theory.     

The structure of the multi-instanton gas

We parametrize the q-instanton solutions of an SU(2) gauge theory in terms of the positions of 2q constituent particles or “instanton quarks”. Explicit computations of lowest order quantum fluctuations about the q = 1 and q = 2 solutions show that the short-distance interaction between instanton quark pairs is logarithmic. Extending these interactions to arbitrary q, we describe the multi-instanton gas as a plasma of instanton quarks.     

SU(l|m|n)阶化代数及其规范模型

本文将阶化李代数SU(m|n)扩充到SU(l|m|n),并讨论了它的局部规范理论。找到了规范不变的拉格郎日量,并将SU(2|1)弱电模型推广到统一强、弱、电相互作用模型。 关键词：     

Unification of effective field theories

We consider the renormalization of the coupling constants in theories with extended gauge hierarchies. An effective field theory approach is used to include an interesting class of higher-order effects in the renormalization group formulas. We calculate these corrections for all possible breakdowns of O(10) to SU(3)×SU(2)×U(1).     

Matone's Relation ofN =2 Super Yang-Mills and Spectrum of Toda Chain

InN=2 super Yang-Mills theory, the Matone's relation relates instanton corrections of the prepotential to instanton corrections of scalar field condensation <
Trφ²>. This relation has been proved to hold for Omega deformed theories too, using localization method. In this paper, we first give a case study supporting the relation, which does not rely on the localization technique. Especially, we show that the magnetic expansion also satisfies a relation of Matone's type. Then we discuss implication of the relation for the spectrum of periodic Toda chain, in the context of recently proposed Nekrasov-Shatashvili scheme.     

VARIATIONAL TREATMENT WITH THE THIRD ORDER CORRECTIONS FOR SU(2) LATTICE GAUGE THEORY

The variational method in Lagrangian formalism in lattice gauge theory in cumulant expansion is discussed. For SU(2) pure gauge Wi1son model to the third order corrections we show that there is no phase transition in four dimensions, but there exists the first order phase transition in five dimensions at β_C=0.838 in agreement with the Monte Carlo results. The physical meaning of higher order corrections is discussed.     

Nekrasov functions and exact Bohr-Sommerfeld integrals

In the case of SU(2), associated by the AGT relation to the 2d Liouville theory, the Seiberg-Witten prepotential is constructed from the Bohr-Sommerfeld periods of 1d sine-Gordon model. If the same construction is literally applied to monodromies of exact wave functions, the prepotential turns into the one-parametric Nekrasov prepotential \(\mathcal{F} (a, \epsilon_{1}) \) with the other epsilon parameter vanishing, ?₂ = 0, and ?₁ playing the role of the Planck constant in the sine-Gordon Shrödinger equation, ? = ?₁. This seems to be in accordance with the recent claim in [1] and poses a problem of describing the full Nekrasov function as a seemingly straightforward double-parametric quantization of sine-Gordon model. This also provides a new link between the Liouville and sine-Gordon theories.     

On the possibility of the dual Meissner effect induced by instantons

The classical Yang-Mills equation of motion is numerically investigated in the Lorentz gauge for a SU(2) gauge group. The color-electric field of two point-like charges is studied in the “empty” vacuum and in a state with an instanton present. The major effect for a fixed orientation of the instanton is that the color-electric field lines are expelled or attracted from the instanton region depending on the orientation of the instanton. If over the orientations of the instanton is averaged, this effect drops out. In this case of a random instanton orientation, we find that the external color-electric field is expelled from the instanton core. The origin of this effect is discussed. Received: 5 September 1997 / Published online: 23 February 1998     

WDVV equation and triple-product relation

We study the relation between the WDVV equations and the τ-function of the noncommutative KP (NCKP) hierarchy. WDVV-like equations (Hirota triple-product relation) in the noncommutative context appear as a consequence of the nontrivial equation for τ-function of the NC KP hierarchy, while the prepotential in the Seiberg–Witten (SW) theory has been identified to the τ-function of the Whitham hierarchy. We show that the spectral curve for the SW theory is the same as the Toda-chain hierarchy. We also show explicitly that Whitham hierarchy includes commutative Toda/KP hierarchy. Further, we comment on the origin of the Hirota triple-product relation in the context of the SW theory.     

High precision mean-field results for lattice gauge theories: with special attention paid to the gauge dependence of mean-field perturbation theory to finite order

We do mean-field perturbation theory for U(1) lattice gauge theory in the axial gauge, and evaluate corrections from fluctuations up to fourth order for the free energy and plaquette energy. Comparing with similar results previously obtained in the Feynman gauge we find, to those orders studied, a gauge dependence of the size of the first correction term neglected with one exception. This gauge dependence decreases rapidly as the order of the approximation is increased. To any finite order, results in axial gauge are better approximations than results in the Feynman gauge. We speculate why. Assuming it to be generally true, we evaluate the first correction beyond the one-loop mean-field approximation to the free energy of SU(2) gauge theory with Wilson action in the axial gauge. This correction brings the mean-field result very close to Monte Carlo results for β > 1.6. It also makes the mean-field result identical, within a narrow margin, to ressumed strong coupling results in the interval 1.6 < β < 2.4, thus showing the absence of a phase transition.For both groups studied, we find that the asymptotic series of mean-field perturbation theory give much better approximations than do ordinary weak coupling series.     

Towards a realistic model of Higgsless electroweak symmetry breaking

We present a 5D gauge theory in warped space based on a bulk SU(2)L x SU(2)R x U(1)(B-L) gauge group where the gauge symmetry is broken by boundary conditions. The symmetry breaking pattern and the mass spectrum resemble that in the standard model (SM). To leading order in the warp factor the rho parameter and the coupling of the Z (S parameter) are as in the SM, while corrections are expected at the level of a percent. From the anti-de Sitter (AdS) conformal field theory point of view the model presented here can be viewed as the AdS dual of a (walking) technicolorlike theory, in the sense that it is the presence of the IR brane itself that breaks electroweak symmetry, and not a localized Higgs on the IR brane (which should be interpreted as a composite Higgs model). This model predicts the lightest W, Z, and gamma resonances to be at around 1.2 TeV, and no fundamental (or composite) Higgs particles.     

On the stochastic confinement in SU(2) lattice gauge theory

We extend the Olesen approach to confinement, originally proposed for SU(∞) gauge theory, to the SU(2) group. We perform Monte Carlo calculations of the spectral density, which describes the distribution of eigenvalues of the Wilson loop in the SU(2) lattice gauge theory (LGT), for square loops up to size 4 × 4. Our results indicate the onset of disorder in the 4-dimensional LGT so that at weak coupling confinement is due to non-abelian fluctuations of the gauge field. We describe the Monte Carlo data by formulae of the 2-dimensional LGT with some effective coupling constant. We formulate how this effective coupling constant should depend on the size of the loop in order that the Olesen hypothesis about the dimensional reduction (i.e. approximate reduction of the 4-dimensional LGT to the effective 2-dimensional LGT in the confinement region) would be held. Using the strong-coupling expansion where the Olesen hypothesis holds in fourteen orders, we perform quantitative estimates. The density of simple vortices up to size 4 × 4 is calculated. A connection between the definitions of spectral densities for the SU(∞) and SU(2) groups is considered. Explicit formulae for the spectral density in the 2-dimensional SU(2) LGT are derived.