Calorons, Nahm's equations on S^1 and bundles over P^1xP^1

The moduli space of solutions to Nahm's equations of rank (k,k+j) on the circle, and hence, of SU(2) calorons of charge (k,j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on P^1xP^1 trivialized at infinity with c_2=k and equipped with a flag of degree j along P^1x{0}. An explicit matrix description of these spaces is given by a monad constructio

QCD Chiral restoration at finite TT under the Magnetic field: Studies based on the instanton vacuum model

We investigate the chiral restoration at finite temperature $(T)$ under the strong external magnetic field $\vec{B}=B_{0}\hat{z}$ of the SU(2) light-flavor QCD matter. We employ the instanton-liquid QCD vacuum configuration accompanied with the linear Schwinger method for inducing the magnetic field. The Harrington-Shepard caloron solution is used to modify the instanton parameters, i.e. the average instanton size $(\bar{\rho})$ and inter-instanton distance $(\bar{R})$, as functions of $T$. In addition, we include the meson-loop corrections (MLC) as the large-$N_{c}$ corrections because they are critical for reproducing the universal chiral restoration pattern. We present the numerical results for the constituent-quark mass as well as chiral condensate which signal the spontaneous breakdown of chiral-symmetry (SB$\chi$S), as functions of $T$ and $B$. Besides we find that the changes for the $F_\pi$ and $m_\pi$ due to the magnetic field is relatively small, in comparison to those caused by the finite $T$ effect.Comment: 4 pages, 1 table, 6figs. arXiv admin note: significant text overlap with arXiv:1103.605

On the stability of Dirac sheet configurations

Using cooling for SU(2) lattice configurations, purely Abelian constant magnetic field configurations were left over after the annihilation of constituents that formed metastable Q=0 configurations. These so-called Dirac sheet configurations were found to be stable if emerging from the confined phase, close to the deconfinement phase transition, provided their Polyakov loop was sufficiently non-trivial. Here we show how this is related to the notion of marginal stability of the appropriate constant magnetic field configurations. We find a perfect agreement between the analytic prediction for the dependence of stability on the value of the Polyakov loop (the holonomy) in a finite volume and the numerical results studied on a finite lattice in the context of the Dirac sheet configurations

Thermodynamic gauge-theory cascade

It is proposed that the cooling of a thermalized SU($N$) gauge theory can be formulated in terms of a cascade involving three effective theories with successively reduced (and spontaneously broken) gauge symmetries, SU($N$) $\to$ U(1)$^{N-1}$ $\to$ Z$_N$. The approach is based on the assumption that away from a phase transition the bulk of the quantum interaction inherent to the system is implicitly encoded in the (incomplete) classical dynamics of a collective part made of low-energy condensed degrees of freedom. The properties of (some of the) statistically fluctuating fields are determined by these condensate(s). This leads to a quasi-particle description at tree-level. It appears that radiative corrections, which are sizable at large gauge coupling, do not change the tree-level picture qualitatively. The thermodynamic self-consistency of the quasi-particle approach implies nonperturbative evolution equations for the associated masses. The temperature dependence of these masses, in turn, determine the evolution of the gauge coupling(s). The hot gauge system approaches the behavior of an ideal gas of massless gluons at asymptotically large temperature. A negative equation of state is possible at a stage where the system is about to settle into the phase of the (spontaneously broken) Z$_N$ symmetry.Comment: 25 pages, 6 figures, 1 reference added, minor corrections in text, errors in Sec. 3.2 corrected, PRD versio

Hyperbolic calorons, monopoles, and instantons

We construct families of SO(3)-symmetric charge 1 instantons and calorons on the space H^3 x R. We show how the calorons include instantons and hyperbolic monopoles as limiting cases. We show how Euclidean calorons are the flat space limit of this family.Comment: 11 pages, no figures 1 reference added Published version available at: http://www.springerlink.com/content/k0j4815u54303450

Comments on Condensates in Non-Supersymmetric Orbifold Field Theories

Non-supersymmetric orbifolds of N=1 super Yang-Mills theories are conjectured to inherit properties from their supersymmetric parent. We examine this conjecture by compactifying the Z_2 orbifold theories on a spatial circle of radius R. We point out that when the orbifold theory lies in the weakly coupled vacuum of its parent, fractional instantons do give rise to the conjectured condensate of bi-fundamental fermions. Unfortunately, we show that quantum effects render this vacuum unstable through the generation of twisted operators. In the true vacuum state, no fermion condensate forms. Thus, in contrast to super Yang-Mills, the compactified orbifold theory undergoes a chiral phase transition as R is varied.Comment: 10 Pages. Added clarifying comments, computational steps and a nice pretty pictur

On topological charge carried by nexuses and center vortices

In this paper we further explore the question of topological charge in the center vortex-nexus picture of gauge theories. Generally, this charge is locally fractionalized in units of 1/N for gauge group SU(N), but globally quantized in integral units. We show explicitly that in d=4 global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy $\Pi_3(S^3)\simeq Z$; this homotopy insures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy $\Pi_2(S^2)\simeq Z$, a homotopy usually associated with 't Hooft-Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We find a new type of nexus which can carry topological charge by linking to vortices or carry d=3 Chern-Simons number without center vortices present; the Chern-Simons number is connected with twisting and writhing of field lines, as the author had suggested earlier. In general, no topological charge in d=4 arises from these specific static configurations, since the charge is the difference of two (equal) Chern-Simons number, but it can arise through dynamic reconnection processes. We complete earlier vortex-nexus work to show explicitly how to express globally-integral topological charge as composed of essentially independent units of charge 1/N.Comment: Revtex4; 3 .eps figures; 18 page

SU(2) Calorons and Magnetic Monopoles

We investigate the self-dual Yang-Mills gauge configurations on $R^3\times S^1$ when the gauge symmetry SU(2) is broken to U(1) by the Wilson loop. We construct the explicit field configuration for a single instanton by the Nahm method and show that an instanton is composed of two self-dual monopoles of opposite magnetic charge. We normalize the moduli space metric of an instanton and study various limits of the field configuration and its moduli space metric.Comment: 17 pages, RevTex, 1 Figur

Covariant derivative expansion of fermionic effective action at high temperatures

We derive the fermionic contribution to the 1-loop effective action for A_4 and A_i fields at high temperatures, assuming that gluon fields are slowly varying but allowing for an arbitrary amplitude of A_4.Comment: RevTex 4, 11 pages, 3 figures. Version 2: Typos corrected; magnetic fields restricted to parallel sector. Version accepted for publication in PR