From QCD to Dual Superconductivity to Effective String Theory

We show how an effective field theory of long distance QCD, describing a dual superconductor, can be expressed as an effective string theory of superconducting vortices. We use the semiclassical expansion of this effective string theory about a classical rotating string solution in any spacetime dimension D to obtain the semiclassical meson energy spectrum. We argue that the experimental data on Regge trajectories along with numerical simulations of the heavy quark potentials provide good evidence for an effective string description of long distance QCD.Comment: Talk given at the 5th International Conference on Quark Confinement and the Hadron Spectrum, Gargnano, Italy, September 200

Effective Field Theory for Long Strings

In previous work we used magnetic SU(N) gauge theory with adjoint representation Higgs scalars to describe the long distance quark-antiquark interaction in pure Yang-Mills theory, and later to obtain an effective string theory. The empirically determined parameters of the non-Abelian effective theory yielded $Z_N$ flux tubes resembling those of the Abelian Higgs model with Landau-Ginzburg parameter equal to $1/\sqrt{2}$, corresponding to a superconductor on the border between type I and type II. However, the physical significance of the differences between the Abelian and the $Z_N$ vortices was not elucidated and no principle was found to fix the value of the 'Landau-Ginzburg parameter' $\kappa$ of the non-Abelian theory determining the structure of the $Z_N$ vortices. Here we reexamine this point of view. We propose a consistency condition on $Z_N$ vortices underlying a confining string. This fixes the value of $\kappa$. The transverse distribution of pressure $p(r)$ in the resulting $Z_N$ flux tubes provides a physical picture of these vortices which differs essentially from that of the vortices of the Abelian Higgs model. We speculate that this general picture is valid independent of the details of the effective magnetic gauge theory from which it was obtained. Long wavelength fluctuations of the axis of the $Z_N$ vortices lead from an effective field theory to an effective string theory with the Nambu-Goto action. This effective string theory depends on a single parameter, the string tension $\sigma$. In contrast, the effective field theory has a second parameter, the intrinsic width 1/M of the flux tube.Comment: 8 page, 2 figure

A universal solution

The phenomenon of an implicit function which solves a large set of second order partial differential equations obtainable from a variational principle is explicated by the introduction of a class of universal solutions to the equations derivable from an arbitrary Lagrangian which is homogeneous of weight one in the field derivatives. This result is extended to many fields. The imposition of Lorentz invariance makes such Lagrangians unique, and equivalent to the Companion Lagrangians introduced in [baker].Comment: arxiv version is already officia

Stable divisorial gonality is in NP

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph $G$ can be defined with help of a chip firing game on $G$. The stable divisorial gonality of $G$ is the minimum divisorial gonality over all subdivisions of edges of $G$. In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer $k$ belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consist of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with $n$ vertices is bounded by $2^{p(n)}$ for a polynomial $p$