QCD strings as a constrained Grassmannian σ model

Abstract

We present calculations for the effective action of string world sheet in R 3 and R 4 utilizing its correspondence with the constrained Grassmannian σ-model. Minimal surfaces describe the dynamics of open strings while harmonic surfaces describe that of closed strings. The one-loop effective action for these are calculated with instanton and anti-instanton background, representing N-string interactions at the tree level. The effective action is found to be the partition function of a classical modified Coulomb gas in the confining phase, with a dynamically generated mass gap.

Quantum effects for extrinsic geometry of strings via the generalized Weierstrass representation

“…At p = 0 one gets a minimal surface (H = 0) and the formulae (1)- (2) are reduced to the classical Weierstrass formulae for minimal surfaces. The generalized Weierstrass representation (1)- (2) is equivalent to another Weierstrass type representation which was proposed in [19] and has been used within the Gauss map approach in the papers [12]- [13]. The equivalence is established by simple formulae [20] …”

“…which however convert the linear system (2) into a nonlinear one which appeared in [19] and, consequently, in [12]- [13]. The fact that in the generalized Weierstrass representation (1)- (2) the functions ψ and ϕ are constrained by the linear equation (2) (instead of a nonlinear one of [18]) is one of its advantages.…”

“…Within this approach both the Nambu-Goto and Polyakov actions can be written in terms of constrained Kähler σ-model action. The use of the Gauss map makes easier the calculations of quantum effects induced by extrinsic geometry [12]- [13]. Unfortunately nonlinear constraints associated with the Gauss map for nonminimal surfaces give rise to serious computational difficulties.…”

“…[2]- [4]). Effects generated by an extrinsic geometry term have been analyzed by several authors [5]- [13]. A canonical description of the strings world-sheets by means of coordinates − → X of the target space in which strings are assumed to evolve was not very successful.…”

“…[14]). A different approach has been developed in the papers [12]- [13]. It is based on the description of the strings world-sheets via the Gauss map for surfaces conformally immersed into the Euclidean spaces.…”