Non-critical confining strings and the renormalization group

Abstract

String vacua for non critical strings satisfying the requirements of Zig-Zag invariance are constructed. The Liouville mode is shown to play the rôle of scale in the Renormalization Group operation. Differences and similarities with the D-brane near horizon approach to non supersymmetric gauge theories are discussed as well.

Static gauge potential from noncritical strings

“…with Dirichlet boundary conditions at the endpoints of the string, r(0) =r(1) = Λ (6) The resulting embedding then resembles half a topological cylinder, with its length along the temporal direction, and bounded by r = Λ (cf. Figure), in such a way that the total action is proportional to the span of the temporal variable, S ∼ T .…”

Confining strings, Wilson loops and extra dimensions

“…This point will be clarified latter. The other situation, when the quark-antiquark pair lies at the same point of the 21-dimensional hyper-plane H has already discussed in references [3,4] (and in our context also in [1]). We will use it as a check of consistency in the limit when |∆ y| → 0, in the previous case.…”

“…It has been considered in the framework of the Renormalization Group approach to the string theory developed byÁlvarez and Gómez [3,4]. In [1] it has been calculated the Wilson loops and analyzed the features of the metric which is a solution of the vanishing string theory sigma model β-function equations, at leading order in α…”

“…Using the Liouville ansatz [3,6] those equations are trivially satisfied by any dilaton field. In the metric we identify 4 of the 26 coordinates as the Euclidean (Σ) (or Minkowski (M 4 )) spacetime, where the gauge theory lives.…”

“…We will return to this point at the end of the paper. 6 Please notice that in that follows we will use three different scales: the dimensionless l related to the extra dimensions on H, the above mentioned l c (which in [3,4] is related to the inverse of Λ QCD ), and also l s related to the string constant α ′ = l We start from the Nambu-Goto action…”