Instantons, fluxons, and open gauge string theory

Abstract

Abstract:We use the exact instanton expansion to illustrate various string characteristics of noncommutative gauge theory in two dimensions. We analyse the spectrum of the model and present some evidence in favour of Hagedorn and fractal behaviours. The decompactification limit of noncommutative torus instantons is shown to map in a very precise way, at both the classical and quantum level, onto fluxon solutions on the noncommutative plane. The weak-coupling singularities of the usual Gross-Taylor string partition function for QCD on the torus are studied in the instanton representation and its double scaling limit, appropriate for the mapping onto noncommutative gauge theory, is shown to be a generating function for the volumes of the principal moduli spaces of holomorphic differentials. The noncommutative deformation of this moduli space geometry is described and appropriate open string interpretations are proposed in terms of the fluxon expansion.

Localization for Yang-Mills Theory on the Fuzzy Sphere

“…The fluxon moduli space M 0 contains orbifold singularities arising from the fixed points of the S N -action on (S 2 ) N , which occur whenever two or more fluxon locations coincide. This is analogous to the vacuum solution of two-dimensional U (N ) gauge theory on a noncommutative torus wherein the moduli space of constant curvature connections is the symmetric product orbifold Sym N (T 2 ) [32], and there is a natural correspondence between two-dimensional noncommutative instantons and fluxons [42]. In the present case the U (N ) action on the fluxon configuration space (3.10) also has additional fixed points.…”

Double scaling string theory of QCD in two dimensions

“…The Morita equivalence rescales the Yang-Mills coupling as g 2 → g 2 /N and the area as A → A/N 2 in order to ensure that the action functionals map into each other, so that the large N limit requires a 't Hooft scaling and a weak-coupling limit. For example, gauge theory on the noncommutative plane R 2 Θ can be induced from gauge theory on the noncommutative torus T 2 1/N in the limit N → ∞, A → ∞ with Θ = A/2π N fixed [4,5]. This is a "double scaling limit" in which, due to above rescalings, the parameter µ = g 2 Θ = N 2 g 2 A/2π is held fixed.…”

“…Applying the Morita transformations to (1) and taking the double-scaling limit, one can derive the fluxon expansion of gauge theory on the noncommutative plane whose partition function is given by [4,5] …”

“…The saddle-point equation is Li 1/2 (−z) = − N λ/2π. Its solution can be found as a 1 N expansion by writing z = e x with x = x −1 N + k∈N 0 x k N k and using the asymptotic expansion of the polylogarithm function given by [5] …”

“…By using the leading singular behaviour of the polylogarithm function Li α (z) ≃ Γ(1−α)(− ln z) α−1 near its branch point z → 1 − [5], and matching with the strong-coupling expansion (9), we can determine the coefficients α h , β h of the asymptotic growth explicitly and draw our first main conclusion: The double scaling limit of the gauge theory free energy is the generating function for the asymptotic Hurwitz numbers with…”