The large N expansion in hyperbolic sigma models

Abstract

Articles you may be interested inScalar mesons in a linear sigma model with (axial-)vector mesons AIP Conf. Proc. 1520, 226 (2013); 10.1063/1.4795961Supercharges in the hyper-Kähler with torsion supersymmetric sigma models A symmetric approach to the massive nonlinear sigma model Invariant correlation functions for SO͑1,N͒ hyperbolic sigma models are investigated. The existence of a large N asymptotic expansion is proven on finite lattices of dimension d Ն 2. The unique saddle point configuration is characterized by a negative gap vanishing at least like 1 / V with the volume. Technical difficulties compared to the compact case are bypassed using horospherical coordinates and the matrix-tree theorem.

Perturbative and nonperturbative correspondences between compact and noncompact sigma-models

“…This also implies that the solutions must be translation invariant, despite the preferred role of the x = x 0 equation. As mentioned earlier, the fact that the moments of W + [H] have an asymptotic expansion follows from [16]; for W − [H] this will be shown in [4] Given the uniqueness of the recursion one can readily re-derive the result of section 3.4. From (3.36) and (2.10) it is clear that the involution …”

“…However it is crucial that the asymptotic expansions are known to exist beforehand in both systems independently. The lattice formulation is especially suited to address this and we shall see that on a finite lattice the relevant asymptotic expansions do exist; for the perturbative one in section 2 below and for the large N expansion in a separate paper [4]. The perturbative correspondence simply involves a sign flip of the coefficients and in a formal expansion (dimensional regularization and minimal subtraction) has been noted to low orders in [5,6] and in the literature on Riemannian sigma models.…”

“…we allowed for a nontrivial xdependence in the extremal configurations ω x , x = x 0 . We anticipate now from [4] that the only solution of (3.54) giving rise to a translation invariant n x ·n y | N =∞ = −λD xy is constant, i.e.…”

Erratum to: “Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models” [Nucl. Phys. B 720 (2005) 235]

“…Specifically we take φ ∈ L 1 and call Ω ∈ L ∞ a generalized ground state of T if (φ, (T − T )Ω) = 0 for all φ ∈ L 1 . The set of generalized ground states forms a linear subspace of L ∞ which we call the ground state sector G(T) of T. The existence of generalized ground states which are moreover strictly positive L ∞ functions is guaranteed by a general result [37], a special case of which we describe here.…”

“…The property (3.32) guarantees that to our transfer operator T 2 (with only essential spectrum) one can associate a compact transfer operator (T 2 ) 0 (with only discrete spectrum) whose unique ground state wave function is the Ω 0 (n) featuring in Theorem 2; see [37] for details.…”