Scalar fields on λ-deformed cosets

Abstract

We study dynamics of scalar fields on a large class of geometries described by integrable sigma models. Although equations of motion are not separable due to absence of isometries and Killing tensors, we completely determine the spectra using algebraic and group-theoretic methods.

On conformal blocks, crossing kernels and multi-variable hypergeometric functions

“…It is interesting to note that similar multi-variable generalization of hypergeometric equations were also used to describe the propagation of scalar field in the integrable deformations of AdSp × S p geometries[22]. It was also brought to our attention after this work is completed, when study conformal correlation functions constrained by conformal Ward identities in the momentum space, expressions in terms of multi-variable hypergeometric functions can also naturally arise, as discussed in[23] 4.…”

Asymmetric λ–deformed cosets

“…The λ-deformation of the cosets SO(n + 1)/SO(n) with λ-deformations for n = 2, 3, 4, 5 in the vector gauge cases have been constructed in [6,7]. These corresponding deformed geometries can be promoted to integrable backgrounds of string theory and their dynamical properties are also analyzed in [15].…”

“…( 3.24) This additional Z 2 transformation is non-trivial for this model. One way to show this is to solve the spectrum of scalar on the deformed geometry following [15]. We will give a simple example in the next section.…”

“…Even though the geometry corresponding to λ-deformed SO(4)/SO (3) coset has no isometries, an algebraic method based on group theory is proposed in [15]. In this appendix, we use their method to give a simple example showing that the Z 2 transformation on the parameter λ is physical.…”

“…In this appendix, we use their method to give a simple example showing that the Z 2 transformation on the parameter λ is physical. According to [15], the scalar field equation on the deformed geometry reduces to a second order differential equation known as the Heun's equation for polynomial ansatz Q(z):…”