Scalar boundary conditions in hyperscaling violating geometry

Jian-Pin Wu,Xiao-Mei Kuang

doi:10.1016/j.physletb.2015.11.046

Abstract

We study the possible boundary conditions of scalar field modes in a hyperscaling violation (HV) geometry with Lifshitz dynamical exponent z(z≥1) and hyperscaling violation exponent θ (θ≠0). For the case with θ>0, we show that in the parameter range 1≤z≤2, −z+d−1<θ≤(d−1)(z−1) or z>2, −z+d−1<θ≤d−1, the boundary conditions have different types, including the Neumann, Dirichlet and Robin conditions, while in the range θ≤−z+d−1, only Dirichlet type condition can be set. In particular, we further confirm that the mass of the scalar field does not play any role in determining the possible boundary conditions for θ>0, which has been addressed in Ref. [1]. Meanwhile, we also carry out the parallel investigation in the case with θ<0. We find that for m2<0, three types of boundary conditions are available, but for m2>0, only one type is available.

Highlights

The use of holographic duality into study of strongly-coupled field theories [2, 3] has produced substantial progress in reproducing and understanding phenomena from relativistic systems like QCD [4], and from the non-relativistic strongly interacting condensed matter systems [5,6,7,8]. These applications have provoked interest in holographic realization of symmetry groups that go beyond of relativistic conformal symmetry. These include in particular the Schrodinger symmetry[9, 10], Lifshitz symmetry[11,12,13,14,15,16,17,18,19,20,21,22,23,24] and HV symmetry [1, 25,26,27,28,29], which exhibit in common the anisotropic scaling characterized by the dynamic critical exponent z > 1 between time and space coordinates on the boundary
By applying this proposal into the scalar field( electromagnetic and gravitational perturbations) in the global anti-de Sitter space in [45], they found that the boundary conditions ranging from ranging from Dirichlet to Robin to Neumann conditions are all possible depending on the effective mass of the scalar field
We examined the normalizablity and studied the possible boundary conditions of scalar field in hyperscaling violating geometry

Summary

INTRODUCTION

The use of holographic duality into study of strongly-coupled field theories [2, 3] has produced substantial progress in reproducing and understanding phenomena from relativistic systems like QCD [4], and from the non-relativistic strongly interacting condensed matter systems [5,6,7,8]. These applications have provoked interest in holographic realization of symmetry groups that go beyond of relativistic conformal symmetry. The discussions on the square integrability of solutions at infinity are presented in Appendix B

SCALAR BOUNDARY CONDITIONS IN HV GEOMETRY

Boundary conditions

CONCLUSIONS AND DISCUSSIONS