Phases of theories with fermions in AdS

Abstract

We study the phases of Yukawa theories at weak coupling and the Gross-Neveu models in AdS spaces at zero and finite temperature. Following the method used in [15], we first compute the one-loop partition functions, using the generalized eigenfunctions of the Dirac and Laplace operators on Euclidean AdS in the Poincaré coordinates. These functions satisfy desired periodicities under thermal identification. The method replicates results for partition functions known in the literature. We then study the phases of these field theories with fermions as regions in the corresponding parameter spaces at zero temperature. The phases and the corresponding phase boundaries are further identified as a function of the mass-squared of the scalar field and temperature for the Yukawa theories. While for the Gross-Neveu models, the changes in the phases as a function of the fermionic mass and the coupling constant at finite temperature are discussed. The Gross-Neveu-Yukawa model is studied for AdS4. We also note certain deviations from phases of these theories in flat space.

On partition functions and phases of scalars in AdS

“…Like the scalar the result for vector trace is known in the literature as mentioned in the introduction. In the following few sections, working in Poincaré coordinates, we reproduce these results for the trace (2.15) for the vectors at zero and finite temperature using an alternate method similar to that of scalars in [23] and spinors in [24].…”

“…The above expression is obtained after performing the integral over ⃗ x which gives the momentum delta function. We now follow the steps in [23,24] to simplify the above expression which include using the following identity…”

“…See for example [4]. With this aspect in mind, in [23] and [24] we have analyzed phases of various theories containing scalars and fermions in AdS which show several novel features as compared to their flat space counterparts. One of the aims of this paper is to extend the analysis to theories with vector field.…”

“…For scalars the partition function was calculated by introducing a method in [23]. Another purpose of this paper is to utilize the same procedure as for scalars and for fermions in [24] to compute the partition function for vectors using the Poincaré coordinates. In the initial parts of the paper we derive the one-loop determinant for Euclidean AdS d+1 by finding the eigenfunctions of the Laplacian for vectors.…”

“…However unlike in flat space, in AdS the thermal contribution is not proportional to the volume of the underlying space. These divergences at zero temperature are taken care of as was done in [23] and [24]. The ultraviolet divergences in the effective potential are regularized by analytically continuing the result in the number of dimensions.…”