Discontinuities of free theories on AdS2

Abstract

The partition functions of free bosons as well as fermions on AdS2 are not smooth as a function of their masses. For free bosons, the partition function on AdS2 is not smooth when the mass saturates the Breitenlohner-Freedman bound. We show that the expectation value of the scalar bilinear on AdS2 exhibits a kink at the BF bound and the change in slope of the expectation value with respect to the mass is proportional to the inverse radius of AdS2. For free fermions, when the mass vanishes the partition function exhibits a kink. We show that expectation value of the fermion bilinear is discontinuous and the jump in the expectation value is proportional to the inverse radius of AdS2. We then show the supersymmetric actions of the chiral multiplet on AdS2 × S1 and the hypermultiplet on AdS2 × S2 demonstrate these features. The supersymmetric backgrounds are such that as the ratio of the radius of AdS2 to S1 or S2 is dialled, the partition functions as well as expectation of bilinears are not smooth for each Kaluza-Klein mode on S1 or S2. Our observation is relevant for evaluating one-loop partition function in the near horizon geometry of extremal black holes.