Casimir energy of hyperbolic orbifolds with conical singularities

Abstract

In this article, we obtain the explicit expression of the Casimir energy for 2-dimensional Clifford-Klein space forms in terms of the geometrical data of the underlying spacetime with the help of zeta-regularization techniques. The spacetime is geometrically expressed as a compact hyperbolic orbifold surface that may have finitely many conical singularities. In computing the contribution to the energy from a conical singularity, we derive an expression of an elliptic orbital integral as an infinite sum of special functions. We prove that this sum converges exponentially fast. Additionally, we show that under a natural assumption (known to hold asymptotically) on the growth of the lengths of primitive closed geodesics of the $(2, 3, 7)$-triangle group orbifold its Casimir energy is positive (repulsive).