Casimir and free energy for free fields and holographic theories in 2+1 on curved spaces

We investigate putting 2+1 free and holographic theories on a product of time with a curved compact 2-d space. We then vary the geometry of the space, keeping the area fixed, at zero/finite temperature, and measure the Casimir/free energy respectively. I will begin by discussing the free theory for a Dirac fermion or scalar field on deformations of the round 2-sphere. I will discuss how the Dirac theory may arise in physical systems such as monolayer graphene. For small deformations we solve analytically using perturbation theory. For large deformations we use novel numerical methods to compute these energies for specific deformations, including ones that drive the space to become singular. I will give evidence that the round sphere globally maximises the Casimir/free energy, which may imply geometric instabilities for spheres of such mono layer materials, although probably not graphene. We then discuss the analog for a holographic theory by studying its gravity dual. Here we use gravity techniques to analytically prove at zero temperature the round sphere maximises the Casimir energy. We discuss attempts to show the same for free energy at finite temperature. And finally I will report on on-going numerical gravity calculations of the Casimir energy for specific  deformations of the round sphere which yield an unexpected result.