Black Hole Determinants and Quasinormal Modes

We derive an expression for functional determinants in thermal spacetimes as a product over the corresponding quasinormal modes. As simple applications we give efficient computations of scalar determinants in thermal AdS, BTZ black hole and de Sitter spacetimes. We emphasize the conceptual utility of our formula for discussing ‘1/N ’ corrections to strongly coupled field theories via the holographic correspondence. 1 Determinants and quasinormal modes In a semiclassical quantisation of gravity the partition function can be written schematically as Z = ∑ g⋆ det ( −∇2g⋆ )±1 e−SE [g⋆] . (1) Here g⋆ are saddle points of the Euclidean gravitational action SE. We use g⋆ to collectively denote the metric and any other nonzero fields. The det ( −∇g⋆ ) term schematically denotes the product of determinants of all the operators controlling fluctuations about the background solution g⋆. As usual, the determinants appear on the numerator or denominator depending on whether the fluctuations are fermionic or bosonic respectively. This formula is readily generalised to correlators of operators by including appropriate sources in the gravitational action. Originally developed with a view to elucidating quantum gravitational effects in our universe [1], the semiclassical approach to quantum gravity has gained a new lease of life through the holographic correspondence [2, 3]. This is because the semiclassical limit in gravity corresponds to the large N limit of a dual gauge theory. Almost all works have restricted attention to the leading order large N result, the exponent in (1). Exceptions to this last statement include [4, 5, 6, 7] who studied an effect due to the determinant term in pure Anti-de Sitter space. In our recent paper [8] we have stressed the fact that the semiclassical determinant term in (1), a 1/N# correction, will be important in the framework of ‘applied holography’. This is the case because there are interesting effects, in for instance thermodynamics and transport, which are either entirely absent at leading order or else swamped by physical processes that are not those of prime relevance. In general it is technically challenging to compute determinants in nontrivial background spacetimes. Even when the Euclidean saddles are homogeneous the calculation can be involved. For example, [9] computed the gravitational determinant on S2 × S2 to obtain the semiclassical rate of nucleation of black holes in de Sitter spacetime. For most cases of interest, often black hole spacetimes, the eigenvalues of the fluctuation operator cannot be determined explicitly and an evaluation of the determinant is difficult without resort to WKB or other approximations (although see for instance [10] and citations thereof). In this paper we present and derive a formula for determinants in black hole backgrounds, indeed finite temperature spacetimes more generally, in terms of the quasinormal modes of the spacetime. For (complex) bosonic fields the expression appears in section 3, equation (40), and takes the form 1 det ( −∇g⋆ ) = e ∏