Nuclear Physics B146 (1978) 90-108 © North-Holland Publishing Company QUANTIZING GRAVITATIONAL INSTANTONS * G.W. GIBBONS and M.J. PERRY ** Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK Received 4 July 1978 Gravitational instantons are complete non-singular positive definite metrics satisfying the Einstein equations with or without cosmological constant. We treat quantum fluctua- tions about such solutions and introduce a useful decomposition of the space of gravita- tional degrees of freedom in terms of which the one-loop corrections take an especially simple form. We treat in detail the question of zero modes, and the fluctuations about de Sitter space. 1. Introduction In the semiclassical approach to quantum field theory and functional integrals, one begins by finding stationary points of the classical Euclidean action and per- forming an expansion about them. Such classical solutions are generally referred to as instantons. In gravity theory, an instanton may be defined to be a complete non- singular Riemannian space satisfying the Einstein equations with or without a A term, and their one-loop corrections an application of the theory of small deforma- tions of such spaces. In this paper, we shall treat both deformations leaving the clas- sical action invariant, and quadratic fluctuations in terms of determinants. It repre- sents an extension of a previous paper on this topic [1 ], in that the effect of the non-vanishing A term is taken into account, and the resulting functional determi- nants are considerably simplified by the use of a certain decomposition of the gravi- tational degrees of freedom. The reason for introducing a A term is that it arises both in some supergravity theories, and in the "volume canonical ensemble" approach to fluctuations in space-time topology [2]. We shall mainly be concerned with the partition function, or vacuum persistance amplitude, Z. This is defined as a functional integral =fO[g] exp(-iI[g]), (1.1) Z where I is the action for gravitation [1 ]. + 1---~-S, TrK dE (1.2) I= l--~- f ( R - ' 2 A ) ( - g ) l / 2 d 4 x 8nK 16n~ M ~,,, * Work partially supported by US Department of Energy Grant no. EY-76-S-02-2220. ** Present address: Institute for Advanced Study, Princeton, NJ 08540, USA. 90