Green functions and dimensional reduction of quantum fields on product manifolds

Abstract

We discuss Euclidean Green functions on product manifolds P = N × M. We show that if M is compact and N is not compact then the Euclidean field on P can be approximated by its zero mode which is a Euclidean field on N . We estimate the remainder of this approximation. We show that for large distances on N the remainder is small. If P = R D−1 × S β , where S β is a circle of radius β, then the result reduces to the well-known approximation of the D dimensional finite temperature quantum field theory by D − 1 dimensional one in the high temperature limit. Analytic continuation of Euclidean fields is discussed briefly.

Universal properties of the near-horizon optical geometry

“…The situation for AdS is somewhat more interesting than that for dS because there exist three equivalence classes of timelike, future directed, hypersurface orthogonal Killing vectors under the action of SO (3,2). To see this we take AdS to be a hyperboloid in E 3,2 :…”

“…For this reason the near horizon geometry of extreme black holes has received a great deal of attention, since it universally behaves like AdS 2 × M n−2 , where M n−2 is typically an n − 2 dimensional Einstein space and the symmetry is enhanced from R to SO(2, 1). By contrast the universal near horizon optical geometry of non-extreme horizons with its enhanced conformal symmetry has largely been ignored (notable exceptions are [1,2]). In other words, little has been done to exploit the fact that near a non-extreme horizon of a static black hole with metric ds 2 = −V 2 dt 2 + γ ij dx i dx j , (1.1) i = 1, 2, .…”

“…However this work, mainly concentrates on the planar approximation to the horizon geometry and does not make use of detailed concepts and ideas of hyperbolic geometry. Closer to what we are interested in is the work of Haba [2] which considers scalar fields near a Killing horizon using an optical geometry approach and constructs approximate Green's functions in cases where the horizon is not necessarily spherical. This approach is more in tune with our philosophy of seeking universal properties.…”

Euclidean scalar Green's functions near the black hole and black brane horizons

“…Now, E = p 2 0 + p 2 1 and dν = dp 0 dp 1 in equations ( 10)- (11). When ω k = 0 then the solution of equation ( 22) is the plane wave…”

“…However, we obtain its exact form. In section 4 we discuss an application of our earlier results [11] on the dimensional reduction of the Green's functions. In section 5 we apply the method to black brane solutions of string theory [12,13] in 10 and 11 dimensional supergravity which at the horizon have the geometry of AdS p × S q .…”