Operads and quantum gravity

Abstract

In this article we try to explain and extend a statement due to Maxim Kontsevich back in 1999 that the Holography Principle in physics should be related to the (higher dimensional) Deligne Conjecture in mathematics. This seems to suggest that the little d -discs operad (or equivalently the notion of a d -algebra) gives a new way to understand the mathematical aspects of quantum gravity using holography. The strategy is as follows: we would like to learn something about quantum gravity in ( d + 1) dimensions: we use holography to reduce our original problem to a CFT in d -dimensions. The deep origin of this dimensional reduction lies on the fact that it is the area and not the volume which appears in the formula giving the entropy of black holes as described long ago by Hawking. Then we use d -algebras (i.e. the little d -discs operad) to study our d -dim CFT. The possible relation between d -dim CFT and d -algebras comes from the lesson we have learnt from strings (namely the 2-dim CFT case): the space of physical states in closed string field theory (i.e. the BRST cohomology) has a natural Gerstenhaber algebra structure and this by Cohen's theorem is related to the little 2-discs operad. The proposal then is that the relation might hold in higher than 2 dimensions. This approach is algebraic although it would have been much more satisfactory if we could generalise Segal's geometric approach to CFT in higher than 2 dimensions. Hopefully the article is mathematically self-contained.