Cosmological Solution in M - theory on S 1 / Z 2

We provide the first example of a cosmological solution of the Horava-Witten supergravity. This solution is obtained by exchanging the role of time with the radial coordinate of the transverse space to the five-brane soliton. On the boundary this corresponds to rotating an instanton solution into a tunneling process in a space with Lorentzian signature, leading to an expanding universe. Due to the freedom to choose different non-trivial Yang-Mills backgrounds on the boundaries, the two walls of the universe ( visible and hidden worlds) expand differently. However at late times the anisotropy is washed away by gravitational interactions. e-mail: [email protected] Recent years have lead to surprising new developments in our attempt to build a fundamental theory. In particular it has been conjectured that some (yet unknown) M-theory unifies all the known vacuua of string theory as well as eleven dimensional supergravity. These vacuua are believed to be interconnected through a network of dualities. Many pieces of evidence have been gathered in this direction. One of the simplest vacuua to analyze, for providing new phenomenological or cosmological scenarios, is the strongly coupled limit of heterotic string theory. Horava and Witten have identified it with a Z2 orbifold [1]. In the infrared limit, it has the description of an eleven dimensional supergravity on a manifold with boundaries. Gravitational anomalies are canceled by coupling the supergravity in each boundary with an E8 super–Yang–Mills theory. Unlike usual orbifold compactifications of string theory, in the Horava-Witten orbifold the communication between the two boundaries is due only to gravitational interactions. There is no gauge bosons propagating in the bulk. This leads to the picture of two parallel worlds (the observable and the hidden world) communicating through gravity. It has been pointed out recently that cosmological solutions can be generated from pbrane solutions [2–4]. The process consists in the exchange the role of time with the radial coordinate of the transverse space[4]. In this note we will use a similar approach to obtain a cosmological solution to supergravity theory on an eleven dimensional manifold with boundaries. A large number of cosmological models have been derived in the past from a variety of vacuua of M-theory. These include solutions of type II strings with both R-R and NS-NS background fields as well as weakly coupled heterotic string compactifications [2–21]. We will promote the discussion to the level of the strongly coupled limit of heterotic string theory. In Horava-Witten model the eleven dimensional bulk theory is described by a graviton supermultiplet. Its degrees of freedom are the metric gIJ , the three–form gauge potential CIJK with the associated field strength components GIJKL and the gravitino. Here I, J ... take values 0, 1, ..., 9, 11. In addition on each boundary lives a super–Yang–Mills ten–dimensional supermultiplet with a gauge field AA and the corresponding gauginos (A,B... = 0, 1, ..., 9). The gauge field strength is denoted by F a(1,2) AB . The bosonic part of the corresponding lagrangian (up to two derivatives) takes the form: