Friday, May 3, 2013

1305.0383 (Madhavan Venkatesh)

An Algebraic Topological Construct of Classical Loop Gravity and the
prospect of Higher Dimensions

Madhavan Venkatesh
In this paper, classical gravity is reformulated in terms of loops, via an algebraic topological approach. The main component is the loop group, whose elements consist of pairs of cobordant loops. A Chas-Sullivan product is described on the cobordism, and three other products, namely the 'vertical', 'horizontal' and 'total' products are re-introduced. (They have already been defined in an earlier paper by the author). A loop calculus is introduced on the space of loops, consisting of the loop variation functional,the loop derivative, Mandelstam derivative, and what the author wishes to call, the Gambini-Pullin contact functional. The loop derivative happens to be a generator of the group of loops, and the Gambini-Pullin functional is an infinitesimal generator of diffeomorphisms. A toy model of gravity is formulated in terms of the above, and it is proven that the total product provides for the Maurer-Cartan structure of the space. Further, new quantities labeled as 'momenta', 'velocity' and 'energy' are introduced in terms of the loop products and loop derivatives. The prospect of re-constructing general relativity in higher dimensions is explored, with the 'momenta' and 'velocity' being the basic loop variables. In the process, it is proven that the 'momenta' and 'velocity' behave as cobordant loops in higher dimensions.
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