turkmath.org

This talk, which essentially consists of three parts, serves as a conceptional introduction to the formulation of Einstein gravity in the context of derived algebraic geometry. The upshot is as follows: we shall first outline how to describe the notion of a (pre-)stack X, by using the functor-of-points type approach, manifestly given as a certain groupoid-valued sheaf over a site C, and present main ingredients of the homotopy theory of stacks in a relatively succinct and naive way. In that respect, one in fact requires to adopt certain simplicial techniques in order to recast the notion of a stack in the language of homotopy theory. This homotopical treatment, on the other hand, is essentially based on so-called the model structure on the 2-category Grpds of groupoids. In the second part of the talk, we shall revisit main aspects of 2+1 dimensional vacuum Einstein gravity on a pseudo-Riemannian manifold M especially in the context of Cartan geometry, and investigate, in the case of M=Σ×(0,∞) with vanishing cosmological constant and Σ being a closed Riemann surface of genus g>1, the equivalence of the quantum gravity with a gauge theory established in the sense that the moduli space E(M) of such a 2+1 dimensional Einstein gravity is isomorphic to that of flat Cartan ISO(2,1)-connections, denoted by Mflat. As an analyzing a classical field theory with an action functional S boils down to the study of the moduli space of solutions to the corresponding field equations, the notion of a stack in fact provides an alternative and elegant way of recording and organizing the moduli data. In the final part, we shall briefly discuss (i) how to construct the appropriate stacks associated to E(M) and Mflat respectively, and (ii) how to extend the isomorphism that essentially captures the equivalence of the quantum gravity with a gauge theory in the above setup to an isomorphism of associated stacks.