Stacky formulations of Einstein gravity

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2021-5-28
Berktav, Kadri İlker
This is a thesis on higher structures in geometry and physics. Indeed, the current work involves an extensive and relatively self-contained investigation of higher categorical and stacky structures in (vacuum) Einstein gravity with vanishing cosmological constant. In the first three chapters of the thesis, we shall provide a realization of the moduli space of Einstein’s field equations as a certain higher space (a stack). In this part of the thesis, the first aim is to present the construction of the moduli stack of vacuum Einstein gravity with vanishing cosmological constant in an n-dimensional setup. In particular, we shall be interested in the moduli space of 3D Einstein gravity on specific Lorentzian spacetimes. With this spirit, the second goal of this part is to show that once it exists, the equivalence of 3D quantum gravity with gauge theory in a particular setup, in fact, induces an isomorphism between the corresponding moduli stacks where the setup involves Lorentzian spacetimes of the form Mx R with M being a closed Riemann surface of genus g > 1. For our purposes, we shall employ a particular treatment that is essentially based on a formulation of stacks in the language of homotopy theory. The remainder of the thesis, on the other hand, is designed as a detailed survey on formal moduli problems, and it is particularly devoted to formalizing specific Einstein gravities in the language of formal moduli problems and L_infinity-algebras. Such an approach allows us to encode further higher structures in the theory if needed. To be more precise, this leads to the realization of the space of fields as a certain higher/derived stack (a formal moduli problem) endowed with more sensitive higher structure (encoding the possible higher symmetries/equivalences in the theory) once we ask the theory to possess higher symmetries. As a particular example, we use this approach to formulate specific 3D Einstein-Cartan-Palatini gravity. In addition, using local models for such higher structures and the algebra of functions on these higher spaces, we intend to study the algebraic structure of observables of 3D Einstein-Cartan-Palatini gravity as well.
Citation Formats
K. İ. Berktav, “Stacky formulations of Einstein gravity,” Ph.D. - Doctoral Program, Middle East Technical University, 2021.