We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow on Riemannian manifolds. First we use BRST quantization to construct a topological Lifshitz-type theory for only the spatial metric, with spatial diffeomorphism invariance and no gauge symmetry, associated with Hamilton's Ricci flow: Hamilton's flow equation appears as the localization equation of the primitive theory. Then we extend the primitive theory by gauging foliation-preserving spacetime symmetries. Crucially, all our theories are required to exhibit an ${\cal N}=2$ extended BRST symmetry. First, we gauge spatial diffeomorphisms, and show that this gives us access to the mathematical technique known as the DeTurck trick. Finally, we gauge foliation-preserving time reparametrizations, both with the projectable and nonprojectable lapse function. The path integral of the full theory is localized to the solutions of Ricci-type flow equations, generalizing those of Perelman. The role of Perelman's dilaton is played by the nonprojectable lapse function. Perelman's ${\cal F}$-functional appears as the superpotential of our theory. Since there is no spin-statistics theorem in nonrelativistic quantum field theory, the two supercharges of our gravity theory do not have to be interpreted as BRST charges and, after the continuation to real time, the theory can be studied as a candidate for nonrelativistic quantum gravity with propagating bosonic and fermionic degrees of freedom.

We consider (effective) Quantum General Relativity coupled to the Standard Model (QGR-SM) and clarify whether graviton-ghosts couple to matter particles. To this end, we examine the corresponding BRST and anti-BRST symmetries, which are generated by infinitesimal diffeomorphisms and infinitesimal gauge transformations. In particular, we study their properties and relations: We find that all differentials mutually anticommute, which implies that they form a double complex. In particular, we introduce the total BRST differential as the sum of the diffeomorphism and gauge BRST differentials and similarly the total anti-BRST differential as the sum of the respective anti-BRST differentials. Furthermore, we identify the functionals in particle fields that are (co)cycles up to total derivatives with respect to the diffeomorphism differentials as scalar tensor densities of weight one: This implies that graviton-ghosts decouple from matter particles if and only if the Yang--Mills gauge fixing Lagrange density has said tensor density weight. Moreover, we discuss the relevant gauge fixing fermions: Starting from the de Donder and Lorenz gauge fixing conditions, we introduce a total gauge fixing fermion that generates the complete gauge fixing and ghost Lagrange density of QGR-SM. Finally, we show that the BRST cocomplexes are isomorphic to their corresponding anti-BRST complexes via ghost conjugation. Notably, this relates the BRST cohomologies to their respective anti-BRST homologies.

We address the question of counting maps between projective spaces such that images of cycles on the source intersect cycles on the target. In this paper we do it by embedding maps into quasimaps that form a projective space of their own. When a quasimap is not a map, it contains freckles (studied earlier) and/or scars, appearing when the complex dimension of the source is greater than one. We consider a lot of examples showing that freckle/scar calculus (using excess intersection theory) works. We also propose the “smooth conjecture” that may lead to computation of the number of maps by an integral over the space of quasimaps.

We study aspects of an equivalent relation of the charge completeness in six-dimensional (6D) $\mathcal{N}=(1,0)$ supergravity theory and a standard assumption on the global structure of the gauge group involving F-theory geometry, recently proved by Morrison and Taylor. We constructed and analyzed a novel 6D supergravity theory, realized as F-theory, on an elliptically fibered Calabi-Yau 3-fold. Our construction yields a novel 6D theory with Mordell-Weil torsion $\mathbb{Z}_4\oplus\mathbb{Z}_4$. Furthermore, we deduce the gauge group and matter fields arising in the 6D F-theory model on the constructed elliptically fibered Calabi-Yau 3-fold. We also discuss the relations of the 6D F-theory model constructed in this study to stable degeneration and the dual heterotic string.