Gravity in $${\varvec{d=2+\epsilon }}$$ dimensions and realizations of the diffeomorphisms group

We discuss two separate realizations of the diffeomorphism group for metric gravity, which give rise to theories that are classically equivalent, but quantum mechanically distinct. We renormalize them in $$d=2+\epsilon $$ d = 2 + ϵ dimensions, developing a new procedure for dimensional continuation of metric theories and highlighting connections with the constructions that previously appeared in the literature. Our hope is to frame candidates for ultraviolet completions of quantum gravity in $$d>2$$ d > 2 and give some perturbative mean to assess their existence in $$d=4$$ d = 4 , but also to speculate on some potential obstructions in the continuation of such candidates to finite values of $$\epsilon $$ ϵ . Our results suggest the presence of a conformal window in d which seems to extend to values higher than four.

Short survey on the existence of slices for the space of Riemannian metrics

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.

Nonstandard Action of Diffeomorphisms and Gravity’s Anti-Newtonian Limit

A tensor calculus adapted to the Anti-Newtonian limit of Einstein gravity is developed. The limit is defined in terms of a global conformal rescaling of the spatial metric. This enhances spacelike distances compared to timelike ones and in the limit effectively squeezes the lightcones to lines. Conventional tensors admit an analogous Anti-Newtonian limit, which however transforms according to a non-standard realization of the spacetime Diffeomorphism group. In addition to the type of the tensor the transformation law depends on, a set of integer-valued weights is needed to ensure the existence of a nontrivial limit. Examples are limiting counterparts of the metric, Einstein, and Riemann tensors. An adapted purely temporal notion of parallel transport is presented. By introducing a generalized Ehresmann connection and an associated orthonormal frame compatible with an invertible Carroll metric, the weight-dependent transformation laws can be mapped into a universal one that can be read off from the index structure. Utilizing this ‘decoupling map’ and a realization of the generalized Ehresmann connection in terms of scalar field, the limiting gravity theory can be endowed with an intrinsic Levi–Civita type notion of spatio-temporal parallel transport.

On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure

A class of spatially one-dimensional completely integrable Chaplygin hydrodynamic systems was studied within framework of Lie-algebraic approach. The Chaplygin hydrodynamic systems were considered as differential systems on the torus. It has been shown that the geometric structure of the systems under analysis has strong relationship with diffeomorphism group orbits on them. It has allowed to find a new infinite hierarchy of integrable Chaplygin like hydrodynamic systems.

Representations of the infinite-dimensional p-adic affine group

We introduce an infinite-dimensional [Formula: see text]-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However, it is possible to define its action on some classes of functions.

The universal n-pointed surface bundle only has n sections

The classifying space BDiff[Formula: see text] of the orientation-preserving diffeomorphism group of a surface [Formula: see text] of genus [Formula: see text] fixing [Formula: see text] points pointwise has a universal bundle [Formula: see text] The [Formula: see text] fixed points provide [Formula: see text] sections [Formula: see text] of [Formula: see text]. In this paper we prove a conjecture of R. Hain that any section of [Formula: see text] is homotopic to some [Formula: see text]. Let [Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points on [Formula: see text]. As part of the proof of Hain’s conjecture, we prove a result of independent interest: any surjective homomorphism [Formula: see text] is equal to one of the forgetful homomorphisms [Formula: see text], possibly post-composed with an automorphism of [Formula: see text]. We also classify sections of the universal hyperelliptic surface bundle.