Torus actions, maximality, and non-negative curvature

Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

Critical lumpy black holes in AdSp×Sq

In this paper we study lumpy black holes with AdSp × Sq asymptotics, where the isometry group coming from the sphere factor is broken down to SO(q). Depending on the values of p and q, these are solutions to a certain Supergravity theory with a particular gauge field. We have considered the values (p, q) = (5, 5) and (p, q) = (4, 7), corresponding to type IIB supergravity in ten dimensions and eleven-dimensional supergravity respectively. These theories presumably contain an infinite spectrum of families of lumpy black holes, labeled by a harmonic number ℓ, whose endpoints in solution space merge with another type of black holes with different horizon topology. We have numerically constructed the first four families of lumpy solutions, corresponding to ℓ = 1, 2+, 2− and 3. We show that the geometry of the horizon near the merger is well-described by a cone over a triple product of spheres, thus extending Kol’s local model to the present asymptotics. Interestingly, the presence of non-trivial fluxes in the internal sphere implies that the cone is no longer Ricci flat. This conical manifold accounts for the geometry and the behavior of the physical quantities of the solutions sufficiently close to the critical point. Additionally, we show that the vacuum expectation values of the dual scalar operators approach their critical values with a power law whose exponents are dictated by the local cone geometry in the bulk.

Matrix group actions on product of spheres

Let [Formula: see text] be the special linear group over integers and [Formula: see text] [Formula: see text], or [Formula: see text] products of spheres and tori. We prove that any group action of [Formula: see text] on [Formula: see text] by diffeomorphims or piecewise linear homeomorphisms is trivial if [Formula: see text] This confirms a conjecture on Zimmer’s program for these manifolds.

Holographic QFTs on S2×S2, spontaneous symmetry breaking and Efimov saddle points

Holographic CFTs and holographic RG flows on space-time manifolds which are d-dimensional products of spheres are investigated. On the gravity side, this corresponds to Einstein-dilaton gravity on an asymptotically AdSd+1 geometry, foliated by a product of spheres. We focus on holographic theories on S2× S2, we show that the only regular five-dimensional bulk geometries have an IR endpoint where one of the sphere shrinks to zero size, while the other remains finite. In the Z2-symmetric limit, where the two spheres have the same UV radii, we show the existence of a infinite discrete set of regular solutions, satisfying an Efimov-like discrete scaling. The Z2-symmetric solution in which both spheres shrink to zero at the endpoint is singular, whereas the solution with lowest free energy is regular and breaks Z2 symmetry spontaneously. We explain this phenomenon analytically by identifying an unstable mode in the bulk around the would-be Z2-symmetric solution. The space of theories have two branches that are connected by a conifold transition in the bulk, which is regular and correspond to a quantum first order transition. Our results also imply that AdS5 does not admit a regular slicing by S2× S2.

SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

A Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n(p) is determined, which completes the list of (non)triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n − 1) → SO(2n) in the sense of James at any prime p.