More Like this

1. Torus actions, maximality, and non-negative curvature

   https://doi.org/10.1515/crelle-2021-0035  

   Escher, Christine; Searle, Catherine (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract 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 showthe 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. 

   more » « less
2. Almost non-negatively curved 4-manifolds with torus symmetry

   https://doi.org/10.1090/proc/15093  

   Harvey, John; Searle, Catherine (January 2020, Proceedings of the American Mathematical Society)

   Wei, Guofang (Ed.)

   We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry. 

   more » « less
3. Pumping with symmetry

   https://doi.org/10.1209/0295-5075/ad3053  

   Iglesias_Martínez, J A; Kadic, M; Laude, V; Prodan, E (April 2024, Europhysics Letters)

   Abstract Re-configurable materials and meta-materials can jump between space symmetry classes during their deformations. Here, we introduce the concept of singular symmetry enhancement, which refers to an abrupt jump to a higher symmetry class accompanied by an un-avoidable reduction in the number of dispersion bands of the excitations of the material. Such phenomenon prompts closings of some of the spectral resonant gaps along singular manifolds in a parameter space. In this work, we demonstrate that these singular manifolds can carry topological charges. As a concrete example, we show that a deformation of an acoustic crystal that encircles aconfiguration of an array of cavity resonators results in an adiabatic cycle that carries a Chern number in the bulk and displays Thouless pumping at the edges. This points to a very general guiding principle for recognizing cyclic adiabatic processes with high potential for topological pumping in complex materials and meta-materials, which rests entirely on symmetry arguments. 

   more » « less
4. A CFT distance conjecture

   https://doi.org/10.1007/JHEP10(2021)070  

   Perlmutter, Eric; Rastelli, Leonardo; Vafa, Cumrun; Valenzuela, Irene (October 2021, Journal of High Energy Physics)

   A bstract We formulate a series of conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in d > 2 spacetime dimensions. We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric. Our central conjecture is that all theories at infinite distance possess an emergent higher-spin symmetry, generated by an infinite tower of currents whose anomalous dimensions vanish exponentially in the distance. Stated geometrically, the diameter of a non-compact conformal manifold must diverge logarithmically in the higher-spin gap. In the holographic context our conjectures are related to the Distance Conjecture in the swampland program. Interpreted gravitationally, they imply that approaching infinite distance in moduli space at fixed AdS radius, a tower of higher-spin fields becomes massless at an exponential rate that is bounded from below in Planck units. We discuss further implications for conformal manifolds of superconformal field theories in three and four dimensions. 

   more » « less
5. C ¹ actions on manifolds by lattices in Lie groups

   https://doi.org/10.1112/S0010437X22007278  

   Brown, Aaron; Damjanović, Danijela; Zhang, Zhiyuan (March 2022, Compositio Mathematica)

   In this paper we study Zimmer's conjecture for $$C^{1}$$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in $${\rm SL}(n, {{\mathbb {R}}})$$ , the dimensional bound is sharp. 

   more » « less