2019/12/24 by Yuri Ximenes Martins, Martins, Yuri Ximenes, Rodney Josué Biezuner +1
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geodesy #Geology #Geometric Analysis and Curvature Flows #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Mathematics #gr-qc #math-ph #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.1912.11198
Essentially a reader-friendly version of arXiv:1808.09249 endowed with further results and discussion
arxiv created 2019/12/24 · openalex publication_date 2019/12/24 · arxiv updated 2019/12/25 · openalex created_date 2020/01/10 · openalex updated_date 2026/08/06
These are notes for a short course and some talks gave at Departament of Mathematics and at Departament of Physics of Federal University of Minas Gerais, based on the author's paper arXiv:1808.09249. Some new information and results are also presented. Unlike the original work, here we try to give a more physical emphasis. In this sense, we present obstructions to realize gravity, modeled by the tetradic Einstein-Hilbert-Palatini (EHP) action functional, in a general geometric setting. In particular, we show that if spacetime has dimension n≥4, then the cosmological constant plays no role in any "concrete geometries" other than Lorentzian. If n≥6, then the entire EHP theory is trivial, meaning that Lorentzian geometry is (essentially) the only "concrete geometry" in which gravity (i.e, the EHP action functional) makes sense. Examples of "concrete geometries" include those locally modeled by group reductions H\hookrightarrow O(k;A) for some k and some algebra A, so that Riemannian geometry, Hermitian geometry, Kähler geometry and symplectic geometry, as well as Type II geometry, Hitchin's generalized complex geometry and G2-geometry are included. We also study EHP theory in "abstract geometries", such as graded geometry (and hence supergeometry), and we show how the obstruction results extend to this context. We construct two theories naturally associated to EHP, which we call the geometric/algebraic dual of EHP, and we analyze the effect of the obstructions in these new theories. Finally, we speculate (and provide evidence for) the existence of a "universal obstruction condition".