Geometry

Abstract

: We study the noncommutative Riemannian geometry of the alternating group A 4 = ( Z 2 × Z 2 ) >⊳ Z 3 using the recent formulation for finite groups in [2]. We find a unique ‘Levi-Civita’ connection for the invariant metric, and find that it has Ricci-flat but nonzero Riemann curvature. We show that it is the unique Ricci-flat connection on A 4 with the standard framing (we solve the vacuum Einstein’s equation). We also propose a natural Dirac operator for the associated spin connection and solve the Dirac equation. Some of our results hold for any finite group equipped with a cyclic conjugacy class of 4 elements. In this case the exterior algebra Ω( A 4 ) has dimensions 1 : 4 : 8 : 11 : 12 : 12 : 11 : 8 : 4 : 1 with top-form 9-dimensional. We also find the noncommutative cohomology H 1 ( A 4 ) = C .