Conformal invariance, geometry-matter duality and number theory

Abstract:
Conserved currents  for conformal Killing vectors in Riemannian geometry are discovered. Vector and traceless symmetric tensor have a highly nonlinear structure in which the role of dimensionality space-time are analogical to the role gauge coupling constants in Yang-Mills theory. Crucial observations for these geometrical currents are: 1) the equality their physical dimensionality with vector and tensor energy-momentum dimensionality in  Standard Model and gravity theories in any
space-time dimension; 2) traceless geometrical stress-energy tensor,which suggest some deep connection with the spinor structure of space-time. Geometrical currents are some analog of the neutral currents Standard Model. Interesting observation on the role of conformal invariance in crritical phenomena and quantum field theory in different critical dimensions for supergravity, superstring, (Super) Grand Unified Theories are discussed.