Bogolyubov Institute for Theoretical Physics,

14b Metrologochna Str., 03143 Kyiv, UKRAINE

E-mail: mmtpitp@gluk.org

**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.