Ivan Avramidi
Department of Mathematics, New Mexico Tech, Socorro, NM 87801
Spectral asymptotics in matrix geometry
We describe a “matrix” generalization of Riemannian geometry when
instead of a Riemannian metric there is a matrix valued self-adjoint
symmetric two-tensor that plays a role of a “non-commutative” metric.
We construct invariant first-order and second-order self-adjoint
elliptic partial differential operators, that can be called
“non-commutative” Dirac operator and non-commutative Laplace
operator.We construct the corresponding heat kernel for the non-commutative
Laplace type operator and compute its first two spectral invariants. A
linear combination of these two spectral invariants gives a functional
that can be considered as a non-commutative generalization of the
Einstein-Hilbert action. The extremals of this functional naturally
define the “non-commutative” generalization of Einstein spaces.