Double Conformal Invariants and the Wodzicki Residue

TL;DR

This paper introduces new double conformal invariants for real and complex manifolds using the Wodzicki residue and differential operators, with explicit computations in flat cases.

Contribution

It constructs novel double conformal invariants for manifolds employing the Wodzicki residue and differential operators, extending conformal invariance concepts.

Findings

Computed the invariants explicitly in flat cases.

Extended the invariants to complex manifolds using the  operator.

Provided examples where the invariants are explicitly calculated.

Abstract

For compact real manifolds, a new double conformal invariant is constructed using the Wodzicki residue and the $d$ operator in the framework of Connes. In the flat case, we compute this double conformal invariant, and in some special cases, we also compute this double conformal invariants. For complex manifolds, a new double conformal invariant is constructed using the Wodzicki residue and the $\bar{\partial}$ operator in the same way, and this double conformal invariant is computed in the flat case.