Local invariants of conformally deformed non-commutative tori II: Multiple operator integrals

We explicitly compute the local invariants (heat kernel coefficients) of a conformally deformed non-commutative d-torus using multiple operator integrals. We derive a recursive formula that easily produces an explicit expression for the local invariants of any order k and in any dimension d. Our recursive formula can conveniently produce all formulas related to the modular operator, which before were obtained in incremental steps for d∈{2,3,4} and k∈{0,2,4}. We exemplify this by writing down some known (k=2, d=2) and some novel (k=2, d≥3) formulas in the modular operator.