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 $d\in \{2,3,4\}$ and $k\in \{0,2,4\}$. We exemplify this by writing down some known ($k=2$, $d=2$) and some novel ($k=2$, $d\ge 3$) formulas in the modular operator.