In the recent development of modular geometry on toric noncommutative manifolds (Connes-Moscovici 2014), metrics are parametrized by self-adjoint elements in the ambient C*-algebra, whose exponential are called Weyl factors. Local invariants, such as the Riemannian curvature, are encoded in the coefficients of certain heat kernel expansion. The new ingredient, purely due to noncommutativity, is the the inner automorphism generated by the Weyl factor, whose corresponding derivation can be viewed as a noncommutative differential. From analytic point of view, curvature is designed to measure the commutators of covariant derivatives. In this talk, we will discuss some intriguing spectral functions which define the interplay between the inner automorphisms and the classical differentials. I recently found that hypergeometric functions and its multivariable generalization are the building blocks. Geometric applications such as Gauss-Bonnet theorem lead to some functional relations/equations between them which are still begging for more conceptual understanding.
