Fu and Yau constructed the first smooth family of gauge bundles over a class of non-Kähler, complex 3-folds that are solutions to Strominger’s system, the heterotic supersymmetry constraints with non-zero $H$-flux. In this paper, we begin the study of the massless spectrum arising from compactification using this construction by counting zero modes of the linearized equations of motion for the gaugino in the supergravity approximation. We rephrase the question in terms of a cohomology problem and show that for a trivial gauge bundle, this cohomology reduces to the Dolbeault cohomology of the 3-fold, which we then compute.

We establish Sakakibara’s differential equations [M. Sakakibara, On the differential equations of the characters for the renormalization group, Mod. Phys. Lett. A 19 (2004), 1453.] in a matrix setting for the counter term (respectively renormalized character) in Connes-Kreimer’s Birkhoff decomposition in any connected graded Hopf algebra, thus including Feynman rules in perturbative renormalization as a key example.