arXiv2026
We develop the Dirac formalism for infinite-dimensional quadratic $\mathbb{Z}$-graded color Lie algebras with finite-dimensional components. Cubic Dirac operators are defined in completions of the quantum Weil algebra determined by the $\mathbb{Z}$-grading. The same grading fixes the normal-ordering convention. Normal ordering introduces a cohomological obstruction to the construction, measured by a color analogue of the Kac-Peterson class. When this class is trivial, we construct cubic and relative cubic Dirac operators satisfying the expected invariance properties and Parthasarathy-type square formulas. We further extend the Chern-Weil homomorphism to completed $\mathfrak{g}$-differential algebras and use it to identify the classical precursor of the cubic Dirac operator with the Chern-Simons element associated with the invariant quadratic polynomial determined by the quadratic structure. As applications, we consider symmetrizable Kac-Moody superalgebras. In this setting, the Kac-Peterson class is trivial, with primitive given by the Weyl vector, which yields the linear correction defining the cubic Dirac operator. We then use the relative Dirac operator to extract representation-theoretic information from highest weight supermodules. As an explicit example, for the affine Kac-Moody superalgebra associated with $\mathfrak{osp}(1\vert 2n)$, we compute the kernel of $\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}$ on integrable highest weight supermodules. Finally, for unitarizable highest weight supermodules, we explain why the usual Dirac inequality is not available in the affine setting.