Dirac operators for infinite-dimensional color Lie algebras

Abstract

We construct cubic Dirac operators and relative cubic Dirac operators for infinite-dimensional quadratic $\mathbb{Z}$-graded color Lie algebras with finite-dimensional components. These operators are defined in completions of the quantum Weil algebra determined by the $\mathbb{Z}$-grading. The same grading fixes the normal-ordering convention. The failure of the normally ordered Casimir to be central, and of the normally ordered cubic Dirac operator to be $\mathfrak{g}$-invariant, is measured by a color analogue of the Kac-Peterson class. If this class is trivial, the Casimir admits a central correction and the cubic Dirac operator admits a corrected $\mathfrak{g}$-invariant form. For the corrected (relative) cubic Dirac operators, we establish Parthasarathy-type square formulas. We also extend the Chern-Weil homomorphism to completed $\mathfrak{g}$-differential algebras and identify the classical element whose quantization is the cubic Dirac operator with the Chern-Simons element associated with the quadratic invariant polynomial defined by $B$. As applications, we consider symmetrizable Kac-Moody superalgebras. In this setting the Kac-Peterson class is trivial, with primitive given by the Weyl vector. For the affine Kac-Moody superalgebra associated to $\mathfrak{osp}(1\vert 2n)$, we compute $\mathrm{ker}\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}^{2}$ on integrable highest weight supermodules. We then apply the relative square formula to $\omega$-unitarizable highest weight supermodules and obtain a Dirac inequality giving necessary conditions for unitarity. Finally, under assumptions satisfied by Kac-Moody superalgebras such as $\widehat{\mathfrak{sl}}(m\vert n)$, we identify the Dirac kernel with Lie superalgebra cohomology.