Higher bracket structure of density operators in Weyl fermion systems and topological insulators

We study the algebraic structure of electron density operators in gapless Weyl fermion systems in $d=3,5,7,\cdots$ spatial dimensions and in topological insulators (without any protecting symmetry) in $d=4,6,8,\cdots$ spatial dimensions. These systems are closely related by the celebrated bulk-boundary correspondence. Specifically, we study the higher bracket -- a generalization of commutator for more than two operators -- of electron density operators in these systems. For topological insulators, we show that the higher-bracket algebraic structure of density operators structurally parallels with the Girvin-MacDonald-Platzman algebra (the $W_{1+\infty}$ algebra), the algebra of electron density operators projected onto the lowest Landau level in the quantum Hall effect. By the bulk-boundary correspondence, the bulk higher-bracket structure mirrors its counterparts at the boundary. Specifically, we show that the density operators of Weyl fermion systems, once normal-ordered with respect to the ground state, their higher bracket acquires a c-number part. This part is an analog of the Schwinger term in the commutator of the fermion current operators. We further identify this part with a cyclic cocycle, which is a topological invariant and an element of Connes' noncommutative geometry.