1 Erratum to: Ann Glob Anal Geom DOI 10.1007/s10455-013-9369-x

The Hermitian structure used for a flag manifold \(G/T\) in Sect. 4 is not a Kähler structure unless \(G\) is a product of \(SU(2)\)’s. This is because the connection used is the canonical connection on the homogeneous space \(G/T\), which has non-zero torsion since \([\mathfrak{t }^\perp , \mathfrak{t }^\perp ] \not \subseteq \mathfrak{t }\). Indeed, using the metric to lower indices, the torsion at \(eT\) is the 3-form given by

$$\begin{aligned} (X,Y, Z) \mapsto -g_0([X,Y], Z), \quad X, Y, Z \in \mathfrak{t }^\perp \simeq T_{eT} G/T\quad [1]. \end{aligned}$$

Such a structure on \(G/T\) turns it into what is called a “Kähler with torsion (KT)” manifold. The general results and constructions established in the previous sections carry over unchanged to KT manifolds except for Proposition 3 which states that the symplectic Dirac operators \(D\) and \(\tilde{D}\) are formally self-adjoint. For connections with torsion and parallel complex structure, a sufficient condition for \(D\) and \(\tilde{D}\) to be self-adjoint is the vanishing of the torsion vector field, defined by

$$\begin{aligned} \mathcal{T } = \sum _{j=1}^n \mathbf{T}(a_j,b_j), \end{aligned}$$

where \(\mathbf{T}\) is the torsion of \(\nabla \) and \(\{a_1,\ldots ,a_n, b_1,\ldots ,b_n\}\) is a symplectic frame [2]. In the case of flag manifolds, a symplectic basis at \(eT\) is proportional to \(\{Z_\alpha , Z_{-\alpha }\}\) where \(Z_\alpha \) is a root vector for \(\alpha \). Since \(\mathbf{T}(Z_\alpha , Z_{-\alpha }) = -[Z_\alpha , Z_{-\alpha }]_{\mathfrak{t }^\perp } = 0\), we see that \(\mathcal{T }\) vanishes. Thus the results concerning flag manifolds are correct using the symplectic Dolbeault operators corresponding to their KT structures.