Dostálová, Marie; Somberg, Petr Symplectic twistor operator and its solution space on \({\mathbb {R}}^2\). (English) Zbl 1299.53115 Arch. Math., Brno 49, No. 3, 161-185 (2013). The symplectic twistor operator \(T_s\) on a symplectic spin manifold \(M\) of dimension \(2n\) is a first-order differential operator acting on smooth symplectic spinors \(\mathcal {S}=\mathcal {S}_-\oplus \mathcal {S}_+\) which is invariant with respect to the metaplectic group \(\operatorname {Mp}(2n,\mathbb {R})\). The authors study in detail the case \(M=\mathbb {R}^2\). They give an explicit formula for \(T_s\), and then they compute the space of its solutions by analytical and combinatorial techniques of the metaplectic Howe duality and algebraic Weyl algebra. Concretely, the polynomial solution space of \(T_s\) is described explicitly via representatives in the solution space of the symplectic Dirac operator \(D_s\). It is remarkable that these representatives are far more complicated for polynomials with values in \(\mathcal {S}_-\) than for polynomials with values in \(\mathcal {S}_+\). Reviewer: Aleš Návrat (Brno) Cited in 1 Document MSC: 53C27 Spin and Spin\({}^c\) geometry 53D05 Symplectic manifolds (general theory) 81R25 Spinor and twistor methods applied to problems in quantum theory Keywords:symplectic spin geometry; metaplectic Howe duality; symplectic twistor operator; symplectic Dirac operator PDFBibTeX XMLCite \textit{M. Dostálová} and \textit{P. Somberg}, Arch. Math., Brno 49, No. 3, 161--185 (2013; Zbl 1299.53115) Full Text: DOI arXiv