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Symplectic twistor operator and its solution space on \({\mathbb {R}}^2\). (English) Zbl 1299.53115

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 \(\mathcal {S}_+\).

MSC:

53C27 Spin and Spin\({}^c\) geometry
53D05 Symplectic manifolds (general theory)
81R25 Spinor and twistor methods applied to problems in quantum theory
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