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In his book Mickelsson notices that the infinite-dimensional Grassmannianmanifold of Segal and Wilson admits a Spin^c structure and after this henaturally considers the problem of defining a Dirac operator on it. Mickelssongives a possible candidate for such an operator but unfortunately it proves outto be badly diverging and he leaves it as an open problem to introduce propermodifications to his original construction in order to obtain a well-defined(unbounded) operator with expected properties.  Using fermionic Fock space representations of the restricted unitary groupassociated to a polarized Hilbert space, introduced in the well-known bookwritten by Pressley and Segal on loop groups, we construct a well-definedcandidate for a Dirac type operator on the restricted Grassmannian manifoldacting on a relevant space of spinors. As our main result we show that ouroperator is an unbounded symmetric operator with finite-dimensional kernel.

Dirac operator on the restricted Grassmannian manifold