Dirac operator on the restricted Grassmannian manifold

TL;DR

This paper constructs a well-defined Dirac-type operator on the infinite-dimensional restricted Grassmannian manifold using fermionic Fock space representations, addressing previous divergence issues and establishing key operator properties.

Contribution

It introduces a new Dirac operator on the restricted Grassmannian that is mathematically well-defined and symmetric, improving upon Mickelsson's earlier divergent candidate.

Findings

Operator is an unbounded symmetric operator

Kernel of the operator is finite-dimensional

Constructed operator addresses divergence problems

Abstract

In his book Mickelsson notices that the infinite-dimensional Grassmannian manifold of Segal and Wilson admits a Spin^c structure and after this he naturally considers the problem of defining a Dirac operator on it. Mickelsson gives a possible candidate for such an operator but unfortunately it proves out to be badly diverging and he leaves it as an open problem to introduce proper modifications 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 group associated to a polarized Hilbert space, introduced in the well-known book written by Pressley and Segal on loop groups, we construct a well-defined candidate for a Dirac type operator on the restricted Grassmannian manifold acting on a relevant space of spinors. As our main result we show that our operator is an unbounded symmetric operator with finite-dimensional kernel.