Schubert Calculus and the Heisenberg Algebra

We show that the Hilbert space with basis indexed by infinite permutations and the cohomology ring of the infinite flag variety can be seen as representations of the Heisenberg algebra, which are isomorphic using the back-stable Schubert polynomials. We give a model for infinite permutations as certain two dimensional fermions, generalizing the Maya diagram construction for partitions. Under this framework, the pipedream model for Schubert polynomials can be viewed as the Hamiltonian time evolution of the 2D fermions.