Infinite-genus surfaces and the universal Grassmannian

TL;DR

This paper constructs states for infinite-genus Riemann surfaces in the Hilbert space of free field theory, demonstrating their inclusion in the Grassmannian and exploring implications for string perturbation theory.

Contribution

It introduces a framework for analyzing infinite-genus surfaces within the Grassmannian, linking them to string theory and boundary flux concepts.

Findings

Infinite-genus surfaces are embedded in the Grassmannian.

A subset of $O_{HD}$ surfaces corresponds to Grassmannian elements.

Flux through boundaries relates to string perturbation domain.

Abstract

Correlation functions can be calculated on Riemann surfaces using the operator formalism. The state in the Hilbert space of the free field theory on the punctured disc, corresponding to the Riemann surface, is constructed at infinite genus, verifying the inclusion of these surfaces in the Grassmannian. In particular, a subset of the class of $O_{HD}$ surfaces can be identified with a subset of the Grassmannian. The concept of flux through the ideal boundary is used to study the connection between infinite-genus surface and the domain of string perturbation theory. The different roles of effectively closed surfaces with Dirichlet boundaries in a more complete formulation of string theory are identified.