Vertex algebras and coordinate rings of semi-infinite flags

TL;DR

This paper explores the structure of the coordinate ring of semi-infinite flag varieties using vertex algebras, revealing new algebraic relations and connections to affine Kac-Moody representations.

Contribution

It establishes a vertex algebra framework for the coordinate ring of semi-infinite flags and derives semi-infinite Plücker relations.

Findings

The coordinate ring is given a vertex algebra structure.

Semi-infinite Plücker relations are derived.

The graded structure relates to affine Kac-Moody representations.

Abstract

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type $ADE$ carries a structure of $P/Q$-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of $P/Q$-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Pl\"ucker-type relations in the homogeneous coordinate ring.