$q$-identities and affinized projective varieties, I. Quadratic monomial ideals

TL;DR

This paper introduces the concept of affinized projective varieties and demonstrates how to derive q-identities through Hilbert series calculations, focusing on varieties linked to quadratic monomial ideals, with applications in quantum systems.

Contribution

It defines affinized projective varieties and connects their Hilbert series to q-identities, specifically for quadratic monomial ideals, revealing new algebraic and physical insights.

Findings

Derived q-identities from Hilbert series of affinized varieties.

Applied identities to systems of quasi-particles with null-states.

Interpreted identities as sums of quasi-particle Fock space characters.

Abstract

We define the concept of an affinized projective variety and show how one can, in principle, obtain q-identities by different ways of computing the Hilbert series of such a variety. We carry out this program for projective varieties associated to quadratic monomial ideals. The resulting identities have applications in describing systems of quasi-particles containing null-states and can be interpreted as alternating sums of quasi-particle Fock space characters.