On Non Commutative Calabi-Yau Hypersurfaces

TL;DR

This paper explores the algebraic structure of non-commutative Calabi-Yau hypersurfaces, deriving new algebra representations and extending results to higher dimensions using algebraic geometry techniques.

Contribution

It introduces new representations of non-commutative Calabi-Yau algebras by varying Calabi-Yau charges and extends the framework to higher-dimensional cases.

Findings

Derived new representations of ${ ext{A}}_{nc}(5)$

Extended algebraic structures to higher dimensions

Constructed representations preserving Calabi-Yau conditions

Abstract

Using the algebraic geometry method of Berenstein et al (hep-th/0005087), we reconsider the derivation of the non commutative quintic algebra ${\mathcal{A}}_{nc}(5)$ and derive new representations by choosing different sets of Calabi-Yau charges ${C_{i}^{a}}$. Next we extend these results to higher $d$ complex dimension non commutative Calabi-Yau hypersurface algebras ${\mathcal{A}}_{nc}(d+2)$. We derive and solve the set of constraint eqs carrying the non commutative structure in terms of Calabi-Yau charges and discrete torsion. Finally we construct the representations of ${\mathcal{A}}_{nc}(d+2) $ preserving manifestly the Calabi-Yau condition $ \sum_{i}C_{i}^{a}=0$ and give comments on the non commutative subalgebras.