Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space

TL;DR

This paper constructs a new class of noncommutative spaces called soft noncommutative schemes via toric geometry and models D-branes on these spaces, linking algebraic geometry with string theory concepts.

Contribution

It introduces the concept of soft noncommutative schemes via toric geometry and develops the mathematical framework for D-branes on such spaces, including embeddings of Calabi-Yau spaces.

Findings

Construction of soft noncommutative schemes via toric geometry

Embedding of Calabi-Yau spaces into these noncommutative schemes

Development of invertible sheaves and twisted sections on noncommutative spaces

Abstract

A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of `soft noncommutative toric schemes' associated to a (simplicial, maximal cone of index $1$) fan, `invertible sheaves' on such a noncommutative space, and `twisted sections' of an invertible sheaf are developed and Azumaya schemes with a fundamental module as the world-volumes of D-branes are reviewed. Two guiding questions, Question 3.12 (soft noncommutative Calabi-Yau spaces and their mirror) and Question 4.2.14 (generalized matrix models), are presented.