Twisted sigma-model solitons on the quantum projective line

TL;DR

This paper develops a framework for analyzing twisted sigma-model solitons on the quantum projective line using noncommutative geometry, leading to explicit solutions and a new understanding of self-duality equations in this setting.

Contribution

It introduces a novel approach using twisted Hochschild and cyclic cocycles to derive self-duality equations and constructs explicit nontrivial solutions on the quantum projective line.

Findings

Derived self-duality equations for twisted sigma-models.

Established a lower bound for the action functional via positivity.

Constructed explicit nontrivial solutions on the quantum projective line.

Abstract

On the configuration space of projections in a noncommutative algebra, and for an automorphism of the algebra, we use a twisted Hochschild cocycle for an action functional, and a twisted cyclic cocycle for a topological term. The latter is Hochschild-cohomologous to the former and positivity in twisted Hochschild cohomology results into a lower bound for the action functional. While the equations for the critical points are rather complicate, the use of the positivity and the bound by the topological term leads to self-duality equations (thus yielding twisted noncommutative sigma-model solitons, or instantons). We present explicit non trivial solutions on the quantum projective line.