Boundary states for branes with non trivial homology in constant closed and open background

TL;DR

This paper computes boundary states for bosonic strings on a torus with non-trivial homology in constant backgrounds, revealing the necessity of twisted sectors and the role of F in geometric embedding.

Contribution

It introduces a method to construct boundary states with non-trivial homology in constant backgrounds, highlighting the importance of twisted sectors and the distinction between F and F+B.

Findings

Boundary states require twisted closed sectors in non-trivial open backgrounds.

Only F, not F+B, determines the geometric embedding of Dp branes.

Closed and open strings reside on different, relatively twisted tori.

Abstract

For the bosonic string on the torus we compute boundary states describing branes with not trivial homology class in presence of constant closed and open background. It turns out that boundary states with non trivial open background generically require the introduction of non physical ``twisted'' closed sectors, that only $F$ and not ${\cal F}=F+B$ determines the geometric embedding for $Dp$ branes with $p<25$ and that closed and open strings live on different tori which are relatively twisted and shrunk. Finally we discuss the T-duality transformation for the open string in a non trivial background.