Drinfel'd Doubles, Twists and All That... in Stringy Geometry and M Theory

TL;DR

This paper extends the mathematical framework of Drinfel'd doubles and bialgebroids to include twists and more general vector bundle pairs, with applications to string theory dualities.

Contribution

It introduces twisted compatibility conditions for proto bialgebroids and generalizes Drinfel'd doubles to arbitrary vector bundles, advancing the mathematical tools for string theory dualities.

Findings

Extended the notion of proto bialgebroids with H- and R-twists.

Derived new compatibility conditions for twisted algebroid structures.

Analyzed examples relevant to physics and mathematics literature.

Abstract

Drinfel'd double of Lie bialgebroids plays an important role in T-duality of string theories. In the presence of $H$ and $R$ fluxes, Lie bialgebroids should be extended to proto Lie bialgebroids. For both cases, the pair is given by two dual vector bundles, and the Drinfel'd double yields a Courant algebroid. However for U-duality, more complicated direct sum decompositions that are not described by dual vector bundles appear. In a previous work, we extended the notion of a Lie bialgebroid for vector bundles that are not necessarily dual. We achieved this by introducing a framework of calculus on algebroids and examining compatibility conditions for various algebroid properties in this framework. Here our aim is two-fold: extending our work on bialgebroids to include both $H$- and $R$-twists, and generalizing proto Lie bialgebroids to pairs of arbitrary vector bundles. To this end, we analyze various algebroid axioms and derive twisted compatibility conditions in the presence of twists. We introduce the notion of proto bialgebroids and their Drinfel'd doubles, where the former generalizes both bialgebroids and proto Lie bialgebroids. We also examine the most general form of vector bundle automorphisms of the double, related to twist matrices, that generate a new bracket from a given one. We analyze various examples from both physics and mathematics literatures in our framework.