Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

NS-NS fluxes in Hitchin's generalized geometry

Ian Ellwood

TL;DR

The paper develops a T-duality covariant generalization of NS-NS flux within Hitchin's generalized geometry by defining a generalized flux $\mathsf{H}$ via a modified Nijenhuis operator on generalized vectors. Flux is integrated over a generalized 3-cycle $(\Sigma,\Omega)$ with a fiber-condition ensuring the integral reduces to ordinary forms and captures $H$, geometric, and $Q$-flux in appropriate limits. The framework is connected to a generalized Bismut connection, showing the usual torsion vanishes while a torsion-like quantity reproduces the flux, and is illustrated through explicit examples. This approach provides a covariant description of diverse flux backgrounds and offers a path toward understanding non-geometric spaces in a unified formalism with potential string-theoretic insights.

Abstract

The standard notion of NS-NS 3-form flux is lifted to Hitchin's generalized geometry. This generalized flux is given in terms of an integral of a modified Nijenhuis operator over a generalized 3-cycle. Explicitly evaluating the generalized flux in a number of familiar examples, we show that it can compute three-form flux, geometric flux and non-geometric Q-flux. Finally, a generalized connection that acts on generalized vectors is described and we show how the flux arises from it.

NS-NS fluxes in Hitchin's generalized geometry

TL;DR

The paper develops a T-duality covariant generalization of NS-NS flux within Hitchin's generalized geometry by defining a generalized flux via a modified Nijenhuis operator on generalized vectors. Flux is integrated over a generalized 3-cycle with a fiber-condition ensuring the integral reduces to ordinary forms and captures , geometric, and -flux in appropriate limits. The framework is connected to a generalized Bismut connection, showing the usual torsion vanishes while a torsion-like quantity reproduces the flux, and is illustrated through explicit examples. This approach provides a covariant description of diverse flux backgrounds and offers a path toward understanding non-geometric spaces in a unified formalism with potential string-theoretic insights.

Abstract

The standard notion of NS-NS 3-form flux is lifted to Hitchin's generalized geometry. This generalized flux is given in terms of an integral of a modified Nijenhuis operator over a generalized 3-cycle. Explicitly evaluating the generalized flux in a number of familiar examples, we show that it can compute three-form flux, geometric flux and non-geometric Q-flux. Finally, a generalized connection that acts on generalized vectors is described and we show how the flux arises from it.

Paper Structure

This paper contains 20 sections, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of generalized geometry
  3. T-duality and generalized geometry
  4. Symmetries of $\mathsf{E} = T\oplus T^*$
  5. The canonical inner product and the metric $\mathsf{G}$
  6. The Courant-bracket and the Nijenhuis operator
  7. Defining the generalized NS-NS flux
  8. The generalized $p$-cycle
  9. The measure for integration
  10. Integration over the generalized measure
  11. The fiber condition
  12. The generalized version of NS-NS flux
  13. Special cases of the generalized-flux formula
  14. Three-form flux
  15. Geometric flux
  16. ...and 5 more sections

Figures (1)

  • Figure 1: In order to define integration over our generalized $p$-cycle, we demand that it take the form of a $T^{p-q}$ fibered over a base space. We further demand that nothing depend on the coordinates of the torus and that the vector parts of the generalized vielbein span the tangent bundle of the $T^{p-q}$. Finally, we assume that the forms live in the span of the $d\xi^i$ for $i \in \{1,\ldots,q\}$.