Generalised geometry #1: Generalised tangent bundle

1. Introduction

The motivation for generalised geometry as first formulated in [Hitc03], [Hitc05], and [Gual04] was to combine complex and symplectic manifolds into a single, common framework. In the sense of Hitchin’s formulation, which follows Courant and Dorfman, generalised geometry has deep application in physics since emphasis is placed on adapting description of the physical motion of extended objects (i.e., strings). In this way, one can view generalised geometry as analogous to how traditional geometry is adapted to the physical motion of point-particles. There are also more general forms of generalised geometries, which can be thought of as further extended and adapted geometries to describe higher dimensional objects such as membranes (and hence also M-theory). These notions of geometry, which we can organise under the conceptual umbrella of extended geometries, correlate closely with the study of extended field theories that captures both Double Field Theory (DFT) and Exceptional Field Theory (EFT).

In these notes, interest in generalised geometry begins with the way in which generalised and extended geometry makes manifest hidden symmetries in string / M-theory. In particular, emphasis is on obtaining a deeper understanding and sense of mathematical intuition for the structure of generalised diffeomorphisms and gauge symmetries. The purpose was to then extend this emphasis to a study of the gauge structure of DFT, which is well known to be closely related with generalised geometry but in fact extends beyond it. We won’t get into this last concern in these notes; it is merely stated to make clear the original motivation for reviewing the topics.

Given that generalised geometry inspired the seminal formulations of DFT, it is no coincidence that what we observe in a detailed review of generalised geometry is the way in which the metric and p-form potentials are explicitly combined into a single object that acts on an enlarged space. This enables a description of diffeomorphisms and gauge transformations of the graviton and Kalb-Ramond B-field in a combined way. In fact, one of Hitchin’s motivations for the introduction of generalised geometry was to give a natural geometric meaning to the B-field. As will become clear in a later note, a key observation in this regard is that the automorphism group of the Courant algebroid ${TM \oplus T^{\star}M}$ is the semidirect product of the group of diffeomorphisms and B-field transformations. We will then study the structure of this group.

Remark 1 (Generalised geometry, branes, and SUGRA) Although not a focus of these notes, it is worth mentioning that generalised geometry in the sense of Hitchin is an important framework for studying branes and also T-dualities, including mirror symmetry. It also offers a powerful collection of tools to study Calabi-Yau manifolds, particularly generalised Calabi-Yau, proving important in the search for more realistic flux compactifications.

2. Generalised tangent bundle

The main objects to study on generalised geometry are Courant algebroids. But before we reach this stage, there are two fundamental structures of generalised geometry that we must first define: 1) the generalised tangent bundle and, 2) the Courant bracket. In this note, we introduce the generalised tangent bundle. Then in the following notes we explore the properties of this structure and the related extension of linear algebra to generalised linear algebra. This brings us to finally study the Courant bracket, its properties and symmetries, before we study Courant algebroids and generalised diffeomorphisms.

Definition 1 (Generalised bundle) The generalised tangent bundle is obtained by replacing the standard tangent bundle ${T}$ of a D-dimensional manifold ${M}$ with the following generalised analogue

$\displaystyle E \cong TM \oplus T^{\star}M. \ \ \ \ \ (1)$

The generalised tangent bundle ${E}$ is therefore a direct sum of the tangent bundle ${TM}$ and co-tangent bundle ${T^{\star}M}$ . As we will learn, the bundle ${E}$ has a natural symmetric form with respect to which both ${TM}$ and ${T^{\star}M}$ are maximally isotropic.

Remark 2 (Notation) Often in these notes we will use ${E}$ and ${TM \oplus T^{\star}M}$ interchangeably, which should be clear in the given context.

The generalised bundle (1) fits the following exact short sequence

$\displaystyle 0 \longrightarrow T^{\star}M \hookrightarrow E \overset{\rho}{\longrightarrow} TM \longrightarrow 0, \ \ \ \ \ (2)$

which, later on, we’ll see is the sort of sequence that describes an exact Courant algebroid.

Remark 3 (Early comment on Courant algebroids) As we will study in a later entry, it is the view afforded by generalised geometry that the bundle ${E}$ is in fact an extension of ${TM}$ by ${T^{\star}M}$ , and so it is a direct example of a Courant algebroid such that, in the exact sequence (2), the Courant algebroid has a symmetric form plus other structure (e.g., the Courant bracket) that makes it isomorphic to ${E}$ . This is true for suitable isotropic splittings of the exact sequence, an example of which is called a Dirac structure.

The sections of ${E}$ are non-trivial sections of ${TM \oplus T^{\star}M}$ . This means that, unlike in standard geometry and how we typically consider vector fields as sections of ${TM}$ only, we now consider elements of the non-trivial sections

$\displaystyle X = x + \xi, Y = y + \varepsilon, \ x,y \in \Gamma(TM), \ \xi, \varepsilon \in \Gamma(T^{\star}M), \ \ \ \ \ (3)$

where ${x, y}$ are vector parts and ${\xi, \varepsilon}$ 1-form parts.

The set of smooth sections ${C^{\infty}(M)}$ of the bundle ${E}$ are denoted by ${\Gamma(E)}$ such that the set of smooth vector fields is denoted by ${\Gamma(TM)}$ and the set of smooth 1-forms by ${\Gamma(T^{\star}M)}$ .

Remark 4 (Sequence and string background fields) For the sequence (2), note that in the map ${\rho}$ there exist sections ${\sigma}$ that are given by rank 2 tensors, which can then be split into symmetric and antisymmetric parts, ${\sigma_{\mu \nu} = g_{\mu \nu} + b_{\mu \nu}}$ . The sections of ${E}$ describe infinitesimal symmetries of these fields, as they are encoded in a generalised vector field ${X}$ capturing infinitesimal diffeomorphisms and a 1-form ${\xi}$ describing the b-field gauge symmetry.

References

[Gual04] M. Gualtieri. Generalized complex geometry [PhD thesis]. arXiv: 0401221[math.DG]. [Gua11] Marco Gualtieri. Generalized complex geometry. Ann. of Math. (2), 174(1):75–123, 2011. url: https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p03-s.pdf. [Hitc03] N. Hitchin. Generalized Calabi–Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281. arXiv: 0209099 [math.DG]. [Hitc05] N. Hitchin. Brackets, forms and invariant functionals. arXiv: 0508618 [math.DG]. [Hitc10] N. Hitchin. Lectures on generalized geometry. arXiv: 1008.0973 [math.DG]. [Rub18] R. Rubio. Generalised geometry: An introduction [lecture notes]. url: https://mat.uab.cat/~rubio/gengeo/Rubio-GenGeo.pdf

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