Generalised geometry #4: The Courant bracket and the Jacobiator

The Courant bracket In addition to the generalised tangent bundle, the next fundamental structure of generalised geometry is the bilinear, skew-symmetric bracket called the Courant bracket. The Courant bracket is defined on the sections of $latex {E = TM \oplus T^{\star}M}&fg=000000$ such that it is the generalised analogue of a standard Lie bracket for vector-fields …

Generalised geometry #3: Symmetries

When doing generalised linear algebra, we want to study transformations that preserve the canonical pairing from the last note (of signature $latex {O(d,d)}&fg=000000$): $latex \displaystyle O(V \oplus V^{\star}) = \{A \in GL(V \oplus V^{\star}): \langle A_v, A_w \rangle = \langle v, w \rangle \ \text{for all} \ v,w \in V \oplus V^{\star}. \} \ \ …

Generalised geometry #2: Generalised vector space and bilinear form

Generalised linear algebra In the first note we introduced one of the fundamental structures of generalised geometry, namely the generalised tangent bundle $latex {E \simeq TM \oplus T^{\star}M}&fg=000000$. In the extension of the standard tangent bundle $latex {TM}&fg=000000$ to $latex {TM \oplus T^{\star}M}&fg=000000$, we are simultaneously extending linear algebra to some notion of generalised linear …

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 …

(n-1)-thoughts, n=6: Asperger’s and writing, Lie 2-algebroids, linguistics, and summer reading

Asperger's, studying, and writing As a person with autistic spectrum disorder (ASD), I've learned that writing plays an important and meaningful role in my life. I write a lot. By 'a lot' I mean to define it as a daily activity. Sometimes I will spend my entire morning and afternoon writing. Other times I will …

The duality symmetric string: A return to Tseytlin

The case of the duality symmetric string is a curious one (in a recent post we began discussing this string in the context of building toward a study of duality symmetric M-theory). In this essay, which may serve as the first of a few on the topic, I want to offer an introduction to some …

‘The unreasonable effectiveness of string theory in mathematics’: Emergence, synthesis, and beauty

As I noted the other day, there were a number of interesting talks at String Math 2020. I would really like to write about them all, but as I am short on time I want to spend a brief moment thinking about one talk in particular. Robbert Dijkgraaf's presentation, 'The Unreasonable Effectiveness of String Theory …

Literature: Duality Symmetric String and the Doubled Formalism

When it comes to a T-duality invariant formulation of string theory, there are two primary actions that are useful to study as a point of entry. The first is Tseytlin's non-covariant action. It is found in his formulation of the duality symmetric string, which presents a stringy extension of the Floreanini-Jackiw Lagrangians for chiral fields. …

Thinking About the Strong Constraint in Double Field Theory

I've been thinking a lot lately about the strong (or section) constraint in Double Field Theory. In this post, I want to talk a bit about this constraint. Before doing so, perhaps a lightning review of some other aspects of DFT might be beneficial, particularly in contextualising why the condition appears in the process of …

Generalised Geometry, Non-commutativity, and Emergence

The project I am working on for my dissertation has to do with the notion of emergent de Sitter space. Of course an ongoing problem in string theory concerns whether asymptotic de Sitter spacetime can exist as a solution, and needless to say this question serves as one motivation for the research. With what appears …