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 …

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When doing generalised linear algebra, we want to study transformations that preserve the canonical pairing from the last note (of signature $latex {O(d,d)}&fg=000000$): $latex \displaystyle O(V \oplus V^{\star}) = \{A \in GL(V \oplus V^{\star}): \langle A_v, A_w \rangle = \langle v, w \rangle \ \text{for all} \ v,w \in V \oplus V^{\star}. \} \ \ … Continue reading Generalised geometry #3: Symmetries Generalised geometry #2: Generalised vector space and bilinear form Generalised linear algebra In the first note we introduced one of the fundamental structures of generalised geometry, namely the generalised tangent bundle$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 …

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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 …

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(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 …

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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 …

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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. …

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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 …