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}. \} \ \ …

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Book review: Fantastic numbers and where to find them

My PhD supervisor, Tony, has published a book. It's titled, Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity. Full disclosure: I read one of the earliest drafts, which must have been about two years ago. It was quite enjoyable witnessing the book develop, hearing about new chapter plans, and …

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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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Cosmological constant, the duality symmetric string, and Atkin-Lehner symmetry

I was going through one of my notebooks and I came across a page with several comments on old papers by Arkady Tseytlin [1] and Gregory Moore [3], respectively. The notes must have been written last autumn at the start of the academic year, because it was around this time my supervisor and I were …

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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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Double Field Theory as the double copy of Yang-Mills

1. Introduction A few weeks ago I came across this paper [DHP] on Double Field Theory and the double copy of Yang-Mills. Its result is most curious. As a matter of introduction, recall how fundamental interactions in nature are governed by two kinds of theories: On the one hand, Einstein's theory of relativity. On the …

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