Number theory in physics

This summer I was expecting to be working mainly on an extended field theory and geometry project as well as pushing toward the conclusion of a study on double sigma models. But somewhat unexpectedly I've found myself also working on some interesting things at the interface of number theory and physics. It has to do …

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}. \} \ \ …

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 …

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 …

Notes on string theory #3: Nambu-Goto action

1. Introduction I haven't been keeping up with this as much as I would like, mainly because I have been busy. But I am committed to continuing to reupload many of my notes on Polchinski's textbooks. It is fun for me to go through it all again in my spare time, and I've noticed that …

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 …

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 …

Back home by the North Sea

As we usually do at the end of term, Beth and I have returned home to North Norfolk. Throughout the academic year, I greatly miss the North Sea from coast to countryside. In the past I've written about the coastal footpaths we've come to know intimately; the secret gardens and ponds off the path, with …

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 …

Notes on string theory #2: The relativistic point particle (pp. 9-11)

1. Introduction In Chapter 1 of Polchinski's textbook, we start with a discussion on the relativistic point particle (pp. 9-11). String theory proposes that elementary particles are not pointlike, but rather 1-dimensional extended objects (i.e., strings). In fact, string theory (both the bosonic string in Volume 1 of Polchinski and the superstring that comprises much …