Generalised geometry #9: Generalised diffeomorphisms and double field theory

Generalised geometry: A study of generalised diffeomorphisms and gauge transformations We finally arrive at one of the most important properties exhibited by generalised geometry: namely, the manner in how generalised structures preserve a change of $latex {T}&fg=000000$ to $latex {TM \oplus T^{\star}M}&fg=000000$. As discussed in the previously, this means that we have a pairing of …

Power and thrones: Systemic crises of the late Middle Ages

One of the periods of human history that I most enjoy studying is the late Middle Ages, especially from a Western European perspective. There is a certain fecundity (in terms of ideas and the generation of concepts) about this period, which I think may be traced to a few principal roots. From these roots we …

Generalised geometry #8: Complex and symplectic structures

Before moving on to the important topic of generalised diffeomorphisms, not much has been said yet about extensions to complex geometry. As a short note, it is worth mentioning a few details as generalised complex structures are a primary subject of study in the context of the generalised geometry programme. These are structures on $latex …

Generalised geometry #7: Generalised metric

Generalised metrics and generalised Riemannian geometry We can introduce further structures on the generalised bundle $latex {E \simeq TM \oplus T^{\star}M}&fg=000000$ leading to further geometries. The first is the natural metric on $latex {E}&fg=000000$. The second is a generalisation of the Riemannian metric and is therefore called the generalised metric. Upon introducing this object, it …

Generalised geometry #6: Dirac structures and subspaces

We're almost at the point where we can start considering physics perspectives in relation to the mathematical structures of generalised geometry. But we still have some more definitions to review before arriving at such considerations. In this note, we consider Dirac structures. As a gentle introduction, a Dirac structure is given by an orthogonal involution …

Leibniz integral rule #2: Differentiating under the integral sign (special case)

We discussed the general case of the Leibniz integral rule, otherwise known as differentiating under the integral, and reviewed its derivation. The special case of this rule, in which the limits of integration are constants as opposed to differentiable functions, is much simpler and can be easily related to the fundamental theorem of calculus. The …

Leibniz integral rule #1: Feynman integration, otherwise known as differentiating under the integral (general case)

1. Introduction [This is a repost of an old blog post, the latex of which had broken when moving my blog some years ago. I repost it and its accompanying post (to follow) as I am collating blog entries on integral methods and tricks to list elsewhere.] Evaluating integrals has over time become one of …

Generalised geometry #5: Twisted by a gerbe

We can extend the theory of the generalised tangent bundle in two important ways: 1) Twisting by a gerbe, 2) Modifying the Courant bracket and exterior derivatives to twisted versions. Twisting by a gerbe In physics we think of a gerbe as a differential geometric object rather than an object of algebraic geometry. They can …

Notes on string theory #6: Enter into the light-cone

Now that we've studied the $latex {S_{NG}}&fg=000000$ and $latex {S_{P}}&fg=000000$ forms of bosonic string action, we turn our attention to the fact that the string will fluctuate. In the next sections of Polchinski's textbook (1.3-1.4), we will study the spectrum of open string fluctuations before moving to the case of closed strings. In doing so, …

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 …