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 …

Thinking about a series of posts on number theory and physics

Lately, I've been thinking of writing a blog series on number theory and physics. I've been thinking a lot about this topic of late, or at least about a very small corner of what is an incredibly broad area of research. In fact, the whole business of working to find a canonical class of regulators …

Notes on string theory #5: Modifying the Polyakov action (cosmological constant and Ricci scalar)

We come to the conclusion of Section 1.2 in Polchinski's textbook. In this note we'll discuss how there are two possible modifications that we can make to the Polyakov action (see last note) that preserve Poincaré invariance. The first is a cosmological constant term on the worldsheet. The second modification involves the scalar curvature $latex …

Notes on string theory #4: Polyakov action

In our ongoing read through of Polchinski's textbook, we left off on page 12 having studied the first principle Nambu-Goto action $latex {S_{NG}}&fg=000000$ for the string. We have glimpsed early on why string theory is a generalisation - or, one could also say, deformation - of point particle theory. The generalisation from point particles to …

Holiday reading list 2022

The end of term is here, and with that comes a much needed winter break. One thing that I enjoy prior to Christmas break is compiling a list of books for my holiday reading. It gets me excited for the holiday season. I also enjoy sharing books, and have grown to like the idea of …

A rare glimpse of climate debate within the finer margins of reason

https://www.youtube.com/watch?v=5Gk9gIpGvSE It seems in the last decade especially narratives about climate have become increasingly saturated with a certain distinguishable hysteria, which, I would argue, is detrimental to rational discourse on what is undoubtedly an important issue of our time. It doesn't help that popular media coverage on climate science is generally poor, if not altogether …

Tameness and Quantum Field Theory

A paper by Michael R. Douglas, Thomas W. Grimm, and Lorenz Schlechter appeared on the archive yesterday. Admittedly, I haven't had time yet to properly work through it. From what I quickly skimmed last night, the paper mainly concentrates on the notion of tameness and how tame classes of functions can be applied in the …

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}. \} \ \ …