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, …

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 …

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 …

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

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 …

Learning M-theory: Gauge theory of membranes, brane intersections, and the self-dual string

I've been learning a lot about M-theory. It's such a broad topic that, when people ask me 'what is M-theory?', I continue to struggle to know where to start. Right now, much of my learning is textbook and I have more questions than answers. I naturally take the approach of first wanting as broad and …

Notes on string theory #1: The non-relativistic string

The study of waves has far reaching applications throughout physics (and across the sciences). Fundamental physics is no exception. Essential for understanding waves is understanding oscillation, and as one will recall from classical mechanics a concept of fundamental importance in this regard is the harmonic oscillator. A simple harmonic oscillator can then also be generalised …