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As I was preparing to leave for holiday, three papers appeared of significant interest. I haven’t had a chance to work through them all yet, between being strict with my holiday time and with String 2021 ongoing, but I felt motivated over a cup of tea to take note:

Heterotic duels of M-theory

A nice paper by Bobby Samir Acharya,  Alex Kinsella, and David R. Morrison on the non-perturbative heterotic duels of M-theory was released. This is of particular interest to me as it relates to the wider study of the non-perturbative aspects of M/heterotic duality. 

This duality was discovered in the mid 90’s in which one can take M-theory compactified on a  and find it relates to the  heterotic theory compactified on a three-torus. When you look at the 4D picture, we may instead compactify M-theory on a  manifold (equipped with a K3 fibration), which is a seven-dimensional Riemannian manifold that is special because it comes with the holonomy group in the exceptional simple Lie group . For the , it gets compactified on a Calabi-Yau threefold equipped with a three-torus. I haven’t had a chance to read through and consider the paper in any great detail, but it is noticeable that it starts with a similar set-up, taking low-energy M-theory with  orbifolds as the choice of compactification, with choice of equipped K3-fiberation to enable comparison with the dual heterotic string spectrum. A key observation, I take it, is that for the heterotic background there is a subtlty with the gauge bundle on  such that, when it comes to the non-perturbative physics, there are point-like instantons on orbifold points of the geometry. This is where things get both interesting and complicated, and I’m not sure in what way these instanton effects in the spectrum relate to M-branes. I am keen to read the second half of the study.

Higgs mass in string theory

Another paper that appeared looks at calculating the Higgs mass. It’s by Steven Abel and Keith R. Dienes. This paper is quite the joy, and I’m sure anyone with interest in string theory will enjoy it over a cup of tea. Abel and Dienes harnesses the powers of the world-sheet theory to perform some proper stringy calculations, developing a framework that presents a relationship between the Higgs mass and the cosmological constant. What is neat about the computation is that this connection is generic for all closed string theories and provides a bit of a platform for future studies on gauge hierarchy problems.

Double sigma models and geometric quantisation 

With a rush of papers leading up to my holiday, this one immediately caught my attention and got me excited. Luigi Alfonsi and David Berman study geometric quantisation in double field theory and double sigma models. From what I have seen, it is grand. 

I was actively thinking about quantisation of double sigma models, as this is one area in which I have been working. In fact, I recall a few discussions a year or more ago about a project looking into the quantisation of the doubled string. In parts, from working in the area, what we see in this paper is kind of what one would expect in that, to start, the zero-mode sector for the closed string is intrinsically non-commutative. This alone is an interesting fact with some deep implications. Commonly, in the set-up where the target-space is treated as a phase-space, one will also equip a symplectic form , and one will can construct a theory with an action following Tseytin (we talked about this in a past post). What is found with the inclusion of  is an interesting connection with Born geometry (maybe I’ll write about this in a future post) and, furthermore, one will often find discussion on symplectic structures as it relates to Poisson geometry which has some deep relation with T-duality. 

In short, in the quantisation procedure there is a choice of polarisation, and the authors want to make a choice of polarisation in conjunction with the strategy for geometric quantisation. What happens, in any case, is that T-duality will give polarisations. And then what one wants to study is the noncommutative algebra associated to the doubled phase space. What the paper shows is that there are, in essence, two types of quantisations going on, because there is one coming from the usual phase space and then another from the duality frame (i.e., what in the formalism is understood in terms of the Lagrangian submanifold).

A deeper idea here has to do with the doubled phase space and para-Hermitean geometry, which I think I’ve mentioned a wee bit in the past. On that note, it is also interesting to think about the findings in this paper as it relates to the idea of metastring theory and quantisation.

As an aside, I’ve been working on a draft essay about a series of papers by Luigi. I wanted to write a bit about double sigma models and double field theory before finishing this essay, with a mind toward giving the reader some reference. They are fantastic papers on the global double space of double field theory, among other things. I also have Luigi’s PhD thesis on hand, which I think is great. There is a lot to discussed here in the context of the doubled geometry of double sigma models and higher structures.

Strings 2021

• The annual string conference, Strings 2021, is ongoing (21 June – 2 July). It’s always an event that I look forward to, as it brings together the entire string theory community. Among a large list of great and usual names, my eye immediate caught an anomalous speaker amongst the expected and anticipated: namely, Roger Penrose. I will be most eager to hear what he has to say during his presentation on Friday 2, July. The topic is on gravitational singularities. There are of course a number of talks that I am looking forward to – too many to list! For now, here is the schedule with list of speakers, including links to notes and recordings. If I find the time and motivation, I’ll write a summary of my favourite talks next week.