• Bokeh

Le bokeh est un effet utilisé en photographie dans lequel un plan de l'image est volontairement flou, afin de faire ressortir un autre plan qui, lui, est net. On peut se demander quels sont les différents paramètres influant sur l'effet de bokeh. Je présente ici une formule simple pour quantifier les effets des paramétres standards, dans le cadre de l'optique géométrique, et en supposant que l'objectif est assimilable à une lentille mince.

• F-theory

Consider type IIB superstring theory. We learn at school that the possible D-branes in this theory will have an odd number os space-like dimensions. Here I want to focus on the D7 brane. One particular feature is that the transverse space is two-dimensional, and because of this low dimensionality, the effect of the presence of a D7 can be felt at large distance. We can see this under two viewpoints : the electric potential sourced by the brane, and the metric retroaction (being a BPS object, these two aspects are related for the D7 brane). In both cases, the behavior at large distance is controlled by the Green function of the transverse Laplace operator, which in two dimensions is a logarithm of the distance to the brane. Because of that, one can compute that a D7 creates a deficit angle at large distance, and it turns out this angle is $\pi /6$.

• Conformal manifold and rings

Just two pages on conformal manifolds and two rings of superconformal primary operators, following the first section of a recent paper by Gerchkovitz, Gomis, Ishtiaque, Karasik, Komargodski and Pufu.

• Hanany-Witten and 3d Mirror Symmetry

Here are a few more notes on mirror symmetry, following Hanany and Witten.

• E-string theory

This is based on the recent paper by Kim, Razamat, Vafa and Zafrir.

• Résidus et Loi de Réciprocité Quadratique

Considérons un nombre premier impair $p$ et calculons $a^2$, pour tout $a \in \mathbb{Z}_p-\{0\}$. Par exemple, pour $p=41$, on obtient $$1,4,9,16,25,36,8,23,40,18,39,21,5,32,20,10,2,37,33,31,31,33,37,2,10,20,32,5,21,39,18,40,23,8,36,25,16,9,4,1$$ (j'identifie ici un élément de $\mathbb{Z}_p$ avec un de ses représentants). Evidemment, comme $a^2 = (-a)^2$, la suite de nombres obtenue est palindromique, et comme $p$ est impair, chaque nombre apparaît un nombre pair de fois. Ordonnons la suite : $$1,1,2,2,4,4,5,5,8,8,9,9,10,10,16,16,18,18,20,20,21,21,23,23,25,25,31,31,32,32,33,33,36,36,37,37,39,39,40,40$$ On observe que tous les nombres apparaissent en fait exactement deux fois. Cela s'explique facilement : le polynôme $X^2 - A$ a au plus deux solutions dans le corps $\mathbb{Z}_p$, pour tout $A$ fixé. On en déduit que $\mathbb{Z}_p-\{0\}$ peut être partitionné en deux parties de même cardinal $(p-1)/2$, à savoir les nombres qui sont un carré, et qu'on appelle les résidus quadratiques, ici $$1,2,4,5,8,9,10,16,18,20,21,23,25,31,32,33,36,37,39,40 \, ,$$ et les autres, qui sont n'en sont pas, et qu'on appelle les non-résidus quadratiques, ici $$3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, \ 35, 38 \, .$$ Mais comment savoir a priori si un nombre donné $A$ est un résidu quadratique ou non ?

• BPS particles and chiral algebras

Many results in 4d N=2 theories come from the combination of two principles: renormalization group and supersymmetry. Computing a protected observable in the UV and the IR gives different expressions for the same quantity, and this can be interesting. Let us see an example, following a talk Clay Cordova gave at String-Maths 2017 (video available on the website of the conference).

• Basics of three-dimensional mirror symmetry

Three-dimensional mirror symmetry is the statement that some $3d$ $\mathcal{N}=4$ theories at non-trivial IR renormalization group fixed point can have a dual description that satisfies some characteristic properties. Namely, this mirror symmetry relates two $3d$ $\mathcal{N}=4$ theories, in such a way that

• The Higgs and Coulomb branches are exchanged by mirror symmetry;
• Quantum effects in one theory arise classically in the other, and visa-versa;
• The equality of the global symmetries at the IR fixed points may involve the appearance of a hidden symmetry.

• Chiral algebras in higher dimensional supersymmetric conformal field theories

In any $\mathcal{N}=2$ supersymmetric conformal field theory in four dimensions, one can find on any $\mathbb{R}^2$ plane a chiral algebra (similar to what exists naturally in two-dimensional CFTs). This chiral algebra has the structure of a $\mathcal{W}$-algebra. This construction also applies to six-dimensional $\mathcal{N}=(2,0)$ theories. Here are some manuscript notes, based on the following papers : Four dimensions; Six dimensions; $\mathcal{W}$-algebras.

• Renormalization group flow in diverse dimensions

Thomas Dumitrescu gave a very clear summary of the status of several conjectures and theorems concerning the renormalization group flow in various dimensions at Strings today. These are usually known as c-theorems, a-theorems or F-theorems. Here are some notes that summarize some aspects of Thomas' talk.