
Holographic Duals of 3d Sfold CFTs
Some notes about the recent paper by Benjamin Assel and Alessandro Tomasiello, Holographic duals of 3d Sfold CFTs.

Coulomb branches with complex singularities
Last week, Philip Argyres and Mario Martone published a very interesting paper, Coulomb branches with complex singularities. Some of the results are similar to those presented in my last publication, with Alessandro Pini and Diego RodriguezGomez, The Importance of Being Disconnected, A Principal Extension for Serious Groups.

Les neurosciences peuventelles éclairer l'éducation?
Conférence de Franck Ramus à l'ENS :

Finite temperature holography
The $AdS/CFT$ correspondence is one of the central discoveries of the last decades in highenergy physics. As we know, in one of its incarnation it relates a certain supergravity theory on $AdS_5 \times S^5$ with $\mathcal{N}=4$ SYM with gauge group $SU(N)$ in the large $N$ limit and large 't Hooft coupling. But what's the relation to the real world? $\mathcal{N}=4$ SYM is an elegant theory, but certainly very different from what we find in experiments, so it seems holography (at least in this restricted sense) allows us to gain little if not no knowledge about the real world.

Exact correlators on the Wilson Loop in $\mathcal{N}=4$ SYM
I'm talking here about the paper which appeared last week, by Simone Giombi and Shota Komatsu.

Vandermonde Gaussian Integral
This is just to signal the formula for an integral that is useful in many matrix model computations: $$\int_{\mathbb{R}^n} \mathrm{d} \Lambda \Delta (\Lambda)^{\beta} \mathrm{exp} \left(  \frac{\mathrm{Tr} \, \Lambda^2 }{g^2}\right) = \frac{g^{n+ \beta n(n1)/2} \pi^{n/2}}{2^{\beta n(n1)/4}} \prod\limits_{j=1}^n \frac{\Gamma \left(1+ j \frac{\beta}{2} \right)}{\Gamma \left(1+ \frac{\beta}{2} \right)} \, . $$ In this formula, $\beta = 1,2,4$ determines which Gaussian ensemble is used (respectively Orthogonal, Unitary and Symplectic), $\Lambda = \mathrm{diag} (\lambda_1 , \dots , \lambda_n)$ is a diagonal matrix, the measure is $\mathrm{d} \Lambda = \mathrm{d} \lambda_1 \dots \mathrm{d} \lambda_n$ and $$\Delta (\Lambda) = \prod\limits_{1 \leq i < j \leq n} (\lambda_i  \lambda_j)$$ is the Vandermonde determinant.

Matrix Integrals
Our aim in this note will be to say something about integrals of the form $$Z = \int_E \mathrm{d}M \, e^{\mathrm{Tr} V (M)} \, , $$ where for simplicity we will consider $V(M)$ to be a polynomial in $M$. We will follow closely the excellent review by Bertrand Eynard, Taro Kimura and Sylvain Ribault called Random matrices, available here.

Homology and Volumes of compact Lie groups
In this post, I would like to show a nice relation between some Lie groups and products of spheres of odd dimensions. This can be made precise in the context of homology, and is useful to compute the volumes of compact Lie groups.

Characters of finite groups
Consider a finite group $G$. In this note, I will define a (complex) representation of $G$ to be any group homomorphism $\rho : G \rightarrow GL(n,\mathbb{C})$. Working on the field of complex numbers instead of an arbitrary field has many advantages:
 The concepts of reducible representation (one that is equivalent to a block upper triangular representation) and decomposable representation (one that is equivalent to a direct sum of nontrivial representations) coincide.
 Any representation is equivalent to a direct sum of irreducible representations.
 Any representation is equivalent to a unitary representation.

Large Charge CFT
(draft)
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