We demonstrate that weakly coupled, large $N, d$-dimensional $SU(N)$ gauge theories on a class of compact spatial manifolds (including $S^{d-1}\times {\rm time}$) undergo deconfinement phase transitions at temperatures proportional to the inverse length scale of the manifold in question. The low temperature phase has a free energy of order one, and is characterized by a stringy (Hagedorn) growth in its density of states. The high temperature phase has a free energy of order $N^2$. These phases are separated either by a single first order transition that generically occurs below the Hagedorn temperature or by two continuous phase transitions, the first of which occurs at the Hagedorn temperature. These phase transitions could perhaps be continuously connected to the usual flat space deconfinement transition in the case of confining gauge theories, and to the Hawking-Page nucleation of $AdS_5$ black holes in the case of the $\mathcal{N}=4$ supersymmetric Yang-Mills theory. We suggest that deconfinement transitions may generally be interpreted in terms of black hole formation in a dual string theory. Our analysis proceeds by first reducing the Yang-Mills partition function to a $(0+0)$-dimensional integral over a unitary matrix $U$, which is the holonomy (Wilson loop) of the gauge field around the thermal time circle in Euclidean space; deconfinement transitions are large $N$ transitions in this matrix integral.

In [K] all the possibilities for the non-positive part $\mathfrak{g}_{\le 0}=\bigoplus_{j=-h}^{0}\mathfrak{g}_{j}$ of the associated graded Lie superalgebra $\mathfrak{g}=\bigoplus_{j\ge -h}\mathfrak{g}_j$ of a simple linearly compact Lie superalgebra $L$, for a "good" choice of its filtration, were obtained. In [CK2] the transitive $\mathbb{Z}$-graded $\mathfrak{g}$ with those $\mathfrak{g}_{\le 0}$ were classified, and in [CK1] and [K], in order to reconstruct $L$ from $\mathfrak{g}$, their filtered deformations were classified. However, some cases had been inadvertently omitted. In this note we take the opportunity to amend this.

We present a countably infinite number of new explicit co-homogeneity one Sasaki-Einstein metrics on $S^2\times S^3$ of both quasi-regular and irregular type. These give rise to new solutions of type IIB supergravity which are expected to be dual to $\mathcal{N}=1$ superconformal field theories in four dimensions with compact or non-compact $R$-symmetry and rational or irrational central charges, respectively.

This papers is concerned with multisymplectic formalisms which are the frameworks for Hamiltonian theories for fields theory. Our main purpose is to study the observable $(n-1)$-forms which allows one to construct observable functionals on the set of solutions of the Hamilton equations by integration. We develop here two different points of view: generalizing the law $\{p,q\} = 1$ or the law $dF/dt =\{H,F\}$. This leads to two possible definitions; we explore the relationships and the differences between these two concepts. We show that — in contrast with the de Donder-Weyl theory — the two definitions coincides in the Lepage-Dedecker theory.