Archimedean cohomology revisited

Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the complex of nearby cycles in the geometry of a degeneration over a disk, and we recall Deninger's approach to the archimedean cohomology through an interpretation as global sections of a analytic Rees sheaf. We show that action of the Lefschetz, the log of monodromy and the log of Frobenius on the archimedean cohomology combine to determine a spectral triple in the sense of Connes. The archimedean part of the Hasse-Weil L-function appears as a zeta function of this spectral triple. We also outline some formal analogies between this cohomological theory at arithmetic infinity and Givental's homological geometry on loop spaces.Comment: 28 pages LaTeX 3 eps figure

A note on the dual of N=1 super Yang-Mills theory

We refine the dictionary of the gauge/gravity correspondence realizing N=1 super Yang-Mills by means of D5-branes wrapped on a resolved Calabi-Yau space. This is done by fixing an ambiguity on the correct interpretation of the holographic dual of the running gauge coupling and amounts to identify a specific 2-cycle in the dual ten-dimensional supergravity background. In doing so, we also discuss the role played in this context by gauge transformations in the relevant seven-dimensional gauged supergravity. While all nice properties of the duality are maintained, this modification of the dictionary has some interesting physical consequences and solves a puzzle recently raised in the literature. In this refined framework, it is also straightforward to see how the correspondence naturally realizes a geometric transition.Comment: 11 pages, latex; minor changes and typos correcte

Skew Howe duality and random rectangular Young tableaux

We consider the decomposition into irreducible components of the external power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(\lambda,\mu)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$\mu=\lambda'$. We show that the Young diagram $\lambda$ which corresponds to a randomly selected irreducible component $(\lambda,\lambda')$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a random Young tableau of rectangular shape with $m$ rows and $n$ columns. This observation allows treatment of the asymptotic version of this decomposition in the limit as $m,n,p\to\infty$ tend to infinity.Comment: 17 pages. Version 2: change of title, section on bijective proofs improve