Quantum parameter space in super Yang-Mills, II

In [1] (hep-th/0211069), the author has discussed the quantum parameter space of the N=1 super Yang-Mills theory with one adjoint Higgs field Phi, tree-level superpotential W_tree = m (Phi^2)/2 + g (Phi^3)/3$, and gauge group U(Nc). In particular, full details were worked out for U(2) and U(3). By discussing higher rank gauge groups like U(4), for which the classical parameter space has a large number of disconnected components, we show that the phenomena discussed in [1] are generic. It turns out that the quantum space is connected. The classical components are related in the quantum theory either through standard singularities with massless monopoles or by branch cuts without going through any singularity. The branching points associated with the branch cuts correspond to new strong coupling singularities, which are not associated with vanishing cycles in the geometry, and at which glueballs can become massless. The transitions discussed recently by Cachazo, Seiberg and Witten are special instances of those phenomena.Comment: 12 pages including 2 large figure

Consistency conditions in the chiral ring of super Yang-Mills theories

Starting from the generalized Konishi anomaly equations at the non-perturbative level, we demonstrate that the algebraic consistency of the quantum chiral ring of the N=1 super Yang-Mills theory with gauge group U(N), one adjoint chiral superfield X and N_f<=2N flavours of quarks implies that the periods of the meromorphic one-form Tr dz/(z-X) must be quantized. This shows in particular that identities in the open string description of the theory, that follow from the fact that gauge invariant observables are expressed in terms of gauge variant building blocks, are mapped onto non-trivial dynamical equations in the closed string description.Comment: 26 pages, 1 figure; v2: typos corrected, published versio

Renormalization of the Non-Linear Sigma Model in Four Dimensions. A two-loop example

The renormalization procedure of the non-linear SU(2) sigma model in D=4 proposed in hep-th/0504023 and hep-th/0506220 is here tested in a truly non-trivial case where the non-linearity of the functional equation is crucial. The simplest example, where the non-linear term contributes, is given by the two-loop amplitude involving the insertion of two \phi_0 (the constraint of the non-linear sigma model) and two flat connections. In this case we verify the validity of the renormalization procedure: the recursive subtraction of the pole parts at D=4 yields amplitudes that satisfy the defining functional equation. As a by-product we give a formal proof that in D dimensions (without counterterms) the Feynman rules provide a perturbative symmetric solution.Comment: Latex, 3 figures, 19 page

Non-perturbative double scaling limits

Recently, the author has proposed a generalization of the matrix and vector models approach to the theory of random surfaces and polymers. The idea is to replace the simple matrix or vector (path) integrals by gauge theory or non-linear sigma model (path) integrals. We explain how this solves one of the most fundamental limitation of the classic approach: we automatically obtain non-perturbative definitions in non-Borel summable cases. This is exemplified on the simplest possible examples involving O(N) symmetric non-linear sigma models with N-dimensional target spaces, for which we construct (multi)critical metrics. The non-perturbative definitions of the double scaled, manifestly positive, partition functions rely on remarkable identities involving (path) integrals.Comment: 18 pages, one figur

Super Yang-Mills, Matrix Models and Geometric Transitions

I explain two applications of the relationship between four dimensional N=1 supersymmetric gauge theories, zero dimensional gauged matrix models, and geometric transitions in string theory. The first is related to the spectrum of BPS domain walls or BPS branes. It is shown that one can smoothly interpolate between a D-brane state, whose weak coupling tension scales as Nc or 1/gs, and a closed string solitonic state, whose weak coupling tension scales as Nc^2 or 1/gs^2. This is part of a larger theory of N=1 quantum parameter spaces. The second is a new purely geometric approach to sum exactly over planar diagrams in zero dimension. It is an example of open/closed string duality.Comment: 11 pages, 2 figures, .cls files included; to appear in the proceedings of the Strings 2004 conference in Pari

Weyl and Marchaud derivatives: a forgotten history

In this paper we recall the contribution given by Hermann Weyl and Andr\'e Marchaud to the notion of fractional derivative. In addition we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics.Comment: arXiv admin note: text overlap with arXiv:1705.00953 by other author

Early physics with top quarks at the LHC

The ATLAS and CMS experiments are now in their final installation phase and will be soon ready to study the physics of proton-proton collisions at the Large Hadron Collider. The LHC, by producing 2 $t\bar{t}$ events per second, will provide more than 8 million top events a year at start-up. In this paper, particular emphasis is given to the $t\bar{t}$ physics studies that can be performed at the beginning of the LHC running, with a limited amount of integrated luminosity ($\le$10 fb$^{-1}$).Comment: Proceedings of Moriond QCD 2007. Luminosity contribution to error on top pair production cross-section has been changed from 0.5% to 5.0