Microscopic quantum superpotential in N=1 gauge theories

We consider the N=1 super Yang-Mills theory with gauge group U(N), adjoint chiral multiplet X and tree-level superpotential Tr W(X). We compute the quantum effective superpotential W_mic as a function of arbitrary off-shell boundary conditions at infinity for the scalar field X. This effective superpotential has a remarkable property: its critical points are in one-to-one correspondence with the full set of quantum vacua of the theory, providing in particular a unified picture of solutions with different ranks for the low energy gauge group. In this sense, W_mic is a good microscopic effective quantum superpotential for the theory. This property is not shared by other quantum effective superpotentials commonly used in the literature, like in the strong coupling approach or the glueball superpotentials. The result of this paper is a first step in extending Nekrasov's microscopic derivation of the Seiberg-Witten solution of N=2 super Yang-Mills theories to the realm of N=1 gauge theories.Comment: 23 pages, 1 figure; typos corrected, version to appear in JHE

Flexible synthesis of spirocyclic pyrans and piperidines

Spirocyclic piperidines and spirocyclic pyrans are prevalent throughout nature, often appearing in natural products which exhibit exciting biological activities. Notable examples of spirocyclic piperidine-containing biologically active natural products are halichlorine, pinnaic acid and tauropinnaic acid. Despite their structural similarity, halichlorine and the pinnaic acids were isolated from separate organisms; halichlorine was isolated from extracts of the marine sponge Halichondria okadai while both pinnaic acid and tauropinnaic acid were isolated from the Okinawan bivalve mollusc Pinna muricata. The complex hybrid molecule polymaxenolide contains a representative spirocyclic pyran core. The biological profile of polymaxenolide is not yet known, however its hybrid origins have rendered it a target of significant interest. The work described herein details the development of a methodology capable of accessing both spirocyclic pyran and spirocyclic piperidine core structures from a common cyclic tertiary furfuryl alcohol intermediate. The key spirocycle forming step involves the oxidative rearrangement of cyclic tertiary furfuryl alcohols and amines for the synthesis of spirocyclic pyrans and piperidines, respectively. Efforts towards the synthesis of a complex, africanane-derived Southern fragment, with the intention of applying this methodology towards the synthesis of polymaxenolide are reported. This methodology has been further elaborated to complete an asymmetric synthesis of the upper framework of an oxa-analogue of pinnaic acid. The potential for a spectator protecting group free synthesis of pinnaic acid was also explored and the synthesis of an advanced intermediate is also reported

Glueball operators and the microscopic approach to N=1 gauge theories

We explain how to generalize Nekrasov's microscopic approach to N=2 gauge theories to the N=1 case, focusing on the typical example of the U(N) theory with one adjoint chiral multiplet X and an arbitrary polynomial tree-level superpotential Tr W(X). We provide a detailed analysis of the generalized glueball operators and a non-perturbative discussion of the Dijkgraaf-Vafa matrix model and of the generalized Konishi anomaly equations. We compute in particular the non-trivial quantum corrections to the Virasoro operators and algebra that generate these equations. We have performed explicit calculations up to two instantons, that involve the next-to-leading order corrections in Nekrasov's Omega-background.Comment: 38 pages, 1 figure and 1 appendix included; v2: typos and the list of references corrected, version to appear in JHE

On the Geometry of Super Yang-Mills Theories: Phases and Irreducible Polynomials

We study the algebraic and geometric structures that underly the space of vacua of N=1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a powerful method to analyse the possible smooth interpolations between different classical limits in the gauge theory. As an application in U(Nc) theories, we provide a simple and completely general proof of the fact that confining and Higgs vacua are in the same phase when fundamental flavors are present, by finding an irreducible polynomial equation satisfied by the glueball operator. We also derive the full phase diagram for the theory with one adjoint when Nc is less than or equal to 7 using computational algebraic geometry programs.Comment: 87 pages; v2: typos and eq. (4.44) correcte

The Magnetohydrodynamic Kelvin-Helmholtz Instability: A Three-Dimensional Study of Nonlinear Evolution

We investigate through high resolution 3D simulations the nonlinear evolution of compressible magnetohydrodynamic flows subject to the Kelvin-Helmholtz instability. We confirm in 3D flows the conclusion from our 2D work that even apparently weak magnetic fields embedded in Kelvin-Helmholtz unstable plasma flows can be fundamentally important to nonlinear evolution of the instability. In fact, that statement is strengthened in 3D by this work, because it shows how field line bundles can be stretched and twisted in 3D as the quasi-2D Cat's Eye vortex forms out of the hydrodynamical motions. In our simulations twisting of the field may increase the maximum field strength by more than a factor of two over the 2D effect. If, by these developments, the Alfv\'en Mach number of flows around the Cat's Eye drops to unity or less, our simulations suggest magnetic stresses will eventually destroy the Cat's Eye and cause the plasma flow to self-organize into a relatively smooth and apparently stable flow that retains memory of the original shear. For our flow configurations the regime in 3D for such reorganization is $4\lesssim M_{Ax} \lesssim 50$, expressed in terms of the Alfv\'en Mach number of the original velocity transition and the initial Alfv\'en speed projected to the flow plan. For weaker fields the instability remains essentially hydrodynamic in early stages, and the Cat's Eye is destroyed by the hydrodynamic secondary instabilities of a 3D nature. Then, the flows evolve into chaotic structures that approach decaying isotropic turbulence. In this stage, there is considerable enhancement to the magnetic energy due to stretching, twisting, and turbulent amplification, which is retained long afterwards. The magnetic energy eventually catches up to the kinetic energy, and the nature of flows become magnetohydrodynamic.Comment: 11 pages, 12 figures in degraded jpg format (2 in color), paper with original quality figures available via ftp at ftp://ftp.msi.umn.edu/pub/users/twj/mhdkh3dd.ps.gz or ftp://canopus.chungnam.ac.kr/ryu/mhdkh3dd.ps.gz, to appear in The Astrophysical Journa

Simple matrix models for random Bergman metrics

Recently, the authors have proposed a new approach to the theory of random metrics, making an explicit link between probability measures on the space of metrics on a Kahler manifold and random matrix models. We consider simple examples of such models and compute the one and two-point functions of the metric. These geometric correlation functions correspond to new interesting types of matrix model correlators. We study a large class of examples and provide in particular a detailed study of the Wishart model.Comment: 23 pages, IOP Latex style, diastatic function Eq. (22) and contact terms in Eqs. (76, 95) corrected, typos fixed. Accepted to JSTA

Towards an embedding of Graph Transformation in Intuitionistic Linear Logic

Linear logics have been shown to be able to embed both rewriting-based approaches and process calculi in a single, declarative framework. In this paper we are exploring the embedding of double-pushout graph transformations into quantified linear logic, leading to a Curry-Howard style isomorphism between graphs and transformations on one hand, formulas and proof terms on the other. With linear implication representing rules and reachability of graphs, and the tensor modelling parallel composition of graphs and transformations, we obtain a language able to encode graph transformation systems and their computations as well as reason about their properties