A unified approach on Springer fibers in the hook, two-row and two-column cases

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.Comment: 42 page

Geometric Satake, Springer correspondence, and small representations

For a simply-connected simple algebraic group $G$ over \C, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$, generalizing a well-known fact about $GL_n$. Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.Comment: Version 2: minor revisions, 33 page

The Structure of n-Point One-Loop Open Superstring Amplitudes

In this article we present the worldsheet integrand for one-loop amplitudes in maximally supersymmetric superstring theory involving any number n of massless open string states. The polarization dependence is organized into the same BRST invariant kinematic combinations which also govern the leading string correction to tree level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling number S_3^{n-1} of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang Mills theory and on the other hand the color tensors at quadratic alpha prime order of the color dressed tree amplitude.Comment: 75 pp, 8 figs, harvmac TeX, v2: published versio

Superconformal Flavor Simplified

A simple explanation of the flavor hierarchies can arise if matter fields interact with a conformal sector and different generations have different anomalous dimensions under the CFT. However, in the original study by Nelson and Strassler many supersymmetric models of this type were considered to be 'incalculable' because the R-charges were not sufficiently constrained by the superpotential. We point out that nearly all such models are calculable with the use of a-maximization. Utilizing this, we construct the simplest vector-like flavor models and discuss their viability. A significant constraint on these models comes from requiring that the visible gauge couplings remain perturbative throughout the conformal window needed to generate the hierarchies. However, we find that there is a small class of simple flavor models that can evade this bound.Comment: 43 pages, 1 figure; V3: small corrections and clarifications, references adde

Brans-Dicke supergravity and the Lambda naturalness problem

The successful $\Lambda CDM$ cosmological model requires a small but nonzero $\Lambda$ which appears to have an unnaturally small value compared to the supersymmetry breaking scale, typically $O(10^{-60}) m_{3/2}^4$ for $m_{3/2} \sim 10 TeV$. We explore the possibility of solving this naturalness problem in a special class of no-scale supergravity models which arise from a supersymmetric version of Brans-Dicke gravity, in which the Volkov and Brans-Dicke multiplets may be identified. These may be embedded in compactified string models, with the Brans-Dicke scalar given by the combination of dilaton and compactification breathing modes which leaves the 4 dimensional gauge couplings fixed. Assuming that 4 dimensional physics has an approximate symmetry under changes in this mode broken only by couplings between the low energy and gravitational or string sectors, the main one loop contribution to $\Lambda$ cancels between Brans-Dicke and gravitational (conformal compensator) F terms, and the leading contributions to $\Lambda$ now appear to be $O(m_{3/2}^8 m_{p}^{-4)}$, enabling a natural reconciliation between observational and particle physics estimates for $\Lambda$. The Brans-Dicke scalar has a range $O(m_{3/2}^{-1})$, lifting observational constraints on scalar gravity in this scenario