Vertex Operator Algebras Associated to Type G Affine Lie Algebras

In this paper, we study representations of the vertex operator algebra $L(k,0)$ at one-third admissible levels $k= -5/3, -4/3, -2/3$ for the affine algebra of type $G_2^{(1)}$. We first determine singular vectors and then obtain a description of the associative algebra $A(L(k,0))$ using the singular vectors. We then prove that there are only finitely many irreducible $A(L(k,0))$-modules from the category $\mathcal O$. Applying the $A(V)$-theory, we prove that there are only finitely many irreducible weak $L(k,0)$-modules from the category $\mathcal O$ and that such an $L(k,0)$-module is completely reducible. Our result supports the conjecture made by Adamovi{\'c} and Milas in \cite{AM}.Comment: 30 page

Exact Post Model Selection Inference for Marginal Screening

We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response $y$, conditional on the model being selected (``condition on selection" framework). This allows us to construct valid confidence intervals and hypothesis tests for regression coefficients that account for the selection procedure. In contrast to recent work in high-dimensional statistics, our results are exact (non-asymptotic) and require no eigenvalue-like assumptions on the design matrix $X$. Furthermore, the computational cost of marginal regression, constructing confidence intervals and hypothesis testing is negligible compared to the cost of linear regression, thus making our methods particularly suitable for extremely large datasets. Although we focus on marginal screening to illustrate the applicability of the condition on selection framework, this framework is much more broadly applicable. We show how to apply the proposed framework to several other selection procedures including orthogonal matching pursuit, non-negative least squares, and marginal screening+Lasso

Light Gravitinos at Colliders and Implications for Cosmology

Light gravitinos, with mass in the eV to MeV range, are well-motivated in particle physics, but their status as dark-matter candidates is muddled by early-Universe uncertainties. We investigate how upcoming data from colliders may clarify this picture. Light gravitinos are produced primarily in the decays of the next-to-lightest supersymmetric particle, resulting in spectacular signals, including di-photons, delayed and non-pointing photons, kinked charged tracks, and heavy metastable charged particles. We find that the Tevatron with 20/fb and the 7 TeV LHC with 1/fb may both see evidence for hundreds of light-gravitino events. Remarkably, this collider data is also well suited to distinguish between currently viable light-gravitino scenarios, with striking implications for structure formation, inflation, and other early-Universe cosmology.Comment: 12 pages, 6 figures. To be submitted to Phys. Rev. D