We study certain six dimensional theories arising on (p, q) brane webs living on R × S. These brane webs are dual to toric elliptically fibered Calabi-Yau threefolds. The compactification of the space on which the brane web lives leads to a deformation of the partition functions equivalent to the elliptic deformation of the Ding-Iohara algebra. We compute the elliptic version Dotsenko-Fateev integrals and show that they reproduce the instanton counting of the six dimensional theory.

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces, including the Weil-Petersson metric, the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric. We prove the dual Nakano negativity of the Weil-Petersson metric. As applications of these results we deduce certain important results about the <i>L</i> ²-cohomology groups of the logarithmic tangent bundle over the compactified moduli spaces.

Factor modeling is an essential tool for exploring intrinsic dependence structures among high-dimensional random variables. Much progress has been made for estimating the covariance matrix from a high-dimensional factor model. However, the blessing of dimensionality has not yet been fully embraced in the literature: much of the available data are often ignored in constructing covariance matrix estimates. If our goal is to accurately estimate a covariance matrix of a set of targeted variables, shall we employ additional data, which are beyond the variables of interest, in the estimation? In this article, we provide sufficient conditions for an affirmative answer, and further quantify its gain in terms of Fisher information and convergence rate. In fact, even an oracle-like result (as if all the factors were known) can be achieved when a sufficiently large number of variables is used. The idea of using data as much as possible

We present a closed form expression for the semiclassical OPE coefficients that are universal for all 2D CFTs with a “weak” light spectrum, by taking the semiclassical limit of the fusion kernel. We match this with a properly regularized and normalized bulk action evaluated on a geometry with three conical defects, analytically continued in the deficit angles beyond the range for which a metric with positive signature exists. The analytically continued geometry has a codimension-one coordinate singularity surrounding the heaviest conical defect. This singularity becomes a horizon after Wick rotating to Lorentzian signature, suggesting a connection between universality and the existence of a horizon.

In this paper, we solve a problem of Kobayashi posed in (Complex Finsler vector bundles, American Mathematical Society, Providence, 1996) by introducing a Donaldson type functional on the space F + ( E ) of strongly pseudo-convex complex Finsler metrics on <i>E</i>a holomorphic vector bundle over a closed Khler manifold <i>M</i>. This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has FinslerEinstein metrics on <i>E</i> as its only critical points, at which this functional attains the absolute minimum.