Non-perturbative volume-reduction of large-N QCD with adjoint fermions

We use nonperturbative lattice techniques to study the volume-reduced "Eguchi-Kawai" version of four-dimensional large-N QCD with a single adjoint Dirac fermion. We explore the phase diagram of this single-site theory in the space of quark mass and gauge coupling using Wilson fermions for a number of colors in the range 8 <= N <= 15. Our evidence suggests that these values of N are large enough to determine the nature of the phase diagram for N-->oo. We identify the region in the parameter space where the (Z_N)^4 center-symmetry is intact. According to previous theoretical work using the orbifolding paradigm, and assuming that translation invariance is not spontaneously broken in the infinite-volume theory, in this region volume reduction holds: the single-site and infinite-volume theories become equivalent when N-->oo. We find strong evidence that this region includes both light and heavy quarks (with masses that are at the cutoff scale), and our results are consistent with this region extending towards the continuum limit. We also compare the action density and the eigenvalue density of the overlap Dirac operator in the fundamental representation with those obtained in large-N pure-gauge theory.Comment: 49 pages, 23 figures. v2: Clarified connection of ZN symmetry realization and the validity of reduction in the abstract, quantified what we mean by "heavy quarks" in abstract, updated discussion on Refs [12,14,15], added a discussion on the kappa dependence of the physical mass, extended discussion on what might happen in the continuum and at N=oo, updated ref'

Secret-Sharing for NP

A computational secret-sharing scheme is a method that enables a dealer, that has a secret, to distribute this secret among a set of parties such that a "qualified" subset of parties can efficiently reconstruct the secret while any "unqualified" subset of parties cannot efficiently learn anything about the secret. The collection of "qualified" subsets is defined by a Boolean function. It has been a major open problem to understand which (monotone) functions can be realized by a computational secret-sharing schemes. Yao suggested a method for secret-sharing for any function that has a polynomial-size monotone circuit (a class which is strictly smaller than the class of monotone functions in P). Around 1990 Rudich raised the possibility of obtaining secret-sharing for all monotone functions in NP: In order to reconstruct the secret a set of parties must be "qualified" and provide a witness attesting to this fact. Recently, Garg et al. (STOC 2013) put forward the concept of witness encryption, where the goal is to encrypt a message relative to a statement "x in L" for a language L in NP such that anyone holding a witness to the statement can decrypt the message, however, if x is not in L, then it is computationally hard to decrypt. Garg et al. showed how to construct several cryptographic primitives from witness encryption and gave a candidate construction. One can show that computational secret-sharing implies witness encryption for the same language. Our main result is the converse: we give a construction of a computational secret-sharing scheme for any monotone function in NP assuming witness encryption for NP and one-way functions. As a consequence we get a completeness theorem for secret-sharing: computational secret-sharing scheme for any single monotone NP-complete function implies a computational secret-sharing scheme for every monotone function in NP

New Dependencies of Hierarchies in Polynomial Optimization

We compare four key hierarchies for solving Constrained Polynomial Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams (SA) hierarchies. We prove a collection of dependencies among these hierarchies both for general CPOPs and for optimization problems on the Boolean hypercube. Key results include for the general case that the SONC and SOS hierarchy are polynomially incomparable, while SDSOS is contained in SONC. A direct consequence is the non-existence of a Putinar-like Positivstellensatz for SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent. Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that provides a O(n) degree bound.Comment: 26 pages, 4 figure

ER Stress-Induced eIF2-alpha Phosphorylation Underlies Sensitivity of Striatal Neurons to Pathogenic Huntingtin

A hallmark of Huntington's disease is the pronounced sensitivity of striatal neurons to polyglutamine-expanded huntingtin expression. Here we show that cultured striatal cells and murine brain striatum have remarkably low levels of phosphorylation of translation initiation factor eIF2 alpha, a stress-induced process that interferes with general protein synthesis and also induces differential translation of pro-apoptotic factors. EIF2 alpha phosphorylation was elevated in a striatal cell line stably expressing pathogenic huntingtin, as well as in brain sections of Huntington's disease model mice. Pathogenic huntingtin caused endoplasmic reticulum (ER) stress and increased eIF2 alpha phosphorylation by increasing the activity of PKR-like ER-localized eIF2 alpha kinase (PERK). Importantly, striatal neurons exhibited special sensitivity to ER stress-inducing agents, which was potentiated by pathogenic huntingtin. We could strongly reduce huntingtin toxicity by inhibiting PERK. Therefore, alteration of protein homeostasis and eIF2 alpha phosphorylation status by pathogenic huntingtin appears to be an important cause of striatal cell death. A dephosphorylated state of eIF2 alpha has been linked to cognition, which suggests that the effect of pathogenic huntingtin might also be a source of the early cognitive impairment seen in patients