Universal Bounds in Even-Spin CFTs

We prove using invariance under the modular $S$- and $ST$-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges $c,\tilde{c}>1$) contains a primary operator with dimension $\Delta_1$ satisfying $0 < \Delta_1 < (c+\tilde{c})/24 + 0.09280...$ After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS$_3$/CFT$_2$ dictionary, the bound on $\Delta_1$ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant $\Lambda < 0$ can be no heavier than $1/(8G_N)+O(\sqrt{-\Lambda})$; the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.Comment: arXiv admin note: text overlap with arXiv:0902.2790 by other authors; author note: this work is an extension of arXiv:0902.2790, please refer to it for additional details..new version has corrected typos and reference

Bounds on Operator Dimensions in 2D Conformal Field Theories

We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on the conformal dimension $\Delta_2$ of the next-to-lowest nontrival primary operator in unitary two-dimensional conformal field theories without chiral primary operators. The bound we find is of the same form as found for $\Delta_1$: $\Delta_2 \leq c_{tot}/12 + O(1)$. We find a similar bound on the conformal dimension $\Delta_3$, and present a method for deriving bounds on $\Delta_n$ for any $n$, under slightly modified assumptions. For asymptotically large $c_{tot}$ and fixed $n$, we show that $\Delta_n \leq \frac{c_{tot}}{12}+O(1)$. We conclude with a brief discussion of the gravitational implications of these results.Comment: Corrected typos; revised arguments (adding detail) for clarity, results unchange

UNIVERSAL CONSTRAINTS ON 2D CFTS AND 3D GRAVITY

We study constraints imposed on a general unitary two-dimensional conformal field theory by modular invariance. We begin with a review of previous bounds on the conformal dimension Delta1 of the lowest primary operator assuming unitarity, a discrete spectrum, modular invariance, cL, cR \u3e 1, and no extended chiral algebra. We then obtain bounds on the conformal dimensions Delta2, Delta3 using no additional assumptions. We also show that in order to find a bound for Delta4 or higher Deltan, we need to assume a larger minimum value for ctot that grows logarithmically with n. We next extend the previous results to remove the requirement that our two-dimensional conformal field theories have no extended chiral algebra. We then show that modular invariance also implies an upper bound on the total number of states of positive energy less than ctot=24 (or equivalently, states of conformal dimension between ctot=24 and ctot=12), in terms of the number of negative energy states. Finally, we consider the case where the CFT has a gravitational dual and investigate the gravitational interpretation of our results. Using the AdS3/CFT2 correspondence, we obtain an upper bound on the lightest few massive excitations (both with and without the constraint of no chiral primary operators) in a theory of 3D matter and gravity with Lambda \u3c 0. We show our results are consistent with facts and expectations about the spectrum of BTZ black holes in 2+1 gravity. We then discuss the upper and lower bounds on number of states and primary operators in the dual gravitational theory, focusing on the case of AdS3 pure gravity

Analysis of Factors Affecting Farmers’ Willingness to Adopt Switchgrass Production

In the United States, biomass is the largest source of renewable energy accounting for over 3 percent of the energy consumed domestically and is currently the only source for liquid, renewable, transportation fuels. Continued development of biomass as a renewable energy source is being driven in large part by the Energy Independence and Security Act of 2007, which mandates that by 2022 at least 36 billion gallons of fuel ethanol be produced, with at least 16 billion gallons being derived from cellulose, hemi-cellulose, or lignin. However, the market for cellulosic biofuels is still under development. As such, little is known about producer response to feedstock prices paid for dedicated energy crops. While there have been some studies done on factors that determine farmers’ willingness to produce switchgrass, these have been very regional in nature. This study will provide information regarding potential switchgrass adoption by agricultural producers in twelve southeastern states. The objectives of this research are 1) to determine the likelihood of farmers growing switchgrass as a biomass feedstock and the acres they would be willing to devote to switchgrass production and 2) to evaluate some of the factors that are likely to influence these decisions, including the price of switchgrass.Switchgrass, Farmer Adoption, Crop Production/Industries, Research and Development/Tech Change/Emerging Technologies, Resource /Energy Economics and Policy, Q12, Q16,