Takiff superalgebras and Conformal Field Theory

A class of non-semisimple extensions of Lie superalgebras is studied. They are obtained by adjoining to the superalgebra its adjoint representation as an abelian ideal. When the superalgebra is of affine Kac-Moody type, a generalisation of Sugawara's construction is shown to give rise to a copy of the Virasoro algebra and so, presumably, to a conformal field theory. Evidence for this is detailed for the extension of the affinisation of the superalgebra gl(1|1): Its highest weight irreducible modules are classified using spectral flow, the irreducible supercharacters are computed and a continuum version of the Verlinde formula is verified to give non-negative integer structure coefficients. Interpreting these coefficients as those of the Grothendieck ring of fusion, partial results on the true fusion ring and its indecomposable structures are deduced.Comment: 25 page

Logarithmic Conformal Field Theory: Beyond an Introduction

This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW model on the Lie supergroup GL(1|1). It concludes with a general discussion of the so-called staggered modules that give these theories their logarithmic structure, before outlining a proposed strategy to understand more general logarithmic conformal field theories. Throughout, the emphasis is on continuum methods and their generalisation from the familiar rational case. In particular, the modular properties of the characters of the spectrum play a central role and Verlinde formulae are evaluated with the results compared to the known fusion rules. Moreover, bulk modular invariants are constructed, the structures of the corresponding bulk state spaces are elucidated, and a formalism for computing correlation functions is discussed.Comment: Invited review by J Phys A for a special issue on LCFT; v2 updated references; v3 fixed a few minor typo

From Jack polynomials to minimal model spectra

In this note, a deep connection between free field realisations of conformal field theories and symmetric polynomials is presented. We give a brief introduction into the necessary prerequisites of both free field realisations and symmetric polynomials, in particular Jack symmetric polynomials. Then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods. While these results on the representation theory of the minimal models are all known, this note exploits the full power of Jack polynomials to present significant simplifications of the original proofs in the literature.Comment: 14 pages, corrected typos and added comment on connections to the AGT conjecture in introduction, version to appear in J. Phys.

The Verlinde formula in logarithmic CFT

In rational conformal field theory, the Verlinde formula computes the fusion coefficients from the modular S-transformations of the characters of the chiral algebra's representations. Generalising this formula to logarithmic models has proven rather difficult for a variety of reasons. Here, a recently proposed formalism (arXiv:1303.0847 [hep-th]) for the modular properties of certain classes of logarithmic theories is reviewed, and refined, using simple examples. A formalism addressing fusion rules in simple current extensions is also reviewed as a means to tackle logarithmic theories to which the proposed modular formalism does not directly apply.Comment: 12 pages, proceedings article for the 30th ICGTMP (Ghent, 2014); v2 fixed an erroneous statement pointed out by Antun Milas; v3 made a few minor clarifications to discussion and added a couple of ref

Non-Chiral Logarithmic Couplings for the Virasoro Algebra

This Letter initiates the study of what we call non-chiral staggered Virasoro modules, indecomposable modules on which two copies of the Virasoro algebra act with the two zero-modes acting non-semisimply. This is motivated by the "puzzle" recently reported in arXiv:1110.1327 [math-ph] involving a non-standard measured value, meaning that the value is not familiar from chiral studies, for the "b-parameter" (logarithmic coupling) of a c=0 bulk conformal field theory. Here, an explanation is proposed by introducing a natural family of bulk modules and showing that the only consistent, non-standard logarithmic coupling that is distinguished through structure is that which was measured. This observation is shown to persist for general central charges and a conjecture is made for the values of certain non-chiral logarithmic couplings.Comment: 10 pages; v2: 11 pages, some modifications to introduction, added conclusions and reference

The Overlap Package

Camera traps - cameras linked to detectors so that they fire when an animal is present - are a major source of information on the abundance and habitat preferences of rare or shy forest animals. Modern cameras record the time of the photo, and the use of this to investigate diel activity patterns was immediately recognised (Gri?ffiths and van Schaik, 1993). Initially this resulted in broad classfication of taxa as diurnal, nocturnal, crepuscular, or cathemeral (van Schaik and Gri?ths, 1996). More recently, researchers have compared activity patterns among species to see how overlapping patterns may relate to competition or predation (Linkie and Ridout, 2011; Carver et al., 2011; Ramesh et al., 2012; Carter et al., 2012; Kamler et al., 2012; Ross et al., 2013). Ridout and Linkie (2009) presented methods to fit kernel density functions to times of observations of animals and to estimate the coe?cient of overlapping, a quantitative measure ranging from 0 (no overlap) to 1 (identical activity patterns). The code they used forms the basis of the overlap package. Although motivated by the analysis of camera trap data, overlap could be applied to data from other sources such as data loggers, provided data collection is carried out around the clock. Nor is it limited to diel cycles: tidal cycles or seasonal cycles, such as plant flowering or fruiting or animal breeding seasons could also be investigated

Controls on ERS altimeter measurements over ice sheets: Footprint-scale topography, backscatter fluctuations, and the dependence of microwave penetration depth on satellite orientation

We consider the reliability of radar altimeter measurements of ice sheet elevation and snowpack properties in the presence of surface undulations. We demonstrate that over ice sheets the common practice of averaging echoes by aligning the first return from the surface at the origin can result in a redistribution of power to later times in the average echo, mimicking the effects of microwave penetration into the snowpack. Algorithms that assume the topography affects the radar echo shape in the same way that waves affect altimeter echoes over the ocean will therefore lead to biased estimates of elevation. This assumption will also cause errors in the retrieval of echo-shape parameters intended to quantify the penetration of the microwave pulse into the snowpack. Using numerical simulations, we estimate the errors in retrievals of extinction coefficient, surface backscatter, and volume backscatter for various undulating topographies. In the flatter portions of the Antarctic plateau, useful estimates of these parameters may be recovered by averaging altimeter echoes recorded by the European Remote Sensing satellite (ERS-1). By numerical deconvolution of the average echoes we resolve the depths in the snowpack at which temporal changes and satellite travel-direction effects occur, both of which have the potential to corrupt measurements of ice sheet elevation change. The temporal changes are isolated in the surface-backscatter cross section, while directional effects are confined to the extinction coefficient and are stable from year to year. This allows the removal of the directional effect from measurement of ice-sheet elevation change