Just Another Gibbs Additive Modeller: Interfacing JAGS and mgcv

The BUGS language offers a very flexible way of specifying complex statistical models for the purposes of Gibbs sampling, while its JAGS variant offers very convenient R integration via the rjags package. However, including smoothers in JAGS models can involve some quite tedious coding, especially for multivariate or adaptive smoothers. Further, if an additive smooth structure is required then some care is needed, in order to centre smooths appropriately, and to find appropriate starting values. R package mgcv implements a wide range of smoothers, all in a manner appropriate for inclusion in JAGS code, and automates centring and other smooth setup tasks. The purpose of this note is to describe an interface between mgcv and JAGS, based around an R function, `jagam', which takes a generalized additive model (GAM) as specified in mgcv and automatically generates the JAGS model code and data required for inference about the model via Gibbs sampling. Although the auto-generated JAGS code can be run as is, the expectation is that the user would wish to modify it in order to add complex stochastic model components readily specified in JAGS. A simple interface is also provided for visualisation and further inference about the estimated smooth components using standard mgcv functionality. The methods described here will be un-necessarily inefficient if all that is required is fully Bayesian inference about a standard GAM, rather than the full flexibility of JAGS. In that case the BayesX package would be more efficient.Comment: Submitted to the Journal of Statistical Softwar

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.

Modular Transformations and Verlinde Formulae for Logarithmic (p+,p−)(p_+,p_-)-Models

The $(p_+,p_-)$ singlet algebra is a vertex operator algebra that is strongly generated by a Virasoro field of central charge $1-6(p_+-p_-)^2/p_+p_-$ and a single Virasoro primary field of conformal weight $(2p_+-1)(2p_--1)$. Here, the modular properties of the characters of the uncountably many simple modules of each singlet algebra are investigated and the results used as the input to a continuous analogue of the Verlinde formula to obtain the "fusion rules" of the singlet modules. The effect of the failure of fusion to be exact in general is studied at the level of Verlinde products and the rules derived are lifted to the $(p_+,p_-)$ triplet algebras by regarding these algebras as simple current extensions of their singlet cousins. The result is a relatively effortless derivation of the triplet "fusion rules" that agrees with those previously proposed in the literature.Comment: 22 pages, v2 minor changes; added ref

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

On the extended W-algebra of type sl_2 at positive rational level

The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu's c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra) and classify all simple M_{p_+,p_-}-modules.Comment: 64 pages, 2 figures; version to appear in journal, Int Math Res Notices (2014

Bosonic Ghosts at c=2c=2 as a Logarithmic CFT

Motivated by Wakimoto free field realisations, the bosonic ghost system of central charge $c=2$ is studied using a recently proposed formalism for logarithmic conformal field theories. This formalism addresses the modular properties of the theory with the aim being to determine the (Grothendieck) fusion coefficients from a variant of the Verlinde formula. The key insight, in the case of bosonic ghosts, is to introduce a family of parabolic Verma modules which dominate the spectrum of the theory. The results include S-transformation formulae for characters, non-negative integer Verlinde coefficients, and a family of modular invariant partition functions. The logarithmic nature of the corresponding ghost theories is explicitly verified using the Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 17 pages, one figure; v2: added refs and rewrote a little of the parabolic subalgebra discussion in Sec. 3 (no change to results); v3: added a sketch proof of Prop. 1, several clarifications and a few more refs (again, no change to results

Relaxed singular vectors, Jack symmetric functions and fractional level sl^(2)\widehat{\mathfrak{sl}}(2) models

The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels $k \in \mathbb{Q}$. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level $\widehat{\mathfrak{sl}}(2)$ models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.Comment: 33 pages; v2: corrected typos and added reference

Singular vectors for the WNW_N algebras

In this paper, we use free field realisations of the A-type principal, or Casimir, $W_N$ algebras to derive explicit formulae for singular vectors in Fock modules. These singular vectors are constructed by applying screening operators to Fock module highest weight vectors. The action of the screening operators is then explicitly evaluated in terms of Jack symmetric functions and their skew analogues. The resulting formulae depend on sequences of pairs of integers that completely determine the Fock module as well as the Jack symmetric functions.Comment: 18 page

Coset Constructions of Logarithmic (1,p)-Models

One of the best understood families of logarithmic conformal field theories is that consisting of the (1,p) models (p = 2, 3, ...) of central charge c_{1,p} = 1 - 6 (p-1)^2 / p. This family includes the theories corresponding to the singlet algebras M(p) and the triplet algebras W(p), as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realized through a coset construction. The W^(2)_n algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of affine sl(n)_k, generalising the Bershadsky-Polyakov algebra W^(2)_3. Inspired by work of Adamovic for p=3, vertex algebras B_p are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p <= 5, the algebra B_p is a homomorphic image of W^(2)_{p-1} at level -(p-1)^2 / p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p>5 as well. The triplet algebra W(p) is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra M(p) is similarly realised inside B_p. As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p=2 and 3.Comment: 22 pages, v2: clarified our definition of Feigin-Semikhatov algebras and added ref