Generating-function method for fusion rules

This is the second of two articles devoted to an exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper focuses on fusion rules, using the machinery developed for tensor products in the companion article. Although the Kac-Walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor-product generating function, we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor-product elementary couplings. A set of inequalities involving the level are derived from this set using Farkas' lemma. These inequalities, taken in conjunction with the inequalities defining the tensor products, define what we call the fusion basis. Given this basis, the machinery of our previous paper may be applied to construct the fusion generating function. New generating functions for sp(4) and su(4), together with a closed form expression for their threshold levels are presented.Comment: Harvmac (b mode : 47 p) and Pictex; to appear in J. Math. Phy

Generating-function method for tensor products

This is the first of two articles devoted to a exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product (infinite-level) limit of fusions rules. We start by reviewing Sharp's character method. An alternative approach to the construction of tensor-product generating functions is then presented which overcomes most of the technical difficulties associated with the character method. It is based on the reformulation of the problem of calculating tensor products in terms of the solution of a set of linear and homogeneous Diophantine equations whose elementary solutions represent ``elementary couplings''. Grobner bases provide a tool for generating the complete set of relations between elementary couplings and, most importantly, as an algorithm for specifying a complete, compatible set of ``forbidden couplings''.Comment: Harvmac (b mode : 39 p) and Pictex; this is a substantially reduced version of hep-th/9811113 (with new title); to appear in J. Math. Phy

Joint constraints on reionization: a framework for combining the global 21cm signal and the kinetic Sunyaev-Zel'dovich effect

Recent measurements from the CMB and from high-redshift galaxy observations have placed rough constraints on the midpoint and duration of the Epoch of Reionization. Detailed measurements of the ionization history remain elusive, although two proposed probes show great promise for this purpose: the 21cm global signal and the kinetic Sunyaev-Zel'dovich (kSZ) effect. We formally confirm the common assumption that these two probes are highly complementary, with the kSZ being more sensitive to extended ionization histories and the global signal to rapidly evolving ones. We do so by performing a Karhunen-Lo\`{e}ve (KL) transformation, which casts the data in a basis designed to emphasize the information content of each probe. We find that reconstructing the ionization history using both probes gives significantly more precise results than individual constraints, although carefully chosen, physically motivated priors play a crucial part in obtaining a bias-free reconstruction. Additionally, in the KL basis, measurements from one probe can be used to detect the presence of residual systematics in the other, providing a safeguard against systematics that would go undetected when data from each probe is analyzed in isolation. Once detected, the modes contaminated by systematics can be discarded from the data analysis to avoid biases in reconstruction

Affine su(3) and su(4) fusion multiplicities as polytope volumes

Affine su(3) and su(4) fusion multiplicities are characterised as discretised volumes of certain convex polytopes. The volumes are measured explicitly, resulting in multiple sum formulas. These are the first polytope-volume formulas for higher-rank fusion multiplicities. The associated threshold levels are also discussed. For any simple Lie algebra we derive an upper bound on the threshold levels using a refined version of the Gepner-Witten depth rule.Comment: 16 pages, LaTe

Fermionic characters for graded parafermions

Fermionic-type character formulae are presented for charged irreduciblemodules of the graded parafermionic conformal field theory associated to the coset $osp(1,2)_k/u(1)$. This is obtained by counting the weakly ordered `partitions' subject to the graded $Z_k$ exclusion principle. The bosonic form of the characters is also presented.Comment: 24 p. This corrects typos (present even in the published version) in eqs (4.4), (5.23), (5.24) and (C.4

Automorphisms of the affine SU(3) fusion rules

We classify the automorphisms of the (chiral) level-k affine SU(3) fusion rules, for any value of k, by looking for all permutations that commute with the modular matrices S and T. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. When k is divisible by 3, the automorphism group (Z_2) is generated by the charge conjugation C. If k is not divisible by 3, the automorphism group (Z_2 x Z_2) is generated by C and the Altsch\"uler--Lacki--Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here for SU(3) can be applied to other algebras.Comment: 21 pages, plain TeX, DIAS-STP-92-4

Berenstein-Zelevinsky triangles, elementary couplings and fusion rules

We present a general scheme for describing su(N)_k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way a complete solution of su(4)_k fusion rules is obtained.Comment: 14 page

Extended radio emission in the galaxy cluster MS 0735.6+7421 detected with the Karl G. Jansky Very Large Array

MS 0735.6+7421 ($z = 0.216$) is a massive cool core galaxy cluster hosting one of the most powerful active galactic nuclei (AGN) outbursts known. The radio jets of the AGN have carved out an unusually large pair of X-ray cavities, each reaching a diameter of $200$ kpc. This makes MS 0735.6+7421 a unique case to investigate active galactic nuclei feedback processes, as well as other cluster astrophysics at radio wavelengths. We present new low-radio-frequency observations of MS 0735.6+7421 taken with the Karl G. Jansky Very Large Array (VLA): 5 hours of P-band ($224-480$ MHz) and 5 hours of L-band ($1-2$ GHz) observations, both in C configuration. Our VLA P-band ($224-480$ MHz) observations reveal the presence of a new diffuse radio component reaching a scale of $\sim$ $900$ kpc in the direction of the jets and of $\sim$ $500$ kpc in the direction perpendicular to the jets. This component is centered on the cluster core and has a radio power scaled at $1.4$ GHz of $P_{1.4\text{ GHz}} = (4\pm2)\times 10^{24}$ WHz$^{-1}$. Its properties are consistent with those expected from a radio mini-halo as seen in other massive cool core clusters, although it may also be associated with radio plasma that has diffused out of the X-ray cavities. Observations at higher spatial resolution are needed to fully characterize the properties and nature of this component. We also suggest that if radio mini-halos originate from jetted activity, we may be witnessing the early stages of this process.Comment: 11 pages, 7 figures, submitted to MNRA