Higher Spin Square Research Articles

Symmetry algebras of stringy cosets

We find the symmetry algebras of cosets which are generalizations of the minimal-model cosets, of the specific form frac{mathrm{SU}{(N)}_ktimes mathrm{SU}{(N)}_{mathrm{ell}}}{mathrm{SU}{(N)}_{k+mathrm{ell}}} . We study this coset in its free field limit, with k, ℓ → ∞, where it reduces to a theory of free bosons. We show that, in this limit and at large N, the algebra {mathcal{W}}_{infty}^eleft[1right] emerges as a sub-algebra of the coset algebra. The full coset algebra is a larger algebra than conventional mathcal{W} -algebras, with the number of generators rising exponentially with the spin, characteristic of a stringy growth of states. We compare the coset algebra to the symmetry algebra of the large N symmetric product orbifold CFT, which is known to have a stringy symmetry algebra labelled the ‘higher spin square’. We propose that the higher spin square is a sub-algebra of the symmetry algebra of our stringy coset.

On tensionless string field theory in AdS3

We report on progress in formulating a field theory of tensionless strings in AdS3, starting from the dual large-N symmetric orbifold CFT. We propose a set of field equations which are gauge invariant under the higher spin algebra of the theory, the ‘Higher Spin Square’. The massless higher spin sector is captured by a Chern-Simons gauge field, while the matter sector is described by unfolded equations similar to those appearing in Vasiliev theory. Our equations incorporate the full perturbative spectrum of the theory, including states coming from the twisted sectors, and capture some of the interactions fixed by gauge invariance. We also discuss the spectrum of the bulk theory and explain how linearization around AdS3 gives rise to the expected set of decoupled wave equations. Our results can be generalized to describe bulk duals of other large-N symmetric orbifolds.

The higher spin rectangle

The chiral algebra of the symmetric product orbifold of a single-boson CFT corresponds to a “higher spin square” algebra in the large N limit. In this note, we show that a symmetrized collection of N bosons defines a similar structure that we refer to as the higher spin rectangle algebra. We explore the relation of this algebra to the higher spin square algebra. The existence of such a truncated algebra hints at bulk theories interpolating between Vasiliev higher spin theory and string theory.

A high-spin square planar iron(ii)-siloxide and its tetrahedral allogon – structural and spectroscopic models of Fe-zeolite sites

The famous α-Fe active sites in Fe-zeolites have recently been revealed to correspond to mononuclear high-spin iron(ii) centres in square planar coordination environments. Here we report a first iron siloxide complex which represents a faithful structural and spectroscopic model of such sites. Notably, also an allogon with a distorted structure exists and could be crystallised.

String theory as a higher spin theory

The symmetries of string theory on ${\rm AdS}_3 \times {\rm S}^3 \times \mathbb{T}^4$ at the dual of the symmetric product orbifold point are described by a so-called Higher Spin Square (HSS). We show that the massive string spectrum in this background organises itself in terms of representations of this HSS, just as the matter in a conventional higher spin theory does so in terms of representations of the higher spin algebra. In particular, the entire untwisted sector of the orbifold can be viewed as the Fock space built out of the multiparticle states of a single representation of the HSS, the so-called `minimal' representation. The states in the twisted sector can be described in terms of tensor products of a novel family of representations that are somewhat larger than the minimal one.

On matter coupled to the higher spin square

Gaberdiel and Gopakumar recently proposed that the tensionless limit of string theory on takes the form of a higher spin theory with a gauge algebra that is referred to as the higher spin square (hss). In this note, we formulate the linearized Vasiliev-type equations which describe a matter field coupled to the hss. We study the particle spectrum of this field and show that it accounts for the entire untwisted sector of the dual symmetric orbifold CFT, thereby confirming a conjecture by Gaberdiel and Gopakumar. In doing so, we pinpoint the group-theoretic data which determine the spectrum of a matter field coupled to a general higher spin algebra, which we illustrate by revisiting the theory based on the algebra.

Stringy symmetries and the higher spin square

Tensionless string theory on , as captured by a free symmetric product orbifold, has a large set of conserved currents which can be usefully organized in terms of representations of a supersymmetric higher spin algebra. In this paper we focus on the single particle currents which generate the asymptotic stringy symmetry algebra on , and whose wedge modes describe the unbroken gauge symmetries of string theory in this background. We show that this global subalgebra contains two distinct higher spin algebras that generate the full algebra as a ‘higher spin square’. The resulting unbroken stringy symmetry algebra is exponentially larger than the two individual higher spin algebras.

Synthesis and characterisation of cobalt(II), cobalt(III) and rhodium(III) complexes ofN, N?, N?, N?-tetra(2-cyanoethyl) cyclam [TCEC] andN, N?, N?, N?-tetra)3-aminopropyl) cyclam [TAPC

Complexes of cobalt(II), cobalt(III) and rhodium(III) with TCEC and TAPC have been synthesised. TCEC with cobalt(II) gave [Co(TCEC)Br]Br and [Co(TCEC)Cl]Cl, five coordinate high spin square pyramid complexes, but the corresponding cobalt(III) complex could not be characterised. Rhodium(III) gave a six coordinate [Rh(TCEC)Cl2]Cl complex, in which the two coordinated chlorides have acis-geometry and the four pendant arms lie on one side of the N4 plane with none of the —C≡N groups coordinated TAPC on the other hand gives the cobalt(III) complex, [Co(TAPC)Br]Br2, in which one of the amino groups of the four pendant arms is coordinated to cobalt. Rhodium(III) with TAPC gave [Rh(TAPC)Cl]Cl2 in which one axial site is occupied by the amino group of one of the pendant arms and the other by Cl−.

Complexes of copper(II), nickel(II) and cobalt(II) with 2-fluoro,5-nitroaniline and 4-fluoro,2-nitroaniline

Preparation of new complexes of 2 - fluoro,5 - nitroaniline (2-F,5-NA) and 4 - fluoro,2 - nitroaniline (4-F,2-NA) of the type ML n X 2· mH 2O (where M = Cu(II), Ni(II), Co(II); L = 2-F,5-NA; 4-F,2-NA; n = 2,4,6; X = Cl −, Br −, SCN −, ClO 4 −; m = 0–2) are described and their i.r., electronic spectra, and magnetic moments are reported. The ligands act always as monodentate O-bonded. The compounds are generally high spin, non electrolytes, and present square planar, tetrahedral and hexacoordinate structures.

New high-spin square pyramidal and octahedral complexes of nickel(II)

New high-spin square pyramidal and octahedral complexes of nickel(II)

Mössbauer Spectroscopy of Some Iron Porphyrins

THE Mossbauer parameters of various iron porphyrin complexes have been investigated as model systems for haem proteins1,2. Topics which require further consideration are the effect of porphyrin structure, basicity and esterification on the Mossbauer spectra. It has been shown that the basicity towards protons of meso and diacetyl deuterohaemins differs by a factor of one hundred whereas their isomer shifts (δ0) and quadrupole splittings (ΔE) were insensitive to this change2. We have studied the water soluble N-methyl substituted meso tetrapyridyl porphine (unpublished results of Fleischer and Hambright), which seems to be the most acidic porphyrin known. The (pK3+pK4) of this compound is the same as the unmethylated derivative3, whereas pK2=12.9. No other porphyrin has a measurable pK2 in aqueous solution. Table 1 shows that the isomer shift of this structurally different haemin compound and protohaemin chloride are almost identical This indicates4 in both a 5 per cent 4s electron contribution to the d5 configuration of Fe(III), and that regardless of structure or basicity, the metal–ligand bonding in these high spin square pyramid chelates is primarily ionic. It is also noted that low spin octahedral hemichromes have approximately the same sigma electron density1 at the iron nucleus (δ0≅ 0.041 cm/sec) as the high spin haemin chlorides, whereas ΔE of the low spin derivatives is, as expected, larger (ΔE≅0.21 cm/sec) than that of the high spin compounds. This unexplained similarity is novel in that typical inorganic low spin complexes have more sigma density than high spin varieties4, presumably because of their increased pi bonding capacities. Thus the similarity in δ0 values might indicate a limited role for pi bonding in low spin ferric porphyrin systems.