The N=1\mathcal{N}=1 algebra W∞[μ]\mathcal{W}_\infty[\mu] and its truncations

The main objective of this work is to construct and classify the most general classical and quantum $\mathcal{N}=1$ $\mathcal{W}_\infty$-algebras generated by the same spins as the singlet algebra of $N$ fermions and $N$ bosons in the vector representation of $O(N)$ in the $N\to\infty$ limit. This type of algebras appears in a recent $\mathcal{N}=1$ version of the minimal model holography. Our analysis strongly suggests that there is a one parameter family $\mathcal{W}_\infty[\mu]$ of such algebras at every given central charge. Relying on this assumption, we identify various truncations of $\mathcal{W}_\infty[\mu]$ with, on the one hand, (orbifolds of) the Drinfel'd-Sokolov reductions of the Lie superalgebras $B(n,n)$, $B(n-1,n)$, $D(n,n)$ and $D(n+1,n)$, and, on the other hand, (orbifolds of) three $\mathcal{N}=1$ cosets. After a closer inspection we show that these cosets can be realized as a Drinfel'd-Sokolov reduction of $B(n,n)$, $D(n,n)$ and $D(n+1,n)$. We then discuss the implications of our findings for the quantum version of the $\mathcal{N}=1$ minimal model holography.Comment: old section 5.4 removed, 46 page

Extended supersymmetry in AdS_3 higher spin theories

We determine the asymptotic symmetry algebra (for fields of low spin) of the $M\times M$ matrix extended Vasiliev theories on AdS$_3$ and find that it agrees with the $\mathcal{W}$-algebra of their proposed coset duals. Previously it was noticed that for $M=2$ the supersymmetry increases from $\mathcal{N}=2$ to $\mathcal{N}=4$. We study more systematically this type of supersymmetry enhancements and find that, although the higher spin algebra has extended supersymmetry for all $M\geq 2$, the corresponding asymptotic symmetry algebra fails to be superconformal except for $M=2$, when it has large $\mathcal{N}=4$ superconformal symmetry. Moreover, we find that the Vasiliev theories based on $shs^E\! \left( \mathcal{N} \vert 2, \mathbb{R} \right)$ are special cases of the matrix extended higher spin theories, and hence have the same supersymmetry properties.Comment: 23 page

Even spin minimal model holography

The even spin W^e_\infty algebra that is generated by the stress energy tensor together with one Virasoro primary field for every even spin s \geq 4 is analysed systematically by studying the constraints coming from the Jacobi identities. It is found that the algebra is characterised, in addition to the central charge, by one free parameter that can be identified with the self-coupling constant of the spin 4 field. We show that W^e_\infty can be thought of as the quantisation of the asymptotic symmetry algebra of the even higher spin theory on AdS_3. On the other hand, W^e_\infty is also quantum equivalent to the so(N) coset algebras, and thus our result establishes an important aspect of the even spin minimal model holography conjecture. The quantum equivalence holds actually at finite central charge, and hence opens the way towards understanding the duality beyond the leading 't Hooft limit.Comment: 32 pages, v2: reference added, minor changes in tex

Edge states and conformal boundary conditions in super spin chains and super sigma models

The sigma models on projective superspaces CP^{N+M-1|N} with topological angle theta=pi mod 2pi flow to non-unitary, logarithmic conformal field theories in the low-energy limit. In this paper, we determine the exact spectrum of these theories for all open boundary conditions preserving the full global symmetry of the model, generalizing recent work on the particular case M=0 [C. Candu et al, JHEP02(2010)015]. In the sigma model setting, these boundary conditions are associated with complex line bundles, and are labelled by an integer, related with the exact value of theta. Our approach relies on a spin chain regularization, where the boundary conditions now correspond to the introduction of additional edge states. The exact values of the exponents then follow from a lengthy algebraic analysis, a reformulation of the spin chain in terms of crossing and non-crossing loops (represented as a certain subalgebra of the Brauer algebra), and earlier results on the so-called one- and two-boundary Temperley Lieb algebras (also known as blob algebras). A remarkable result is that the exponents, in general, turn out to be irrational. The case M=1 has direct applications to the spin quantum Hall effect, which will be discussed in a sequel.Comment: 50 pages, 18 figure

Yangian Superalgebras in Conformal Field Theory

Quantum Yangian symmetry in several sigma models with supergroup or supercoset as target is established. Starting with a two-dimensional conformal field theory that has current symmetry of a Lie superalgebra with vanishing Killing form we construct non-local charges and compute their properties. Yangian axioms are satisfied, except that the Serre relations only hold for a subsector of the space of fields. Yangian symmetry implies that correlation functions of fields in this sector satisfy Ward identities. We then show that this symmetry is preserved by certain perturbations of the conformal field theory. The main example are sigma models of the supergroups PSL(N|N), OSP(2N+2|2N) and D(2,1;\alpha) away from the WZW point. Further there are the OSP(2N+2|2N) Gross-Neveu models and current-current perturbations of ghost systems, both for the disc as world-sheet. The latter we show to be equivalent to CP^{N-1|N} sigma models, while the former are conjecturally dual to supersphere sigma models

On the coset duals of extended higher spin theories

We study the holographic duality between the M x M matrix extension of Vasiliev higher spin theories on AdS3 and the large N limit of SU(N+M)/SU(N) x U(1) type cosets. We present a simplified proof for the agreement of the spectra and clarify the relation between this duality and the version in which the cosets are replaced by Kazama-Suzuki models of Grassmannian type.Comment: 27 pages, 1 tabl

Smart Specialisation, seizing new industrial opportunities

This study offers a novel analytical approach to inform the regional search for new industrial opportunities, as promoted by smart specialisation in the EU Cohesion policy context. The analysis departs from the challenges of practicing smart specialisation and its entrepreneurial discovery process in a dynamic perspective. It argues that the adoption of a dynamic approach to identify new opportunities implies mapping regional business and innovation assets as well as, assessing their position within the global technological and industrial landscape. The study brings a case study of Lombardy region, spurring the S3 Lab initiative (in collaboration with Baden-Württemberg, Catalonia and Lapland), together with a comparative analysis of its technological profile. The empirical study combines patent data from OECD REGPAT and territorial proprietary micro-data from Lombardy region on firm creation in emerging industries (EI) – new industrial sectors or existing sectors evolving into new industries (European Cluster Observatory). These industries represent a priority area for Lombardy's innovation-led development strategy. The initial observations confirm the importance of such industries in the region; they represent more than one-third of employment, almost a half of the regional value-added and feature together the majority of start-ups, suggesting the relevance of the regional strategic development choices. Also, in terms of productive advantages, Lombardy ranks high in some key EI. The mapping of technological competences through patent indicators, e.g. specialisation, diversification and ability to specialise in fast-growing and niche fields gives relevant insights on the technological potential of the region, providing further guidance for better targeted interventions.JRC.B.3-Territorial Developmen

ANALISIS CURRENT RATIO DAN QUICK RATIO UNTUK MENILAI KINERJA KEUANGAN PADA PT BANK RAKYAT INDONESIA (PERSERO) TBK

Tujuan penelitian ini dilakukan adalah untuk menganalisa current ratio dan quick ratio pada PT Bank Rakyat Indonesia (Persero) Tbk, dengan metode analisis deskriptif kuantitatif dan alat analisis ratio dari annual report PT Bank Rakyat Indonesia (Persero) Tbk periode 2017-2021. Hasil penelitian menunjukkan bahwa current ratio periode 2017-2022 belum dikatakan baik karena belum memenuhi standar Bank Indonesia, hal ini disebabkan karena terjadinya penurunan nilai aset produktif dan penghapusan kredit. Sedangkan quick ratio pada periode 2017-2021 mengalami fluktuasi, walaupun current ratio rendah tapi quick ratio tinggi menunjukkan tidak adanya investasi besar dalam kas, bank, efek maupun piutang sehingga rasio antara kas, dan efek yang segera bisa direalisasi bila diperlukan dengan total utang jangka pendek, sehingga dapat disimpulkan bahwa semakin tinggi quick ratio, maka semakin besar pula kemampuan bank dalam memenuhi kewajiban terhadap para deposan (likuid) karena telah memenuhi standar Bank Indonesia (BI) sebesar 4,05%.   &nbsp