Takiff superalgebras and Conformal Field Theory

Abstract

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.

25 pages

References in corpus (8)

• Associative-algebraic approach to logarithmic conformal field theories
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• From Percolation to Logarithmic Conformal Field Theory
• Logarithmic observables in critical percolation
• Branes in the GL(1|1) WZNW-Model
• The GL(1|1)-symplectic fermion correspondence
• Conformal invariance and quantum integrability of sigma models on symmetric superspaces
• Conformal superspace sigma-models

Cited by in corpus (18)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• Schur-Weyl Duality for Heisenberg Cosets
• Modular Data and Verlinde Formulae for Fractional Level WZW Models II
• Assembling integrable sigma-models as affine Gaudin models
• Boundary algebras and Kac modules for logarithmic minimal models
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• An admissible level $\widehat{\mathfrak{osp}}\left(1|2\right)$-model: modular transformations and the Verlinde formula
• Galilean contractions of $W$-algebras
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Modularity of Bershadsky-Polyakov minimal models
• On conformal field theories based on Takiff superalgebras
• Higher-order Galilean contractions
• On a class of conformal $\mathcal{E}$-models and their chiral Poisson algebras
• Multi-graded Galilean conformal algebras
• Asymmetric Galilean conformal algebras
• On the Integrable Structure of the SU(2) Wess-Zumino-Novikov-Witten Model