The GL(1|1) WZW model: From Supergeometry to Logarithmic CFT

Abstract

We present a complete solution of the WZW model on the supergroup GL(1|1). Our analysis begins with a careful study of its minisuperspace limit (``harmonic analysis on the supergroup''). Its spectrum is shown to contain indecomposable representations. This is interpreted as a geometric signal for the appearance of logarithms in the correlators of the full field theory. We then discuss the representation theory of the gl(1|1) current algebra and propose an Ansatz for the state space of the WZW model. The latter is established through an explicit computation of the correlation function. We show in particular, that the 4-point functions of the theory factorize on the proposed set of states and that the model possesses an interesting spectral flow symmetry. The note concludes with some remarks on generalizations to other supergroups.

arXiv comment34 pages

References in corpus (35)

• Hidden Symmetries of the AdS_5 x S^5 Superstring
• Type IIB superstring action in AdS_5 x S^5 background
• Logarithmic Operators in Conformal Field Theory
• Conformal Field Theory of AdS Background with Ramond-Ramond Flux
• Bits and Pieces in Logarithmic Conformal Field Theory
• Superstring Theory on AdS_2 x S^2 as a Coset Supermanifold
• Symplectic Fermions
• An algebraic approach to logarithmic conformal field theory
• A Local Logarithmic Conformal Field Theory
• Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions
• gl(N|N) Super-Current Algebras for Disordered Dirac Fermions in Two Dimensions
• T-duality for torus bundles via noncommutative topology
• Topological strings on noncommutative manifolds
• Logarithmic Operators and Hidden Continuous Symmetry in Critical Disordered Models
• Non-unitary Conformal Field Theory and Logarithmic Operators for Disordered Systems
• Fusion rules and logarithmic representations of a WZW model at fractional level
• $PSL(n|n)$ Sigma Model as a Conformal Field Theory
• Supermanifolds, Rigid Manifolds and Mirror Symmetry
• S and T matrices for the super $U(1,1)$ WZW model. Application to surgery and 3-manifold invariants based on the Alexander Conway polynomial
• Towards a Field Theory of the Plateau Transition
• Bethe Ansatz for a Quantum Supercoset Sigma Model
• The Mini-Superspace Limit of the SL(2,C)/SU(2)-WZNW Model
• Conformal Field Theory at central charge c=0 and Two-Dimensional Critical Systems with Quenched Disorder
• Mirror Symmetry and Supermanifolds
• Continuum Limit of the Integrable sl(2/1) 3-\bar{3} Superspin Chain
• Representation theory of sl(2|1)
• c-Theorem for Disordered Systems
• Higher-Level Appell Functions, Modular Transformations, and Characters
• Sigma-models having supermanifolds as target spaces
• Conformal field theory of the integer quantum Hall plateau transition
• On mysteriously missing T-duals, H-flux and the T-duality group
• Logarithmic lift of the su(2)_{-1/2} model
• Integrable quantum field theories with OSP(m/2n) symmetries
• Two-dimensional Conformal Field Theories on AdS_{2d+1} Backgrounds
• On representations of super coalgebras

Cited by in corpus (96)

• Associative-algebraic approach to logarithmic conformal field theories
• Logarithmic Conformal Field Theory: Beyond an Introduction
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• Logarithmic Conformal Field Theory: a Lattice Approach
• Modular Data and Verlinde Formulae for Fractional Level WZW Models II
• The WZNW model on PSU(1,1|2)
• Coset Constructions of Logarithmic (1,p)-Models
• On the SU(2|1) WZW model and its statistical mechanics applications
• From boundary to bulk in logarithmic CFT
• The integer quantum Hall plateau transition is a current algebra after all
• Oscillator Construction of su(n|m) Q-Operators
• The logarithmic triplet theory with boundary
• Continuum limit and symmetries of the periodic gl(1|1) spin chain
• Conformal Sigma-Models on Supercoset Targets
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• Free fermion resolution of supergroup WZNW models
• A modular invariant bulk theory for the c=0 triplet model
• Branes in the GL(1|1) WZNW-Model
• Instantons beyond topological theory II
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Conformal field theory at central charge c=0: a measure of the indecomposability (b) parameters
• Boundary Spectra in Superspace Sigma-Models
• The GL(1|1)-symplectic fermion correspondence
• The Sigma Model on Complex Projective Superspaces
• On the Percolation BCFT and the Crossing Probability of Watts
• Superspace conformal field theory
• Conformal invariance and quantum integrability of sigma models on symmetric superspaces
• Takiff superalgebras and Conformal Field Theory
• Structure constants of the OSP(1|2) WZNW model
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Gaussian free fields at the integer quantum Hall plateau transition
• Poisson-Lie T-dual sigma models on supermanifolds
• String States on AdS_3 x S^3 from the Supergroup
• Massless particles on supergroups and AdS3 x S3 supergravity
• Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
• Geometry of branes on supergroups
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• An admissible level $\widehat{\mathfrak{osp}}\left(1|2\right)$-model: modular transformations and the Verlinde formula
• Boundary Correlators in Supergroup WZNW Models
• The massless string spectrum on AdS_3 x S^3 from the supergroup
• The conformal current algebra on supergroups with applications to the spectrum and integrability
• Superconformal Blocks: General Theory
• Dissipation-driven integrable fermionic systems: from graded Yangians to exact nonequilibrium steady states
• New boundary conditions for the c=-2 ghost system
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Super Poisson-Lie symmetry of the GL(1|1) WZNW model and worldsheet boundary conditions
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Supergroup - extended super Liouville correspondence
• Finite dimensional vertex
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• From world-sheet supersymmetry to super target spaces
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• WZW models as mutual super Poisson-Lie T-dual sigma models
• Invariants and reduced matrix elements associated with the Lie superalgebra gl(m|n)
• W-Algebras Extending Affine gl(1|1)
• The gl(1|1) Lie superbialgebras
• Toward logarithmic extensions of ^sl(2)_k conformal field models
• Conformal superspace sigma-models
• Free field realization of current superalgebra $gl(m|n{)}_k$
• N=2 Superconformal Symmetry in Super Coset Models
• Lax Operator and superspin chains from 4D CS gauge theory
• Towards the Construction of Local Logarithmic Conformal Field Theories
• Anomalous Dimensions in Deformed WZW Models on Supergroups
• Yang-Baxter deformation of WZW model based on Lie supergroups: The cases of $GL(1|1)$ and $(C^3+A)$
• $gl(4|4)$ current algebra: free field realization and screening currents
• On conformal field theories based on Takiff superalgebras
• Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model
• A hierarchy of WZW models related to super Poisson-Lie T-duality
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain
• Non-chiral current algebras for deformed supergroup WZW models
• Edge states and conformal boundary conditions in super spin chains and super sigma models
• Spectra of Coset Sigma Models
• Yangian Superalgebras in Conformal Field Theory
• Representations of the Nappi--Witten vertex operator algebra
• Differential operator realizations of superalgebras and free field representations of corresponding current algebras
• Mock Modular Index of M2-M5 Brane System
• Models for Modules
• The plane-wave limit of ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathrm{S}}^3\times {\mathrm{S}}^1$
• Dyson-Maleev representation of nonlinear sigma-models
• Reduced Wigner coefficients for Lie superalgebra gl(m|n) corresponding to unitary representations and beyond
• Free field realization of the $osp(2n|2n)$ current algebra
• Consistently Constrained SL(N) WZWN Models and Classical Exchange Algebra
• Fermionic Coset, Critical Level W^(2)_4-Algebra and Higher Spins
• Kappa symmetric OSp(2|2) WZNW model
• Free field realization of the exceptional current superalgebra $\hat{D(2,1;\a)}_k$
• The Mock Modular Data of a Family of Superalgebras
• Staggered modules of $N=2$ superconformal minimal models
• Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
• On the gl(1|1) Wess-Zumino-Witten Model
• Harmonic Analysis in d-dimensional Superconformal Field Theory
• A solvable non-unitary fermionic long-range model with extended symmetry
• More on (gauged) WZW models over low-dimensional Lie supergroups and their integrable deformations
• High-Gradient Operators in the psl(2|2) Gross-Neveu Model
• Topological Field Theory and Stochastic Dynamics
• Geometric construction of D-branes in WZW models
• The free field representation for the $GL(1|1)$ WZW model revisited