The WZNW model on PSU(1,1|2)

TL;DR

This work provides a complete, covariant solution of the WZNW model on the supergroup $PSU(1,1|2)$ at the NSNS (Wess-Zumino) point, revealing that the full state space does not factorize into bosonic and fermionic sectors but couples to produce a logarithmic CFT with indecomposable representations. It builds the rigorous representation theory for $\mathfrak{psl}(2|2)$ and its affine extension, including Kac modules, atypicals, and projective covers, and then constructs a free-field realization of the affine superalgebra to obtain the spectrum, characters, and vertex operators. The minisuperspace analysis shows a non-diagonalizable Laplacian with Jordan blocks, which is inherited by the full field theory and drives the logarithmic structure. A Racah-Speiser-type algorithm for Casimir-decomposition of the state space is developed, enabling explicit branching of bulk states into $\mathfrak{psl}(2|2)$ representations and preparing the ground for studying RR deformations. Collectively, these results supply a robust framework for AdS$_3$ backgrounds with mixed NSNS/RR flux, and illuminate how bosonic-fermionic coupling yields projective representations that control the spectrum and correlation functions.

Abstract

According to the work of Berkovits, Vafa and Witten (hep-th/9902098), the non-linear sigma model on the supergroup PSU(1,1|2) is the essential building block for string theory on AdS(3)xS(3)xT(4). Models associated with a non-vanishing value of the RR flux can be obtained through a psu(1,1|2) invariant marginal deformation of the WZNW model on PSU(1,1|2). We take this as a motivation to present a manifestly psu(1,1|2) covariant construction of the model at the Wess-Zumino point, corresponding to a purely NSNS background 3-form flux. At this point the model possesses an enhanced psu(1,1|2) current algebra symmetry whose representation theory, including explicit character formulas, is developed systematically in the first part of the paper. The space of vertex operators and a free fermion representation for their correlation functions is our main subject in the second part. Contrary to a widespread claim, bosonic and fermionic fields are necessarily coupled to each other. The interaction changes the supersymmetry transformations, with drastic consequences for the multiplets of localized normalizable states in the model. It is only this fact which allows us to decompose the full state space into multiplets of the global supersymmetry. We analyze these decompositions systematically as a preparation for a forthcoming study of the RR deformation.

Figures (2)

• Figure 1: The moduli space of string theory on ${\text{PSU}(1,1|2)}$. The vertical axis gives the normalization of the kinetic term, the horizontal the normalization of the Wess-Zumino term. The lines with $\lambda\neq0$ correspond to marginal deformations of the WZNW model.
• Figure 2: The root diagram of ${\text{psl}(2|2)}$.

Theorems & Definitions (3)

• Lemma 1
• proof
• Conjecture 1