The BRST Charge for the $\hat{D}(2,1;\a)$ Non-Linear Quasi-Superconformal Algebra

TL;DR

This paper constructs the quantum BRST charge for the complex non-linear $ ext{D}(2,1; ext{a})$ superconformal algebra, revealing unique solutions and proposing a new string theory with non-linear supersymmetry and non-unitary matter.

Contribution

It provides the first explicit construction of the quantum BRST charge for the $ ext{D}(2,1; ext{a})$ algebra and identifies the unique solution for nilpotency, suggesting a novel string theory framework.

Findings

Quantum BRST charge constructed for $ ext{D}(2,1; ext{a})$ algebra.

Unique solution at $k^+=k^-=-2$ for nilpotency.

Proposal of a new string theory with non-linear supersymmetry.

Abstract

The quantum BRST charge for the most general, two-dimensional, non-linear, $N=4$ quasi-superconformal algebra $\hat{D}(1,2;\a)$, whose linearisation is the so-called `large' $N=4$ superconformal algebra, is constructed. The $\hat{D}(1,2;\a)$ algebra has $\Hat{su(2)}_{k^+}\oplus \Hat{su(2)}_{k^-}\oplus\Hat{u(1)}$ Ka\v{c}-Moody component, and $\a=k^-/k^+$. As a pre-requisite to our construction, we check the $\hat{D}(1,2;\a)$ Jacobi identities and construct a classical BRST charge. Then, we analyse the quantum BRST charge nilpotency conditions and find the only solution, $k^+=k^-=-2$. The $\hat{D}(1,2;1)$ algebra is actually isomorphic to the $SO(4)$-based Bershadsky-Knizhnik non-linear quasi-superconformal algebra. We argue about the existence of a new string theory with (i) the non-linearly realised $N=4$ world-sheet supersymmetry, (ii) non-unitary matter in a $\hat{D}(1,2;\a)$ representation of $k=-2$ and $c=-6$, and (iii) negative `critical dimension'.