Integrable $\mathbb{Z}_2^2$-graded super-Liouville Equation and Induced $\mathbb{Z}_2^2$-graded super-Virasoro Algebra

TL;DR

This paper develops a new framework for $ olinebreak bZ_2^2$-graded super-Liouville equations and extends the super-Virasoro algebra, employing graded zero-curvature formalism and Hamiltonian reduction techniques.

Contribution

It introduces a $bZ_2^2$-graded extension of super-Liouville equations and derives new $bZ_2^2$-graded super-Virasoro algebra extensions using algebraic and geometric methods.

Findings

Derived a $bZ_2^2$-graded super-Liouville equation.

Constructed solutions and Bäcklund transformations for the equation.

Obtained three new $bZ_2^2$-graded super-Virasoro algebra extensions.

Abstract

We present a framework for enlarging the construction of $\mathbb{Z}_2^2$-graded classical Toda theory from the class of $\mathbb{Z}_2^2$-graded Lie algebras to the class of $\mathbb{Z}_2^2$-graded Lie superalgebras. This scheme is applied to derive a $\mathbb{Z}_2^2$-graded extension of the super-Liouville equation based on a $\mathbb{Z}_2^2$-graded extension of $\mathfrak{osp}(1|2).$ The mathematical tools employed in this work are a $\mathbb{Z}_2^2$-graded version of the zero-curvature formalism and of the Polyakov's soldering procedure. It is demonstrated that both methods yield the same $\mathbb{Z}_2^2$-graded super-Liouville equation. An algebraic construction of solutions to the resulting equations is also presented, together with their B\"acklund transformations. Furthermore, three distinct new $\mathbb{Z}_2^2$-graded extensions of the super-Virasoro algebra are obtained via Hamiltonian reduction of the WZNW currents defined for $\mathbb{Z}_2^2$-$\mathfrak{osp}(1|2).$