Classification of Conformal Supersymmetric Deformations of Schwarzschild Spacetime via Super Poisson-Lie T-duality/Plurality

This paper investigates conformal supersymmetric deformations of Schwarzschild spacetime through the framework of super Poisson-Lie T-duality/plurality. We first study super non-Abelian T-duality in $σ$-models featuring a Schwarzschild metric coupled to two fermionic fields. This is realized by describing the original four-dimensional geometry and its dual counterpart via semi-Abelian Drinfeld superdoubles generated by the Lie superalgebras $({\C}^3 +{\A})$, ${\C}^3 \oplus {\A}_{1,1}$ and $(2{ \A}_{1,1}+2{ \A})^0$, supplemented by two spectator fields. We enforce the vanishing of one-loop beta-function equations for both original and dual models to guarantee UV finiteness at quantum level; this procedure necessitates the inclusion of the dilaton field alongside the metric and the Kalb-Ramond field ($B$-field) already present in the classical action. We further study the curvature invariants, singularity structure, and superisometries, and compare the resulting supergeometries, with particular emphasis on their fermionic parts and $B$-fields. Starting from the decompositions of semi-Abelian Drinfeld superdoubles associated with the $({\C}^3 +{\A})$ and ${\C}^3 \oplus {\A}_{1,1}$ Lie superalgebras, we derive the conformal duality/plurality chains for four-dimensional string backgrounds, characterized by a Schwarzschild metric coupled to two fermions. Our results span an interesting spectrum of super Poisson-Lie T-dual $σ$-models described by Lie superalgebras with two bosonic and two fermionic generators. These are all non-trivial and interesting examples of super Poisson-Lie T-dual models which help in the intent of providing a general classification of four-dimensional geometries describing supergravity backgrounds.