Supersymmetric Backgrounds in $(1+1)$ Dimensions and Inhomogeneous Field Theory

We find a $(1+1)$-dimensional metric solution for a background hosting various supersymmetric field theories with a single non-chiral real supercharge. This supersymmetric background is globally hyperbolic even though it contains a naked null singularity. In this regard, we show that scalar wave propagation on the background is well-defined and so the curvature singularity is a {\it mild} one. Taking inspiration from our previous work, we relate the field theory on this curved background to some classes of $(1+1)$-dimensional inhomogeneous field theory in the supersymmetric setup. Utilizing our supersymmetric background, we elucidate the limitations of canonical quantization and highlight the conceptual advantages of the algebraic approach to quantization.
Comment: V1: 1+46 pages, 10 figures. V2: Minor typos fixed, V3: Added some references with major improvement, 1+48 pages,V4: Minor corrections, Published version, 1+50 pages