On symmetric simplicial (super)string backgrounds, (super-)WZW defect fusion and the Chern-Simons theory

The super-$\sigma$-model of dynamics of the super-charged loop in an ambient supermanifold in the presence of worldsheet defects of arbitrary topology is formalised within Gaw\c{e}dzki's higher-cohomological approach, drawing inspiration from the precursor arXiv:0808.1419 [hep-th]. A distinguished class of the corresponding backgrounds (supertargets with additional bicategorial supergeometric data), organised into simplicial hierarchies, is considered. To these, configurational (super)symmetry of the bulk field theory is lifted coherently, whereby the notion of a maximally (super)symmetric background, and in particular that of a simplicial Lie background, arises as the target structure requisite for the definition of the super-$\sigma$-model with defects fully transmissive to the currents of the bulk (super)symmetry. The formal concepts are illustrated in two settings of physical relevance: that of the WZW $\sigma$-model of the bosonic string in a compact simple 1-connected Lie group and that of the GS super-$\sigma$-model of the superstring in the Minkowski super-space. In the former setting, the structure of the background is fixed through a combination of simplicial, symmetry(-reducibility) and cohomological arguments, and a novel link between fusion of the maximally symmetric WZW defects of Fuchs et al. and the 3$d$ CS theory with timelike Wilson lines with fixed holonomy is established. Moreover, a purely geometric interpretation of the Verlinde fusion rules is proposed. In the latter setting, a multiplicative structure compatible with supersymmetry is shown to exist on the GS super-1-gerbe of arXiv:1706.05682 [hep-th], and subsequently used in a novel construction of a class of maximally (rigidly) supersymmetric bi-branes whose elementary fusion is also studied.