Topological semiinfinite tensor (super)modules

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl(V circle plus Pi V) and osp(V circle plus Pi V) associated with a Tate space V. Here V circle plus Pi V is a Z/2Z-graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories (T) over cap (gl(V)), (T) over cap (o(V)) and (T) over cap (sp(V)), and show that these categories are anti-equivalent to respective previously studied categories T-gl(V), T-o(V), T-sp(V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to (T) over cap (gl(V)), (T) over cap (o(V)) and (T) over cap (sp(V)). Moreover, the categories T-o(V) and T-sp(V) are known to be equivalent, and this implies the equivalence of the categories (T) over cap (o(V)) and (T) over cap (sp(V)). After introducing a supersymmetric setting, we establish the equivalence of the category (T) over cap (gl(V)) with the category (T) over cap (gl(V circle plus Pi V)), and the equivalence of both categories T-o(V) and (T) over cap (sp(V)) with (T) over cap (osp(V circle plus Pi V)). (c) 2023 Elsevier Inc. All rights reserved.