À la recherche de l'âme tordue

par Prof. Rafal Suszek (Warsaw)

mercredi 17 mai 2023, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Bâtiment Braconnier)

The fundamental role of higher geometry – that of (n-)gerbes realising integral classes in the de Rham cohomology of manifolds and of the attendant higher categories – in the description of dynamics of (extended) distributions of charge and Maxwell-type gauge fields – such as σ-models and certain topological field theories – has, by now, been well established. Higher geometry (HG) determines a (pre-)quantisation of the dynamics, captures its rigid and gauge symmetries as well as dualities (such as, e.g., T-duality), models defects etc.

 

In the talk, a generalisation shall be presented of Murray's formulation of HG to the Berezin-Leïtes-Kostant Z/2Z-graded geometry with Lie-supergroup `symmetries' – a natural setting of the so-called super-σ-models of Green and Schwarz. The generalisation, taking as its point of departure the supersymmetric refinement of the de Rham cohomology, calls for a conceptual reworking of the differential-geometric tools employed in the un-graded setting – a replacement, i.a., of the usual Čech and Beilinson-Deligne techniques by purely algebraic ones based on the Kostant-Leïtes approach to Lie supergroups and the cohomology of their Lie superalgebras. And yet, by the end of the day, the ensuing HG objects – the super-n-gerbes – are seen to encode/resolve, just as their Murray's predecessors, a nontrivial `topology' – that of the Soul of certain non-Rothstein superorbifolds of their bases. Thus, they provide an intricate reinterpretation of the aforementioned supersymmetric refinement of the de Rham cohomology, consistent with Freed's functorial definition of the underlying superfield theories. This shall be demonstrated with the help of a suitable adaptation of the equivariantisation techniques and of the gauge-defect construction for the two-dimensional σ-model worked out, once upon a temps perdu, in a joint venture with Gawędzki, Runkel and Waldorf.