Supermanifolds (Cambridge Monographs on Mathematical Physics)

A detailed account of the modern theory of supermanifolds, Bryce DeWitt's monograph gives a rigorous treatment of the super-analogs of the basic structures of ordinary manifold theory. It opens with analysis over supernumbers, Berezin integration, supervector spaces and the superdeterminant, then applies them to supermanifolds, with super-analogs of Lie derivatives, connections, curvature, geodesics and Killing flows. Further material covers super Lie groups and super Lie algebras, with complete descriptions of all the simple super Lie groups. Applications include path integrals for fermionic quantum systems, super Hilbert spaces and simple supersymmetric models mixing bosons and fermions. The revised second edition adds a chapter on dynamical systems in which the topology of the configuration supermanifold matters. Many exercises complement the text.

It turns up on the four shelves of books Eric Weinstein shared on Twitter as the ones worth bringing home from his office.