De Witt Supermanifolds and Infinitedimensional Ground Rings

Batchelor compared graded manifolds with Rogers supermanifolds (for B L = Λ R L , L < ∞) using algebraic techniques. We investigate the extension to supermanifolds over an infinite‐dimensional ground ring B (not necessarily of ‘Banach‐Grassman’ type). An ‘abstract’ supermanifold over B is a Z 2 ‐ringed space ( S ,T s ∞ ) with an appropriate local modelling property (soT s ∞is not necessarily a sheaf of B ‐valued functions). We show that the rule (X, A ) → Alg( A (X), B) (algebraic representation) gives an equivalence between graded manifolds and abstract de Witt supermanifolds if and only if B is reflexive in the sense that B = B o ∗ . We recover Batchelor's results for B = B L with the fine topology since ( T s ∞ ) is abstract de Witt if and only if S is H ∞ ‐de Witt (that is,T s ∞=H s ∞ ), provided that L exceeds the odd dimension of S . We can extend to L = ∞ provided that B ∞ is taken to be the (formal) Grassmann algebra B ∞ = Λ R ∞ with its Frechet topology rather than Rogers's Banach‐Grassmann subalgebra BG. The algebra B ∞ is reflexive but BG is not. These results become relevant for comparing de Witt supermanifolds with Jadczyk‐Pilch supermanifolds when we take a formal L → ∞ limit.