On the structure of DeWitt supermanifolds |
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| Institution: | 1. Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2–4, 34014 Trieste, Italy;2. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146, Genova, Italy;3. School of Mathematical and Computing Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand;1. Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA;2. CERN TH Division, CH-1211 Geneva 23, Switzerland;1. Department of Mathematics, Swansea University, Swansea University Bay Campus, Fabian Way, Swansea SA1 8EN, UK;2. Department of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland;1. Ludwig-Maximilians-Universität München, Fakultät für Physik, Arnold Sommerfeld Center for Theoretical Physics, D-80333 München, Germany;2. Theoretische Physik 1, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany;3. Centre de Physique Théorique (CPT), UMR 7332 CNRS/Aix-Marseille Univ./Univ. du Sud Toulon-Var, Marseille, France;4. IFIC, Universitat de València – CSIC, Apt. Correus 22085, E-46071 València, Spain;1. Department of Neurology, Washington University School of Medicine, St. Louis, MO 63110, USA;2. Department of Radiology, Washington University School of Medicine, St. Louis, MO 63110, USA;3. Department of Anatomy and Neurobiology, Washington University School of Medicine, St. Louis, MO 63110, USA;4. Department of Bioengineering, Washington University School of Medicine, St. Louis, MO 63110, USA;5. Department of Neuroscience, University of Padua, Padua, Italy;6. Padua Neuroscience Center, Padua, Italy |
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| Abstract: | We show that for a wide and most natural class of (possibly infinite-dimensional) Grassmannian algebras of coefficients, the structure sheaf of every smooth DeWitt supermanifold is acyclic (i.e. its cohomology vanishes in positive degree). This result was previously known for finite-dimensional ground algebras and is new even for the original DeWitt algebra of supernumbers /GL∞. From here we deduce that (equivalence classes of) smooth DeWitt supermanifolds over a fixed ground algebra and of graded smooth manifolds are in a natural bijection with each other. However, contrary to what was stated previously by some authors, this correspondence fails to be functorial; so it happens, for instance, for Rogers' ground algebra B∞. Finally, we observe that every DeWitt super Lie group is a deformation of a graded Lie group over the spectrum Spec /GL of the ground algebra. |
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