Loop spaces and connections
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Publication:2892846
DOI10.1112/JTOPOL/JTS007zbMATH Open1246.14027arXiv1002.3636OpenAlexW2164791290MaRDI QIDQ2892846
Author name not available (Why is that?)
Publication date: 25 June 2012
Published in: (Search for Journal in Brave)
Abstract: We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well known connection between rotation of loops and the de Rham differential. Our main result, a categorification of the geometric description of cyclic homology, relates S^1-equivariant quasicoherent sheaves on the loop space of a smooth scheme or geometric stack X in characteristic zero with sheaves on X with flat connection, or equivalently D_X-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover D_X-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their Omega S^3-equivariance.
Full work available at URL: https://arxiv.org/abs/1002.3636
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