Loop spaces and connections

By David Ben-Zvi, D. Nadler. 2010

94 citations recorded by the source corpus.

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 S1‐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 DX‐modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover DX‐modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration Ω S3→Ω S2→S1, and relate Ω S2‐equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their Ω S3‐equivariance.

View Paper: https://www.semanticscholar.org/paper/57d926ace95163ee62e1db53d667cbb441c0a1b7

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Citations in the corpus

Derived Algebraic Geometry (2014, background): “4) appear in [14]: TX/k[−1] is identified with the Lie algebra of the derived loop stack L(X) −→ X , but assume that the relations between Lie algebras and formal groups extend to the general setting of derived Artin stack (which is today not yet fully established).”
A Darboux theorem for derived schemes with shifted symplectic structure (2013, unknown): “See To¨en and Vezzosi [30] and Ben-Zvi and Nadler [4].”
On the equivalence between Lurie's model and the dendroidal model for infinity-operads (2013, background): “It plays an important role in current advances in derived algebraic geometry, specifically chiral homology [17, 28], geometric representation theory [1, 5, 6, 18, 19, 20] and mirror symmetry [32].”
Derived algebraic geometry (2014, background): “4) appear in [14]: TX/k[−1] is identified with the Lie algebra of the derived loop stack L(X) −→ X , but assume that the relations between Lie algebras and formal groups extend to the general setting of derived Artin stack (which is today not yet fully established).”