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Flat Z-graded connections and loop spaces ; Schatz, Florian E-print/Working paper (2015) The pull back of a flat bundle E→X along the evaluation map π:LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally ... [more ▼] The pull back of a flat bundle E→X along the evaluation map π:LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally generalizes to flat Z-graded connections on X. Our main result is that the restriction of this holonomy automorphism to the based loop space ΩX of X provides an A-infinity quasi-equivalence between the dg category of flat Z-graded connections on X and the dg category of representations of C(ΩX), the dg algebra of singular chains on ΩX. [less ▲] Detailed reference viewed: 67 (8 UL)Higher holonomies: comparing two constructions ; Schatz, Florian in Differential Geometry & its Applications (2015), 40 We compare two different constructions of higher dimensional parallel transport. On the one hand, there is the two dimensional parallel transport associated to 2-connections on 2-bundles studied by Baez ... [more ▼] We compare two different constructions of higher dimensional parallel transport. On the one hand, there is the two dimensional parallel transport associated to 2-connections on 2-bundles studied by Baez-Schreiber, Faria Martins-Picken and Schreiber-Waldorf. On the other hand, there are the higher holonomies associated to flat superconnections as studied by Igusa, Block-Smith and Arias Abad-Schätz. We first explain how by truncating the latter construction one obtains examples of the former. Then we prove that the 2-dimensional holonomies provided by the two approaches coincide. [less ▲] Detailed reference viewed: 74 (4 UL)Reidemeister torsion for flat superconnections ; Schatz, Florian in Journal of Homotopy & Related Structures (2014), 9(2), 579-606 We use higher parallel transport -- more precisely, the integration A-infinity-functor constructed in -to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger ... [more ▼] We use higher parallel transport -- more precisely, the integration A-infinity-functor constructed in -to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger-Müller theorem, namely that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu. [less ▲] Detailed reference viewed: 46 (1 UL)Holonomies for connections with values in L_infty algebras ; Schatz, Florian in Homology, Homotopy & Applications (2014), 16(1), 89-118 Given a flat connection on a manifold M with values in a filtered L-infinity-algebra g, we construct a morphism, generalizing the holonomies of flat connections with values in Lie algebras. The ... [more ▼] Given a flat connection on a manifold M with values in a filtered L-infinity-algebra g, we construct a morphism, generalizing the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheim's A-infinity version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space Rd, and to generalizations of the holonomy representations of braid groups. [less ▲] Detailed reference viewed: 52 (1 UL)The A_infty de Rham theorem and integration of representations up to homotopy ; Schatz, Florian in International Mathematics Research Notices (2013), 2013(16), 3790-3855 We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A-infinity functor from the representations up to homotopy of a Lie algebroid A to those of its ... [more ▼] We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an A-infinity functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an A-infinity version of de Rham's theorem due to Gugenheim. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa and Block-Smith, to the case of certain differential graded manifolds. [less ▲] Detailed reference viewed: 35 (2 UL)Deformations of Lie brackets and representations up to homotopy ; Schatz, Florian in Indagationes Mathematicae (2011), 22 We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on ... [more ▼] We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets. [less ▲] Detailed reference viewed: 41 (1 UL) |
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