Yet another construction of the central extension of the loop group

Central figure-8 cross-cuts make surfaces cylindrical

We prove: If a complete connected smooth surface M in euclidean 3-space has general position, intersects some plane along a clean figure-8 (a loop with total curvature zero) and all compact intersections with planes have central symmetry, then M is a (geometric) cylinder over some central figure-8. On the way, we establish interesting facts about centrally symmetric loops in the plane; for instance, a clean loop with even rotation number 2k can never be central unless it passes through its center exactly twice and k=0.

面积保持微分同胚环的中心扩张及其模糊球极限

Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits

研究面积保持微分同胚环的中心扩张，并揭示其在大k极限下的模糊球极限性质。

On sigma model RG flow, "central charge" action and Perelman's entropy

Zamolodchikov's c-theorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be gradient one to all loop orders. However, the monotonicity of the flow of the target-space metric is not obvious since the metric on the space of metric-dilaton couplings is indefinite. To leading (one-loop) order when the RG flow is simply the Ricci flow the monotonicity was proved by Perelman (math.dg/0211159) by constructing an ``entropy'' functional which is essentially the metric-dilaton action extremised with respect to the dilaton with a cond…

Transgressive loop group extensions

A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry over the Lie group. We show how transgressive central extensions can be characterized in a loop-group theoretical way, in terms of loop fusion and thin homotopy equivariance.

源于环路群扩张的 Lie 2-群

Lie 2-groups from loop group extensions

本文利用环路群在其中心扩张上的共轭作用，给出了 string 2-group 作为严格 Fréchet Lie 2-群的简单构造，并推广至半单且单连通 Lie 群的严格 2-群扩张。

Loop groups and string topology

Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.