Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

May 8, 2008

On Lie oo-Theory

Posted by Urs Schreiber

Taking a day off at HIM (will report tomorrow on what Liang Kong has been teaching us about vertex operator algebras from Segal’s CFT axioms, using theorems by Huang), today I am giving a talk in Hamburg in our series on BRST-BV formalism (as you will have guessed), the goal being to illuminate the geometric \infty-categorical meaning of the BRST complex regarded as a L(ie)L(ie) \infty-algebroid:

The things I’ll say and draw to the board are those at the beginning of section 1 of

On action Lie \infty-groupoids and action Lie \infty-algebroids
(pdf)

Abstract. We discuss actions of Lie nn-groups and the corresponding action Lie nn-groupoids; discuss actions of Lie nn-algebras (L L_\infty-algebras) and the corresponding action Lie nn-algebroids; and discuss the relation between the two by integration and differentiation.

As an example of interest, we discuss the BRST complex that appears in quantum field theory. We describe it as the Chevalley-Eilenberg algebra of the Lie nn-algebroid which linearizes the action nn-groupoid of a gauge nn-group acting on the space of fields.

This identifies the ghosts-of-ghosts of degree kk as the cotangents to the space of kk-morphisms of this action nn-groupoid.

Several separate aspects of what we say here are essentially “well known” to those who know it well. But a coherent description as attempted here is certainly missing in the literature and deserves to be better known.

Posted at May 8, 2008 11:53 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1677

0 Comments & 6 Trackbacks

Read the post HIM Trimester Geometry and Physics, Week 1
Weblog: The n-Category Café
Excerpt: On vertex operator algebras, operads and Segals QFT axioms. On integration over supermanifolds, supergroupoids and the two notions of supercategories.
Tracked: May 14, 2008 6:40 PM
Read the post A Groupoid Approach to Quantization
Weblog: The n-Category Café
Excerpt: On Eli Hawkins' groupoid version of geometric quantization.
Tracked: June 12, 2008 5:49 PM
Read the post An Exercise in Groupoidification: The Path Integral
Weblog: The n-Category Café
Excerpt: A remark on the path integral in view of groupoidification and Sigma-model quantization.
Tracked: June 13, 2008 6:26 PM
Read the post Eli Hawkins on Geometric Quantization, I
Weblog: The n-Category Café
Excerpt: Some basics and some aspects of geometric quantization. With an emphasis on the geometric quantization of duals of Lie algebras and duals of Lie algebroids.
Tracked: June 20, 2008 5:06 PM
Read the post Block on L-oo Module Categories
Weblog: The n-Category Café
Excerpt: On Jonathan Block's concept of modules over differential graded algebras.
Tracked: June 30, 2008 11:44 PM
Read the post Smooth Differential Graded Algebra?
Weblog: The n-Category Café
Excerpt: What is the right definition of generalized smooth differential graded-commutative algebras?
Tracked: September 9, 2008 11:03 AM

Post a New Comment