New preprint: From Loop Groups to 2-Groups (and the String Group)
Posted by Urs Schreiber
I am happy to be able to announce a new preprint:
J. Baez, A. Crans, U. Schreiber & D. Stevenson
From Loop Groups to 2-Groups
math.QA/0504123.
Abstract:
We describe an interesting relation between Lie 2-algebras, the Kac–Moody central extensions of loop groups, and the group
. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2-group is a categorified version of a Lie group. If is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras each having as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on . There appears to be no Lie 2-group having as its Lie 2-algebra, except when . Here,
however, we construct for integral an infinite-dimensional Lie
2-group whose Lie 2-algebra is equivalent to . The objects of are based paths in , while the automorphisms of any object form the level- Kac–Moody central extension of the loop group . This 2-group is closely related to the th power of the canonical gerbe over . Its nerve gives a topological group
that is an extension of by . When ,
can also be obtained by killing the third homotopy group of . Thus, when , is none other than .
[Update: I am aware of that problem with the incorrectly-displayed TeX code above. I am hoping to find the solution to that problem soon.]
There are two central theorems here:
1) The weak Lie 2-algebras defined in HDA VI are equivalent to infinite-dimensional strict Fréchet Lie 2-algebras . These are related to the Kac-Moody central extension of the loop algebra and come from infinite-dimensional Fréchet Lie 2-groups .
2) The so-called ‘nerve’ of is, for , the topological group .
The conclusion of the paper is the following:
We have seen that the Lie 2-algebra is equivalent to an infinite-dimensional Lie 2-algebra , and that when is an integer, comes from an infinite-dimensional Lie 2-group . Just as the Lie 2-algebra is built from the simple Lie algebra and a shifted version of :
(1)
the Lie 2-group is built from and another Lie 2-group:
(2)
whose geometric realization is a shifted version of :
(3)
None of these exact sequences split; in every case an interesting
cocycle plays a role in defining the middle term. In the first case,
the Jacobiator of is .
In the second case, composition of morphisms is defined using
multiplication in the level- Kac–Moody central extension of
, which relies on the Kac–Moody cocycle
. In the third case, is the total
space of a twisted -bundle over whose Dixmier–Douady
class is . Of course, all these cocycles
are different manifestations of the fact that every simply-connected
compact simple Lie algebra has .
We conclude with some remarks of a more speculative nature.
There is a theory of ‘2-bundles’ in which a Lie 2-group plays
the role of structure group
[3,
4]. Connections on
2-bundles describe parallel transport of 1-dimensional
extended objects, e.g. strings. Given the importance of the
Kac–Moody extensions of loop groups in string theory, it is
natural to guess that connections on 2-bundles with structure
group will play a role in this theory.
The case when and is particularly interesting,
since then . In this case we suspect that
-bundles on a spin manifold with structure -group can be thought as substitutes for principal -bundles on
. It is interesting to think about ‘string structures’
[16] on from this perspective: given a principal -bundle
on (thought of as a -bundle with only identity morphisms)
one can consider the obstruction problem of trying to lift the structure
-group from to . There should be a single topological
obstruction in to finding a lift, namely the characteristic
class . When this characteristic class vanishes, every principal
-bundle on should have a lift to a -bundle on
with structure -group . It is tempting to conjecture that the
geometry of these -bundles is closely related to the enriched
elliptic objects of Stolz and Teichner
[20].
Posted at April 7, 2005 9:28 AM UTC
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/547
Re: New preprint: From Loop Groups to 2-Groups (and the String Group)
I guess it is time to learn about these things. Maybe I can start with an embarrassingly elementary question: what is this String group good for? I understand (to some extent) the reason why a physicist wants to kill the first homotopy group of SO(n) — it is because the physicist wants to be able to define the parallel transport of fermion fields. When it is possible, lifting from SO(n) to Spin(n) involves (roughly) fixing a Z_2-valued ambiguity for every 1-cycle of spacetime, which one describes as the choice of periodic or antiperiodic boundary conditions for the fermions. Is there an analogous “problem” that one encounters in string theory and gets solved by lifting to String(n)? Is there a Z-valued ambiguity for something associated with 3-cycles of spacetime?
Read the post
Remarks on String(n)-Connections
Weblog: The String Coffee Table
Excerpt: Is Stolz/Teichner's String-connection a 2-connection in an associated 2-bundle?
Tracked: March 11, 2006 3:13 PM
Read the post
Stevenson, Henriques on String(n)
Weblog: The String Coffee Table
Excerpt: Stevenson and Henriques have articles on the String group.
Tracked: March 23, 2006 9:27 AM
Read the post
Categorified Gauge Theory in Chicago
Weblog: The String Coffee Table
Excerpt: Conference with emphasis on categorified gauge theory, covering gerbes and the String group.
Tracked: April 4, 2006 10:12 AM
Read the post
Jurco on Gerbes and Stringy Applications
Weblog: The String Coffee Table
Excerpt: Jurco reviews some facts concerning nonabelian gerbes in string theory.
Tracked: April 19, 2006 11:38 AM
Read the post
10D SuGra 2-Connection
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra governing 10-dimensional supergravity.
Tracked: August 28, 2006 3:51 PM
Read the post
Puzzle Pieces falling into Place
Weblog: The n-Category Café
Excerpt: On the 3-group which should be underlying Chern-Simons theory.
Tracked: September 28, 2006 3:29 PM
Read the post
Dijkgraaf-Witten Theory and its Structure 3-Group
Weblog: The n-Category Café
Excerpt: The idea of Dijkgraaf-Witten theory and its reformulation in terms of parallel volume transport with respect to a structure 3-group.
Tracked: November 6, 2006 8:25 PM
Read the post
Some Conferences
Weblog: The n-Category Café
Excerpt: A conference on bundles and gerbes, another one on topology, and comments on associated 2-vector bundles and String connections.
Tracked: April 19, 2007 8:56 PM
Read the post
Zoo of Lie n-Algebras
Weblog: The n-Category Café
Excerpt: A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
Tracked: May 10, 2007 6:21 PM
Read the post
On BV Quantization. Part I.
Weblog: The n-Category Café
Excerpt: On BV-formalism applied to Chern-Simons theory and its apparent relation to 3-functorial extentended QFT.
Tracked: August 17, 2007 10:42 PM
Read the post
Lie n-Algebra Cohomology
Weblog: The n-Category Café
Excerpt: On characteristic classes of n-bundles.
Tracked: September 7, 2007 6:00 PM
Read the post
Cohomology of the String Lie 2-Algebra
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra cohomlogy of the String Lie 2-algebra and its relation to twisted K-theory.
Tracked: October 2, 2007 10:45 PM
Read the post
On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 4:34 PM
Read the post
Construction of Cocycles for Chern-Simons 3-Bundles
Weblog: The n-Category Café
Excerpt: On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
Tracked: February 12, 2008 1:03 PM
Re: New preprint: From Loop Groups to 2-Groups (and the String Group)
I guess it is time to learn about these things. Maybe I can start with an embarrassingly elementary question: what is this String group good for? I understand (to some extent) the reason why a physicist wants to kill the first homotopy group of SO(n) — it is because the physicist wants to be able to define the parallel transport of fermion fields. When it is possible, lifting from SO(n) to Spin(n) involves (roughly) fixing a Z_2-valued ambiguity for every 1-cycle of spacetime, which one describes as the choice of periodic or antiperiodic boundary conditions for the fermions. Is there an analogous “problem” that one encounters in string theory and gets solved by lifting to String(n)? Is there a Z-valued ambiguity for something associated with 3-cycles of spacetime?