String group explained
In topology, a branch of mathematics, a string group is an infinite-dimensional group
introduced by as a
-connected cover of a
spin group. A
string manifold is a
manifold with a lifting of its
frame bundle to a string group bundle. This means that in addition to being able to define
holonomy along paths, one can also define holonomies for surfaces going between strings. There is a short
exact sequence of
topological groups
0 → {\displaystyleK(Z,2)} → \operatorname{String}(n) → \operatorname{Spin}(n) → 0
where
is an
Eilenberg–MacLane space and
is a spin group. The string group is an entry in the
Whitehead tower (dual to the notion of
Postnikov tower) for the
orthogonal group:
… → \operatorname{Fivebrane}(n)\to\operatorname{String}(n) → \operatorname{Spin}(n) → \operatorname{SO}(n) → \operatorname{O}(n)
It is obtained by killing the
homotopy group for
, in the same way that
is obtained from
by killing
. The resulting manifold cannot be any finite-dimensional
Lie group, since all finite-dimensional compact Lie groups have a non-vanishing
. The fivebrane group follows, by killing
.
More generally, the construction of the Postnikov tower via short exact sequences starting with Eilenberg - MacLane spaces can be applied to any Lie group G, giving the string group String(G).
Intuition for the string group
The relevance of the Eilenberg–MacLane space
lies in the fact that there are the homotopy equivalences
for the
classifying space
, and the fact
. Notice that because the complex spin group is a
group extension0\toK(Z,1)\to\operatorname{Spin}C(n)\to\operatorname{Spin}(n)\to0
the String group can be thought of as a "higher" complex spin group extension, in the sense of
higher group theory since the space
is an example of a higher group. It can be thought of the topological realization of the
groupoid
whose object is a single point and whose morphisms are the group
. Note that the homotopical degree of
is
, meaning its homotopy is concentrated in degree
, because it comes from the
homotopy fiber of the map
\operatorname{String}(n)\to\operatorname{Spin}(n)
from the Whitehead tower whose homotopy cokernel is
. This is because the homotopy fiber lowers the degree by
.
Understanding the geometry
The geometry of String bundles requires the understanding of multiple constructions in homotopy theory,[1] but they essentially boil down to understanding what
-bundles are, and how these higher group extensions behave. Namely,
-bundles on a space
are represented geometrically as
bundle gerbes since any
-bundle can be realized as the homotopy fiber of a map giving a homotopy square
\begin{matrix}
P&\to&*\\
\downarrow&&\downarrow\\
M&\xrightarrow{}&K(Z,3)
\end{matrix}
where
. Then, a string bundle
must map to a spin bundle
which is
-equivariant, analogously to how spin bundles map equivariantly to the frame bundle.
Fivebrane group and higher groups
The fivebrane group can similarly be understood[2] by killing the
\pi7(\operatorname{Spin}(n))\cong\pi7(\operatorname{O}(n))
group of the string group
using the Whitehead tower. It can then be understood again using an exact sequence of
higher groups0\toK(Z,6)\to\operatorname{Fivebrane}(n)\to\operatorname{String}(n)\to0
giving a presentation of
\operatorname{Fivebrane}(n)
it terms of an iterated extension, i.e. an extension by
by
. Note map on the right is from the Whitehead tower, and the map on the left is the homotopy fiber.
See also
External links
- From Loop Groups to 2-groups - gives a characterization of String(n) as a 2-group
Notes and References
- Jurco. Branislav. August 2011. Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry. International Journal of Geometric Methods in Modern Physics. 08. 5. 1079–1095. 10.1142/S0219887811005555. 0219-8878. math/0510078. 2011IJGMM..08.1079J . 1347840.
- Sati. Hisham. Schreiber. Urs. Stasheff. Jim. November 2009. Fivebrane Structures. Reviews in Mathematical Physics. 21. 10. 1197–1240. 10.1142/S0129055X09003840. 0805.0564. 2009RvMaP..21.1197S . 13307997. 0129-055X.
