The String Coffee Table

I am now going to list some literature in which the above facts can be found discussed. Before doing so I would like to emphasize, though, that the situation sketched above remains at this point a little mysterious in that there is so far no transparent picture available which explains the relation between the need for $\widehat{L\mathrm{SO}(n)}$ on loop space $LM$ to the need for that strange group $\mathrm{String}(n)$ on spacetime itself. It just happens to work out this way.

In the remainder I therefore want to briefly review what the results of our paper have to say about this question, and how the above mystery is at least to large parts resolved by regarding strings as categorified points.

But first some literature:

One of the earliest discussions of these issues is given in

T. Killingback
World-Sheet Anomalies and Loop Geometry
Nucl.Phys.B288:578,1987

Killingback discusses the spinning point particle, the spinning string, the obstructions to lifting the $L\mathrm{SO}(n)$-bundle on loop space to the central extension and the relation to the perturbative anomalies of the effective field theory of the heterotic string.

As a supplement to this I can recommend the nice and more detailed discussion of the relation between $H^3(LM;\mathbb{Z})$ and $p_1/2$ as given in

M. Murray & D. Stevenson
Higgs Fields, Bundle Gerbes and String Structures
math.DG/0106179

A more detailed discussion of the nature of Dirac operators on loop space with a review of Killinback’s results is given in

E. Witten
The Index of the Dirac Operator in Loop Space
Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology, Princeton, N.J., Sep 1986.

which has the companion paper

E. Witten
Elliptic Genera and Quantum Field Theory
Commun.Math.Phys.109:525,1987

Concerning the group $\mathrm{String}(n)$ I can only point to

S. Stolz & P. Teichner
What is an Elliptic Object?
in:
U. Tillmann (ed.)
Topology, Geometry and Quantum Field Theory
Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal
London Mathematical Society Lecture Notes Series 308 Cambridge University Press (2004)
online available here

A discussion of the group $\mathrm{String}(n)$ and of string structures is given on the top of p. 5 of this text and then in the beginning of section 5 on pp. 65. The ‘killing’ of homotopy groups is discussed on p. 65, the definition of $\mathrm{String}(n)$ by means of an exact sequence is discussed on p. 66, and the relation to the Pontryagin class is discussed on p. 67.

To conclude, I’ll briefly sketch aspects of the category-theoretic picture behind all this, as discussed in

J. Baez, A. Crans, U. Schreiber & D. Stevenson
From Loop Groups to 2-Groups
math.QA/0504123.

Like a point particle couples to a bundle, a string should couple to a categorified bundle, called a 2-bundle. Indeed, it is for instance well known that the Kalb-Ramond $B$-field that the string couples to really is the 2-connection of an abelian gerbe, which is nothing but an abelian 2-bundle.

The structure group (‘gauge group’) of a 2-bundle is a 2-group, which is the categorification (‘stringification’) of an ordinary group. So-called strict 2-groups are determined in terms of crossed modules of two ordinary groups.

Hence it is natural to ask if there is a crossed module involving the group $\widehat{L\mathrm{SO}(n)}$ that is relevant for spinning strings, such that we get a 2-bundle with structure 2-group from it which would describe the parallel transport of spinning strings.

This indeed seems to be possible.

In the above paper it is shown that there are 2-groups called $\mathcal{P}_k \mathrm{Spin}(n)$ and $\mathcal{L}_k \mathrm{Spin}(n)$ of this kind which fit into a strict exact sequence of 2-groups

One can show that the lift of an ordinary $\mathrm{Spin}(n)$-bundle $E \to M$ to a 2-bundle with structure 2-group $\mathcal{P}_k \mathrm{Spin}(n)$ should be obstructed precisely by $\frac{1}{2}p_1(M)$. (The proof of this is sort of obvious given certain ingredients which I haven’t discussed here, but since it is not written down yet in fully rigorous form I am forced to say ‘should’ for now.)

Furthermore, one can show that $\mathcal{P}_k\mathrm{Spin}(n)$ knows all about $\mathrm{String}(n)$:

There is a notion called taking the geometric realization of the nerve of a category. This amounts essentially to building a simplicial space $|C|$ in which to every tuple of $n$ composable morphisms in the category $C$ there is an $n$-simplex in $|C|$.

(For example when $C = G$ is just a group regarded as a category with only one object and all morphisms invertible, then $|C| \simeq BG$ is nothing but the classifying space of $G$.)

When $C$ is a topological category, the space $|C|$ inherits a topology and becomes a topological space. When $|C|$ is also a 2-group, $|C|$ becomes a topological group.

(So for instance if $G$ is abelian we can regard it as a strict 2-group with the group of objects being trivial, and then $BG$ is indeed itself an abelian group.)

Moreover, the operation $|\cdot|$ extends to a functor from the category of 2-groups to that of topological spaces. Hence we can apply it to the above exact sequence and obtain an exact sequence of topological groups:

The fun thing is now that one can show that $|\mathcal{L}_k \mathrm{Spin}(n)|$ is an Eilenberg-MacLane space $K(\mathbb{Z},2)$. Hence if I write this as

then comparison with the exact sequence

displayed in part I shows that this implies that hence $|\mathcal{P}_k\mathrm{String}(n)|$ is nothing but $\mathrm{String}(n)$

Note that while $\mathrm{String}(n)$ is only a topological group and defined only somewhat indirectly, the 2-group $\mathcal{P}_k \mathrm{Spin}(n)$ is Lie, meaning that every operation in it is indeed smooth, and furthermore it directly involves $\widehat{L\mathrm{SO}(n)}$ (in fact $\widehat{L\mathrm{Spin}(n)}$).

Hence it seems that $\mathcal{P}_{(k=1)}\mathrm{Spin}(n)$-2-bundles are the geometrical bridge between $\widehat{L\mathrm{SO}(n)}$-bundles on loop space and $\mathrm{String}(n)$-bundles on $M$ itself.

This again suggests that the notion of parallel transport of spinning strings is captured by the surface holonomy available in 2-bundles.

Interestingly, while the 2-group $\mathcal{P}_k\mathrm{Spin}(n)$ is infinite dimensional and requires a little bit of technology to get under control, its Lie 2-algebra is equivalent in the categorical sense to a very simple weak Lie 2-algebra called $\mathfrak{spin}_k$, as we show in the last section of the above paper.

Using the technology of weak nonabelian Deligne hypercohomology mentioned in this recent entry and detailed in these notes it is possible to set up the infinitesimal version of 2-bundles with weak gauge Lie 2-algebra $\mathfrak{spin}_k(n)$. This allows to compute the nonabelian Deligne hypercohomolohy class of these ‘infinitesimal 2-bundles’ and hence find a classification for them. I’d expect this to be the Pontryagin class once again. But this remains to be studied.