The String Coffee Table

First recall the situation for $\mathrm{Spin}(n)$. A (Riemannian) manifold $M$ is spin or admits a spin structure if spinning particles can consistently propagate on it.

This is the case iff an $\mathrm{SO}(n)$-bundle

over the manifold can be lifted to a $\mathrm{Spin}(n)$-bundle, where $\Spin(n)$ is (of course) the central extension of $\mathrm{SO}(n)$ by $\mathbb{Z}_2$:

And this is the case iff $M$ is orientable and the second Stiefel-Whitney class $w_2(E) \in H^1(M;\mathbb{Z}/2)$ vanishes.

The situation for $\mathrm{String}(n)$ is similar, but with everything lifted by one dimension. A manifold is string or admits a string structure if spinning strings can consistently propagate on it.

This is the case iff a principal loop-group $L\mathrm{SO}(n)$-bundle

over the free loop space $LM$ can be lifted to a $\widehat{L\mathrm{SO}(n)}$-bundle, where $\widehat{L\mathrm{SO}(n)}$ is a (Kac-Moody-)central extension of $L\mathrm{SO}(n)$ by $U(1)$:

And this is the case iff the so-called string class of $LM$ in $H^3(LM;\mathbb{Z})$ vanishes.

These two conditions on the topology of $LM$ can equivalently be formulated in terms of $M$ itself:

1) The vanishing of the string class in $H^3(LM;\mathbb{Z})$ is equivalent to the vanishing of the Pontryagin class $\frac{1}{2}p_1(E)$ of any vector bundle associated to a principal $\mathrm{Spin}(n)$-bundle $E \to M$.

The string class in $H^3(LM;\mathbb{Z})$ is obtained from the Pontryagin class $p_1/2$ by transgression. This means that it is represented by the 3-form

where $\xi$ is a representative of $p_1/2$, $\mathrm{ev}^*$ is the pull-back by the evaluation map

and $\int_\gamma$ denotes the integral over the $S^1$-factor in $LM \times S^1$.

2) This again is equivalent to the existence of a lift of the structure group of $E$ from $\mathrm{Spin}(n)$ to the topological group called $\mathrm{String}(n)$.

The group $\mathrm{String}(n)$ (or rather a ‘realization’ thereof) is defined as a topological group all of whose homotopy groups equal those of $\mathrm{Spin}(n)$, except for the third one, which has to vanish for $\mathrm{String}(n)$:

Why this makes sense is best made plausible by looking at the first few homotopy groups of $\mathrm{O}(n)$. For $n \gt 8$ they start as follows:

We get from $\mathrm{O}(n)$ to $\mathrm{SO}(n)$ by ‘killing’ the 0th homotopy group, i.e. by going to the connected component.

We get from $\mathrm{SO}(n)$ to $\mathrm{Spin}(n)$ by ‘killing’ the 1st homotopy group, i.e. by going to the universal cover.

We get from $\mathrm{Spin}(n)$ to $\mathrm{String}(n)$ by ‘killing’ the next nonvanishing homotopy group, which is the 3rd.

Since every simple Lie group has $\pi_3 = \mathbb{Z}$ it follows that $\mathrm{Spin}(n)$ cannot be a Lie group. It is only a topological group. (Meaning that it is a topological space on which the group multiplication acts as a continuous map, but that there is no smooth structure with respect to which the group operation were smooth.)

It can be shown and is well known that an equivalent way to define (a realization of) the group $\mathrm{String}(n)$ is as the topological group which makes this sequence of groups exact:

Here $K(\mathbb{Z},2)$ denotes (a realization of) the Eilenberg-MacLane space $K(\mathbb{Z},2)$, which is by definition a topological space all whose homotopy groups vanish, except for the second one, which is isomorphic to $\mathbb{Z}$. In general

by definition.

The importance of string structures in string theory results from the fact that superstrings are nothing but ‘spinning strings’, i.e. fermions on loop space, and that their quantum equations of motion are nothing but a generalized Dirac equation on loop space. (The 0-mode of the worldsheet supercharge is a generalized Dirac(-Ramond) operator on loop space (for the closed string).)

It hence follows by the above discussion that superstrings can propagate consistently only on manifolds which are string, just like an ordinary point-like fermion can propagate consistently only on a manifold that is spin.

More technically, the wavefunction of a point-like fermion is really a section of a $\widehat{\mathrm{SO}(n)} \simeq \mathrm{Spin}(n)$-bundle and hence such a bundle needs to exist over spacetime in order for the fermion to exists.

Similarly, the wavefunction of a fermionic string (‘spinning string’) is really a section of a $\widehat{L\mathrm{SO}(n)}$-bundle over loop space, and hence such a bundle needs to exist over the loop space over spacetime for fermionic strings to exist.

For instance the worldsheet supercharge of the heterotic string is a Dirac operator on loop space for fermions that are also ‘charged’ under an $\mathrm{SO}(32)$- or $E_8 \times E_8$-bundle

In $K$-theory one can form the difference bundle

where $T$ is the tangent bundle and the condition for this bundle to admit a string structure is that the Pontryagin class vanishes, i.e. that

This is in fact the relation which follows from the cancellation of the perturbative anomaly of the effective $\mathrm{SO}(32)$- or $E_8 \times E_8$-field theory obtained from these strings. Hence this famous anomaly is related to the fact that heterotic strings are spinors on loop space.

(This entry is continued in part II.)