nLab
Fivebrane structure

The notion of Fivebrane structure is the next higher analog of that of spin structure and string structure.

Recall from the discussion there that a string structure on manifold $X$ with spin structure is a lift $\hat{g}$ of the classifying map $g:X\to ℬ\mathrm{Spin}(n)$ of the tangent bundle associated to a Spin group-principal bundle through the next step in the Whitehead tower of $O(n)$, called $ℬ\mathrm{String}(n)$ – the delooping of the String group:

$$\begin{array}{ccc}& & ℬ\mathrm{String}(n)\\ & \hat{g}\nearrow & \downarrow \\ X& \stackrel{g}{\to }& ℬ\mathrm{Spin}(n)\end{array}.$$\array{ && \mathcal{B} String(n) \\ & {\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B}Spin(n) } \,.

The names “Spin” and “String” both derive from the role these structures play in quantum field theory: a spin structure is required on $X$ for it to serve as a target space for spinning particles (superparticles), while a string structure is required for it to serves as a target for “spinning strings” – superstrings – (see heterotic string theory? for more). Topologists just say (said) $O(n)⟨2⟩$ for $\mathrm{Spin}(n)$ and $O(n)⟨6⟩$ for $\mathrm{String}(n)$, respectively.

They wrote $O(n)⟨8⟩$ for the next step in the Whitehead tower of $O(n)$.

It was Hisham Sati who first realized that a lift of the tangent bundle $TX$ to this highly connected structure group is related to $X$ serving as a target for “spinning 5-branes” – super-5-branes – in what is called dual heterotic string theory?. Following the history of the term String group he gave the topological group $O(n)⟨8⟩$ the name Fivebrane group: $\mathrm{Fivebrane}(n)$.

Accordingly, a Fivebrane structure(n) on a manifold $X$ with string structure is a lift of $\hat{g}:X\to ℬ\mathrm{String}(n)$ to $\hat{\hat{g}}$

$$\begin{array}{ccc}& & ℬ\mathrm{Fivebrane}(n)\\ & \hat{g}\nearrow & \downarrow \\ X& \stackrel{g}{\to }& ℬ\mathrm{String}(n)\end{array}.$$\array{ && \mathcal{B} Fivebrane(n) \\ & {\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B}String(n) } \,.

In the article

• H. Sati, U. Schreiber, J. Stasheff, Fivebrane structures , Reviews in Math. Phys. (arXiv)

the physical interpretation of this topological lift was established by comparison with known quantum anomaly cancellaton conditions in dual heterotic string theory?.

Thee term “Fivebrane” apparently quickly caught on in the mathematical community, for instance in

• Christopher L. Douglas, André G. Henriques, Michael A. Hill, Homological obstructions to string orientations (arXiv)

Since gauge theory is not just about principal bundles, but about principal bundles with connection, what matters in physics is not just the topological Spin-, String- and Fivebrane Structures, but their refinement to differential nonabelian cohomology. The full picture of such differential Fivebrane structures in heterotic String theory is described at twisted differential String- and Fivebrane structures.

further references

• blog entry: Dual formulation of String theory and Fivebrane structures