nLab Fivebrane structure

Idea

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 $g : X \to B Spin(n)$ of the tangent bundle associated to a Spin group-principal bundle through the next step in the Whitehead tower of $O(n)$, called $B String(n)$ – the delooping of the String group:

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

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) $O(n)\langle 2\rangle$ for $Spin(n)$ and $O(n)\langle 6\rangle$ for $String(n)$, respectively.

They wrote $O(n)\langle 8\rangle$ for the next step in the Whitehead tower of $O(n)$ (note that this is only the next step for $n \gt 6$; for lower $n$ there are intermediate steps, as can be seen in the table at orthogonal group).

It was Hisham Sati who first realized that a lift of the tangent bundle $T X$ 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 $O(n)\langle 8\rangle$ the name Fivebrane group: $Fivebrane(n)$.

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

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

The obstruction class to this lift is a fractional multiply of the second Pontrjagin class. Namely the generator of $H^8(B String, \mathbb{Z})$ is $\frac{1}{6}p_2$,

$$\array{ B String &\stackrel{\frac{1}{6} p_2}{\to}& B^8 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 6}} \\ B SO &\stackrel{p_2}{\to}& B^8 \mathbb{Z} } \,.$$

(stated in SSS2, then in DHH, also follows from the index theory argument leading to (3.3) in Witten 96).

The Fivebrane group is the loop space object of the corresponding homotopy fiber

$$\array{ B Fivebrane &\to& * \\ \downarrow && \downarrow \\ B String &\stackrel{\frac{1}{6} p_2}{\to}& B^7 U(1)& }$$

and so, by the universal property of the homotopy pullback, String-structures $\hat g$ lift to Fivebrane structures precisely if $\frac{1}{6}p_2(\hat g)$ is trivial in cohomology

$$\array{ && B Fivebrane &\to& * \\ & {}^{\mathllap{\hat \hat g}}\nearrow & \downarrow && \downarrow \\ X &\stackrel{\hat g}{\to}& B String &\stackrel{\frac{1}{6}p_2}{\to}& B^7 U(1) } \,.$$

In (SSS2) the physical interpretation of this topological lift was established by comparison with known quantum anomaly cancellation conditions in dual heterotic string theory.

The term “Fivebrane” apparently quickly caught on in the mathematical community, for instance in (DouglasHenriquesHill).

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. See differential fivebrane structure.

Related concepts

• spin structure, spin connection;

• string structure, differential string structure;

• fivebrane structure, differential fivebrane structure

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$\vdots$
	$\downarrow$
ninebrane 10-group	$\mathbf{B}Ninebrane$	ninebrane structure	third fractional Pontryagin class
	$\downarrow$
fivebrane 6-group	$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
general linear group	$\mathbf{B}GL$	smooth manifold

(all hooks are homotopy fiber sequences)

References

The notion was introduced in:

• Hisham Sati, Urs Schreiber, Jim Stasheff, Fivebrane structures, Reviews in Mathematical Physics, 21 10 (2009) 1197-1240 [arXiv:0805.0564, doi:10.1142/S0129055X09003840]

Brief mentioning in:

• Chris Douglas, André Henriques, Michael Hill, Homological obstructions to string orientations, Int. Math. Res. Notices 18 (2011) 4074-4088 [arXiv:0810.2131, doi:10.1093/imrn/rnq237]

On the differential refinement (the fivebrane version of differential string structure):

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures [arXiv:0910.4001]

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech Cocycles for Differential characteristic Classes [arXiv:1011.4735]

The M-theoretic version of Fivebrane structure (“M5-brane-structure”), and rigorously derived from 5-brane anomaly cancellation via Hypothesis H:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: Exp. 3.2, Rem. 4.3, Thm. 4.8 in: Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane, Comm. Math. Phys. 384 403–432 (2021) [arXiv:1906.07417, doi:10.1007/s00220-021-03951-0]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: around (12) in Twisted cohomotopy implies twisted String structure on M5-branes, J. Mathematical Physics 62 (2021) 042301 [arXiv:2002.11093, doi:10.1063/5.0037786]

Applications of Fivebrane structures:

• Boris Botvinnik, Mohammed Labbi, Highly connected manifolds of positive $p$-curvature, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 [arXiv:1201.1849, doi:10.1090/S0002-9947-2014-05939-4]