Fivebrane structure in nLab
Article Images
Context
Cohomology
Special and general types
-
group cohomology, nonabelian group cohomology, Lie group cohomology
-
-
cohomology with constant coefficients / with a local system of coefficients
Special notions
Variants
-
differential cohomology
Operations
Theorems
Higher spin geometry
spin geometry, string geometry, fivebrane geometry …
Ingredients
Spin geometry
String geometry
Fivebrane geometry
Ninebrane geometry
String theory
Ingredients
Critical string models
Extended objects
Topological strings
Backgrounds
Phenomenology
Contents
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 with spin structure is a lift of the classifying map of the tangent bundle associated to a Spin group-principal bundle through the next step in the Whitehead tower of , called – the delooping of the String group:
The names “Spin” and “String” both derive from the role these structures play in quantum field theory: a spin structure is required on 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) for and for , respectively.
They wrote for the next step in the Whitehead tower of (note that this is only the next step for ; for lower 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 to this highly connected structure group is related to 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 the name Fivebrane group: .
Accordingly, a Fivebrane structure(n) on a manifold with string structure is a lift of to
The obstruction class to this lift is a fractional multiply of the second Pontrjagin class. Namely the generator of is ,
(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
and so, by the universal property of the homotopy pullback, String-structures lift to Fivebrane structures precisely if is trivial in cohomology
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.
-
fivebrane structure, differential fivebrane structure
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 -curvature, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 [arXiv:1201.1849, doi:10.1090/S0002-9947-2014-05939-4]
Last revised on January 11, 2023 at 10:35:13. See the history of this page for a list of all contributions to it.