nLab nonabelian differential cohomology

Idea

What may be called nonabelian differential cohomology – in combination of nonabelian cohomology and differential cohomology – is a notion of connections on higher bundles with higher gauge group being a possibly non-abelian n-group.

Early motivation was the observation (SSS12) that with the Green-Schwarz mechanism imposed, the B-field in heterotic string theory combines with the ordinary non-abelian gauge field into a non-abelian higher connection (a twisted differential string structure), and analogously so after lifting the situation to the C-field in Hořava-Witten theory (FSS15).

Later, the desire to more accurately model more of the expected subtle topological properties of the C-field (such as the shifted C-field flux quantization combined with Page charge-quantization) led to the hypothesis (“Hypothesis H”) that it ought to be flux quantized in unstable differential 4-Cohomotopy, hence with homotopy type of its higher gauge group being the loop $\infty$-group of the 4-sphere. This Hypothesis H turns out to subtly reproduce the Green-Schwarz mechanism in its lift to Hořava-Witten theory (FSS22) and also that on M5-branes (SS20, FSS21).

Generally, one may understand differential nonabelian cohomology as modelling flux quantized higher gauge fields (SS23).

For example, also the “Hypothesis K” of D-brane charge quantization in topological K-theory is, at face value, a flux quantization (of the RR-field combined with the B-field) in a non-abelian differential cohomology (see the overview here) which however may be and commonly is regarded as twisted abelian namely as twisted differential K-theory, by regarding the B-field as a “background field” and considering the RR-field in its dependence.

Related concepts

• nonabelian cohomology

• differential cohomology

• higher gauge theory

References

Via higher Cartan connections from (adjusted) Weil algebras

Early explorative notes:

• Urs Schreiber, On nonabelian differential cohomology, March 17, 2008, Slides.

The local structure of $L_\infty$-algebra valued differential forms, via dg-algebra homomorphisms out of (“adjusted”) Weil algebras in the de Rham complex of the base manifold:

• Hisham Sati, Urs Schreiber, Jim Stasheff, example 5 in section 6.5.1, p. 54 of $L_\infty$ algebra connections and applications to String- and Chern-Simons n-transport, in: Quantum Field Theory, Birkhäuser (2009) 303-424 [doi:0801.3480, doi:10.1007/978-3-7643-8736-5_17]

The Lie integration of this local structure to globally possibly nontrivial actual connections on higher bundles:

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Čech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Physics 16:1 (2012), 149–250, arXiv:1011.4735, doi:10.1007/BF02104916.

For the application of this construction to modelling the Green-Schwarz mechanism see below.

Via the nonabelian character map

A more axiomatic formulation of differential non-abelian cohomology (not yet proven to subsume the above construction) in non-abelian generalization of the original Hopkins-Singer construction of abelian (namely Whitehead-generalized cohomology) differential cohomology, now using a nonabelian generalization of the Chern-Dold character map:

• Domenico Fiorenza, Urs Schreiber, Hisham Sati, Def. 9.3 (p. 161) in: The Character Map in Non-Abelian Cohomology, World Scientific (2023) [doi:10.1142/13422]

On how this generally serves to reflect flux quantization in higher gauge theories:

• Hisham Sati, Urs Schreiber, Flux Quantization on Phase Space [arXiv:2312.12517]

reviewed in:

• Hisham Sati, Urs Schreiber, Introduction to Hypothesis H (here)

Differential cohomotopy for the 11d SuGra C-field

The example of unstable (and as such non-abelian) differential Cohomotopy (in the context of modeling the supergravity C-field via Hypothesis H):

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, §4 in: The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56 102301 (2015) [arXiv:1506.07557, doi:10.1063/1.4932618]

• Daniel Grady, Hisham Sati, Def. 3.2 in: Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers, Journal of Geometry and Physics 165 (2021) 104203 [arXiv:2001.07640, doi:10.1016/j.geomphys.2021.104203]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Ex. 9.3 in: The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalized, World Scientific (2023) [arXiv:2009.11909, doi:10.1142/13422]

Higher gauge theory of the Green-Schwarz mechanism

Discussion of higher gauge theory modeling the Green-Schwarz mechanisms for anomaly cancellation in heterotic string theory, on M5-branes, and in related systems in terms of some kind of nonabelian differential cohomology (ordered by arXiv time-stamp):

• Hisham Sati, Urs Schreiber, Jim Stasheff, pp. 13 in: $L_\infty$-algebra connections and applications to String- and Chern-Simons $n$-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 [arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17]

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Comm. Math. Phys. 315 (2012) 169-213 (arXiv:0910.4001, doi:10.1007/s00220-012-1510-3)

  (via adjusted Weil algebras, see there for more)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, §3.7, §3.8 in: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory, Adv. Theor. Math. Phys. 18 (2014) 229-321 [arXiv:1201.5277, euclid:atmp/1414414836]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The $E_8$ moduli 3-stack of the C-field in M-theory, Comm. Math. Phys. 333 1 (2015) 117-151 [arXiv:1202.2455, doi:10.1007/s00220-014-2228-1]

• Clay Cordova, Thomas Dumitrescu, Kenneth Intriligator, Exploring 2-Group Global Symmetries, J. High Energ. Phys. 2019 184 (2019) (arXiv:1802.04790, doi:10.1007/JHEP02(2019)184)

• Francesco Benini, Clay Cordova, Po-Shen Hsin, On 2-Group Global Symmetries and Their Anomalies, J. High Energ. Phys. 2019 118 (2019) (arXiv:1803.09336, doi:10.1007/JHEP03(2019)118)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation, Reviews in Mathematical Physics 34 05 (2022) 2250013 [doi:10.1142/S0129055X22500131, arXiv:2008.08544]

• Clay Cordova, Thomas T. Dumitrescu, Kenneth Intriligator, 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories, J. High Energ. Phys. 2021, 252 (2021) (arXiv:2009.00138, doi:10.1007/JHEP04(2021)252)

• Michele Del Zotto, Kantaro Ohmori, 2-Group Symmetries of 6D Little String Theories and T-Duality, Annales Henri Poincaré 22 (2021) 2451–2474 [arXiv:2009.03489, doi:10.1007/s00023-021-01018-3]

• Hisham Sati, Urs Schreiber, The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes [arXiv:2011.06533]

• Hisham Sati, Urs Schreiber, §2.9 in: M/F-Theory as Mf-Theory, Reviews in Mathematical Physics 35 10 (2023) [doi:10.1142/S0129055X23500289, arXiv:2103.01877]

• Yasunori Lee, Kantaro Ohmori, Yuji Tachikawa, Matching higher symmetries across Intriligator-Seiberg duality, J. High Energ. Phys. 2021 114 (2021) [arXiv:2108.05369, doi:10.1007/JHEP10(2021)114]

• Monica Jinwoo Kang, Sungkyung Kang, Central extensions of higher groups: Green-Schwarz mechanism and 2-connections [arXiv:2311.14666]