We combine rational homotopy theory and higher Lie theory to describe the Wess-Zumino-Witten (WZW) term in the M5-brane sigma model. We observe that this term admits a natural interpretation as a twisted 7-cocycle on super-Minkowski spacetime with coefficients in the rational 4-sphere. This exhibits the WZW term as an element in twisted cohomology, with the twist given by the cocycle of the M2-brane. We consider integration of this rational situation to differential cohomology and differential cohomotopy.
REFERENCES
1.
Azcárraga
, J.
and Townsend
, P.
, “Superspace geometry and the classification of supersymmetric extended objects
,” Phys. Rev. Lett.
62
, 2579
–2582
(1989
).2.
Bergshoeff
, E.
, Sezgin
, E.
, and Townsend
, P.
, “Supermembranes and eleven-dimensional supergravity
,” Phys. Lett. B
189
, 75
–78
(1987
).3.
Chryssomalakos
, C.
, de Azcárraga
, J. A.
, Izquierdo
, J. M.
, and Peŕez Bueno
, J. C.
, “The geometry of branes and extended superspaces
,” Nucl. Phys. B
567
, 293
–330
(2000
); e-print arXiv:hep-th/9904137.4.
Bandos
, I.
, Lechner
, K.
, Nurmagambetov
, A.
, Pasti
, P.
, Sorokin
, D.
, and Tonin
, M.
, “Covariant action for the super-fivebrane of M-theory
,” Phys. Rev. Lett.
78
, 4332
(1997
); e-print arXiv:hep-th/9701149.5.
D’Auria
, R.
and Fré
, P.
, “Geometric supergravity in D = 11 and its hidden supergroup
,” Nucl. Phys. B
201
, 101
–140
(1982
).6.
Distler
, J.
and Sharpe
, E.
, “Heterotic compactifications with principal bundles for general groups and general levels
,” Adv. Theor. Math. Phys.
14
, 335
–398
(2010
); e-print arXiv:hep-th/0701244.7.
Fiorenza
, D.
, Rogers
, C. L.
, and Schreiber
, U.
, “L∞-algebras of local observables from higher prequantum bundles
,” Homol., Homotopy Appl.
16
, 107
–142
(2014
); e-print arXiv:1304.6292.8.
Fiorenza
, D.
, Sati
, H.
, and Schreiber
, U.
, “Extended higher cup-product Chern-Simons theories
,” J. Geom. Phys.
74
, 130
–163
(2013
); e-print arXiv:1207.5449.9.
Fiorenza
, D.
, Sati
, H.
, and Schreiber
, U.
, “A higher stacky perspective on Chern-Simons theory
,” in Mathematical Aspects of Quantum Field Theories
, edited byCalaque
, D.
, et al (Springer
, 2014
); e-print arXiv:1301.2580.10.
Fiorenza
, D.
, Sati
, H.
, and Schreiber
, U.
, “Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields
,” Int. J. Geom. Methods Mod. Phys.
12
, 1550018
(2015
); e-print arXiv:1308.5264.11.
Fiorenza
, D.
, Schreiber
, U.
, and Stasheff
, J.
, “Čech-cocycles for differential characteristic classes
,” Adv. Theor. Math. Phys.
16
, 149
–250
(2012
); e-print arXiv:1011.4735.12.
Getzler
, E.
, “Lie theory for nilpotent L∞-algebras
,” Ann. Math.
170
(1
), 271
–301
(2009
); e-print arXiv:math/0404003v4.13.
Henneaux
, M.
and Mezincescu
, L.
, “A sigma model interpretation of Green-Schwarz covariant superstring action
,” Phys. Lett. B
152
, 340
–432
(1985
).14.
Henriques
, A.
, “Integrating L∞-algebras
,” Compos. Math.
144
(4
), 1017
–1045
(2008
); e-print arXiv:math/0603563.15.
Kriz
, I.
and Sati
, H.
, “M-theory, type IIA superstrings, and elliptic cohomology
,” Adv. Theor. Math. Phys.
8
, 345
(2004
); e-print arXiv:hep-th/0404013.16.
Hinich
, V.
, “Descent of Deligne groupoids
,” Int. Math. Res. Not.
5
, 223
–239
(1997
).17.
Lurie
, J.
, “Moduli problems for ring spectra
,” in Proceedings of the International Congress of Mathematicians 2010
(ICM
, 2010
), pp. 1099
–1125
(2011).18.
Nikolaus
, T.
, Schreiber
, U.
, and Stevenson
, D.
, “Principal ∞-bundles—General theory
,” J. Homotopy Relat. Struct.
1
–53
(2014
);Nikolaus
, T.
, Schreiber
, U.
, and Stevenson
, D.
, e-print arXiv:1207.0248.19.
Pridham
, J. P.
, “Unifying derived deformation theories
,” Adv. Math.
224
(3
), 772
-826
(2010
); e-print arXiv:0705.0344.20.
Sati
, H.
, “Geometric and topological structures related to M-branes
,” Proc. Symp. Pure Math.
81
, 181
-236
(2010
); e-print arXiv:1001.5020 [math.DG].21.
22.
Sati
, H.
, Schreiber
, U.
, and Stasheff
, J.
, “L∞ algebra connections and applications to String- and Chern-Simons n-transport
,” in Recent Developments in QFT
(Birkhäuser
, 2009
), pp. 303
-424
; e-print arXiv:0801.3480.23.
Sati
, H.
, Schreiber
, U.
, and Stasheff
, J.
, “Fivebrane structures
,” Rev. Math. Phys.
21
, 1
-44
(2009
); e-print arXiv:0805.0564 [math.AT].24.
25.
Schreiber
, U.
, Structure theory for higher WZW terms, lecture notes accompanying a minicourse at H. Sati (org.),Flavors of cohomology, Pittsburgh, June 2015, available at ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms.26.
Sullivan
, D.
, “Infinitesimal computations in topology
,” Publ. Math. IHES
47
, 269
-331
(1977
).27.
Witten
, E.
, “On holomorphic factorization of WZW and coset models
,” Commun. Math. Phys.
144
, 189
-212
(1992
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.