We prove the non-abelian Poincaré lemma in higher gauge theory in two different ways. That is, we show that every flat local connective structure is gauge trivial. The first method uses a result by Jacobowitz [J. Differ. Geom. 13, 361 (1978)] which states solvability conditions for differential equations of a certain type. The second method extends a proof by Voronov [Proc. Am. Math. Soc. 140, 2855 (2012)] and yields the explicit gauge parameters connecting a flat local connective structure to the trivial one. Finally, we show how higher flatness appears as a necessary integrability condition of a linear system which featured in recently developed twistor descriptions of higher gauge theories.
REFERENCES
1.
2.
H.
Sati
, U.
Schreiber
, and J.
Stasheff
, in Quantum Field Theory
, edited by B.
Fauser
, J.
Tolksdorf
, and E.
Zeidler
(Birkhauser
, 2009
), p. 303
; e-print arXiv:0801.3480 [math.DG].3.
J. C.
Baez
and J.
Huerta
, Gen. Relativ. Gravitation
43
, 2335
(2011
); e-print arXiv:1003.4485 [hep-th].4.
E.
Witten
, in Proceedings of Strings 95
(University of Southern California
, 1995
); e-print arXiv:hep-th/9507121.5.
6.
7.
B.
Jurco
, C.
Saemann
, and M.
Wolf
, J. High Energy Phys.
1504
, 087
(2015
) ; e-print arXiv:1403.7185 [hep-th].8.
9.
10.
K.
Igusa
, e-print arXiv:0912.0249 [math.AT].11.
R. L.
Bryant
, S. S.
Chern
, R. B.
Gardner
, H. L.
Goldschmidt
, and P. A.
Griffiths
, Mathematical Sciences Research Institute Publications
(Springer-Verlag
, New York
, 1991
), Vol. 18
, p. viii+475
.12.
13.
P.
Aschieri
, L.
Cantini
, and B.
Jurčo
, Commun. Math. Phys.
254
, 367
(2005
) ; e-print arXiv:hep-th/0312154.14.
T.
Bartels
, “Higher gauge theory I: 2-Bundles
,” Ph.D. thesis, University of California
, Riverside, CA,2006
; e-print arXiv:math.CT/0410328 [math.CT].15.
J. C.
Baez
and A. D.
Lauda
, Theory Appl. Categories
12
, 423
(2004
) ; e-print arXiv:math.QA/0307200 [math].16.
J. F.
Martins
and R.
Picken
, Differ. Geom. Appl.
29
, 179
(2011
) ; e-print arXiv:0907.2566 [math.CT].17.
18.
D.
Conduché
, J. Pure Appl. Algebra
34
, 155
(1984
).19.
U.
Schreiber
and K.
Waldorf
, Homol., Homotopy Appl.
13
, 143
(2011
) ; e-print arXiv:0802.0663 [math.DG].20.
R. S.
Ward
, Phys. Lett. A
61
, 81
(1977
).21.
J. D.
Stasheff
, Trans. Am. Math. Soc.
108
, 275
(1963
).22.
J. D.
Stasheff
, Trans. Am. Math. Soc.
108
, 293
(1963
).23.
J.
Stasheff
, Quantum Groups (Leningrad, 1990)
, Lecture Notes in Mathematics
Vol. 1510
(Springer
, Berlin
, 1992
), p. 120137
.24.
25.
M.
Markl
, S.
Shnider
, and J.
Stasheff
, Mathematical Surveys and Monographs
(American Mathematical Society
, 2002
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
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