In this article, we treat G2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G2-structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G2-structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry.

1.
Akbulut
,
S.
,
Efe
,
B.
, and
Salur
,
S.
, “
Mirror duality in a Joyce manifold
,”
Adv. Math.
223
,
444
453
(
2010
).
2.
Akbulut
,
S.
and
Salur
,
S.
, “
Mirror duality via G2 and Spin(7) manifolds
,” in
Arithmetic and Geometry Around Quantization
,
Progress in Mathematics
Vol.
279
(
Birkhäuser Boston Inc.
,
Boston, MA
,
2010
), pp.
1
21
.
3.
Arikan
,
M. F.
,
Cho
,
H.
, and
Salur
,
S.
, “
Existence of compatible contact structures on G2-manifolds
,”
Asian J. Math.
17
(
2
),
321
333
(
2013
).
4.
Arikan
,
M. F.
,
Cho
,
H.
, and
Salur
,
S.
, “
Contact structures on G2-manifolds and spin 7-manifolds
,” preprint arXiv:1207.2046 (
2012
).
5.
Arnol'd
,
V.
,
Mathematical Methods of Classical Mechanics
(
Springer
,
1989
), Vol.
60
.
6.
Atiyah
,
M.
and
Witten
,
E.
, “
M-theory dynamics on a manifold of G2 holonomy
,”
Adv. Theor. Math. Phys.
6
,
1
106
(
2002
).
7.
Baez
,
J.
,
Hoffnung
,
A.
, and
Rogers
,
C.
, “
Categorified symplectic geometry and the classical string
,”
Commun. Math. Phys.
293
,
701
725
(
2010
).
8.
Baez
,
J.
and
Rogers
,
C.
, “
Categorified symplectic geometry and the string Lie 2-algebra
,”
Homology, Homotopy Appl.
12
,
221
236
(
2010
).
9.
Brown
,
R.
and
Gray
,
A.
, “
Vector cross products
,”
Comment. Math. Helv.
42
,
222
236
(
1967
).
10.
Bryant
,
R.
, “
Metrics with exceptional holonomy
,”
Ann. Math.
126
(
2
),
525
576
(
1987
).
11.
Bryant
,
R.
, “
Some remarks on G2-structures
,” in
Proceedings of Gökova Geometry-Topology Conference 2005
(
Gökova Geometry/Topology Conference (GGT)
,
Gökova
,
2006
), pp.
75
109
.
12.
Bryant
,
R.
and
Salamon
,
S.
, “
On the construction of some complete metrics with exceptional holonomy
,”
Duke Math. J.
58
,
829
850
(
1989
).
13.
Cabrera
,
F.
,
Monar
,
M.
, and
Swann
,
A.
, “
Classification of G2-structures
,”
J. London Math. Soc.
53
(
2
),
407
416
(
1996
).
14.
Cantrijn
,
F.
,
Ibort
,
A.
, and
de León
,
M.
, “
Hamiltonian structures on multisymplectic manifolds
,”
Rend. Sem. Mat. Univ. Politec. Torino
54
,
225
236
(
1996
).
15.
Cantrijn
,
F.
,
Ibort
,
A.
, and
de León
,
M.
, “
On the geometry of multisymplectic manifolds
,”
J. Austral. Math. Soc. Ser. A
66
,
303
330
(
1999
).
16.
Carathéodory
,
C.
, “
Über die Variationsrechnung bei mehrfachen Integralen
,”
Acta Szeged
4
,
193
216
(
1929
).
17.
Cariñena
,
J. F.
,
Crampin
,
M.
, and
Ibort
,
A.
, “
On the multisymplectic formalism for first order field theories
,”
Differential Geom. Appl.
1
,
345
374
(
1991
).
18.
Cho
,
H.
,
Salur
,
S.
, and
Todd
,
A. J.
, “
A note on closed G2-structures and 3-manifolds
,” preprint arXiv:1112.0830 (
2011
).
19.
Cleyton
,
R.
and
Ivanov
,
S.
, “
On the geometry of closed G2-structures
,”
Commun. Math. Phys.
270
,
53
67
(
2007
).
20.
De Donder
,
T.
, “
Theorie invariantive du calcul des variations
,”
Ann. Math
36
,
607
(
1935
).
21.
Echeverría-Enríquez
,
A.
,
Munoz-Lecanda
,
M. C.
, and
Román-Roy
,
N.
, “
Multivector field formulation of Hamiltonian field theories: Equations and symmetries
,”
J. Phys. A
32
,
8461
(
1999
).
22.
Echeverria-Enriquez
,
A.
,
Munoz-Lecanda
,
M. C.
, and
Roman-Roy
,
N.
, “
Geometry of multisymplectic Hamiltonian first order field theories
,”
J. Math. Phys.
41
,
7402
(
2000
).
23.
Fernández
,
M.
, “
An example of a compact calibrated manifold associated with the exceptional Lie group G2
,”
J. Diff. Geom.
26
,
367
370
(
1987
).
24.
Fernández
,
M.
, “
A family of compact solvable G2-calibrated manifolds
,”
Tohoku Math. J.
39
(
2
),
287
289
(
1987
).
25.
Fernández
,
M.
and
Gray
,
A.
, “
Riemannian manifolds with structure group G2
,”
Ann. Mat. Pura Appl. (4)
132
,
19
45
(
1983
).
26.
Fernández
,
M.
and
Iglesias
,
T.
, “
New examples of Riemannian manifolds with structure group G2
,”
Rend. Circ. Mat. Palermo (2)
35
,
276
290
(
1986
).
27.
Forger
,
M.
,
Paufler
,
C.
, and
Römer
,
H.
, “
The Poisson bracket for Poisson forms in multisymplectic field theory
,”
Rev. Math. Phys.
15
,
705
743
(
2003
).
28.
Forger
,
M.
,
Paufler
,
C.
, and
Römer
,
H.
, “
Hamiltonian multivector fields and Poisson forms in multisymplectic field theory
,”
J. Math. Phys.
46
,
112903
, (
2005
).
29.
Forger
,
M.
and
Römer
,
H.
, “
A Poisson bracket on multisymplectic phase space
,”
Rep. Math. Phys.
48
,
211
218
(
2001
).
30.
Gopakumar
,
R.
and
Vafa
,
C.
, “
M-theory and topological strings – II
,” preprint arXiv:hep-th/9812127 (
1998
).
31.
Gotay
,
M. J.
,
Isenberg
,
J.
, and
Marsden
,
J. E.
, “
Momentum maps and classical relativistic fields; 1, covariant field theory
” (
1998
).
32.
Gray
,
A.
, “
Vector cross products on manifolds
,”
Trans. Am. Math. Soc.
141
,
465
504
(
1969
).
33.
Gukov
,
S.
,
Yau
,
S.-T.
, and
Zaslow
,
E.
, “
Duality and fibrations on G2-manifolds
,”
Turkish J. Math.
27
,
61
97
(
2003
).
34.
Harvey
,
R.
and
Lawson
 Jr.,
H. B.
, “
Calibrated geometries
,”
Acta Math.
148
,
47
157
(
1982
).
35.
Hélein
,
F.
,
Multisymplectic Formalism and the Covariant Phase Space
,
London Mathematical Society Lecture Note Series
Vol.
394
(
2011
), pp.
94
126
.
36.
Hélein
,
F.
and
Kouneiher
,
J.
, “
Finite dimensional Hamiltonian formalism for gauge and quantum field theories
,”
J. Math. Phys.
43
,
2306
(
2002
).
37.
Hélein
,
F.
and
Kouneiher
,
J.
, “
Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl
,”
Adv. Theor. Math. Phys.
8
,
565
601
(
2004
).
38.
Hélein
,
F.
and
Kouneiher
,
J.
, “
The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables
,”
Adv. Theor. Math. Phys.
8
,
735
777
(
2004
).
39.
Ibort
,
A.
, “
Multisymplectic geometry: Generic and exceptional
,” in
Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltrú, 2000)
,
Publ. R. Soc. Mat. Esp.
Vol.
3
(
R. Soc. Mat. Esp.
,
Madrid
,
2001
), pp.
79
88
.
40.
Joyce
,
D.
,
Compact Manifolds with Special Holonomy
,
Oxford Mathematical Monographs
(
Oxford University Press
,
Oxford
,
2000
), pp.
xii+436
.
41.
Joyce
,
D.
,
Riemannian Holonomy Groups and Calibrated Geometry
,
Oxford Graduate Texts in Mathematics
Vol.
12
(
Oxford University Press
,
Oxford
,
2007
), pp.
x+303
.
42.
Karigiannis
,
S.
, “
Deformations of G2- and Spin(7)-structures
,”
Can. J. Math.
57
,
1012
1055
(
2005
).
43.
Leung
,
N. C.
, “
Topological quantum field theory for Calabi-Yau threefolds and G2-manifolds
,”
Adv. Theor. Math. Phys.
6
,
575
591
(
2002
).
44.
Madsen
,
T.
and
Swann
,
A.
, “
Homogeneous spaces, multi-moment maps and (2, 3)-trivial algebras
,”
AIP Conf. Proc.
1360
,
51
(
2011
).
45.
Madsen
,
T.
and
Swann
,
A.
, “
Multi-moment maps
,”
Adv. Math.
229
,
2287
2309
(
2012
).
46.
Madsen
,
T.
and
Swann
,
A.
, “
Closed forms and multi-moment maps
,”
Geom. Dedic.
165
,
25
52
(
2013
).
47.
Marle
,
C.-M.
, “
The Schouten-Nijenhuis bracket and interior products
,”
J. Geom. Phys.
23
,
350
359
(
1997
).
48.
Marsden
,
J.
and
Shkoller
,
S.
, “
Multisymplectic geometry, covariant Hamiltonians, and water waves
,” in
Mathematical Proceedings of the Cambridge Philosophical Society
(
Cambridge Univ Press
,
1999
), Vol.
125
, pp.
553
575
.
49.
McDuff
,
D.
and
Salamon
,
D.
,
Introduction to Symplectic Topology
, 2nd ed., Oxford Mathematical Monographs (
The Clarendon Press Oxford University Press
,
New York
,
1998
), pp.
x+486
.
50.
Michor
,
P.
, “
Remarks on the Schouten-Nijenhuis bracket
,” in
Proceedings of the Winter School on Geometry and Physics (Srní, 1987)
(
Rend. Circ. Mat. Palermo (2) Suppl.
,
1987
), Vol.
16
, pp.
207
215
.
51.
Paufler
,
C.
and
Römer
,
H.
, “
Geometry of Hamiltonian n-vector fields in multisymplectic field theory
,”
J. Geom. Phys.
44
,
52
69
(
2002
).
52.
Pelling
,
S.
and
Rogers
,
A.
, “
Multisymplectic BRST
,”
Int. J. Geom. Methods Mod. Phys.
10
,
1360012
(
2013
).
53.
Román-Roy
,
N.
, “
Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories
,”
SIGMA
5
,
100
(
2009
).
54.
Salamon
,
S.
,
Riemannian Geometry and Holonomy Groups
,
Pitman Research Notes in Mathematics Series
Vol.
201
(
Longman Scientific & Technical
,
Harlow
,
1989
), pp.
viii+201
.
55.
Salur
,
S.
and
Santillan
,
O.
, “
Mirror symmetry aspects for compact G2-manifolds
,” preprint arXiv:0707.1356 (
2007
).
56.
Salur
,
S.
and
Santillan
,
O.
, “
New Spin(7) holonomy metrics admitting G2 holonomy reductions and M-theory/type-IIA dualities
,”
Phys. Rev. D
79
,
086009
(
2009
).
57.
Cannas da Silva
,
A.
,
Lectures on Symplectic Geometry
,
Lecture Notes in Mathematics
Vol.
1764
(
Springer-Verlag
,
Berlin
,
2001
), pp.
xii+217
.
58.
Vaisman
,
I.
,
Lectures on the Geometry of Poisson Manifolds
,
Progress in Mathematics
Vol.
118
(
Birkhäuser Verlag
,
Basel
,
1994
), pp.
viii+205
.
59.
Vankerschaver
,
J.
,
Yoshimura
,
H.
, and
Leok
,
M.
, “
On the geometry of multi-Dirac structures and Gerstenhaber algebras
,”
J. Geom. Phys.
61
,
1415
1425
(
2011
).
60.
Vey
,
D.
, “
The notion of observables in multisymplectic geometry
,” preprint arXiv:1203.5895 (
2012
).
61.
Weyl
,
H.
, “
Geodesic fields in the calculus of variation for multiple integrals
,”
Ann. Math.
36
,
607
629
(
1935
).
62.
Xu
,
P.
, “
Gerstenhaber algebras and BV-algebras in Poisson geometry
,”
Commun. Math. Phys.
200
,
545
560
(
1999
).
You do not currently have access to this content.