We construct explicitly classical and quantum supercharges satisfying the standard |${\cal N} = 4$| supersymmetry algebra in the supersymmetric sigma models describing the motion over hyper-Kähler with torsion manifolds. One member of the family of superalgebras thus obtained is equivalent to the superalgebra derived and formulated earlier in purely mathematical framework.
REFERENCES
1.
P. S.
Howe
and G.
Papadopoulos
, Phys. Lett. B
379
, 80
(1996
);e-print arXiv:hep-th/9602108.
2.
G. W.
Gibbons
, G.
Papadopoulos
, and K. S.
Stelle
, Nucl. Phys. B
508
, 623
(1997
);e-print arXiv:hep-th/9706207.
3.
C. M.
Hull
, e-print arXiv:hep-th/9910028.4.
F.
Delduc
and E.
Ivanov
, Nucl. Phys. B
855
, 815
(2012
);e-print arXiv:1107.1429.
5.
N. E.
Mavromatos
, J. Phys. A
21
, 2279
(1988
).6.
J.-M.
Bismut
, Math. Ann.
284
, 681
(1989
).7.
A. V.
Smilga
, J. Math. Phys.
53
, 042103
(2012
);e-print arXiv:1104.3986.
8.
R.
Jackiw
and C.
Rebbi
, Phys. Rev. D
14
, 517
(1976
);S.
Krivonos
, O.
Lechtenfeld
, and A.
Sutulin
, Phys. Rev. D
81
, 085021
(2010
), e-print arXiv:1001.2659;A. V.
Smilga
, Symmetry, Integr. Geom.: Methods Appl.
7
, 105
(2011
);e-print arXiv:1105.3935.
9.
E.
Witten
, Nucl. Phys. B
188
, 513
(1981
);10.
D. Z.
Freedman
and P. K.
Townsend
, Nucl. Phys. B
177
, 282
(1981
).11.
E. A.
Ivanov
and A. V.
Smilga
, Int. J. Mod. Phys. A
27
, 042103
(2012
);e-print arXiv:1012.2069.
12.
H.
Braden
, Ann. Phys. (N.Y.)
171
, 433
(1986
).13.
H. J.
Grönewold
, Physica (Amsterdam)
12
, 405
(1946
);I. E.
Moyal
, Proc. Cambridge Philos. Soc.
45
, 99
(1949
).14.
A. V.
Smilga
, Nucl. Phys.
B292
, 363
(1987
).15.
See, e.g., Propositions 1.4.23 and 1.4.25 in
L. I.
Nicolaescu
, Notes on Seiberg-Witten Theory
(AMS
, Providence
, 2000
).16.
M.
Verbitsky
, Funct. Anal. Appl.
24
(2
), 70
(1990
);J.-M.
Figueroa-O’Farrill
, C.
Köhl
, and B.
Spence
, Nucl. Phys.
B503
, 614
(1997
);e-print arXiv:hep-th/9705161.
17.
A.
Kirchberg
, J. D.
Lange
, and A.
Wipf
, Ann. Phys.
315
, 467
(2005
);e-print arXiv:hep-th/0401134.
18.
E.
Ivanov
and O.
Lechtenfeld
, J. High Energy Phys.
0309
, 073
(2003
);e-print arXiv:hep-th/0307111.
19.
M.
Konyushikhin
and A.
Smilga
, Phys. Lett. B
689
, 95
(2010
);e-print arXiv:0910.5162.
20.
e-print arXiv:math/0112215.
21.
${\cal N}$
counts the number of real supercharges such that the minimal supersymmetry (involving double degeneracy of all excited states) corresponds to ${\cal N} = 2$
.22.
Unfortunately, mathematicians and physicists use nowadays rather different languages, even when the problems they discuss are identical. In most of the cases, we do not understand each other without translation. This article is written in the mixture of two languages in a hope that it will be understandable to both communities.
23.
One can as well consider the systems involving a non-Abelian field.
24.
25.
In Sec. VII of that paper, the superalgebra involving untwisted ∂ and ∂J was discussed, but a commutator like {∂†, ∂J} vanishes only for a metric with constant determinant g, if the Hermitian conjugation is defined in a standard way with the covariant measure (22).
© 2012 American Institute of Physics.
2012
American Institute of Physics
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