We present simple models of ${\cal N}{=}\,4$ supersymmetric mechanics with ordinary and mirror linear (4, 4, 0) multiplets that give a transparent description of Hyper-Kähler with Torsion (HKT), Clifford Kähler with Torsion (CKT), and Octonionic Kähler with Torsion (OKT) geometries. These models are treated in the ${\cal N}{ = }\,4$ and ${\cal N}{ = }2\,$ superfield approaches, as well as in the component approach. Our study makes manifest that the CKT and OKT supersymmetric sigma models are distinguished from the more simple HKT models by the presence of extra holomorphic torsion terms in the supercharges.
REFERENCES
1.
E.
Witten
, “Dynamical breaking of supersymmetry
,” Nucl. Phys. B
188
, 513
(1981
);2.
J.
Michelson
and A.
Strominger
, “The geometry of (super)conformal quantum mechanics
,” Commun. Math. Phys.
213
, 1
(2000
); e-print arXiv:hep-th/9907191.3.
J.
Michelson
and A.
Strominger
, “Superconformal multi-black-hole quantum mechanics
,” JHEP
09
(1999
) 005
; e-print arXiv:hep-th/9908044.4.
R.
Britto-Pacumio
, J.
Michelson
, A.
Strominger
, and A.
Volovich
, “Lectures on superconformal quantum mechanics and multi-black hole moduli spaces
,” Cargese 1999
, Progress in String Theory and M-theory
, p. 235
, L.
Baulieu
et al (Eds.) (Dordrecht, Netherlands
, Kluwer
, 2001
, 417p); e-print arXiv:hep-th/9911066.5.
J.
Gutowski
and G.
Papadopoulos
, “The dynamics of very special black holes
,” Phys. Lett. B
472
, 45
(2000
); e-print arXiv:hep-th/9910022.6.
J.
Gutowski
and G.
Papadopoulos
, “Moduli spaces for four-dimensional and five-dimensional black holes
,” Phys. Rev. D
62
, 064023
(2000
); e-print arXiv:hep-th/0002242.7.
S.
Fedoruk
, E.
Ivanov
, and O.
Lechtenfeld
, “Superconformal mechanics
,” J. Phys. A
45
, 173001
(2012
); e-print arXiv:1112.1947 [hep-th].8.
R. A.
Coles
and G.
Papadopoulos
, “The geometry of the one-dimensional supersymmetric nonlinear sigma models
,” Class. Quant. Grav.
7
, 427
(1990
).9.
C. M.
Hull
, “The geometry of supersymmetric quantum mechanics
,” Queen Mary and Westfield college, # QMW-99-16, e-print arXiv:hep-th/9910028.10.
G. W.
Gibbons
, G.
Papadopoulos
, and K. S.
Stelle
, “HKT and OKT geometries on soliton black hole moduli spaces
,” Nucl. Phys. B
508
, 623
(1997
); e-print arXiv:hep-th/9706207.11.
A.
Pashnev
and F.
Toppan
, “On the classification of N extended supersymmetric quantum mechanical systems
,” J. Math. Phys.
42
, 5257
(2001
); e-print arXiv:hep-th/0010135.12.
E. A.
Ivanov
and A. V.
Smilga
, “Dirac operator on complex manifolds and supersymmetric quantum mechanics
,” Int. J. Mod. Phys. A
27
, 1230024
(2012
); e-print arXiv:1012.2069 [hep-th].13.
${\cal N}$
counts the number of real supercharges.14.
H. W.
Braden
, “Sigma models with torsion
,” Phys. Lett.
B163
, 171
(1985
);R.
Rohm
and E.
Witten
, “The antisymmetric tensor field in superstring theory
,” Ann. Phys. (N.Y.)
170
, 454
(1986
);T.
Kimura
, “Index theorems of torsional geometries
,” JHEP
08
(2007
) 048
; e-print arXiv:0704.2111 [hep-th].15.
S. A.
Fedoruk
, E. A.
Ivanov
, and A. V.
Smilga
, “Real and complex supersymmetric d = 1 sigma models with torsions
,” Int. J. Mod. Phys. A
27
, 1250146
(2012
); e-print arXiv:1204.4105 [hep-th].16.
B.
Zumino
, “Supersymmetry and Kähler manifolds
,” Phys. Lett. B
87
, 203
(1979
).17.
L.
Alvarez-Gaume
and D. Z.
Freedman
, “Geometrical structure and ultraviolet finiteness in the supersymmetric sigma model
,” Commun. Math. Phys.
80
, 443
(1981
).18.
P. S.
Howe
and G.
Papadopoulos
, “Twistor spaces for HKT manifolds
,” Phys. Lett. B
379
, 80
(1996
); e-print arXiv:hep-th/9602108.19.
A.
Kirchberg
, J. D.
Lange
, and A.
Wipf
, “Extended supersymmetries and the Dirac operator
,” Ann. Phys.
315
, 467
(2005
); e-print arXiv:hep-th/0401134.20.
See, e.g.,
M.
Verbitsky
, “Hyperkähler manifolds with torsion, supersymmetry and Hodge theory
,” Asian J. Math.
6
, 679
(2002
); e-print arXiv:math/0112215.21.
N. E.
Mavromatos
, “A note on the Atiyah-singer index theorem for manifolds with totally antisymmetric Htorsion
,” J. Phys. A
21
, 2279
(1988
);J.-M.
Bismut
, “A local index theorem for non Kähler manifolds
,” Math. Ann.
284
, 681
(1989
).22.,”
F.
Delduc
and E.
Ivanov
, “${\cal N}=4$
mechanics of general (4, 4, 0) multipletsNucl. Phys. B
855
, 815
(2012
); e-print arXiv:1107.1429 [hep-th].23.,”
E.
Ivanov
and O.
Lechtenfeld
, “${\cal N}=4$
supersymmetric mechanics in harmonic superspaceJHEP
09
(2003
) 073
; e-print arXiv:hep-th/0307111.24.
A. S.
Galperin
, E. A.
Ivanov
, V. I.
Ogievetsky
, and E. S.
Sokatchev
, Harmonic Superspace
(Cambridge University Press
, 2001
), p. 306
.25.
M.
Faux
and S. J.
Gates
Jr., “Adinkras: A graphical technology for supersymmetric representation theory
,” Phys. Rev. D
71
, 065002
(2005
); e-print arXiv:hep-th/0408004.26.
S.
Bellucci
, S.
Krivonos
, A.
Marrani
, and E.
Orazi
, “‘Root’ action for
,” ${\cal N}=4$
supersymmetric mechanics theoriesPhys. Rev. D
73
, 025011
(2006
); e-print arXiv:hep-th/0511249.27.
F.
Delduc
and E.
Ivanov
, “Gauging
,” ${\cal N}=4$
supersymmetric mechanicsNucl. Phys. B
753
, 211
(2006
); e-print arXiv:hep-th/0605211.28.
A. V.
Smilga
, “Taming the zoo of supersymmetric quantum mechanical models
,” JHEP
05
(2013
) 119
; e-print arXiv:1301.7438 [hep-th].29.
E.
Ivanov
, O.
Lechtenfeld
, and A.
Sutulin
, “Hierarchy of
,” ${\cal N}=8$
mechanics modelsNucl. Phys.
B790
, 493
(2008
); e-print arXiv:0705.3064 [hep-th].30.,”
S.
Bellucci
, E.
Ivanov
, S.
Krivonos
, and O.
Lechtenfeld
, “${\cal N}=8$
superconformal mechanicsNucl. Phys. B
684
, 321
(2004
); e-print arXiv:hep-th/0312322.31.
S.
Bellucci
, E. A.
Ivanov
, S.
Krivonos
, and O.
Lechtenfeld
, “ABC of
,” ${\cal N}=8, d=1$
supermultipletsNucl. Phys. B
699
, 226
(2004
); e-print arXiv:hep-th/0406015.32.
Z.
Kuznetsova
, M.
Rojas
, and F.
Toppan
, “Classification of irreps and invariants of the N-extended supersymmetric quantum mechanics
,” JHEP
03
(2006
) 098
; Fifth International Conference on Mathematical Methods in Physics IC2006, April 24–28 2006, CBPF, Rio de Janeiro, Brazil, e-print arXiv:hep-th/0511274;F.
Toppan
, “Irreps and off-shell invariant actions of the N-extended supersymmetric quantum mechanics
,” e-print arXiv:hep-th/0610180.33.
The index
${\cal A}$
labeling complex chiral superfields should not be confused with the real 4-vector index A.34.
A. V.
Smilga
, “How to quantize supersymmetric theories
,” Nucl. Phys. B
292
, 363
(1987
).35.
H. J.
Grönewold
, “On the principles of elementary quantum mechanics
,” Physica
12
, 405
(1946
);J. E.
Moyal
, “Quantum mechanics as a statistical theory
,” Proc. Cambridge Philos. Soc.
45
, 99
(1949
).36.
Hereafter, for better readability, the index
${\hat{a}}$
on Cp, q, etc., will be omitted.37.
See, e.g., Eq. in Ref. 12.
38.
Even in the general case, one of the complex structures can be made constant by choosing the appropriate coordinates on the target space, e.g., by passing to the
${\cal N}=2$
superfield formulation.39.
A. V.
Smilga
, “Supercharges in the HKT supersymmetric sigma models
,” J. Math. Phys.
53
, 122105
(2012
); e-print arXiv:1209.0539 [hep-th].40.
M. A.
Konyushikhin
and A. V.
Smilga
, “Self-duality and supersymmetry
,” Phys. Lett. B
689
, 95
(2010
); e-print arXiv:0910.5162 [hep-th].41.
E. A.
Ivanov
, M. A.
Konyushikhin
, and A. V.
Smilga
, “SQM with non-Abelian self-dual fields: Harmonic superspace description
,” JHEP
05
(2010
) 033
; e-print arXiv:0912.3289 [hep-th].42.
E. A.
Ivanov
and J.
Niederle
, “Biharmonic superspace for
,” ${\cal N}=4$
mechanicsPhys. Rev. D
80
, 065027
(2009
); e-print arXiv:0905.3770 [hep-th].43.
F.
Delduc
and E.
Ivanov
, “New model of
,” ${\cal N}=8$
superconformal mechanicsPhys. Lett. B
654
, 200
(2007
); e-print arXiv:0706.2472 [hep-th].44.
G.
’t Hooft
, “Computation of the quantum effects due to a four-dimensional pseudoparticle
,” Phys. Rev. D
14
, 3432
(1976
).45.
A. V.
Belitsky
, S.
Vandoren
, and P.
van Nieuwenhuizen
, “Yang-Mills- and D-instantons
,” Class. Quant. Grav.
17
, 3521
(2000
); e-print arXiv:hep-th/0004186.© 2014 AIP Publishing LLC.
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