We construct explicitly classical and quantum supercharges satisfying the standard |${\cal N} = 4$|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.

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21.
${\cal N}$
N
counts the number of real supercharges such that the minimal supersymmetry (involving double degeneracy of all excited states) corresponds to
${\cal N} = 2$
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.
To avoid confusion, please note that the derivative operator in (24) acts on the coefficients A, Am, Amn, etc., of the expansion of such a wave function over ψm, but not on the variables ψm. This is in contrast to the expressions like (20), where the operator ΠM = −iM acts also on ψN = eAN(xA.
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).
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