4.11 Supergauge Theories
131
The relation (4.318) is crucial. It implies that the background-covariant spacetime derivatives D α ˙
α , unlike of the common covariant derivatives, have non-zero
commutators with spinor background-covariant derivatives D α , ¯
D ˙
α , and, moreover,
that the background strengths could arise during the D-algebra transformations. In
particular, the identity (4.318) results in the following expressions (cf. [106]):
[D α , ¯
D
2
] = −W α + 4i ¯
D
˙
α
D α ˙
α = W α + 4iD α ˙
α
¯
D
˙
α
.
(4.319)
We also must define the (background) covariantly chiral superfields to describe
the coupling of the gauge superfields to matter: the superfield is referred as the
(background) covariantly chiral one if it satisfies the condition ¯
D ˙
α = 0. It is easy
to see that the is related to the usual chiral field 0 as
= e
g ¯
0 .
(4.320)
Really, condition of chirality ¯
D ˙
α 0 = 0 implies in
e
g ¯
¯
D ˙
α e
−g ¯
e
g ¯
0 = 0.
(4.321)
Using the definitions (4.314), (4.320) we arrive just to the condition ¯
D ˙
α = 0.
Now we can develop the perturbative approach for the theory with the action
(4.313). First, we note that this theory possesses the symmetry with respect to the
following gauge transformations:
e
gv
→ e
ig ¯
e
gv e
−ig
,
(4.322)
where is a covariantly chiral parameter, i.e. it satisfies the condition ¯
D ˙
α = 0
(similarly, D α ¯
= 0). Therefore we need to introduce a gauge fixing. The most
natural background covariant gauge fixing term looks like
S g f = −
1
32
tr
d
8 zv{D
2
, ¯
D
2
}v,
(4.323)
which is a covariant generalization of the usual gauge fixing term in the Feynman
gauge. Summarizing the (4.313) and (4.323), we get the following action of the
quantum v field:
S t = S + S g f = −
1
2
tr
d
8 zvv + S int ,
(4.324)
where the S int in the expression above is an interaction part including all W α , ¯
W ˙
α
dependent terms, and = D
m
D m is the covariant d’Alembertian operator. In principle, one can fix other gauges (by introducing the factor ξ
−1 in S g f ), however, even
the problem of finding the propagator for v field appears to be very complicated for
an arbitrary gauge, and up to now, nobody found this propagator in a closed form.
131
The relation (4.318) is crucial. It implies that the background-covariant spacetime derivatives D α ˙
α , unlike of the common covariant derivatives, have non-zero
commutators with spinor background-covariant derivatives D α , ¯
D ˙
α , and, moreover,
that the background strengths could arise during the D-algebra transformations. In
particular, the identity (4.318) results in the following expressions (cf. [106]):
[D α , ¯
D
2
] = −W α + 4i ¯
D
˙
α
D α ˙
α = W α + 4iD α ˙
α
¯
D
˙
α
.
(4.319)
We also must define the (background) covariantly chiral superfields to describe
the coupling of the gauge superfields to matter: the superfield is referred as the
(background) covariantly chiral one if it satisfies the condition ¯
D ˙
α = 0. It is easy
to see that the is related to the usual chiral field 0 as
= e
g ¯
0 .
(4.320)
Really, condition of chirality ¯
D ˙
α 0 = 0 implies in
e
g ¯
¯
D ˙
α e
−g ¯
e
g ¯
0 = 0.
(4.321)
Using the definitions (4.314), (4.320) we arrive just to the condition ¯
D ˙
α = 0.
Now we can develop the perturbative approach for the theory with the action
(4.313). First, we note that this theory possesses the symmetry with respect to the
following gauge transformations:
e
gv
→ e
ig ¯
e
gv e
−ig
,
(4.322)
where is a covariantly chiral parameter, i.e. it satisfies the condition ¯
D ˙
α = 0
(similarly, D α ¯
= 0). Therefore we need to introduce a gauge fixing. The most
natural background covariant gauge fixing term looks like
S g f = −
1
32
tr
d
8 zv{D
2
, ¯
D
2
}v,
(4.323)
which is a covariant generalization of the usual gauge fixing term in the Feynman
gauge. Summarizing the (4.313) and (4.323), we get the following action of the
quantum v field:
S t = S + S g f = −
1
2
tr
d
8 zvv + S int ,
(4.324)
where the S int in the expression above is an interaction part including all W α , ¯
W ˙
α
dependent terms, and = D
m
D m is the covariant d’Alembertian operator. In principle, one can fix other gauges (by introducing the factor ξ
−1 in S g f ), however, even
the problem of finding the propagator for v field appears to be very complicated for
an arbitrary gauge, and up to now, nobody found this propagator in a closed form.