JHEP10(2015)181
For later convenience, we rewrite this Lagrangian as
L
VT
=
Z
d
4
θ f
IJ
−2Σ
I
D
α
V
J
W
T α
+
1
2
∂V
I
D
α
V
J
− ∂D
α
V
I
V
J
W
T α
+ h.c.
+Φ
T
D
α
V
I
W
J
α
+
¯
D
˙α
V
I
¯
W
J ˙α
+ V
I
D
α
W
J
α
+4(
¯
∂V
I
−
¯
Σ
I
)(∂V
J
− Σ
J
) − 2
¯
∂V
I
∂V
J
, (2.9)
where we have dropped total derivatives and used the firs t constraint i n (
2.6). As we have
shown i n ref. [
30], this Lagrangian is invariant under the gauge transformation (2.5)
4
up
to tot al derivatives, and reduces to (
2.2) i n the limit of Φ
T
= 1 and W
T α
= 0, which
corresponds to the case wher e the tensor mul t i pl e t is absent.
The superfields Φ
T
and W
T α
are expressed as
Φ
T
= −2iD
α
¯
D
2
Y
α
+ 2i
¯
D
˙α
D
2
¯
Y
˙α
,
W
T α
= i
¯
D
2
D
α
¯
X + 4
¯
∂Y
α
, (2.10)
where X and Y
α
are complex superfields that are related through
¯
D
2
(D
α
X + 4∂Y
α
) = 0. (2.11)
This relation indicates that Y
α
cannot be a general superfield. The first constraint in (2.6)
is automatically satisfied if (
2.11) is satisfied. Thus, independent constraints are (2.11) and
the second constraint in (
2.6). Note that Φ
T
and W
T α
are the field strength supe r fie l ds of
the “gauge potentials” X and Y
α
, and are invariant under
X → X + ∂V
G
− Σ
G
, Y
α
→ Y
α
−
1
4
D
α
V
G
, (2.12)
where the transformation paramete rs V
G
and Σ
G
are N = 1 real vector and chiral super-
fields, and form a 6D vector multiplet . The transformation (
2.12) is the SUSY extension
of the gauge transformation: B
+
MN
→ B
+
MN
+ ∂
M
λ
N
− ∂
N
λ
M
(λ
M
: real transformation
parameter).
Here we decompose X as
X = s
−1
X
4
− sX
5
, (2.13)
where X
4
and X
5
are real superfields. Then the second equation in (
2.10) and (2.11) are
rewritten as
W
T α
=
¯
D
2
s
−1
D
α
X
4
+ sD
α
X
5
+ 4
s
−1
∂
4
+ s∂
5
Y
α
, (2.14)
and
¯
D
2
s
−1
D
α
X
4
− sD
α
X
5
+ 4∂Y
α
= 0. (2.15)
Using the constrai nt (
2.15), W
T α
is also expresse d as
W
T α
= 2s
−1
¯
D
2
(D
α
X
4
+ 4∂
4
Y
α
) = s
−1
W
4α
+ 8s
−1
∂
4
¯
D
2
Y
α
= 2s
¯
D
2
(D
α
X
5
+ 4∂
5
Y
α
) = sW
5α
+ 8s∂
5
¯
D
2
Y
α
, (2.16)
4
The tensor multiplet (Φ
T
, W
T α
) is invariant under the gauge transformation.
– 5 –
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