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17 Superstring
where
= −1 + −1/2 .
(17.71)
Each gauge parameter satisfies the same conditions as the associated field (in
particular, −1/2 is in the restricted Hilbert space).
17.3.3 Auxiliary Field Approach
The disadvantage of the constraint approach is two-fold. First, it treats both
components of the field on a different footing. Second, the constraint must be
imposed by hand and does not follow from any fundamental principles. Another
possibility is to embed the propagator in a higher-dimensional field space by
introducing additional fields: in this way, the propagator can be inverted without
introducing the inverse of X 0 .
Let us introduce the new field
=
−1 +
−3/2 ,
(17.72)
which satisfies the same conditions as :
b
−
0 |
= L
−
0 |
= 0,
b
+
0 |
= 0.
(17.73)
A tentative kinetic term is then
S 0,2 =
1
2
| c
−
0 c
+
0 L
+
0 G |
−−
| c
−
0 c
+
0 L
+
0 | .
(17.74)
The kinetic operator in matrix form for (
,) reads
K = c
−
0 c
+
0 L
+
0
−G 1
1 0,
(17.75)
and its inverse is
= b
−
0 b
+
0
1
L
+
0
0 1
1 G
.
(17.76)
This reproduces the expected propagator for ((, ,) without needing to invert X 0 .
What is the interpretation of the additional fields? The gauge invariance of the
action is
δ | = Q B | ,
δ|
= Q B |
,
(17.77)