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10 Free BRST String Field Theory
Finally, the gauge fixing condition can be incorporated inside the action by
using a Lagrange multiplier β, which is an auxiliary string field containing also
components of all ghost numbers:
|β =
n∈Z
|β n .
(10.81)
The path integral then reads
Z =
ddβ e
−S[ ,
(10.82)
where
S[, β] =
1
2
|Q B | + +β|b 0 |
(10.83a)
=
n∈Z
1
2
2−n |Q B | n + +β 4−n |b 0 | n
.
(10.83b)
The first term of the action has the same form as the classical action (10.14) but now
includes fields at every ghost number. The complete BV analysis is relegated to the
interacting theory.
Removing the auxiliary field β = 0, one finds that the action is invariant under
the extended gauge transformation
δ | = Q B | ,
(10.84)
where the gauge parameter has also components of all ghost numbers:
| =
n∈Z
| n .
(10.85)
10.4 Spacetime Action
In order to make the string field action more concrete, and as emphasized in Chap. 9,
it is useful to expand the string field in spacetime fields and write the action for the
lowest modes. This also helps to check that the normalization chosen until here
correctly reproduces the standard QFT normalizations. For simplicity, we focus on
the open bosonic string in D = 26.
We build the string field from the vacuum |k, ↓↓ (Chap. 8) by acting with the
ghost positive-frequency modes b −n and c −n and the zero-mode c 0 and scalar
oscillators iα
μ
−n .