15.4 BV Theory
321
BV structure, which explains why the BV quantization Sect. 15.4 is straightforward.
This interplay between gauge invariance, covering of the moduli space, BV and
homotopy algebra is particularly beautiful. It has also been fruitful in constructing
super-SFT.
15.4 BV Theory
As indicated in the previous section (Sect. 15.3), the classical gauge algebra is open
and has field-dependent structure constants. The BV formalism (Appendix C.3) is
necessary to define the theory.
In the BV formalism, the classical action for the physical fields is extended to
the quantum master action by solving the quantum master equation (C.114). It is
generically difficult to build this action exactly, but the discussion of Sect. 10.3 can
serve as a guide: it was found that the free quantum action (with the tower of ghosts)
has exactly the same form as the free classical action (without ghosts). Hence, this
motivates the ansatz that it should be of the same form as the classical action (15.36)
to which are added the counter-terms from (15.24):
S =
1
g 2
s
g≥0
¯
h
g g
2g
s
n≥0
g n
s
n!
V g,n ((
n )
(15.47a)
=
1
2
| c
−
0 Q B | +
g,n≥0
¯
h g g
2g−2+n
s
n!
V g,n ((
n )
(15.47b)
=
1
g 2
s
g,n≥0
¯
h g g
2g−2+n
s
n!
| c
−
0
g,n−1((
n−1 )
,
(15.47c)
but without any constraint on the ghost number of :
∈ H
−
∩ ker L
−
0 .
(15.48)
In order to show that (15.47) is a consistent quantum master action, it is necessary
to show that it solves the master BV equation (C.114):
(S, S) − 2 ¯
hhS = 0.
(15.49)
The first step is to introduce the fields and antifields. In fact, because the CFT ghost
number induces a spacetime ghost number, there is a natural candidate set.
The string field is expanded as (15.1)
| =
r
ψ r |φ r ,
(15.50)
321
BV structure, which explains why the BV quantization Sect. 15.4 is straightforward.
This interplay between gauge invariance, covering of the moduli space, BV and
homotopy algebra is particularly beautiful. It has also been fruitful in constructing
super-SFT.
15.4 BV Theory
As indicated in the previous section (Sect. 15.3), the classical gauge algebra is open
and has field-dependent structure constants. The BV formalism (Appendix C.3) is
necessary to define the theory.
In the BV formalism, the classical action for the physical fields is extended to
the quantum master action by solving the quantum master equation (C.114). It is
generically difficult to build this action exactly, but the discussion of Sect. 10.3 can
serve as a guide: it was found that the free quantum action (with the tower of ghosts)
has exactly the same form as the free classical action (without ghosts). Hence, this
motivates the ansatz that it should be of the same form as the classical action (15.36)
to which are added the counter-terms from (15.24):
S =
1
g 2
s
g≥0
¯
h
g g
2g
s
n≥0
g n
s
n!
V g,n ((
n )
(15.47a)
=
1
2
| c
−
0 Q B | +
g,n≥0
¯
h g g
2g−2+n
s
n!
V g,n ((
n )
(15.47b)
=
1
g 2
s
g,n≥0
¯
h g g
2g−2+n
s
n!
| c
−
0
g,n−1((
n−1 )
,
(15.47c)
but without any constraint on the ghost number of :
∈ H
−
∩ ker L
−
0 .
(15.48)
In order to show that (15.47) is a consistent quantum master action, it is necessary
to show that it solves the master BV equation (C.114):
(S, S) − 2 ¯
hhS = 0.
(15.49)
The first step is to introduce the fields and antifields. In fact, because the CFT ghost
number induces a spacetime ghost number, there is a natural candidate set.
The string field is expanded as (15.1)
| =
r
ψ r |φ r ,
(15.50)