3.4 Measure, Vertices, and BV Action
41
Hence, our BV algebra has an underlying Gerstenhaber algebra equipped with the
bracket {−, −} and multiplication
The additional structure needed to ensure that the decomposition of the moduli
space ˆ
P g,n does not produce an artificial boundary (or multiple covering) is a
boundary operator ∂, together with the BV master equation
∂ν g,n +
1
2
n 1 ≤n 2 ;g 1 ≤g 2
{ν g 1 ,n 1 +1 , ν g 2 ,n 2 +1 } + ¯
hΔν g−1,n+2 = 0
(3.17)
where n = n 1 + n 2 and g = g 1 + g 2 . This geometric decomposition of the moduli
space is manifestly background independent, in particular, independent of the choice
of a metric in M.
Remark 3.2 In string theory, the background dependence enters through the conformal field theory morphism that maps this structure to the BV algebra of the physical
Hilbert space. As explained above, the Polyakov action on a given space-time
defines a conformal field theory. Its BRST quantization introduces the Faddeev–
Popov ghosts c and b. The resulting BRST symmetry is generated by a ghost number
one BRST differential Q. In contrast to the point particle discussed in the first
chapter, in string theory, the construction of Q depends crucially on the choice of a
background. For a generic choice of the metric g on M the path integral measure in
(3.3) fails to be conformally invariant so that the construction presented here cannot
be used.
3.4
Measure, Vertices, and BV Action
In order to complete the construction of the CFT morphism between the geometric
and algebraic BV structures, we need to construct a measure on ˆ
P g,n . One way to
do this is by completing the set of BRST transformations (3.5) by
δ BRST h ab = δh ab , δ BRST δh ab = 0 ,
where h ab is the reference metric and δh ab is a ghost number 1, traceless, symmetric
tensor. This extended set of BRST transformations is a symmetry of the evolution
kernel (3.3) provided we add the term
ΔI =
1
4πi
Σ
√
h δh ab b
ab dτ dσ
to the action in (3.3). To continue, we consider the generating functional
F A (h, δh) =
D[φ, b, c] e
−I −ΔI
A (φ, c),
(3.18)
41
Hence, our BV algebra has an underlying Gerstenhaber algebra equipped with the
bracket {−, −} and multiplication
The additional structure needed to ensure that the decomposition of the moduli
space ˆ
P g,n does not produce an artificial boundary (or multiple covering) is a
boundary operator ∂, together with the BV master equation
∂ν g,n +
1
2
n 1 ≤n 2 ;g 1 ≤g 2
{ν g 1 ,n 1 +1 , ν g 2 ,n 2 +1 } + ¯
hΔν g−1,n+2 = 0
(3.17)
where n = n 1 + n 2 and g = g 1 + g 2 . This geometric decomposition of the moduli
space is manifestly background independent, in particular, independent of the choice
of a metric in M.
Remark 3.2 In string theory, the background dependence enters through the conformal field theory morphism that maps this structure to the BV algebra of the physical
Hilbert space. As explained above, the Polyakov action on a given space-time
defines a conformal field theory. Its BRST quantization introduces the Faddeev–
Popov ghosts c and b. The resulting BRST symmetry is generated by a ghost number
one BRST differential Q. In contrast to the point particle discussed in the first
chapter, in string theory, the construction of Q depends crucially on the choice of a
background. For a generic choice of the metric g on M the path integral measure in
(3.3) fails to be conformally invariant so that the construction presented here cannot
be used.
3.4
Measure, Vertices, and BV Action
In order to complete the construction of the CFT morphism between the geometric
and algebraic BV structures, we need to construct a measure on ˆ
P g,n . One way to
do this is by completing the set of BRST transformations (3.5) by
δ BRST h ab = δh ab , δ BRST δh ab = 0 ,
where h ab is the reference metric and δh ab is a ghost number 1, traceless, symmetric
tensor. This extended set of BRST transformations is a symmetry of the evolution
kernel (3.3) provided we add the term
ΔI =
1
4πi
Σ
√
h δh ab b
ab dτ dσ
to the action in (3.3). To continue, we consider the generating functional
F A (h, δh) =
D[φ, b, c] e
−I −ΔI
A (φ, c),
(3.18)