28
3 String Theory
Fig. 3.1 Closed and open
string
The constraint equation obtained upon varying this action with respect to h now
reads T
φ
ab = 0, where
T
φ
ab = −
4π
√
h
δI [φ, h]
δh ab
(3.2)
is a symmetric and traceless world-sheet (stress) tensor.
Remark 3.1 The world-sheet stress tensor is traceless, since the diagonal part
of h does not appear in S. However, the absence of anomalies for this “Weyl
invariance” implies strong conditions on the manifold M. We will assume that M is
of dimension 26 equipped with a flat metric to avoid this complication.
In analogy with the point particle, the Faddeev–Popov procedure takes care of the
redundancy, due to the world-sheet diffeomorphism invariance of (3.1) generated by
the vector field ξ ,
δφ = −[ξ, φ] ,
δh= −L ξ h .
The resulting gauge-fixed Euclidean evolution kernel is then given by
Z(h, φ f , φ i ) =
φ ∂Σ f =φ f
φ ∂Σ i =φ i
D[φ, b, c] e
−I [φ,b,c,h] ,
(3.3)
where φ f/i are maps from the boundary components ∂Σ f/i to M and
I [φ, b, c, h] =
1
4π
Σ
g(dφ, ∧
∗ dφ) −
i
2π
Σ
√
h b
s
r ∇ s c
r dτ dσ .
(3.4)
Here ∇ is the covariant derivative compatible with the metric h on the world-sheet.
Geometrically, c is an odd vector field on Σ while b is an odd, symmetric traceless
tensor. The residual BRST symmetry is (with ξ → −ic)
δ BRST φ = i[c, φ] , δ BRST b = iT , δ BRST c =
i
2
L c c ,
(3.5)
where T = T φ + T g , with T φ given in (3.2) and T g = L c b is the stress tensor of
the ghost sector. The extra factor of i missing in (2.7) is due to the Wick-rotation
discussed above. This is the appropriate one-dimensional generalization of (2.6)
and (2.7).
3 String Theory
Fig. 3.1 Closed and open
string
The constraint equation obtained upon varying this action with respect to h now
reads T
φ
ab = 0, where
T
φ
ab = −
4π
√
h
δI [φ, h]
δh ab
(3.2)
is a symmetric and traceless world-sheet (stress) tensor.
Remark 3.1 The world-sheet stress tensor is traceless, since the diagonal part
of h does not appear in S. However, the absence of anomalies for this “Weyl
invariance” implies strong conditions on the manifold M. We will assume that M is
of dimension 26 equipped with a flat metric to avoid this complication.
In analogy with the point particle, the Faddeev–Popov procedure takes care of the
redundancy, due to the world-sheet diffeomorphism invariance of (3.1) generated by
the vector field ξ ,
δφ = −[ξ, φ] ,
δh= −L ξ h .
The resulting gauge-fixed Euclidean evolution kernel is then given by
Z(h, φ f , φ i ) =
φ ∂Σ f =φ f
φ ∂Σ i =φ i
D[φ, b, c] e
−I [φ,b,c,h] ,
(3.3)
where φ f/i are maps from the boundary components ∂Σ f/i to M and
I [φ, b, c, h] =
1
4π
Σ
g(dφ, ∧
∗ dφ) −
i
2π
Σ
√
h b
s
r ∇ s c
r dτ dσ .
(3.4)
Here ∇ is the covariant derivative compatible with the metric h on the world-sheet.
Geometrically, c is an odd vector field on Σ while b is an odd, symmetric traceless
tensor. The residual BRST symmetry is (with ξ → −ic)
δ BRST φ = i[c, φ] , δ BRST b = iT , δ BRST c =
i
2
L c c ,
(3.5)
where T = T φ + T g , with T φ given in (3.2) and T g = L c b is the stress tensor of
the ghost sector. The extra factor of i missing in (2.7) is due to the Wick-rotation
discussed above. This is the appropriate one-dimensional generalization of (2.6)
and (2.7).