376
A Conventions
understand this factor for the spatial vector is by considering the electromagnetic
case, where J contains a time derivative.
The term “zero-mode” has two (related) meanings:
1. given an operator D acting on a space of fields ψ(z), zero-modes ψ 0,i (z) of the
operator are all fields with zero eigenvalue Dψ 0,i (z) = 0, i = 1, . . . , dim ker D;
2. the zero-mode of a field expansion ψ =
n ψ n z −n−h is the mode ψ 0 for n = 0:
on the cylinder, it corresponds to the constant term of the Fourier expansion on
the cylinder (hence, a zero-mode of ∂ z according to the previous definition).
A prime indicates that the zero-modes are excluded. For example, det
D is the
product of non-zero eigenvalues, φ is a field without zero-mode and d φ is the
corresponding integration measure.
A.4
Curved Space and Gravity
The covariant derivative is defined by
∇ μ = ∂ μ + μ ,
(A.26)
where μ is the connection. For example, one has for a vector field
∇ μ A
ν
= ∂ μ A
ν
+
ν
μρ A
ρ .
(A.27)
The negative-definite Laplacian (or Laplace–Beltrami operator) is defined by
= g
μν
∇ μ ∇ ν =
1
√
g
∇ μ
√
gg
μν
∇ ν ).
(A.28)
Note that ∇ μ does not contain the Christoffel symbol for the index ν because of the
identity (B.4) (but it contains a connection for any other index of the field). For a
scalar field, both derivatives become simple derivatives.
The energy–momentum tensor is defined by
T μν = −
2λ
√ g
δS
δg μν ,
(A.29)
where λ = 2π for D = 2 and λ = 1 otherwise.