372
A Conventions
The Lorentzian signature is taken to be mostly plus, and the flat Minkowski
metric reads
η μν = diag(−1, 1, . . . , 1
d
).
(A.3)
The flat Euclidean metric is
δ μν = diag(1, . . . , 1
D
).
(A.4)
Similar notations hold for the worldvolume metrics η ab and δ ab . The Levi-Civita
(completely antisymmetric) tensor is normalized by
01 = −
01
= 1.
(A.5)
Wick rotation from Lorentzian time t to Euclidean time τ (either worldsheet or
target spacetime) is defined by
t = −iτ.
(A.6)
Accordingly, contravariant (covariant) vector transforms with the same (opposite)
factor:
V
0
M = −iV
0
E ,
V M,0 = iV E,0 .
(A.7)
Most computations are performed with both spacetime and worldsheet Euclidean
signatures. Expressions are Wick-rotated when needed.
Light-cone coordinates are defined by
x
±
= x
0
± x
1 .
(A.8)
A function depending only on x + (x − ) is said to be left-moving (right-moving)
by analogy with the displacement of a wave. Under analytic continuation, the
left-moving (right-moving) coordinate is mapped to the holomorphic 1 (antiholomorphic) coordinate z (¯ z).
1 The terms of holomorphic are simply used to indicate that the object depends only on z, but not
on ¯
z. Typically, the objects have singularities and are really meromorphic in z.
A Conventions
The Lorentzian signature is taken to be mostly plus, and the flat Minkowski
metric reads
η μν = diag(−1, 1, . . . , 1
d
).
(A.3)
The flat Euclidean metric is
δ μν = diag(1, . . . , 1
D
).
(A.4)
Similar notations hold for the worldvolume metrics η ab and δ ab . The Levi-Civita
(completely antisymmetric) tensor is normalized by
01 = −
01
= 1.
(A.5)
Wick rotation from Lorentzian time t to Euclidean time τ (either worldsheet or
target spacetime) is defined by
t = −iτ.
(A.6)
Accordingly, contravariant (covariant) vector transforms with the same (opposite)
factor:
V
0
M = −iV
0
E ,
V M,0 = iV E,0 .
(A.7)
Most computations are performed with both spacetime and worldsheet Euclidean
signatures. Expressions are Wick-rotated when needed.
Light-cone coordinates are defined by
x
±
= x
0
± x
1 .
(A.8)
A function depending only on x + (x − ) is said to be left-moving (right-moving)
by analogy with the displacement of a wave. Under analytic continuation, the
left-moving (right-moving) coordinate is mapped to the holomorphic 1 (antiholomorphic) coordinate z (¯ z).
1 The terms of holomorphic are simply used to indicate that the object depends only on z, but not
on ¯
z. Typically, the objects have singularities and are really meromorphic in z.