B Summary of Important Formulas
383
The integral of the curvature is a topological invariant
χ g;b :=
1
4π
d
2 σ
√ g R +
1
2π
ds k
= 2 − 2g − b,
(B.11)
called the Euler characteristics and where g is the number of holes and b the number
of boundaries.
B.3
Conformal Field Theory
In two dimensions, the energy–momentum tensor is defined by
T ab = −
4π
√ g
δS
δg ab .
(B.12)
B.3.1 Complex Plane
Defining the real coordinates (x, y) from the complex coordinate on the complex
plane
z = x + iy,
z = x − iy,
(B.13)
we have the formulas:
ds
2
= dx
2
+ dy
2
= dzd¯ z,
g z¯ z =
1
2
,
g zz = g ¯
z¯ z = 0,
(B.14a)
z¯ z =
i
2
,
,
z¯ z
= −2i,
(B.14b)
∂ := ∂ z =
1
2
(∂ x − i∂ y ),
¯
∂ := ∂ ¯
z =
1
2
(∂ x + i∂ y ),
(B.14c)
V
z
= V
x
+ iV
y ,
V
¯
z
= V
x
− iV
y ,
(B.14d)
d
2 x = dxdy =
1
2
d
2 z,
d
2 z = dzd¯ z,
(B.14e)
δ(z) =
1
2
δ
(2) (x),
1 =
d
2 z δ
(2) (z) =
d
2 x δ
(2) (x),
(B.14f)
R
d
2 z (∂ z v
z
+ ∂ ¯
z v
¯
z ) = −i
∂R
dz v
¯
z
− d¯ zv
z
= −2i
∂R
(v z dz − v ¯
z d¯ z).
(B.14g)
383
The integral of the curvature is a topological invariant
χ g;b :=
1
4π
d
2 σ
√ g R +
1
2π
ds k
= 2 − 2g − b,
(B.11)
called the Euler characteristics and where g is the number of holes and b the number
of boundaries.
B.3
Conformal Field Theory
In two dimensions, the energy–momentum tensor is defined by
T ab = −
4π
√ g
δS
δg ab .
(B.12)
B.3.1 Complex Plane
Defining the real coordinates (x, y) from the complex coordinate on the complex
plane
z = x + iy,
z = x − iy,
(B.13)
we have the formulas:
ds
2
= dx
2
+ dy
2
= dzd¯ z,
g z¯ z =
1
2
,
g zz = g ¯
z¯ z = 0,
(B.14a)
z¯ z =
i
2
,
,
z¯ z
= −2i,
(B.14b)
∂ := ∂ z =
1
2
(∂ x − i∂ y ),
¯
∂ := ∂ ¯
z =
1
2
(∂ x + i∂ y ),
(B.14c)
V
z
= V
x
+ iV
y ,
V
¯
z
= V
x
− iV
y ,
(B.14d)
d
2 x = dxdy =
1
2
d
2 z,
d
2 z = dzd¯ z,
(B.14e)
δ(z) =
1
2
δ
(2) (x),
1 =
d
2 z δ
(2) (z) =
d
2 x δ
(2) (x),
(B.14f)
R
d
2 z (∂ z v
z
+ ∂ ¯
z v
¯
z ) = −i
∂R
dz v
¯
z
− d¯ zv
z
= −2i
∂R
(v z dz − v ¯
z d¯ z).
(B.14g)