374
A Conventions
Table A.1 Conventions for the coordinates. The notations are the following (they can slightly vary
depending on the references): the Euclidean time is obtained by the analytic continuation τ = it
(denoted also by τ = σ 0 = σ 2 ), the spatial direction is σ = σ 1 and the light-cone coordinates are
σ ± = t ± σ
Refs
Cylinder
Plane
Light cone
Left-moving
Here, Di Francesco et al.
[5, 8, 14–16, 23, 28, 30]
w = τ + iσ z = e w
w = iσ + , ¯
w = iσ − Holomorphic
Blumenhagen et al.
[2, 4, 13, 22]
w = τ − iσ z = e w
w = iσ − , ¯
w = iσ + Anti-holomorphic
Polchinski [18, 21, 25]
w = σ + iτ z = e −iw w = −σ − , ¯
w = σ + Anti-holomorphic 4
A.2
Operators
Commutators and anti-commutators are denoted by
[A, B] := [A, B] − = AB − BA,
{A, B} := [A, B] + = AB + BA.
(A.16)
The Grassmann parity of a field A is denoted by |A|
|A| =
+1 Grassmann odd,
0
Grassmann even.
(A.17)
Two (anti-)commuting operators satisfy
AB = (−1)
|A||B| BA.
(A.18)
A.3
QFT
Energy is defined as the first component of the momentum vector
p
μ
:= (E, p
i ).
(A.19)
The following notations are used to denote the number of supersymmetries:
(N L , N R ),
N = N L + N R ,
(A.20)
4 In fact, the terms of “left”- and “right”-moving are interchanged in [21, p. 34] to get agreement
with the literature. But, it means that the spatial axis orientation is reversed.
Moreover, concerning [8], the first definition agrees with (6.1) but not with (6.53) since the
definition of ξ (our w) is modified in-between. This explains why the definitions of left- and rightmoving [8, p. 161] do not agree with the one given in the table.
A Conventions
Table A.1 Conventions for the coordinates. The notations are the following (they can slightly vary
depending on the references): the Euclidean time is obtained by the analytic continuation τ = it
(denoted also by τ = σ 0 = σ 2 ), the spatial direction is σ = σ 1 and the light-cone coordinates are
σ ± = t ± σ
Refs
Cylinder
Plane
Light cone
Left-moving
Here, Di Francesco et al.
[5, 8, 14–16, 23, 28, 30]
w = τ + iσ z = e w
w = iσ + , ¯
w = iσ − Holomorphic
Blumenhagen et al.
[2, 4, 13, 22]
w = τ − iσ z = e w
w = iσ − , ¯
w = iσ + Anti-holomorphic
Polchinski [18, 21, 25]
w = σ + iτ z = e −iw w = −σ − , ¯
w = σ + Anti-holomorphic 4
A.2
Operators
Commutators and anti-commutators are denoted by
[A, B] := [A, B] − = AB − BA,
{A, B} := [A, B] + = AB + BA.
(A.16)
The Grassmann parity of a field A is denoted by |A|
|A| =
+1 Grassmann odd,
0
Grassmann even.
(A.17)
Two (anti-)commuting operators satisfy
AB = (−1)
|A||B| BA.
(A.18)
A.3
QFT
Energy is defined as the first component of the momentum vector
p
μ
:= (E, p
i ).
(A.19)
The following notations are used to denote the number of supersymmetries:
(N L , N R ),
N = N L + N R ,
(A.20)
4 In fact, the terms of “left”- and “right”-moving are interchanged in [21, p. 34] to get agreement
with the literature. But, it means that the spatial axis orientation is reversed.
Moreover, concerning [8], the first definition agrees with (6.1) but not with (6.53) since the
definition of ξ (our w) is modified in-between. This explains why the definitions of left- and rightmoving [8, p. 161] do not agree with the one given in the table.