6.4 Operator Formalism and Radial Quantization
127
The identity is invariant under both conjugation
1
‡
= 1
t
= 1.
(6.100)
6.4.4 Mode Expansion
Any field of weight (h, ¯
h) can be expanded in terms of modes O m,n
O(z, ¯
z) =
m,n
O m,n
z m+h ¯
z n+ ¯
h
.
(6.101)
Note that the modes O m,n themselves are operators. The ranges of the two indices
are such that
m + h ∈ Z + ν,
n + ¯
h ∈ Z + ¯
ν,
ν, ¯
ν =
0
periodic,
1/2 anti-periodic.
(6.102)
The values of ν and ¯
ν depend on whether the fields satisfy periodic or antiperiodic boundary conditions on the plane (for half-integer weights, the periodicity
is reversed on the cylinder):
O(e
2π i z, ¯
z) = e
2π iν
O(z, ¯
z),
O(z, e
2π i
¯
z) = e
2π i¯ ν
O(z, ¯
z).
(6.103)
Depending on whether the weights are integers or half-integers, additional terminology is introduced:
• If h ∈ Z + 1/2, then one can choose anti-periodic (Neveu–Schwarz or NS) or
periodic (Ramond or R) boundary conditions on the cylinder (reversed for the
plane):
ν, ¯
ν =
0
NS
1/2 R.
(6.104)
The indices are half-integers (resp. integers) for the NS (R) sector.
• If h ∈ Z, periodic (or untwisted) boundary conditions are more natural, but antiperiodic boundary conditions may also be considered:
ν, ¯
ν =
0
untwisted
1/2 twisted.
(6.105)
The modes of untwisted (resp. twisted) fields have integer (half-integers) indices.