String Theory Demystified

CHAPTER 9 Superstring Theory Continued 181
Light-Cone Gauge
As we found in Chap. 7, the quantum theory will force us to take the number of
space-time dimensions to be D = 10. Since a general Dirac spinor has components
1 2 2 D /
, ..., , in 10 space-time dimensions a general Dirac spinor is going to have 32
components. I am sure the reader found dealing with 4 components in quantum fi eld
theory enough of a headache, what are we going to do with 32 components? Luckily
certain restrictions will cut this down dramatically. The fi rst thing to note is that the
complete action, which is given by adding up Eqs. (9.12), (9.15), and (9.16)
S = S1+ S2
which is invariant under SUSY transformations and the mysterious local Kappa
symmetry only under very specifi c conditions that restrict the number of space-time
dimensions and the type of spinors in the theory. These conditions are given as
follows:
• D = 3 with Majorana fermions.
• D = 4 with Majorana or Weyl fermions.
• D = 6 with Weyl fermions.
• D = 10 with Majorana-Weyl fermions.
It is clear that we don’t live in fl atland, so that rules out the fi rst case. The quantum
theory forces us to take D = 10, which is no surprise since this was explored in
Chap. 7. Therefore the spinors that are relevant to our discussion are Majorana-
Weyl fermions. This helps us in two ways:
• The Majorana condition makes the spinor components real.
• The Weyl condition eliminates half of the components. This leaves us with
a 16-component spinor.
Once again the Kappa symmetry reveals its hand by cutting the number of components
by half. So we are left with an eight component Majorana-Weyl spinor.
With this in mind we will proceed with some aspects of light-cone quantization.
This procedure imposes several conditions. First let’s begin by defi ning light-cone
components of the Dirac matrices. This is done by singling out the µ = 9 component
to make the following defi nitions:
Γ
Γ
+ =
− =
Γ + Γ
2
Γ − Γ
2
0 9
0 9
(9.17)
(9.18)