String Theory Demystified

216 String Theory Demystifi ed
superpartners, the status of string theory will be put in doubt. On the other hand, the
discovery of superpartners does not prove string theory, but would be a good
indication that the theory is on the right track.
Compactifi cation and Quantized Momentum
In this section, for readers who are curious, we briefl y describe a different approach
developed by Gross, Harvey, Matinec, and Rohm where compactifi cation is used to
construct the heterotic string. We follow the description laid out in Kaku (please see
References). In the light-cone gauge, the action for the heterotic string can be
written as
16
1 2 ⎛ i a
I a
− ⎞
S =− d ∂ X ∂ X + ∂ X ∂ X + iS ∂ +∂ S
a i a I
4πα
′ ∫ σ ∑
Γ ( ) τ σ
⎝
⎜
⎠
⎟
I=
i
The approach used here is to compactify the extra bosonic dimensions to generate
the group E ⊗ E . The extra 16 dimensions of the bosonic sector are compactifi ed
8 8
on a lattice. As described in the previous section, the right-moving sector is
supersymmetric. The spinors S a ( τ − σ)
have 8 components (so a = 1,..., 8).
Remember we are in the light-cone gauge, so only consider transverse components.
The index i is used for the space-time components, in the light-cone gauge i = 1,..., 8
as well. The remaining index I is used to run over the lattice used to compactify
the extra 16 dimensions. So it runs over 1 to 16.
The physics is much the same as the previous analysis. Bosonic states X i ( τ + σ)
and X i ( τ − σ)
are included in the left- and right-moving sectors, respectively. The
right-moving sector also includes the fermionic component S a ( τ − σ)
, while the
states X I ( τ + σ)
are in the left-moving sector.
The action is invariant under the supersymmetry transformation:
i + −12
/ i a
δX = ( p ) ε Γ S
a + −12
/
S = i( p ) ΓΓ( ∂ −∂
µ
) X εε
δ − µ τ σ
The following constraints are used to keep each component properly locked
away as a left mover or a right mover:
( ∂ −∂ ) =
τ σ
+
=
X
I
0
a
Γ
S 0