String Theory and M-Theory

252 The heterotic string
7.2 Fermionic construction of the heterotic string
In this section we would like to construct the action for the heterotic string.
The conformal gauge action describing the bosonic string is
S = − 1
d
2π
2 σ∂αXµ∂ α X µ , (7.1)
where the dimension of space-time is D = 26. This is supplemented by
Virasoro constraints for both the left-moving and right-moving modes. The
corresponding conformal gauge action for superstrings in the RNS formalism
is
S = − 1
2π
d 2 σ(∂αXµ∂ α X µ + ¯ ψ µ ρ α ∂αψµ). (7.2)
In this case D = 10, and there are super-Virasoro constraints for both the
left-moving and right-moving modes. The world-sheet fields ψ µ are ten
two-component Majorana spinors. This superstring action has world-sheet
supersymmetry. Space-time supersymmetry arises by including both the R
and NS sectors and imposing the GSO projection, as explained in Chapter 5.
In order to incorporate gauge degrees of freedom, let us consider a slightly
different extension of the bosonic string theory. Specifically, let us add worldsheet
fermions that are singlets under Lorentz transformation in space-time
but which carry some internal quantum numbers. Introducing n Majorana
fermions λA with A = 1, . . . , n, consider the action
S = − 1
2π
d 2 σ ∂αXµ∂ α X µ + ¯ λ A ρ α ∂αλ A . (7.3)
This theory has an obvious global SO(n) symmetry under which the λ A
transform in the fundamental representation and the coordinates X µ are
invariant. Since a fermion contributes half a unit to the central charge, the
requirement that the total central charge should be 26 is satisfied provided
that D + n/2 = 26. This is one way of describing a compactification of the
bosonic string theory to D < 26.
Examining this theory more carefully, one sees that the symmetry is actually
larger than SO(n). Indeed, writing the terms out explicitly in world-
sheet light-cone coordinates gives
S = 1
π
d 2 σ 2∂+Xµ∂−X µ + iλ A −∂+λ A − + iλ A +∂−λ A
+ . (7.4)
Written this way, it is evident that the theory actually has an (unwanted)
SO(n)L × SO(n)R global symmetry under which the left-movers and rightmovers
transform independently. One could try to discard the right-movers,