(Source : Polchinski)
Consider a toroidal compactification for a bosonic closed string.
We make the identification : X∼X+2πR, X being one of the 25 spatial dimensions, say X25
The left and right momenta are :
kL=nR+wRα′=0, kR=nR−wRα′=0
The on-shell mass conditions are written :
m2=k2L+4α′(N−1), m2=k2R+4α′(˜N−1)
From this we get :
0=k2L−k2R+4α′(N−˜N)
Using a "dimensionless" momentum lL,R=kL,R(α′2)12, we get :
0=l2L−l2R+2(N−˜N)
If we compactify 16 dimensions, we will have vectors →lL,→lR, with :
0=→l2L−→l2R+2(N−˜N)
Now, in the heterotic string, we consider only left - movers, so →lR=→0, so we have :
0=→l2L+2(N−˜N)
If we consider a lattice Γ made up with the →lL, we see that it must be a even lattice.
Note :
The expression of the dimensionless momentum may be justifyed by looking at the Operator Product Expansion (OPE) :
XL(z1)XL(z2)∼−α′2lnz12 and XR(z1)XR(z2)∼−α′2lnˉz12
Note that we have also :
:eikLXL(z)+ikRXR(ˉz)::eik′LXL(0)+ik′RXR(ˉ0): ∼zα′kLk′L/2(ˉz)α′kRk′R/2 :ei(kL+k′L)XL(0)+i(kR+k′R)XR(0):
where the z,ˉz term could be written zlLl′LˉzlRl′R
In fact, single-valuedness of the last OPE under a circle means that :
e2iπ(lLl′L−lRl′R)=1, so lLl′L−lRl′R is in Z
This post imported from StackExchange Physics at 2014-03-09 09:11 (UCT), posted by SE-user Trimok