cubic vertex, shown in the paper on Supersymmetric Lightcone World Sheet construction
[7] to be given by
V123 4 g ( K"3 K"2 Knl
V"4n3 3/ 2 +g +k+ 3 + 2k3+ where (2-19)
P3 P2 Pi
K = Pi 1- P1P2 PP2 P2P3 PP3 PPl
Note that here the labels 1,2 and 3 denote the three particles coming into the vertex, and
nk particle's k polarization. The moment qk are those of world sheet strips meeting at
the vertex and each is therefore a difference Pk A qB where A, B are boundaries of a
strip. The following identities hold irrespective of 1, i.e., irrespective of where on the strip
between the boundaries the Aq1 insertion is made
I Dqe-sq
qM q0 / Dq (q, q1_1) e-sq J DqAqle-sq (2-20)
where Sq is the q-action unchanged from the scalar case Eq. (2-3) and the measure Dq
is just as in that case: Dq = d2q, ... d2qM_1. Comparing with Eq. (2-19) we see that
these identities are sufficient for constructing the couplings for the all A-s vertices. With
an addition of more ghost variables like the b, c-s 'l., v are the basis for constructing all the
rational functions required for super-symmetric gauge theory cubic vertices in Lightcone
gauge.
Now the only thing left to explain is how the quartic vertices are handled. A'priori,
this would seem the 1-- -t4 obstacle to finding a world sheet local description of the
theory. The reason is that it seems not possible to recreate such vertices, as four strips of
arbitrary width and height cannot in general be joined in a point. It seems that locality
would have to be abandoned and the strips joined along a whole line. One hope, is that
one could construct local cubic-like insertions which would reproduce the quartic vertices
when occurring on the same time-slice, but even this can a'priori not be guaranteed. It