With the correct linking structure, we use the fact that
SDSe IS SK J DSe ^S SK tab (2-15)
J DSe SI SK = DSeASI SK =0. (2-16)
This ensures that the correct fermionic information travels from vertex to vertex.
Similarly
JDSeaS JK = JDSeASi JK = DSetJi S2K D)SeAJi SK = 0(2-17)
J DSeAJ{ J2 (2-18)
takes care of the vector and scalar particle information. At the vertices, along with other
insertions, Dirac gamma matrices connect the fermionic indices a, b with the vector indices
k. The details are carefully presented in the above mentioned article [7]. It is rather
interesting to notice that, as with the transverse momentum fields q on the world sheet,
the time derivative of the Grassmann's comes up only along boundaries. The mean field
approach, where the boundaries reach a finite density, should therefore exhibit 5 and S
dependence.
Let us next turn to items 4) and 5) from the list of issues we expect from the
Lightcone World Sheet description of Gauge theory. In the scalar theory the plain
Feynman vertices contained no momentum dependent factors but the treatment of the
propagators created factors of 1/M in the vertices. In the Gauge theory however, the
vertices are somewhat more complex. Recalling now the long expression (2-10) for the
Lightcone Hamiltonian, the Feynman rules contain a number of A3 and itAQ cubics as
well as A4, QtA2Q and (QtQ)2 quartics and the derivation of their coupling is found in the
t The case of A = 4 extended supersymmetry requires a slightly different treatment
since then the spinors are simultaneously Majorana and Weyl making Sa and S the same.
This different treatment still produces equivalent equations