QUANTUM STRINGS AND RANDOM SURFACES
241
The reason why the particular combination z^2 = ~ ^ 2 ~
has
appeared in (9.393) is also simple: it is invariant under supersymmetry:
¿¿12 = ¿^1 — ¿^2 — ¿ ^ 1^2 + ^^2^1 = 0
(¿Zi = — eOi, Sz2 = — £02» ^01 = <502 =
Let us now find the expression for the supersymmetric vertex operator,
beginning with the simplest case of the critical dimension D = 10. In
this case, according to the preceeding section we expect superconformal
invariance of the theory, i.e. the fields cp and x should not participate on
the mass shell. The natural guess for V{p) will be:
V(P) =
(9.395)
- i p • ^ r O - 0 e i ( p • J(p • \|/r) - ip •/]}:
Setting / = 0 and integrating on 0 we obtain:
V(J>) = j
(9.396)
As we have noted in the bosonic case, the dimension of :e*^ *: is equal to
p^. Since the free fermions,
^ and
have dimension 1/2, we obtain
the following mass shell condition:
A = p2 + 1 = 2,
= 1
(9.397)
So, the vertex (9.396) still describes a tachyon, but a “better” one than
in the bosonic case (which has
= 2). If we consider an open string,
which amounts to taking only the “left” part of all fields and coordinates, we get:
Lope„(p)
'/'„(z)e' ,ip*(z).
= i
(9.398)
We can be confident that the “old” bosonic tachyon will not appear in
collisions of the new ones, because :e*^ *: is not supersymmetric. Still it
is instructive to see this fact explicitly. Let us consider the operator
product:
(9.399)