QUANTUM STRINGS AND RANDOM SURFACES
243
Massless vectors or tensors can interact without producing ghosts
(and we know that the ghosts are absent for ^ ^ 10 either from (9.390)
or from direct counting of states which we have done in the bosonic
case), only if their effective action is gauge invariant, being of YangMills type for vectors and Einstein’s for tensors. We shall postpone
explicit computation of this effective action, and point out here only one
manifestation of the gauge structure. In the operator product:
r^(p, z)FXk, 0) ^
k)T^(p + A t , 0)
(9.402)
an explicit computation shows that the quantity
coincides
with the triple Yang-Mills vertex. This is an important observation,
since we already know that structure constants in operator products are
just the residues of the poles in scattering amplitudes. A little later we
shall describe an efficient method for their computation.
Up to now, we have spoken of the fermionic string but, clearly, all
particles obtained from the vertex operators (9.400) are bosons.
It is most important that apart from them, the spectrum of the theory
contains fermions appearing as soliton excitations of the string. In
terms of functional integrals, solitons appear owing to special boundary
conditions applied on the fields. In our case, anomalous boundary
conditions have the following meaning. We are dealing with the world
sheet which is a sphere with omitted points, where the external particles
are injected. On such a punctured sphere we are free to chose different
spinor structures for the fields
that is we can consider double-valued
fields, which change sign when going round the injection point.
In order to understand why nontrivial spinor structures are related to
space-time fermions, we have to take a step back and return to the case
of particles. Let us recall, that we described spinors by the anticommuting fields
with the action:
5 =
dt
(9.403)
Why does this action describe a space-time spinor (while the variables
{¡/^ are vectors)? To answer this question let us pass to the Hamiltonian
formalism. Assuming that the dimensionality of space-time is even we
can choose half of
to be coordinates, while the other half are
conjugate momenta. So, let us introduce complex fields:
Then the action takes the form:
d
Jt
C l> adt
(9.404)
243
Massless vectors or tensors can interact without producing ghosts
(and we know that the ghosts are absent for ^ ^ 10 either from (9.390)
or from direct counting of states which we have done in the bosonic
case), only if their effective action is gauge invariant, being of YangMills type for vectors and Einstein’s for tensors. We shall postpone
explicit computation of this effective action, and point out here only one
manifestation of the gauge structure. In the operator product:
r^(p, z)FXk, 0) ^
k)T^(p + A t , 0)
(9.402)
an explicit computation shows that the quantity
coincides
with the triple Yang-Mills vertex. This is an important observation,
since we already know that structure constants in operator products are
just the residues of the poles in scattering amplitudes. A little later we
shall describe an efficient method for their computation.
Up to now, we have spoken of the fermionic string but, clearly, all
particles obtained from the vertex operators (9.400) are bosons.
It is most important that apart from them, the spectrum of the theory
contains fermions appearing as soliton excitations of the string. In
terms of functional integrals, solitons appear owing to special boundary
conditions applied on the fields. In our case, anomalous boundary
conditions have the following meaning. We are dealing with the world
sheet which is a sphere with omitted points, where the external particles
are injected. On such a punctured sphere we are free to chose different
spinor structures for the fields
that is we can consider double-valued
fields, which change sign when going round the injection point.
In order to understand why nontrivial spinor structures are related to
space-time fermions, we have to take a step back and return to the case
of particles. Let us recall, that we described spinors by the anticommuting fields
with the action:
5 =
dt
(9.403)
Why does this action describe a space-time spinor (while the variables
{¡/^ are vectors)? To answer this question let us pass to the Hamiltonian
formalism. Assuming that the dimensionality of space-time is even we
can choose half of
to be coordinates, while the other half are
conjugate momenta. So, let us introduce complex fields:
Then the action takes the form:
d
Jt
C l> adt
(9.404)