250
GAUGE FIELDS AND STRINGS
In principle it is possible to describe this string not by the
fields,
but by the spin fields on the world sheet. In the case of particles this
corresponds to transition from the action (9.317) to (9.335). Unfortunately, since the have anomalous dimension ^/16 they are far from free
fields, and no practically useful description has yet been found. I am
sure it exists, however. The situation is somewhat easier for ^ = 10,
since then the light-cone gauge is possible and the effective dimensionality of the spin becomes ( ^ — 2)/16 = 1/2. In this case, as can be
expected, the spin fields are free and the fermionic string can be
described by the noncovariant Green-Schwarz action.
In the covariant formalism we have to construct the vertex operator
for emission of a space-time fermion. It must be something like
Vp,a -
(9 .4 3 3 )
where a is a spinor index. However, (9.433) is not good by itself, since it
does not have the required dimension 1 for the vertex. The reason is
that, while
has changed the spinor structure for the (/^-fields, it is
necessary to do the same job for the gravitino and ghost fields, which
enter into the functional integral. There are formal constructions for
completing this task, but an appealing derivation from first principles is
still lacking. Still we can compute, if we like, amplitudes containing two
fermions. For this we use bosonic emission vertices
and presume that
has the mode decomposition (9.432). This just
means that we have spin operators at 0 and oo and examine the
scattering of bosons (in arbitrary number) by one fermion. For several
fermions one has either to use the algebraic construction for the
fermionic vertex alluded to above or to pass to the light-cone gauge.
Hopefully, the situation will improve soon.
So in the critical dimension (^ = 10) the fermionic string has the
following properties. Its gound state is massless (being vector for the
open string case and tensor for the closed one) and supersymmetric in
the space-time sense, because the number of states in the Ramond
sector is equal to that in the Neveu-Schwarz sector.
This point must be clarified, since certain projections in both sectors
are needed to reach this conclusion. We have already mentioned them
in passing, but now we will discuss their meaning.
When we worked in the bosonic sector, without spin operators, we
considered only vertices which contain an even number of
or in
GAUGE FIELDS AND STRINGS
In principle it is possible to describe this string not by the
fields,
but by the spin fields on the world sheet. In the case of particles this
corresponds to transition from the action (9.317) to (9.335). Unfortunately, since the have anomalous dimension ^/16 they are far from free
fields, and no practically useful description has yet been found. I am
sure it exists, however. The situation is somewhat easier for ^ = 10,
since then the light-cone gauge is possible and the effective dimensionality of the spin becomes ( ^ — 2)/16 = 1/2. In this case, as can be
expected, the spin fields are free and the fermionic string can be
described by the noncovariant Green-Schwarz action.
In the covariant formalism we have to construct the vertex operator
for emission of a space-time fermion. It must be something like
Vp,a -
(9 .4 3 3 )
where a is a spinor index. However, (9.433) is not good by itself, since it
does not have the required dimension 1 for the vertex. The reason is
that, while
has changed the spinor structure for the (/^-fields, it is
necessary to do the same job for the gravitino and ghost fields, which
enter into the functional integral. There are formal constructions for
completing this task, but an appealing derivation from first principles is
still lacking. Still we can compute, if we like, amplitudes containing two
fermions. For this we use bosonic emission vertices
and presume that
has the mode decomposition (9.432). This just
means that we have spin operators at 0 and oo and examine the
scattering of bosons (in arbitrary number) by one fermion. For several
fermions one has either to use the algebraic construction for the
fermionic vertex alluded to above or to pass to the light-cone gauge.
Hopefully, the situation will improve soon.
So in the critical dimension (^ = 10) the fermionic string has the
following properties. Its gound state is massless (being vector for the
open string case and tensor for the closed one) and supersymmetric in
the space-time sense, because the number of states in the Ramond
sector is equal to that in the Neveu-Schwarz sector.
This point must be clarified, since certain projections in both sectors
are needed to reach this conclusion. We have already mentioned them
in passing, but now we will discuss their meaning.
When we worked in the bosonic sector, without spin operators, we
considered only vertices which contain an even number of
or in