174
GAUGE FIELDS AND STRINGS
where we have accounted for the fact that jS interacts with the x-field
through the Lagrangian:
j
A straightforward estimate of (9.91) gives:
B(q^) = c A V 4- 0(^VA"))
(9.92)
(9.93)
If the constant c in (9.93) does not vanish, then the correlation length
for the j5-field, which is determined by the singularities in ^-space, is of
the order of A“ ^ Therefore, in the generic case c / 0, we can neglect the
influence of jS-fluctuations since:
(9.94)
where A is the volume of our object. In the presence of nontrivial our
computation indicates that we have a term in the induced action
-f ••
(9.95)
which supresses fluctuations of Turning to the case of /-fluctuations,
by similar arguments we find the most singular term in the corresponding action:
5 „ [/ ] = d ■ A"(dt)- ^ j
d-i
(9.96a)
which for d # 0 indicates the irrelevance of /-fluctuations.
Let us notice at this point that the question of whether c and d may be
taken to have generic values or whether one should apply the condition
that either of them be zero is far from trivial. There could exist different
continuum limits for string theory, the simplest one obtained without
extra conditions on c and d, while the others require fine tuning of these
constants. One has to decide what kind of possible continuum limits
have the desired properties and correspond to gauge theory. At present,
this question is unsolved, and we shall mainly investigate the generic
continuum theory, keeping in mind other options.
In the generic case we have shown that the fluctuations of the
Lagrange multiplier may be dropped, and we end up with the following
GAUGE FIELDS AND STRINGS
where we have accounted for the fact that jS interacts with the x-field
through the Lagrangian:
j
A straightforward estimate of (9.91) gives:
B(q^) = c A V 4- 0(^VA"))
(9.92)
(9.93)
If the constant c in (9.93) does not vanish, then the correlation length
for the j5-field, which is determined by the singularities in ^-space, is of
the order of A“ ^ Therefore, in the generic case c / 0, we can neglect the
influence of jS-fluctuations since:
(9.94)
where A is the volume of our object. In the presence of nontrivial our
computation indicates that we have a term in the induced action
-f ••
(9.95)
which supresses fluctuations of Turning to the case of /-fluctuations,
by similar arguments we find the most singular term in the corresponding action:
5 „ [/ ] = d ■ A"(dt)- ^ j
d-i
(9.96a)
which for d # 0 indicates the irrelevance of /-fluctuations.
Let us notice at this point that the question of whether c and d may be
taken to have generic values or whether one should apply the condition
that either of them be zero is far from trivial. There could exist different
continuum limits for string theory, the simplest one obtained without
extra conditions on c and d, while the others require fine tuning of these
constants. One has to decide what kind of possible continuum limits
have the desired properties and correspond to gauge theory. At present,
this question is unsolved, and we shall mainly investigate the generic
continuum theory, keeping in mind other options.
In the generic case we have shown that the fluctuations of the
Lagrange multiplier may be dropped, and we end up with the following