286
GAUGE FIELDS AND STRINGS
where dependence of g^t is fixed by general covariance, while the value
of X is just the position of the logarithmic pole in (10.88) (since 1
represents an infrared cut-off):
(10.90)
1 and I
A, so in the
1
If we do not have a special fine tuning, then ao
continuum limit we have the effective action
S = A I(^^^ v*"
d^i
(10.91)
which we discussed in the previous chapters.
We see that in this case the critical exponent a = 0, because the
change of /io does not have any influence on I. An attempt to make a
small surface tension in this case will be ruined by the violent infrared
fluctuations described by (10.91). From the geometrical point of view,
the Lagrange multiplier 1 plays the role of the inverse correlation length
for the normals of the surface. In the regime described this correlation
length is of the order of the cut-off. The surface is terribly creased.
Perhaps the bosonic tachyon is related to this creasing.
For QCD and for Ising models, creased strings with non vanishing
surface tension are unacceptable. How can this undesirable property be
avoided?
It is clear, that in the purely bosonic case we have to find a version of
the theory with the j?-function having a zero at some point a^. If we
succeed, then a generic ao will be attracted to a^, the correlation length
will be infinite (without a Nambu term in the action) and we will have a
scale invariant theory with anomalous dimensions, one of which will
determine the critical exponent. The creasing will be avoided in this
case.
In four-dimensions there is a good candidate for all that. In this case
we have a specific 0-term which can be added to the action. At 6 = n
there are reasons to expect a scale-invariant theory. The term we are
talking about is the algebraic number of self-intersections, v(S) for our
two-dimensional surface S, immersed into four-dimensional space.
Analytically, it is given by:
fiv ^bhp
The partition function is given by:
(S)
(10.92)
(10.93)
GAUGE FIELDS AND STRINGS
where dependence of g^t is fixed by general covariance, while the value
of X is just the position of the logarithmic pole in (10.88) (since 1
represents an infrared cut-off):
(10.90)
1 and I
A, so in the
1
If we do not have a special fine tuning, then ao
continuum limit we have the effective action
S = A I(^^^ v*"
d^i
(10.91)
which we discussed in the previous chapters.
We see that in this case the critical exponent a = 0, because the
change of /io does not have any influence on I. An attempt to make a
small surface tension in this case will be ruined by the violent infrared
fluctuations described by (10.91). From the geometrical point of view,
the Lagrange multiplier 1 plays the role of the inverse correlation length
for the normals of the surface. In the regime described this correlation
length is of the order of the cut-off. The surface is terribly creased.
Perhaps the bosonic tachyon is related to this creasing.
For QCD and for Ising models, creased strings with non vanishing
surface tension are unacceptable. How can this undesirable property be
avoided?
It is clear, that in the purely bosonic case we have to find a version of
the theory with the j?-function having a zero at some point a^. If we
succeed, then a generic ao will be attracted to a^, the correlation length
will be infinite (without a Nambu term in the action) and we will have a
scale invariant theory with anomalous dimensions, one of which will
determine the critical exponent. The creasing will be avoided in this
case.
In four-dimensions there is a good candidate for all that. In this case
we have a specific 0-term which can be added to the action. At 6 = n
there are reasons to expect a scale-invariant theory. The term we are
talking about is the algebraic number of self-intersections, v(S) for our
two-dimensional surface S, immersed into four-dimensional space.
Analytically, it is given by:
fiv ^bhp
The partition function is given by:
(S)
(10.92)
(10.93)