ATTEMPT AT A SYNTHESIS
265
We see from this formula, that if the perturbation drives the system to a
new fixed point, the central charge C(k^) tends to some new constant
value. Incidentally, since the system relaxes towards smaller value of the
effective action F, this new central charge must be smaller than the
original one (Zamolodchikov theorem).
The situation is rather amusing. Suppose that we start from a string
theory in the critical dimension. Let us assume also, that there are
massless modes which tend to condense, leading us to a new string
theory. We see, that generically this new theory will have a noncritical
central charge. The implication is that the Liouville field, originally
absent, must somehow appear. The only exception to this rule is the
case when all )S-functions are identically zero.
By the way, what goes wrong if we consider formally the dual
amplitudes for noncritical central charge without the Liouville mode?
The answer to this question is that nothing will be wrong at the tree
level (provided that C is less than critical—there will be ghosts
otherwise). However, as we consider unitarity corrections to the
amplitudes, or higher topologies, then we encounter unphysical
singularities in momentum space.
So, we have traced the connection between the j?-functions and the
effective actions for the massless modes of the strings. Applying this
conclusion to the case of the a-model with the metric y^v(x) and the
dilaton (/)(x), we see that the low energy expansion of the S-matrix
elements can be obtained by the loop expansion of the jS-functions.
Moreover, we see that the central charge is given by :
C ,,, = ^ - 26 + r(y(x\(x))
S = F(y»v(^X 0(x))(det
d^x
(here S is the action which generates the S-matrix) and, according to
previous formulas, the fields which minimize S have the property:
F(y®*(x), 0 ‘'^(x)) = const. < 0
Let us explain now the qualitative origin of these results. We have
considered a string in the external fields and then minimized its effective
action. This seems to be a bizzare procedure. The true meaning of it is
the following. The string contains interacting massless modes. Let us
suppose that they tend to form nonzero condensates. Then the picture
of the random world-surface can be viewed as something like a sphere
on which branching polymers, consisting of the propagators of these
massless modes, grow. What we have really done, has been to prescribe
265
We see from this formula, that if the perturbation drives the system to a
new fixed point, the central charge C(k^) tends to some new constant
value. Incidentally, since the system relaxes towards smaller value of the
effective action F, this new central charge must be smaller than the
original one (Zamolodchikov theorem).
The situation is rather amusing. Suppose that we start from a string
theory in the critical dimension. Let us assume also, that there are
massless modes which tend to condense, leading us to a new string
theory. We see, that generically this new theory will have a noncritical
central charge. The implication is that the Liouville field, originally
absent, must somehow appear. The only exception to this rule is the
case when all )S-functions are identically zero.
By the way, what goes wrong if we consider formally the dual
amplitudes for noncritical central charge without the Liouville mode?
The answer to this question is that nothing will be wrong at the tree
level (provided that C is less than critical—there will be ghosts
otherwise). However, as we consider unitarity corrections to the
amplitudes, or higher topologies, then we encounter unphysical
singularities in momentum space.
So, we have traced the connection between the j?-functions and the
effective actions for the massless modes of the strings. Applying this
conclusion to the case of the a-model with the metric y^v(x) and the
dilaton (/)(x), we see that the low energy expansion of the S-matrix
elements can be obtained by the loop expansion of the jS-functions.
Moreover, we see that the central charge is given by :
C ,,, = ^ - 26 + r(y(x\
S = F(y»v(^X 0(x))(det
d^x
(here S is the action which generates the S-matrix) and, according to
previous formulas, the fields which minimize S have the property:
F(y®*(x), 0 ‘'^(x)) = const. < 0
Let us explain now the qualitative origin of these results. We have
considered a string in the external fields and then minimized its effective
action. This seems to be a bizzare procedure. The true meaning of it is
the following. The string contains interacting massless modes. Let us
suppose that they tend to form nonzero condensates. Then the picture
of the random world-surface can be viewed as something like a sphere
on which branching polymers, consisting of the propagators of these
massless modes, grow. What we have really done, has been to prescribe