which was obtained from the Nambu action
254
GAUGE FIELDS AND STRINGS
S = j(det
=
• dt,x - g^t,)}
( 10.2)
by presuming condensation of the Lagrange multiplier,
(A"**) = const, g^^^g"'^
(10.3)
Therefore, in the phase described by (10.3) we have to interpret g^t in
(10.1) as an induced metric d^x-df^x.
Let us consider a scale transformation in space-time: x ^ X x . It is
equivalent to a Weyl transformation on the world sheet:
(10.4)
We have shown in the previous chapter that if the dimension ^ is
critical ( ^ = 26 for the bosonic case and ^ = 10 for the fermionic one)
then the Liouville mode decouples and the theory is Weyl invariant.
The only way to ensure scale invariance in space-time is to have a
massless dilaton in the string spectrum. It is indeed there. Perhaps,
along the same lines one can explain the graviton as well.
With all these massless modes at hand, one has to be able to compute
their effective action in the low energy approximation. In principle,
since we know the rules for computing the 5-matrix through the
averages of vertex operators it is straightforward to go to the low
energy limit in these formulas. However, this direct way is very
cumbersome and not very illuminating. There exists an interesting
alternative which we describe now.
Let us consider the string theory in a curved background:
S = ^ I
8,x' dH
(lo.s)
where
is some fixed metric tensor of the ^-dimensional space. We
are going to show, that the conditions for conformal invariance of the
action (10.5) give equations for the
(and some other fields) which
coincide precisely with the ones obtained from the S-matrix. Therefore,
the problem of the low energy limit reduces to the computation of the
j?-functions for (10.5) treated as a nonlinear (x-model.
Let us assume first, that the world sheet is flat,
Then, the
action
■ s = ,
S^x" 5jX” d^^
( 10.6)
254
GAUGE FIELDS AND STRINGS
S = j(det
=
• dt,x - g^t,)}
( 10.2)
by presuming condensation of the Lagrange multiplier,
(A"**) = const, g^^^g"'^
(10.3)
Therefore, in the phase described by (10.3) we have to interpret g^t in
(10.1) as an induced metric d^x-df^x.
Let us consider a scale transformation in space-time: x ^ X x . It is
equivalent to a Weyl transformation on the world sheet:
(10.4)
We have shown in the previous chapter that if the dimension ^ is
critical ( ^ = 26 for the bosonic case and ^ = 10 for the fermionic one)
then the Liouville mode decouples and the theory is Weyl invariant.
The only way to ensure scale invariance in space-time is to have a
massless dilaton in the string spectrum. It is indeed there. Perhaps,
along the same lines one can explain the graviton as well.
With all these massless modes at hand, one has to be able to compute
their effective action in the low energy approximation. In principle,
since we know the rules for computing the 5-matrix through the
averages of vertex operators it is straightforward to go to the low
energy limit in these formulas. However, this direct way is very
cumbersome and not very illuminating. There exists an interesting
alternative which we describe now.
Let us consider the string theory in a curved background:
S = ^ I
8,x' dH
(lo.s)
where
is some fixed metric tensor of the ^-dimensional space. We
are going to show, that the conditions for conformal invariance of the
action (10.5) give equations for the
(and some other fields) which
coincide precisely with the ones obtained from the S-matrix. Therefore,
the problem of the low energy limit reduces to the computation of the
j?-functions for (10.5) treated as a nonlinear (x-model.
Let us assume first, that the world sheet is flat,
Then, the
action
■ s = ,
S^x" 5jX” d^^
( 10.6)