212
GAUGE FIELDS AND STRINGS
The condition of degeneracy is given by:
3
2(2A + 1)
4A -h c/2
8A " + (c
_ 5 - c ± Vc
= 0
6A
5)A + c/2 = 0
(9.271)
25){c - 1)
16
This result shows, that for a fixed value of the central charge c, we have
an operator family degenerate at the second level, if and only if the
dimension A of the original operator is given by (9.271). The natural
question is what are the conditions for degeneracy at the iVth level? The
simple answer to this question is given by the so called Kac formula.
The result is the following. There are a set of dimensionalities
characterized by two integers n and m. They lead to degeneracy at the
level N = nm, provided that:
c - 1
^nm =
+ 8 (na+ + m
12
12
(9.272)
In the case n = 1, w = 2 the (9.272) is equivalent to (9.271).
Degenerate conformal families are very important. The reason is that
their operator content is in a sense minimal, since many operators can
be set to zero. In string theory, as we shall see, this implies decoupling of
unwanted ghost states, while in statistical physics of phase transitions
all known systems choose to be “minimal” theories or combinations
thereof.
Armed with these results, we are ready to discuss the “no-ghost”
properties of critical strings. By “no-ghost” we mean positivity of the
norm of string states. Actually it is guaranteed by our construction of
Liouville field theory, when ^ < 26. But explicit check at ^ = 26 is
very instructive.
9.10 Physical States of String Theory in the Critical Dimension
In Section 9.9 we presented a set of operators, composed out of the free
field x(z, z), which correspond to the particle states of the string. We
also showed by explicit calculations, that some of these particles
GAUGE FIELDS AND STRINGS
The condition of degeneracy is given by:
3
2(2A + 1)
4A -h c/2
8A " + (c
_ 5 - c ± Vc
= 0
6A
5)A + c/2 = 0
(9.271)
25){c - 1)
16
This result shows, that for a fixed value of the central charge c, we have
an operator family degenerate at the second level, if and only if the
dimension A of the original operator is given by (9.271). The natural
question is what are the conditions for degeneracy at the iVth level? The
simple answer to this question is given by the so called Kac formula.
The result is the following. There are a set of dimensionalities
characterized by two integers n and m. They lead to degeneracy at the
level N = nm, provided that:
c - 1
^nm =
+ 8 (na+ + m
12
12
(9.272)
In the case n = 1, w = 2 the (9.272) is equivalent to (9.271).
Degenerate conformal families are very important. The reason is that
their operator content is in a sense minimal, since many operators can
be set to zero. In string theory, as we shall see, this implies decoupling of
unwanted ghost states, while in statistical physics of phase transitions
all known systems choose to be “minimal” theories or combinations
thereof.
Armed with these results, we are ready to discuss the “no-ghost”
properties of critical strings. By “no-ghost” we mean positivity of the
norm of string states. Actually it is guaranteed by our construction of
Liouville field theory, when ^ < 26. But explicit check at ^ = 26 is
very instructive.
9.10 Physical States of String Theory in the Critical Dimension
In Section 9.9 we presented a set of operators, composed out of the free
field x(z, z), which correspond to the particle states of the string. We
also showed by explicit calculations, that some of these particles