QUANTUM STRINGS AND RANDOM SURFACES
211
In this case we say that the conformal family with A = 0 is degenerate at
the first level. A less trivial example is degeneracy at the second level.
Consider an operator:
X = (aT.,-^bTl,)(l>
and apply the degeneracy conditions:
TiX= T2X = 0
(9.266)
(9.267)
The conditions (9.267) are always sufficient because owing to the
Virasoro algebra (9.264) any T„ can be obtained by combining
and
^2- Applying the first of these equations, and using
[Ti, T_2] = 3T_i
[Ti, T l J = 2(roT_i + r_iTo)
= 2T_i(2To + 1)
we get
Ta = (3a + 2(2A + 1)6)T_ = 0
(9.268)
Or, in other words the degenerate operator must have the form:
3
X = T.
2(2A + 1)
(9.269)
This form, which satisfies (9.268) for any A, is in fact just the projective
invariant operator which can always be formed out of any primary
field. The nontrivial condition, which implies covariance under the
infinite conformal group is the second equation (9.267). Applying it to
(9.266), and using the relations:
[T2,T_2]=4To + i
[T2, T l J = [72, T_ JT_i + 7_i[72, 7_J
= 3(7iT_i + 7_iTi)
= 67o + 67_i7i
we obtain, together with (9.268):
3a + 2(2A + 1)6 = 0
(4 A + Ì ) .
4A + - |a - h 6 A 6 = 0
(9.270)