arXiv:hep-th/9304011 v1 Apr 5 1993

Using the KPZ formula (2.26) written as
∆ X − 1 + 1 8 Q2 = 1 2(
α −
1
2 Q) 2
(4.6)
(as suggested after (3.22)), we obtain the minisuperspace Wheeler–DeWitt equation
(
−(l ∂ ∂l )2 + 4µl 2 + ν 2) ψ(l) = 0 ,
ν = ± 2 γ
(
α −
1
2 Q) . (4.7)
The solution decaying at large lengths is the non-normalizable wavefunction
ψ O (l) ∝ K ν (2 √ µl) , (4.8)
illustrated in fig. 6.
As promised in sec. 3.5, the wavefunctions corresponding to geometries (3.35) appear
naturally in the theory. From the geometrical picture of chapt. 3, it is natural to associate
this geometry with the insertion of a local operator at t = −∞. In [7], Seiberg has
further interpreted the blowup of the wavefunction at short distances as being physically
appropriate. The idea is that the wavefunctions associated to local operators in quantum
gravity should have support on metrics which are infinitesimally small in the physical
metric e γφ ĝ (because they are local).
Remark: In (4.5) we appear to have made an approximation. Astonishingly, matrix
model calculations (for example eq. 10.19 below) confirm that (4.8) is exact. It is not
understood why this should be so.
4.4. LZ states in 2D Quantum Gravity
So far we have discussed only the KPZ states in which the ghost modes are not excited.
These form only part of the full spectrum of the theory, as demonstrated in the continuum
formulation in the work of Lian–Zuckerman [63,64]. Treating the Liouville field as free,
they calculated the semi-infinite (BRST) cohomology of L ⊗ M(p, q), and found that the
cohomology is spanned by operators of the form
O n e α nφ ,
α n
γ = p + q − n
2q
n ≥ 1, ≠ 0 mod p, ≠ 0 mod q , (4.9)
and γ is determined as in (2.19). The operator O n is made of ghosts, matter, and derivatives
of φ. The ghost number of O n depends linearly on n.
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