arXiv:hep-th/9304011 v1 Apr 5 1993

c) Show that the solution corresponds to the choice of functions (in (3.14)):
A(z) = i zν + 1
z ν − 1
B(¯z) = −i ¯z ν + 1
¯z ν − 1 . (3.38)
d) Show that the monodromy when z circles around the puncture is the real elliptic
Möbius transformation
cos πνA(z) − sin πν
A(z) →
sin πνA(z) + cos πν . (3.39)
e) Show that a straight line through z = 0 is conformally mapped into an angle
πν, and hence we must have ν ≥ 0 on geometric grounds.
f) Repeat the above for the hyperbolic and parabolic cases.
3.6. Seiberg bound
Classically, the metrics (3.34, 3.35) are invariant under ν → −ν, E → −E. Quantum
mechanically, the wavefunctions K ν , K iE share this invariance, due to the total reflection
property of the Liouville “wall.” Turning on the wall by setting µ > 0 effectively halves
the states that exist in the (µ = 0) free spectrum.
In the DDK/KPZ formalism described in chapt. 2, on the other hand, the choice of
root in the KPZ formula (2.27) affects the scaling properties of the operator e αφ Φ 0 . Since
the Liouville interaction truncates the spectrum by half, we must choose a root. In [7,8],
it is argued that only those operators with
α ≤ 1 2 Q (3.40)
can exist. This choice of root has many distinguishing properties, some of which will be
noted in later sections:
1) This root gives a smooth semiclassical limit in quantum gravity, as we saw in sec. 2.2.
2) The area element is integrable only for sources satisfying (3.40). A related fact in terms
of deficit angles has appeared in part (e) in the exercise above (following (3.39)).
3) As we will see in the penultimate paragraph of sec. 4.3 , the Wheeler–DeWitt wavefunction
for a local operator in quantum gravity, related to the vertex operator V (φ)
by ψ(φ) = e − 1 2 Qφ V (φ), must be concentrated on l → 0, that is, where φ → −∞.
4) A closely related question is the nature of gravitational dressing (see sec. 2.2). Only
with the choice of root (3.40) do gravitationally dressed relevant/irrelevant operators
grow/decay in the worldsheet infrared.
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