arXiv:hep-th/9304011 v1 Apr 5 1993

c) Three macroscopic loops:
κ
The same techniques as above can be applied to three and four macroscopic loop amplitudes.
The final formulae are rather complicated (see e.g. [91]), while at genus zero there
is considerable simplification.
The entire genus zero amplitude, together with the integer powers of l, is summarized
nicely in terms of macroscopic state wavefunctions [91]:
〈∏ 〉 ∫ ∞ ∏ E i
W (l i , q i ) = dE i K iEi (2 √ µl i ) (E 1 +E 2 +E 3 ) coth ( π
E i − iq i 2 (E 1+E 2 +E 3 ) ) .
i
−∞
Exercise. macroscopic → microscopic
i
Evaluate the integrals in (11.48) using residues by closing the E i contours in the
upper or lower half plane. (Warning: this involves a certain amount of algebra — the
result is in [91].)
(11.48)
The results (11.46),(11.47), and (11.48) contain a wealth of information on the nature
of contact terms and singular geometries in the path integral of the c = 1 theory. Note
especially from (11.47) that the propagator agrees with the naive Wheeler–DeWitt propagator
at low values of |E|, but is quite distinct, indeed exponentially decaying, at large
values of |E|. Correspondingly, in position space the propagator turns out to be smooth at
l 1 = l 2 . From a quantum gravity point of view, the only source of violation of the naive
WdW propagator in the Euclidean quantum gravity path integral is the contribution of
singular geometries such as fig. 20. From a target space point of view, the smoothness
of the propagator suggests the existence of other degrees of freedom. This guess is confirmed
by the pole structure of the propagator manifested in the second line of (11.47).
These extra degrees of freedom are clearly related to the special states. The two ideas: 1)
contributions of singular geometries, and 2) existence of new degrees of freedom related
to special states, are tied together by interpreting the new degrees of freedom in terms
of Liouville boundary operators, or, equivalently, in terms of redundant operators in the
matrix model. Recall from sec. 10.4 that such operators contribute figure eights — the
lattice version of the singular geometry of fig. 20. In [91] this interpretation was elaborated
upon, and partially carried out for the data provided by the three-point function (11.48).
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