arXiv:hep-th/9304011 v1 Apr 5 1993

Fig. 20: Two loops on a continuum surface collide.
10.4. Redundant operators, singular geometries and contact terms
One important and not generally discussed issue is the contribution of singular geometries
to the path integral. In the case of macroscopic loop amplitudes, there are geometries
in which loops collide to make figure-eights as in fig. 20, and as well more complicated geometries.
Our understanding of the contributions of these geometries is very incomplete,
but there is plenty of evidence that such terms are responsible for several peculiarities of
the matrix model answers (e.g. the cosh propagator discovered above) and perhaps lie
at the heart of a geometrical understanding of the Lian–Zuckerman states. See also the
discussion at the end of sec. 11.4 below.
11. Loops and States in the c = 1 Matrix Model
11.1. Definition of the c = 1 Matrix Model
There are several approaches to defining a matrix model for gravity coupled to c = 1
matter. The most direct method is the discretization of the Polyakov path integral for a
one-dimensional target space,
Z qg (κ, g) = ∑ Λ
κ 2h−2 g |Λ|
V ∏
i=1
∫
dX i e − ∑ 〈ij〉 L(X i − X j ) ,
(11.1)
where |Λ| is the number of vertices on the lattice Λ which is summed over Euler character
2 − 2h, and the nearest neighbor interaction L(X i − X j ) between the bosonic fields X i,j
at vertices i, j is summed over links 〈ij〉 between vertices.
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