ABSTRACT
We now consider the case where we have no Gaussian matter fields X i coupled to two-dimensional quantum gravity. The Einstein-Hilbert action is given by (3.4) with https://www.w3.org/1998/Math/MathML"> M = 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math6_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We have already seen that for two-dimensional gravity the curvature term is topological (eqs. (5.38) and (5.39)) and thus does not contribute to any dynamics unless we consider processes where the topology changes. On the other hand we have already discussed the problems with two-dimensional quantum gravity and topology changes in Chapter 5.3, so in the following we are going to restrict ourselves to two-dimensional manifolds which have the topology of the sphere ( https://www.w3.org/1998/Math/MathML"> h = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math6_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ), but with a number n, https://www.w3.org/1998/Math/MathML"> n ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math6_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , of boundaries. It is convenient to associate independent boundary cosmological constants Z i to each boundary i. In this way our (trivial) action will be (dropping the curvature term in the Einstein-Hilbert action) https://www.w3.org/1998/Math/MathML"> S [ g , Λ ] = Λ ∫ d 2 ξ g ( ξ ) , no boundary cosmological constants https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math6_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
