# Quantum gravity on a manifold with a boundary

This is in reference to the section 9.4 General Theory of Random Surfaces in Gauge Fields and Strings, Polyakov. In this particular section and the next one (specialized to 2 dimensional surfaces), path integral formulation is being developed to study, from what I understand, a quantum theory of surfaces (or equivalently, I guesss, gravity). After motivating a renormalizable action which is invariant under coordinate transformations or diffeomorphisms. Now we are restricted to only those diffeomorphisms that don't move the boundary ((9.80), although in the section on 2 dimensional surfaces, you allow for diffeomorphisms which move boundary points to boundary points).

I wanted to know if there is a coordinate independent way to formulate the problem, as should be the case for a theory of gravity. The definition of the boundary, as described in the book, makes use of coordinate dependent statements ((9.80), (9.148)). Moreover, what is wrong if I consider a coordinate transformation that completely changes the boundary? For example, between cartesian and polar coordinates, the boundary of a unit disk changes from $x^2+y^2=1$ to $r=1$, which doesn't in particular seem to satisfy (9.148).

As an aside, when can we not understand a coordinate transformation as a diffeomorphism?

• Comment to the post (v1): It would be good if OP (or somebody else?) could try to make the question formulation self-contained. Nov 1, 2016 at 18:46
• First: why do you assume that a quantum theory of surfaces is a quantum theory of gravity? Second: why do you state that the formulation is not coordinate independent? You stated that the action is invariant under coordinate transformations or diffeomorphism. Third: your statement that the boundary of a unit disk changes is wrong. You just changed the coordinate system, and how to label it, but the coordinate is still the same. Finally: in what kind of problems are you trying to change the boundary? Are you talking about, for example, metrics like FRW?
– CGH
Nov 4, 2016 at 1:19
• The comment that the quantum theory of surfaces is same as quantum gravity is probably true only in two dimensions. More precisely, it is a theory of scalar coupled to 2-d metric. Dec 3, 2016 at 10:17
• My confusion of all the subsequent questions arises basically from my lack of understanding of the similarity/difference between coordinate transformations and diffeomorphisms. As stressed in the book, Eq (9.80), we are particularly restricting ourselves to diffeomorphisms that either die out on the boundary or are tangential to it. So I don't understand what does that mean in terms of the coordinate transformations. Naively, who stops me from putting arbitrary coordinates on my manifold and defining the boundary in terms of the end points in that new label, as @CGH points out. Dec 3, 2016 at 10:29
• I am interested in doing quantum gravity (with some action, say Einstein action (just for the sake of argument, ignore that it is topological), or Jackiw-Teitelboim-dilaton gravity) on right half plane. I am confused about what is the space of metrics that I need to integrate over in my path integral. I think I am right in assuming that whatever that space be, the equivalence class of metrics in that space will be given only by small diffs. Following polyakov, it seems like all such metrics are parametrized by Weyl factor and boundary reparametrizations,Eq (9.156). I want to understand how? Dec 3, 2016 at 10:35