Abstract
We compute the sum over flat surfaces of disc topology with arbitrary number of conical singularities. To that end, we explore and generalize a specific case of the matrix model of dually weighted graphs (DWG) proposed and solved by one of the authors, M. Staudacher and Th. Wynter. Namely, we compute the sum over quadrangulations of the disc with certain boundary conditions, with parameters controlling the number of squares (area), the length of the boundary and the coordination numbers of vertices. The vertices introduce conical defects with angle deficit given by a multiple of π, corresponding to positive, zero or negative curvature. Our results interpolate between the well-known 2d quantum gravity solution for the disc with fluctuating 2d metric and the regime of “almost flat” surfaces with all the negative curvature concentrated on the boundary. We also speculate on possible ways to study the fluctuating 2d geometry with AdS2 background instead of the flat one.
Highlights
Matrix integrals at large size N of the matrices are a formidable tool for counting various types of planar graphs, i.e. “fat” graphs which have a certain two dimensional topology [1,2,3,4]
That allows us to interpolate between the regimes of large curvature fluctuations and the small curvature regime dominated by almost flat (AF) configurations which we describe in more detail later
The way to represent the partition function of the dually weighted graphs (DWG) matrix model is as follows [27]: first we expand the exponent of the second term in the action (2.1) w.r.t. the Schur characters χR[t∗] and GL(N ) characters χR[AM ], where R is a representation of GL(N ), we use the orthogonality property of the characters to integrate over the angular variable
Summary
Matrix integrals at large size N of the matrices are a formidable tool for counting various types of planar graphs, i.e. “fat” graphs which have a certain two dimensional topology [1,2,3,4]. The study of the exact sphere partition function of this DWG model in the continuous limit (i.e. for large quadrangulations with no boundary) showed that there are there two smoothly connected regimes: almost flat manifolds (with very few conical singularities) and the pure 2d gravity regime of [5,6,7, 9, 31]. We compute a number of observables and explore their critical behavior These are the resolvent for a particular kind of correlators with exponential accuracy, its continuum limit corresponding to nearly flat manifolds, and a similar limit for the resolvent describing the disc partition function with a different boundary.
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