Disc partition function of 2d $R^2$ gravity from DWG matrix model

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 $\pi$, 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 $Ad{S}_2$ background instead of the flat one.

arXiv comment43 pages, 3 figures

References in corpus (18)

• Gauge Theory Correlators from Non-Critical String Theory
• HPL, a Mathematica implementation of the harmonic polylogarithms
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals
• Large $N$ Phase Transition in Continuum QCD${}_2$
• JT gravity as a matrix integral
• The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral
• Character Expansion Methods for Matrix Models of Dually Weighted Graphs
• Liouville quantum gravity -- holography, JT and matrices
• Exact Solution of the Six-Vertex Model on a Random Lattice
• Complex Matrix Models and Statistics of Branched Coverings of 2D Surfaces
• Exact Solution of Discrete Two-Dimensional R^2 Gravity
• Dilaton-gravity, deformations of the minimal string, and matrix models
• A Generating Function for Fatgraphs
• Superintegrability of Kontsevich matrix model
• Matrix Models and Deformations of JT Gravity
• Almost Flat Planar Diagrams
• Field Theory as a Matrix Model
• Curvature Matrix Models for Dynamical Triangulations and the Itzykson-DiFrancesco Formula

Cited by in corpus (2)

• Phase transitions for deformations of JT supergravity and matrix models
• A Flow in the Forest