Zero and Infinite Curvature Limits of Hyperbolic Monopoles

We show that the zero curvature limit of the space of hyperbolicmonopoles gives the Euclidean monopoles, settling a conjectureof Atiyah. We also study the infinite curvature limit of thespace of hyperbolic monopoles, and show that the associatedrational maps appear explicitly here. 1991 Mathematics SubjectClassification 81T13, 53C07.     

Topological recursion for Gaussian means and cohomological field theories

We introduce explicit relations between genus-filtrated s-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces M _g,s ^disc (discrete volumes). Using these relations, we express Gaussian means in all orders of the genus expansion as polynomials in special times weighted by ancestor invariants of an underlying cohomological field theory. We translate the topological recursion of the Gaussian model into recurrence relations for the coefficients of this expansion, which allows proving that they are integers and positive. We find the coefficients in the first subleading order for M _g,1 for all g in three ways: using the refined Harer–Zagier recursion, using the Givental-type decomposition of the KPMM, and counting diagrams explicitly.