Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.Comment: minor changes: more references adde

Matrix Models, Complex Geometry and Integrable Systems. I

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of the quasiclassical integrable systems, solved by construction of tau-functions or prepotentials. The complex curves and tau-functions of one- and two- matrix models are discussed in detail.Comment: 52 pages, 19 figures, based on several lecture courses and the talks at "Complex geometry and string theory" and the Polivanov memorial seminar; misprints corrected, references adde

Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies

We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and the correlation functions of the lattice) obey the topological recursion, as usual in matrix models, i.e they are given by the symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor correction