Higher order loop equations for A r and D r quiver matrix models

We use free boson techniques to investigate AD -E-quiver matrix models. Certain higher spin fields in the free boson formulation give rise to higher order loop equations valid at finite N. These fields form a special kind of W-algebra, called Casimir algebra. We compute explicitly the loop equations for A r and D r quiver models and check that at large N they are related to a deformation of the corresponding singular Calabi-Yau geometry. Contents 1. Introduction 1 2. AD -E quiver matrix models and free bosons 3 2.1 Loop equations and matrix models 3 2.2 The AD -E quiver matrix model 4 2.3 A free boson representation 6 2.4 Integration contours 8 2.5 Higher spin currents commuting with screening charges 9 2.6 From Casimir fields to loop equations 11 2.7 Undetermined parameters in the loop equations 14 2.8 The quadratic loop equation as an example 14 3. Examples 15 3.1 A r –quiver model loop equations at finite N 15 3.2 A closer look at the cubic and quartic loop equations 17 3.3 A r –quiver model loop equations at large N 18 3.4 D r –quiver model loop equations at finite N 20 3.5 D r –quiver model loop equations at large N 23 4. Relation to Casimir algebras 23 4.1 Some generalities on W-algebras 24 4.2 Casimir algebras and WZW-models 24 4.3 Casimir fields and Casimir algebras 25