Superintegrability of Kontsevich matrix model
Andrei Mironov, Alexei Morozov
Abstract
Abstract Many eigenvalue matrix models possess a peculiar basis of observables that have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability . As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property $$\langle character\rangle \,\sim character$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>⟨</mml:mo><mml:mi>c</mml:mi><mml:mi>h</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi><mml:mo>⟩</mml:mo><mml:mspace/><mml:mo>∼</mml:mo><mml:mi>c</mml:mi><mml:mi>h</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> . This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields : a special version of complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated Q Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.