Challenges of Matrix Models

Abstract

Brief review of concepts and unsolved problems in the theory of matrix models.Comment: Contribution to Proceedings of Cargese 200

On KP-integrable Hurwitz functions

“…Of course, (1.11) are very special, besides they are τ -functions [1,12,13], they actually belong to the class of matrix model τ -functions [45][46][47][48][49][50][51]. This not-yet-rigourously-defined class is characterized by coexistence of a wide variety of very different representations and properties [52]: 2 Belyi pair describes a complex curve as a covering of CP 1 , ramified at just three points 0, 1, ∞ (the pair is the curve C and the mapping C −→ CP 1 ).…”

“…Still, there are many other ways to demonstrate that Z (2,n) is a τ -function of the Toda lattice hierarchy. The most important is the free-fermion approach of [70] and closely related determinant formulas, see [45][46][47][48][49][50][51]71]. From the point of view of Hurwitz theory, the basic well-known fact is that the character expansion…”

“…We keep the same label C for these operators, to emphasize similarity with (7.4). In fact they belong to the class of the W -operators [45][46][47][48][49][50][51][100][101][102], appearing in description of Kontsevich and multi-matrix models and mnemonically are powers of the currentĴ C defined by (9.22), subjected to peculiar normal ordering, when all theĴ C − operators on the right are simply thrown away, see [100,101] for a detailed description.…”

Explicit examples of DIM constraints for network matrix models

“…(2) by a change of integration variables δx i = x n+1 i in the multiple integral (2.28), [17,18,[73][74][75][76][77][78][79]: in this case we get the identities in a slightly different form:…”

“…In other words, the full symmetry of the Seiberg-Witten theory seems to be the Pagoda triple-affine elliptic DIM algebra (not yet fully studied and even defined), and particular models (brane patterns or Calabi-Yau toric varieties labeled by integrable systems a la [3,4]) are associated with its particular representations. The ordinary DF matrix models arise when one specifies "vertical" and "horizontal" directions, then convolutions of topological vertices can be split into vertex operators and screening charges, and the DIM algebra constraints can be attributed in the usual way [70][71][72][73][74][75][76][77][78][79] to commutativity of screening charges with the action of the algebra in the given representation. Dualities are associated with the change of the vertical/horizontal splitting, or, more general, with the choice of the section, where the algebra acts [80][81][82].…”