Conformal matrix models as an alternative to conventional multi-matrix models

Abstract

We introduce conformal multi-matrix models (CMM) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with c < 1 matter. We define CMM as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP hierarchy and by definition satisfy the discrete W -constraints. We prove that discrete CMM coincide with the (p, q)-series of 2d gravity models in a well-defined continuum limit, thus demonstrating that they provide a proper generalization of Hermitian one-matrix model.

Seiberg-Witten curve via generalized matrix model

“…The matrix model expression of the conformal block on a sphere is given by reinterpreting the Dotsenko-Fateev integral representation of it as the (beta-deformed) matrix integral [34][35][36][37] with a logarithmic potential. When the central charge c = 1 + 6Q 2 (Q = b + 1/b) equals one, or equivalently, b = i, the conformal block is represented by the usual matrix model.…”

Rotational symmetry breaking in multimatrix models

“…U (N ) invariance, where N is the size of the matrix) which determines much of the universal behaviour in the large N limit. This global symmetry is present also in all the most relevant multi-matrix models (Ising model on random lattice [6,7], the Q-state Potts model [8,9,10,11,12,13], chain of matrices [14,15,16,17,18,19], models for coloring problem [20,21,22,23,24], vertex models [25,26,27,28,29], the meander model [30,31], the O(n)-model and some generalizations of it [32,33,34,35,36,37,38,39,40,41], and several others [42,43,44,45,46,47,48,49,50]. The list is not complete).…”

Explicit examples of DIM constraints for network matrix models

“…|vac > between two vacua of operators in the fixed chronological order and in the chiral sector [87,88]. Here V α (z) is a primary field (vertex operator) in the free field c = 1 CFT, T (z) is its stress-energy tensor and Q is the corresponding screening charge [70][71][72], which is the integral Q = x S(x) of the screening current S(x).…”

“…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].…”