Entropy of Operator-Valued Random Variables: A Variational Principle for Large N Matrix Models

Journal of Mathematical Physics

2008

For a class of large-N multi-matrix models, we identify a group G that plays the same role as the group of loops on space-time does for Yang-Mills theory. G is the spectrum of a commutative shuffle-deconcatenation Hopf algebra that we associate to correlations. G is the exponential of the free Lie algebra. The generating series of correlations is a function on G and satisfies quadratic equations in convolution. These factorized Schwinger-Dyson or loop equations involve a collection of Schwinger-Dyson operators, which are shown to be right-invariant vector fields on G, one for each linearly independent primitive of the Hopf algebra. A large class of formal matrix models satisfying these properties are identified, including as special cases, the zero momentum limits of the Gaussian, Chern-Simons and Yang-Mills field theories. Moreover, the Schwinger-Dyson operators of the continuum Yang-Mills action are shown to be right-invariant derivations of the shuffle-deconcatenation Hopf algebra generated by sources labeled by position and polarization.

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Schwinger–Dyson operators as invariant vector fields on a matrix model analog of the group of loops

Journal of Mathematical Physics

2008

“…Meanwhile, there has been a steady stream of developments in matrix models of which we cite a few examples. These include their connections to non-commutative probability theory [9,10,11,12,13,14], the study of multi-matrix symmetry algebras and their connections to spin chains [15,16], exact solutions [17] and their relation to CFT [18] and algebraic geometry and detailed studies of the loop equations [19,20]. Much of the existing literature deals with 1-matrix models or exact solutions for specific observables of carefully chosen multi-matrix models.…”

Section: Introductionmentioning

confidence: 99%

Non-anomalous `Ward' identities to supplement large-Nmulti-matrix loop equations for correlations

Akant

2007

J. High Energy Phys.

This work concerns single-trace correlations of Euclidean multi-matrix models. In the large-N limit we show that Schwinger-Dyson equations (SDE) imply loop equations (LE) and non-anomalous Ward identities (WI). LE are associated to generic infinitesimal changes of matrix variables (vector fields). WI correspond to vector fields preserving measure and action. The former are analogous to Makeenko-Migdal equations and the latter to Slavnov-Taylor identities. LE correspond to leading large-N SDE. WI correspond to 1/N 2 suppressed SDE. But they become leading equations since LE for non-anomalous vector fields are vacuous. We show that symmetries at N = ∞ persist at finite N , preventing mixing with multi-trace correlations. For 1 matrix, there are no non-anomalous infinitesimal symmetries. For 2 or more matrices, measure preserving vector fields form an infinite dimensional graded Lie algebra, and non-anomalous action preserving ones a subalgebra. For Gaussian, Chern-Simons and Yang-Mills models we identify up to cubic non-anomalous vector fields, though they can be arbitrarily non-linear. WI are homogeneous linear equations. We use them with the LE to determine some correlations of these models. WI alleviate the underdeterminacy of LE. Non-anomalous symmetries give a naturalness-type explanation for why several linear combinations of correlations in these models vanish.

“…The Lie algebra is a Cuntz-type algebra which can be thought of as an algebra of vector fields on a noncommutative space. 12,5,6,13,14 However, it is still very challenging to find the proper mathematical framework for these theories and develop approximation methods to solve them even in the large N limit. To develop the necessary tools, it becomes worthwhile to practice on simpler theories whose gauge-invariant phasespace is the coadjoint orbit of a less formidable group.…”

Section: Introductionmentioning

confidence: 99%

2 + 1 Abelian "Gauge Theory" Inspired by Ideal Hydrodynamics

2006

Int. J. Mod. Phys. A

We study a possibly integrable model of Abelian gauge fields on a two-dimensional surface M , with volume form µ. It has the same phase-space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M . Gauge field Poisson brackets differ from the Heisenberg algebra, but are reminiscent of Yang-Mills theory on a null surface. Enstrophy invariants are Casimirs of the Poisson algebra of gauge invariant observables. Some symplectic leaves of the Poisson manifold are identified. The Hamiltonian is a magnetic energy, similar to that of electrodynamics, and depends on a metric whose volume element is not a multiple of µ. The magnetic field evolves by a quadratically nonlinear "Euler" equation, which may also be regarded as describing geodesic flow on SDiff(M, µ). Static solutions are obtained. For uniform µ, an infinite sequence of local conserved charges beginning with the Hamiltonian are found. The charges are shown to be in involution, suggesting integrability. Besides being a theory of a novel kind of ideal flow, this is a toy-model for Yang-Mills theory and matrix field theories, whose gauge-invariant phase-space is conjectured to be a coadjoint orbit of the diffeomorphism group of a noncommutative space.