Narrow Results

We show that for a large class of conformal field theories the 1/N correction to the chiral anomaly of the U(1) R-current can be shown to arise from the D7-branes present in the dual orientifolded orbifold string theories, generalizing a result in the literature for the simplest case. We also study the U(1)_R anomaly for conformal field theories that arise from orientifolds of the conifold. We find agreement between the field- and string-theoretic calculations, confirming a prediction of the AdS/CFT correspondence at order 1/N for string theories on AdS₅ × T¹¹/ℤ₂.

We formulate the planar "large N limit" of matrix models with a continuously infinite number of matrices directly in terms of U(N) invariant variables. Noncommutative probability theory is found to be a good language to describe this formulation. The change of variables from matrix elements to invariants induces an extra term in the Hamiltonian, which is crucial in determining the ground state. We find that this collective potential has a natural meaning in terms of noncommutative probability theory: it is the "free Fisher information" discovered by Voiculescu. This formulation allows us to find a variational principle for the classical theory described by such large N limits. We then use the variational principle to study models more complex than the one describing the quantum mechanics of a single Hermitian matrix (i.e. go beyond the so-called D = 1 barrier). We carry out approximate variational calculations for a few models and find excellent agreement with known results where such comparisons are possible. We also discover a lower bound for the ground state by using the noncommutative analog of the Cramer–Rao inequality.

A novel approach to evaluate the Wilson loops associated with a SU(∞) gauge theory in terms of pure string degrees of freedom is presented. It is based on the Guendelman–Nissimov–Pacheva formulation of composite antisymmetric tensor field theories of area (volume) preserving diffeomorphisms which admit p-brane solutions and which provide a new route to scale-symmetry breaking and confinement in Yang–Mills theory. The quantum effects are discussed and we evaluate the vacuum expectation values (VEV) of the Wilson loops in the large N limit of the quenched reduced SU(N) Yang–Mills theory in terms of a path integral involving pure string degrees of freedom. The quenched approximation is necessary to avoid a crumpling of the string worldsheet giving rise to very large Hausdorff dimensions as pointed out by Olesen. The approach is also consistent with the recent results based on the AdS/CFT correspondence and dual QCD models (dual Higgs model with dual Dirac strings). More general Loop wave equations in C-spaces (Clifford manifolds) are proposed in terms of generalized holographic variables that contain the dynamics of an aggregate of closed branes (p-loops) of various dimensionalities. This allows us to construct the higher-dimensional version of Wilson loops in terms of antisymmetric tensor fields of arbitrary rank which couple to p-branes of different dimensionality.

We study 't Hooft's equation for bound states in two-dimensional multicolor QCD. We consider the case of quarks with equal masses. We derive asymptotic expansions for the spectrum of mesons in different regimes and study their properties.

It is shown how actions corresponding to antisymmetric non-Abelian tensorial gauge field theories of (p+1)-dimensional diffeomorphisms yield p-brane actions associated with their (p+1)-dimensional worldvolume evolution. We conclude with a discussion of how to obtain p-brane actions from the large N limit of covariant matrix models based on generalized hypermatrices. A deformation quantization of Nambu classical mechanics furnishing Nambu quantum mechanics by constructing the n-ary noncommutative product of n functions f₁ •f₂ • ⋯ •f_n, the n-ary version of the Moyal bracket, and the analog of the Weyl–Wigner–Groenowold–Moyal map among operators and c-functions remains an open problem. A solution to this problem will reveal important relations between the physics of p-branes and matrix models based on generalized hypermatrices in the large N limit.

Quantum chromodynamics (QCD) is the theory of the strong interaction. The fundamental particles of QCD, quarks and gluons, carry color charge and form colorless bound states at low energies. The hadronic bound states of primary interest to us are mesons and baryons. A modern approach to computing hadron masses relies on the computationally intensive framework of lattice QCD. In cases where the exact quark composition or other quantum numbers of hadronic states are not precisely known, the prediction of masses from theoretical first principles is especially challenging. We address the problem of creating accurate and interpretable models of hadronic masses without resorting to extensive numerical computations. In this study, we construct a model of hadronic masses using both Bayesian and non-Bayesian techniques in machine learning. From knowledge of the meson spectrum only, neural networks and Gaussian processes predict the masses of baryons with 90.3% and 96.6% accuracy, respectively. We also predict the masses of pentaquarks and other exotic hadrons and demonstrate that machine learning is an effective tool for testing composition hypotheses. Our results surpass the benchmark constituent quark model both in terms of accuracy of predictions and hypothesis testing across all sectors of hadrons. We anticipate that our methods could yield a mass formula for hadrons from quark composition and other quantum numbers.