arXiv subjects

João P. Rodrigues

Publications and source records attributed to João P. Rodrigues.

At least 19 recordsLinked to original sources

arXiv2026

Secondary invariants and non-perturbative states

At finite $N$ the ring of gauge invariant operators is not freely generated. For problems of interest in physics, these rings are Cohen--Macaulay and admit a Hironaka decomposition, in which the full invariant ring is a free module over a polynomial ring generated by the primary invariants. The module basis is given by finitely many secondary invariants. This motivates a physical picture in which the primary invariants are regarded as perturbative degrees of freedom while the secondary invariants are associated with distinguished non-perturbative states or sectors. The purpose of this study is to show that a concrete algebraic version of this picture is visible in simple zero-dimensional matrix integrals.

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arXiv2025

"Glueballs" in the Quantum Mechanics of three large massless Yang-Mills coupled Matrices

We use a loop truncated Jevicki-Sakita effective collective field Hamiltonian to obtain, over a very large range of values of 't Hooft's coupling, and directly in the large N limit, the large N (planar) ground state energy, the planar ground state expectation values of invariant correlators, and the 1/N spectrum of the quantum mechanical system of three massless Yang-Mills coupled matrices. This captures the dynamics of the (residual) gauge invariant sector of the spatially reduced 3+1 dimensional pure Yang-Mills theory, in the large N limit. The large N loop space constraints are handled by the use of master variables. As is the case for two matrices, the method is highly efficient directly in the massless limit, and it reproduces to a very high precision the scaling dependence of physical quantities, determined by their dimensions, on the dimensionful 't Hooft coupling. We obtain the bound state masses of "glueballs", their quantum numbers and ensuing degeneracies.

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arXiv2023

Large N Master Field Optimization: the Quantum Mechanics of two Yang-Mills coupled Matrices

We study the large N dynamics of two massless Yang-Mills coupled matrix quantum mechanics, by minimization of a loop truncated Jevicki-Sakita effective collective field Hamiltonian. The loop space constraints are handled by the use of master variables. The method is successfully applied directly in the massless limit for a range of values of the Yang-Mills coupling constant, and the scaling behaviour of different physical quantities derived from their dimensions are obtained with a high level of precision. We consider both planar properties of the theory, such as the large N ground state energy and multi-matrix correlator expectation values, and also the spectrum of the theory. For the spectrum, we establish that the U(N) traced fundamental constituents remain massless and decoupled from other states, and that bound states develop well defined mass gaps, with the mass of the two degenerate lowest lying bound states being determined with a particularly high degree of accuracy. In order to confirm, numerically, the physical interpretation of the spectrum properties of the U(N) traced constituents, we add masses to the system and show that, indeed, the U(N) traced fundamental constituents retain their "bare masses". For this system, we draw comparisons with planar results available in the literature.

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arXiv2022

Constructing the bulk at the critical point of three-dimensional large $N$ vector theories

In the context of the $AdS_{4}/CFT_{3}$ correspondence between higher spin fields and vector theories, we use the constructive bilocal fields based approach to this correspondence, to demonstrate, at the $IR$ critical point of the interacting vector theory and directly in the bulk, the removal of the $Δ=1$ ($s=0$) state from the higher spins field spectrum, and to exhibit simple Klein-Gordon higher spin Hamiltonians. The bulk variables and higher spin fields are obtained in a simple manner from boundary bilocals, by the change of variables previously derived for the $UV$ critical point (in momentum space), together with a field redefinition.

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arXiv2022

Large N Optimization for multi-matrix systems

In this work we revisit the problem of solving multi-matrix systems through numerical large $N$ methods. The framework is a collective, loop space representation which provides a constrained optimization problem, addressed through master-field minimization. This scheme applies both to multi-matrix integrals ($c=0$ systems) and multi-matrix quantum mechanics ($c=1$). The complete fluctuation spectrum is also computable in the above scheme, and is of immediate physical relevance in the later case. The complexity (and the growth of degrees of freedom) at large $N$ have stymied earlier attempts and in the present work we present significant improvements in this regard. The (constrained) minimization and spectrum calculations are easily achieved with close to $10^4$ variables, giving solution to Migdal-Makeenko, and collective field equations. Considering the large number of dynamical (loop) variables and the extreme nonlinearity of the problem, high precision is obtained when confronted with solvable cases. Through numerical results presented, we prove that our scheme solves, by numerical loop space methods, the general two matrix model problem.

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arXiv2018

Large N bilocals at the infrared fixed point of the three dimensional O(N) invariant vector theory with a quartic interaction

We study the three dimensional O(N) invariant bosonic vector model with a $\fracλ{N}(ϕ^{a}ϕ^{a})^{2}$ interaction at its infrared fixed point, using a bilocal field approach and in an $1/N$ expansion. We identify a (negative energy squared) bound state in its spectrum about the large $N$ conformal background. At the critical point this is identified with the $Δ=2$ state. We further demonstrate that at the critical point the $Δ=1$ state disappears from the spectrum.

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arXiv2015

De Alfaro, Fubini and Furlan from multi Matrix Systems

We consider the quantum mechanics of an even number of space indexed hermitian matrices. Upon complexification, we show that a closed subsector naturally parametrized by a matrix valued radial coordinate has a description in terms of non interacting $s$-state "radial fermions" with an emergent De Alfaro, Fubini and Furlan type potential, present only for two or more complex matrices. The concomitant $AdS_2$ symmetry is identified. The large $N$ description in terms of the density of radial eigenvalues is also described.

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arXiv2015

Large N Matrix Hyperspheres and the Gauge-Gravity Correspondence

The large N dynamics of a subsector of d=0 interacting complex multi matrix systems, which is naturally parametrized by a matrix valued radial coordinate, and which embodies the canonical AdS/CFT relationship between 't Hooft's coupling constant and radius, is obtained. Unlike the case of the single complex matrix, for two or more complex matrices a new repulsive logarithmic potential is present, as a result of which the density of radial eigenvalues has support on an hyper annulus. For the single complex matrix, the integral over the angular degrees of freedom of the Yang-Mills interaction can be carried out exactly, and in the presence of an harmonic potential, the density of radial eigenvalues is shown to be of the Wigner type.

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arXiv2015

Canonical Formulation of $O(N)$ Vector/Higher Spin Correspondence

We discuss the canonical structure of the collective formulation of Vector Model/Higher Spin Duality in AdS$_4$. This involves a construction of bulk AdS Higher Spin fields through a time-like bi-local Map, with a Hamiltonian and canonical structure which are established to all orders in $1/N$.

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arXiv2012

S=1 in O(N)/HS duality

Following the work of Maldacena and Zhiboedov, we study the implementation of the Coleman-Mandula theorem in the free O(N)/Higher Spin correspondence. In the bi-local framework we first define an S-matrix for scattering of collective dipoles. Its evaluation in the case of free UV fixed point theory leads to the result S=1 stated in the title. We also present an appropriate field transformation that is seen to transform away all the non-linear 1/N interactions of this theory. A change of boundary conditions and/or external potentials results in a nontrivial S-matrix.

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arXiv2011

How universal is the Wigner distribution?

We consider Gaussian ensembles of m N x N complex matrices. We identify an enhanced symmetry in the system and the resultant closed subsector, which is naturally associated with the radial sector of the theory. The density of radial eigenvalues is obtained in the large N limit. It is of the Wigner form only for m=1. For m \ge 2, the new form of the density is obtained.

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arXiv2010

AdS_4/CFT_3 Construction from Collective Fields

We pursue the construction of higher-spin theory in AdS_4 from CFT_3 of the O(N) vector model in terms of canonical collective fields. In null plane quantization an exact map is established between the two spaces. The coordinates of the AdS_4 space-time are generated from the collective coordinates of the bi-local field. This, in the light cone gauge, provides an exact one to one reconstruction of bulk AdS_4 space-time and higher-spin fields.

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arXiv2009

Laplacians in polar matrix coordinates and radial fermionization in higher dimensions

We consider the quantum mechanical hamiltonian of two, space indexed, hermitean matrices. By introducing matrix valued polar coordinates, we obtain the form of the laplacian acting on invariant states. For potentials depending only on the eigenvalues of the radial matrix, we establish that the radially invariant sector is equivalent to a system of non interacting 2+1 dimensional fermions, and obtain its density description. For a larger number of matrices, the presence of a repulsive radial inter-eigenvalue potential is identified.

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arXiv2008

Non supersymmetric strong coupling background from the large N quantum mechanics of two matrices coupled via a Yang-Mills interaction

We derive the planar large N non-supersymmetric background of the quantum mechanical hamiltonian of two hermitean matrices coupled via a Yang-Mills interaction, in terms of the density of eigenvalues of one of the matrices. This background satisfies an implicit non linear integral equation, with a perturbative small coupling expansion and a solvable large coupling solution, which is obtained. The energy of system and the expectation value of several correlators are obtained in this strong coupling limit. They are free of infrared divergences.

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arXiv2008

Strongly coupled large N spectrum of two matrices coupled via a Yang-Mills interaction

We consider the large N spectrum of the quantum mechanical hamiltonian of two hermitean matrices coupled via a Yang-Mills interaction. In a framework where one of the matrices is treated exactly and the other is treated as a creation operator impurity, the difference equation associated with the Yang-Mills interaction is derived and solved exactly for two impurities. In this case, the full string tension corrected spectrum depends on two momenta. For a specific value of one of these momenta, the spectrum has the same structure as that of giant magnon bound states. States with general number of impurities are also discussed.

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arXiv1998

Duality and Light Cone Symmetries of the Equations of Motion

The matrix theory description of the discrete light cone quantization of $M$ theory on a $T^{2}$ is studied. In terms of its super Yang- Mills description, we identify symmetries of the equations of motion corresponding to independent rescalings of one of the world sheet light cone coordinates, which show how the $S$ duality of Type IIB string theory is realized as a Nahm-type transformation. In the $M$ theory description this corresponds to a simple $9-11$ flip.

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arXiv1997

Classical integrability of chiral $QCD_{2}$ and classical curves

In this letter, classical chiral $QCD_{2}$ is studied in the lightcone gauge $A_{-}=0$. The once integrated equation of motion for the current is shown to be of the Lax form, which demonstrates an infinite number of conserved quantities. Specializing to gauge group SU(2), we show that solutions to the classical equations of motion can be identified with a very large class of curves. We demonstrate this correspondence explicitly for two solutions. The classical fermionic fields associated with these currents are then obtained.

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arXiv1997

The Dynamics of Classical Chiral $QCD_{2}$ Currents

In this paper the dynamics of the classical chiral $QCD_{2}$ currents is studied. We describe how the dynamics of the theory can be summarized in an equation of the Lax form, thereby demonstrating the existence of an infinite set of conserved quantities. Next, the $r$ matrix of a fundamental Poisson relation is obtained and used to demonstrate that the conserved charges Poisson commute. An underlying diffeomorphism symmetry of the equations of motion which is not a symmetry of the action is used to provide a geometric interpretation for the case of gauge group SU(2). This enables us to show that the solutions to the classical equations of motion can be identified with a large class of curves, to demonstrate an auto-Bäcklund transformation and to demonstrate a non linear superposition principle. A link between the spectral problem for $QCD_{2}$ and the solution to the closed curve problem is also demonstrated. We then go on to provide a systematic inverse scattering treatment. This formalism is used to obtain the reflectionless single boundstate eigenvalue soliton solution.

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