Complex Matrix Model Research Articles

1/2 BPS structure constants and random matrices

We study three point functions of half BPS operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory focusing on correlators of two of the operators with dimension of order ∆ ~ N2 and a light single trace operator. These describe vacuum expectation values of type IIB supergravity modes in LLM backgrounds that do not necessarily preserve the same symmetries as the background solution. We propose a class of complex matrix models that fully capture the combinatorics of the problem, and describe their solution in the large N limit. In simple regimes when the dual description is in terms of widely separated condensates of giant gravitons we find that the models are solvable in the large N and can be approximated by unitary Jacobi ensembles; we describe how these distributions are reproduced in the dual bubbling geometry picture for large droplets. In the case of two eigenvalue droplets the model is exactly solvable at finite N. As a result we compute all half-BPS structure constants of heavy-heavy-light type, and reproduce the formulas found via holographic renormalization in the large N limit. We also comment on structure constants of three heavy operators.

Complete Solution of the LSZ Model via Topological Recursion

We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families of correlation functions, one corresponding to the meromorphic differentials $\omega_{g,n}$ of topological recursion, we obtain Dyson-Schwinger equations that eventually lead to the abstract loop equations being, together with their pole structure, the necessary condition for topological recursion. This strategy to show the exact solvability of the LSZ model establishes another approach towards the exceptional property of integrability in some quantum field theories. We compare differences in the loop equations for the LSZ model (with complex fields) and the Grosse-Wulkenhaar model (with hermitian fieldss) and their consequences for the resulting particular type of topological recursion that governs the models.

Superintegrability for (beta -deformed) partition function hierarchies with W-representations

We construct the (beta -deformed) partition function hierarchies with W-representations. Based on the W-representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur functions and Jack polynomials, respectively. Some well known superintegrable matrix models such as the Gaussian hermitian one-matrix model (in the external field), Ntimes N complex matrix model, beta -deformed Gaussian hermitian and rectangular complex matrix models are contained in the constructed hierarchies.

CFT approach to constraint operators for (β-deformed) hermitian one-matrix models

Since the (β-deformed) hermitian one-matrix models can be represented as the integrated conformal field theory (CFT) expectation values, we construct the operators in terms of the generators of the Heisenberg algebra such that the constraints can be derived by inserting the constructed operators into the integrated expectation values. We also obtain the second order total derivative operators associating with the derived constraint operators and analyze their properties. We explore the intrinsic connection between the derived constraint operators and W-representations of some matrix models. For the Gaussian hermitian one-matrix model in the external field and β-deformed N×N complex matrix model, we investigate the superintegrability and derive the corresponding character expansions from their W-representations. Moreover a conjectured formula for the averages of Jack polynomials in the literature is proved.

Target space entanglement in a matrix model for the bubbling geometry

We study the target space entanglement entropy in a complex matrix model that describes the chiral primary sector in mathcal{N} = 4 super Yang-Mills theory, which is associated with the bubbling AdS geometry. The target space for the matrix model is a two-dimensional plane where the eigenvalues of the complex matrix distribute. The eigenvalues are viewed as the position coordinates of fermions, and the eigenvalue distribution corresponds to a droplet formed by the fermions. The droplet is identified with one that specifies a boundary condition in the bubbling geometry. We consider states in the matrix model that correspond to AdS5× S5, an AdS giant graviton and a giant graviton in the bubbling geometry. We calculate the target space entanglement entropy of a subregion for each of the states in the matrix model as well as the area of the boundary of the subregion in the bubbling geometry, and find a qualitative agreement between them.

Delayed deconfinement and the Hawking-Page transition

We revisit the confinement/deconfinement transition in mathcal{N} = 4 super Yang-Mills (SYM) theory and its relation to the Hawking-Page transition in gravity. Recently there has been substantial progress on counting the microstates of 1/16-BPS extremal black holes. However, there is presently a mismatch between the Hawking-Page transition and its avatar in mathcal{N} = 4 SYM. This led to speculations about the existence of new gravitational saddles that would resolve the mismatch. Here we exhibit a phenomenon in complex matrix models which we call “delayed deconfinement.” It turns out that when the action is complex, due to destructive interference, tachyonic modes do not necessarily condense. We demonstrate this phenomenon in ordinary integrals, a simple unitary matrix model, and finally in the context of mathcal{N} = 4 SYM. Delayed deconfinement implies a first-order transition, in contrast to the more familiar cases of higher-order transitions in unitary matrix models. We determine the deconfinement line and find remarkable agreement with the prediction of gravity. On the way, we derive some results about the Gross-Witten-Wadia model with complex couplings. Our techniques apply to a wide variety of (SUSY and non-SUSY) gauge theories though in this paper we only discuss the case of mathcal{N} = 4 SYM.

Natanzon-Orlov model and refined superintegrability

We reconsider the simple matrix model description of Hurwitz numbers proposed by S. Natanzon and A. Orlov, which uses the superintegrability property of the complex matrix model, and discuss a way of its possible supersymmetric extension to approach spin Hurwitz numbers.

Matrix model partition function by a single constraint

In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of {hat{w}}-operators. In this letter, we demonstrate that even more is true: a singlew-constraint is sufficient to uniquely specify the partition functions provided one assumes that it is a power series in time-variables. This substitutes the previous specifications in terms of two requirements: either a string equation imposed on the KP/Toda tau -function or a pair of Virasoro generators. This mysterious single-entry definition holds for a variety of theories, including Hermitian and complex matrix models, and also matrix models with external matrix: the unitary and cubic Kontsevich models. In these cases, it is equivalent to W-representation and is closely related to super integrability. However, a similar single equation that completely determines the partition function exists also in the case of the generalized Kontsevich model (GKM) with potential of higher degree, when the constraint algebra is a larger W-algebra, and neither W-representation, nor superintegrability are understood well enough.

Performance Analysis of Multi-Task Deep Learning Models for Flux Regression in Discrete Fracture Networks

In this work, we investigate the sensitivity of a family of multi-task Deep Neural Networks (DNN) trained to predict fluxes through given Discrete Fracture Networks (DFNs), stochastically varying the fracture transmissivities. In particular, detailed performance and reliability analyses of more than two hundred Neural Networks (NN) are performed, training the models on sets of an increasing number of numerical simulations made on several DFNs with two fixed geometries (158 fractures and 385 fractures) and different transmissibility configurations. A quantitative evaluation of the trained NN predictions is proposed, and rules fitting the observed behavior are provided to predict the number of training simulations that are required for a given accuracy with respect to the variability in the stochastic distribution of the fracture transmissivities. A rule for estimating the cardinality of the training dataset for different configurations is proposed. From the analysis performed, an interesting regularity of the NN behaviors is observed, despite the stochasticity that imbues the whole training process. The proposed approach can be relevant for the use of deep learning models as model reduction methods in the framework of uncertainty quantification analysis for fracture networks and can be extended to similar geological problems (for example, to the more complex discrete fracture matrix models). The results of this study have the potential to grant concrete advantages to real underground flow characterization problems, making computational costs less expensive through the use of NNs.

Superintegrability of Kontsevich matrix model

Many eigenvalue matrix models possess a peculiar basis of observables that have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property langle characterrangle ,sim character. This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated Q Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.

Complex (super)-matrix models with external sources and q-ensembles of Chern–Simons and ABJ(M) type

The Langmann–Szabo–Zarembo (LSZ) matrix model is a complex matrix model with a quartic interaction and two external matrices. The model appears in the study of a scalar field theory on the non-commutative plane. We prove that the LSZ matrix model computes the probability of atypically large fluctuations in the Stieltjes–Wigert matrix model, which is a q-ensemble describing U(N) Chern–Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices, the LSZ matrix model either describes probabilities of large fluctuations in the Chern–Simons partition function, in the unknot invariant or in the two-unknot invariant. We extend the result to supermatrix models, and show that a generalized LSZ supermatrix model describes the probability of atypically large fluctuations in the ABJ(M) matrix model.

Nonperturbative Jackiw-Teitelboim gravity

Recently, Saad et al. showed how to define the genus expansion of Jackiw-Teitelboim (JT) quantum gravity in terms of a double-scaled Hermitian matrix model. However, the model's nonperturbative sector has fatal instabilities at low energy that they be cured by procedures that render the physics nonunique. This might not be a desirable property for a system that is supposed to capture key features of quantum black holes. Presented here is a model with identical zperturbative physics at high energy that instead has a stable and unambiguous nonperturbative completion of the physics at low energy. An explicit examination of the full spectral density function shows how this is achieved. The new model, which is based on complex matrix models, also allows for the straightforward inclusion of spacetime features analogous to Ramond-Ramond fluxes. Intriguingly, there is a deformation parameter that connects this nonperturbative formulation of JT gravity to one which, at low energy, has features of a super JT gravity.

Complete solution to Gaussian tensor model and its integrable properties

Similarly to the complex matrix model, the rainbow tensor models are superintegrable in the sense that arbitrary Gaussian correlators are explicitly expressed through the Clebsh-Gordan coefficients. We introduce associated (Ooguri-Vafa type) partition functions and describe their W-representations. We also discuss their integrability properties, which can be further improved by better adjusting the way the partition function is defined. This is a new avatar of the old unresolved problem with non-Abelian integrability concerning a clever choice of the partition function. This is a part of the long-standing problem to define a non-Abelian lift of integrability from the fundamental to generic representation families of arbitrary Lie algebras.

Optimization of physicochemical properties of novel multiple nanoemulsion for complex food matrices through iterative mathematical modelling

Unfavorable characteristics of certain complex food matrices, such as unpleasant flavors, chemical instability and/or low bioavailability, present a challenge for the food industry. Nanosystems present a solution to these problems, by encapsulating the matrix of interest. In this work a multiple nanoemulsion that encapsulates honeybee pollen extract has been designed and optimized as a natural complex matrix model. Optimization of size and zeta potential was carried out using an iterative mathematical modeling method through a guided experimental design varying the quantities of raw materials (honeybee pollen extract, Lauroglycol 90®, Pluronic F-68 and Chitosan). A total of 29 formulations were performed, 4 of them in triplicate, and an optimal formulation of size 90 ± 1 nm and a zeta potential of +33 ± 2 mV were obtained at the end of the process. The proposed methodology allowed the optimal combination of quantities of raw materials that simultaneously optimize size and zeta potential to be found.

Tensorial generalization of characters

In rainbow tensor models, which generalize rectangular complex matrix model (RCM) and possess a huge gauge symmetry U(N1) × … × U(Nr), we introduce a new sub-basis in the linear space of gauge invariant operators, which is a redundant basis in the space of operators with non-zero Gaussian averages. Its elements are labeled by r-tuples of Young diagrams of a given size equal to the power of tensor field. Their tensor model averages are just products of dimensions: leftlangle chi {R}_1,dots, {R}_rrightrangle sim {C}_{R_1},dots {,}_{R_r}left({N}_1right)dots {D}_{R_r}left({N}_rright) of representations Ri of the linear group SL(Ni), with {C}_{R_1},dots {,}_{R_r} , made of the Clebsch­Gordan coefficients of representations Ri of the symmetric group. Moreover, not only the averages, but the operators {chi}_{overrightarrow{R}} themselves exist only when these {C}_{overrightarrow{R}} are non-vanishing. This sub-basis is much similar to the basis of characters (Schur functions) in matrix models, which is distinguished by the property \\character) ~ character, which opens a way to lift the notion and the theory of characters (Schur functions) from matrices to tensors. In particular, operators {chi}_{overrightarrow{R}} are eigenfunctions of operators which generalize the usual cut-and­join operators hat{W} ; they satisfy orthogonality conditions similar to the standard characters, but they do not form a full linear basis for all gauge-invariant operators, only for those which have non-vanishing Gaussian averages.

Correlators in the β-deformed Gaussian Hermitian and complex matrix models

We investigate the [Formula: see text]-deformed Gaussian Hermitian and [Formula: see text] complex matrix models which are defined as the eigenvalue integral representations. Their [Formula: see text] constraints are constructed such that the constraint operators yield the same [Formula: see text][Formula: see text]-algebra. When particularized to the Virasoro constraints in the [Formula: see text] constraints, the corresponding constraint operators yield the Witt algebra and null 3-algebra. By solving our Virasoro constraints, we derive the formulas for correlators in these two [Formula: see text]-deformed matrix models, respectively.

On W-representations of β- and q,t-deformed matrix models

W-representation realizes partition functions by an action of a cut-and-join operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for β- and q,t-deformations of the simplest rectangular complex matrix model. In the latter case, instead of the complicated definition in terms of multiple Jackson integrals, we define partition functions as the weight-two series, made from Macdonald polynomials, which are evaluated at different loci in the space of time variables. Resulting expression for the Wˆ operator appears related to the problem of simple Hurwitz numbers (contributing are also the Young diagrams with all but one lines of length two and one). This problem is known to exhibit nice integrability properties. Still the answer for Wˆ can seem unexpectedly sophisticated and calls for improvements. Since matrix models lie at the very basis of all gauge- and string-theory constructions, our exercise provides a good illustration of the jump in complexity between β- and q,t-deformations – which is not always seen at the accidentally simple level of Calogero-Ruijsenaars Hamiltonians (where both deformations are equally straightforward). This complexity is, however, quite familiar in the theories of network models, topological vertices and knots.

Factorial and Economic Evaluation of an Aqueous Two-Phase Partitioning Pilot Plant for Invertase Recovery From Spent Brewery Yeast.

Aqueous two-phase systems (ATPS) have been reported as an attractive biocompatible extraction system for recovery and purification of biological products. In this work, the implementation, characterization, and optimization (operational and economic) of invertase extraction from spent brewery yeast in a semi-automatized pilot plant using ATPS is reported. Gentian violet was used as tracer for the selection of phase composition through phase entrainment minimization. Yeast suspension was chosen as a complex cell matrix model for the recovery of the industrial relevant enzyme invertase. Flow rates of phases did not have an effect, given that a bottom continuous phase is given, while load of sample and number of agitators improved the recovery of the enzyme. The best combination of factors reached a recovery of 129.35 ± 2.76% and a purification factor of 4.98 ± 1.10 in the bottom phase of a PEG-Phosphate system, also resulting in the removal of inhibitor molecules increasing invertase activity as reported by several other authors. Then, an economic analysis was performed to study the production cost of invertase analyzing only the significant parameters for production. Results indicate that the parameters being analyzed only affect the production cost per enzymatic unit, while variations in the cost per batch are not significant. Moreover, only the sample load is significant, which, combined with operational optimization results, gives the same optimal result for operation, maximizing recovery yield (15% of sample load and 1 static mixer). Overall res ults of these case studies show continuous pilot-scale ATPS as a viable and reproducible extraction/purification system for high added-value biological compounds.

Complex Langevin dynamics in large N unitary matrix models

Using complex Langevin dynamics we examine the phase structure of complex unitary matrix models and compare the numerical results with analytic results found at large $N$. The actions we consider are manifestly complex, and thus the dominant contribution to the path integral comes from the space of complexified gauge field configuration. For this reason, the eigenvalues of unitary matrix lie off the unit circle and venture out in the complex plane. One example of a complex unitary matrix model, with Polyakov line as the unitary matrix, is an effective description of a QCD at finite density and temperature with $N$ number of colors and $N_f$ number of quark flavors defined on the manifold $S^1 \times S^3$. A distinct feature of this model, the occurrence of a series of Gross-Witten-Wadia transitions, as a function of the quark chemical potential, is reproduced using complex Langevin simulations. We simulate several other observables including Polyakov lines and quark number density, for large $N$ and $N_f$ and found excellent match with the analytic results.

Spectra of eigenstates in fermionic tensor quantum mechanics

We study the $O(N_1)\times O(N_2)\times O(N_3)$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks $N_i$ are all equal, this model has a large $N$ limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of $SO(N_1)\times SO(N_2)\times SO(N_3)$ invariant states for any set of $N_i$. For equal ranks the number of singlets is non-vanishing only when $N$ is even, and it exhibits rapid growth: it jumps from $36$ in the $O(4)^3$ model to $595354780$ in the $O(6)^3$ model. We derive bounds on the values of energy, which show that they scale at most as $N^3$ in the large $N$ limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order $1/N$. For $N_3=1$ the tensor model reduces to $O(N_1)\times O(N_2)$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU(N_1)\times SU(N_2)\times U(1)$ symmetry. Finally, we study the $N_3=2$ case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O(N_1)\times O(N_2)\times U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large $N$ limits where the ground state energies are of order $N^2$, while the energy gaps are of order $1$.