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We review a cosmological model where the metric determinant plays a dynamical role and present new numerical results on the cancellation of the vacuum energy density including the contribution of a cosmological constant. The action of this model is only invariant under restricted coordinate transformations with unit Jacobian (the same restriction appears in the well-known unimodular-gravity approach to the cosmological constant problem). As to the possible origin of the nonstandard terms in the matter action of the model, we show that these terms can, in principle, arise from the emergent gravity in the IIB matrix model, a nonperturbative formulation of superstring theory., Comment: 27 pages, 2 figures, v4: published version

This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson, which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions., 12 pages, 5 figures

In this work, we present a neural approach to reconstructing rooted tree graphs describing hierarchical interactions, using a novel representation we term the Lowest Common Ancestor Generations (LCAG) matrix. This compact formulation is equivalent to the adjacency matrix, but enables learning a tree's structure from its leaves alone without the prior assumptions required if using the adjacency matrix directly. Employing the LCAG therefore enables the first end-to-end trainable solution which learns the hierarchical structure of varying tree sizes directly, using only the terminal tree leaves to do so. In the case of high-energy particle physics, a particle decay forms a hierarchical tree structure of which only the final products can be observed experimentally, and the large combinatorial space of possible trees makes an analytic solution intractable. We demonstrate the use of the LCAG as a target in the task of predicting simulated particle physics decay structures using both a Transformer encoder and a Neural Relational Inference encoder Graph Neural Network. With this approach, we are able to correctly predict the LCAG purely from leaf features for a maximum tree-depth of $8$ in $92.5\%$ of cases for trees up to $6$ leaves (including) and $59.7\%$ for trees up to $10$ in our simulated dataset., Comment: 14 pages, 6 figures, accepted in Machine Learning: Science and Technology

We study the leading- (LO) and the next-to-leading-order (NLO) stability of multipole perturbations for a static dielectric M2-brane with spherical topology in the 11-dimensional maximally supersymmetric plane-wave background. We observe a cascade of instabilities that originates from the dipole (j=1) and quadrupole (j=2) sectors (the only unstable sectors of the LO) and propagates toward all the multipoles of the NLO.

The bosonic large-$N$ master field of the IIB matrix model can, in principle, give rise to an emergent classical spacetime. The task is then to calculate this master field as a solution of the bosonic master-field equation. We consider a simplified version of the algebraic bosonic master-field equation and take dimensionality $D=2$ and matrix size $N=6$. For an explicit realization of the pseudorandom constants entering this simplified algebraic equation, we establish the existence of a solution and find, after diagonalization of one of the two obtained matrices, a band-diagonal structure of the other matrix., 20 pages, v6: published version

We explore emergent geometry of the spacetime at the microscopic scale by adiabatic transport of a quasi-coherent state of a fermionic string, with quantum spacetime described by the matrix theory (BFSS matrix model). We show that the generator of the Berry phase is the shift vector of the spacetime foliation by spacelike surfaces associated with the quasi-coherent state. The operator-valued generator of the geometric phase of weak adiabatic transport is the Lorentz connection of the emergent geometry which is not torsion free at the microscopic scale. The effects of the torsion seem consistent with the usual interpretation of the Berry curvature as a pseudo magnetic field.

Journal of high energy physics 05(5), 195 (2021). doi:10.1007/JHEP05(2021)195, We study the two-point function of the stress-tensor multiplet of $ \mathcal{N} $ = 4 SYM in the presence of a line defect. To be more precise, we focus on the single-trace operator of conformal dimension two that sits in the 20′ irrep of the $ \mathfrak{so} $(6)$_{R}$ R-symmetry, and add a Maldacena-Wilson line to the configuration which makes the two-point function non-trivial. We use a combination of perturbation theory and defect CFT techniques to obtain results up to next-to-leading order in the coupling constant. Being a defect CFT correlator, there exist two (super)conformal block expansions which capture defect and bulk data respectively. We present a closed-form formula for the defect CFT data, which allows to write an efficient Taylor series for the correlator in the limit when one of the operators is close to the line. The bulk channel is technically harder and closed-form formulae are particularly challenging to obtain, nevertheless we use our analysis to check against well-known data of $ \mathcal{N} $ = 4 SYM. In particular, we recover the correct anomalous dimensions of a famous tower of twist-two operators (which includes the Konishi multiplet), and successfully compare the one-point function of the stress-tensor multiplet with results obtained using matrix-model techniques., Published by SISSA, [Trieste]

We construct the multi-variable realizations of the W 1 + ∞ algebra such that they lead to the W 1 + ∞ n-algebra. Based on our realizations of the W 1 + ∞ algebra, we derive the W 1 + ∞ constraints for the hermitian one-matrix model. The constraint operators yield not only the W 1 + ∞ algebra but also the closed W 1 + ∞ n-algebra.

We consider the sphere free energy $F(b;m_I)$ in $\mathcal{N}=6$ ABJ(M) theory deformed by both three real masses $m_I$ and the squashing parameter $b$, which has been computed in terms of an $N$ dimensional matrix model integral using supersymmetric localization. We show that setting $m_3=i\frac{b-b^{-1}}{2}$ relates $F(b;m_I)$ to the round sphere free energy, which implies infinite relations between $m_I$ and $b$ derivatives of $F(b;m_I)$ evaluated at $m_I=0$ and $b=1$. For $\mathcal{N}=8$ ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives in terms of derivatives of $m_1,m_2$, which were previously computed to all orders in $1/N$ using the Fermi gas method. This allows us to compute $\partial_b^4 F\vert_{b=1}$ and $\partial_b^5 F\vert_{b=1}$ to all orders in $1/N$, which we precisely match to a recent prediction to sub-leading order in $1/N$ from the holographically dual $AdS_4$ bulk theory., Comment: 23 pages

The quantum chaos is related to a Gaussian random matrix model, which shows a dip-ramp-plateau behavior in the spectral form factor for the large size $N$. The spectral form factor of time dependent Gaussian random matrix model shows also dip-ramp-plateau behavior with a rounding behavior instead of a kink near Heisenberg time. This model is converted to two matrix model, made of $M_1$ and $M_2$. The numerical evaluation for finite $N$ and analytic expression in the large $N$ are compared for the spectral form factor., Comment: 38 pages,16 figures

We study real time classical matrix mechanics of a simplified $2\times 2$ matrix model inspired by the black hole evaporation problem. This is a step towards making a quantitative model of real time evaporation of a black hole, which is realized as a bound state of D0-branes in string theory. The model we study is the reduction of Yang Mills in $2+1$ dimension to $0+1$ dimensions, which has been corrected with an additional potential that can be interpreted as a zero point energy for fermions. Our goal is to understand the lifetime of such a classical bound state object in the classical regime. To do so, we pay particular attention to when D-particles separate to check that the "off diagonal modes" of the matrices become adiabatic and use that information to improve on existing models of evaporation. It turns out that the naive expectation value of the lifetime with the fermionic correction is infinite. This is a logarithmic divergence that arises from very large excursions in the separation between the branes near the threshold for classical evaporation. The adiabatic behavior lets us get some analytic control of the dynamics in this regime to get this estimate. This divergence is cutoff in the quantum theory due to quantization of the adiabatic parameter, resulting in a long lifetime of the bound state, with a parametric dependence of order $\log(1/\hbar)$., Comment: 17 pages, 4 figures. v2: added references

In this article, we study in detail the linear dynamics and cubic interactions for any number $N_\mathrm{field}$ of scalar fields during inflation, directly in terms of the observable curvature perturbation $\zeta$ and $N_\mathrm{field}-1$ entropic fluctuations, a choice that is more suitable for analytical works. In the linear equations of motion for the perturbations, we uncover rich geometrical effects beyond terms involving just the scalar curvature of the field space, and that come from the non-canonical kinetic structure of the scalar fields when the dimension of the field space is larger than two. Moreover, we show that a fast rotation of the local entropic basis can result in negative eigenvalues for the entropic mass matrix, potentially destabilising the background dynamics when $N_\mathrm{field} \geqslant 3$. We also explain how to render manifest the sizes of cubic interactions between the adiabatic and the entropic fluctuations, extending a previous work of ours to any number of interacting fields. As a first analytical application of our generic formalism, we derive the effective single-field theory for perturbations up to cubic order when all entropic fluctuations are heavy enough to be integrated out. In a slow-varying limit, we recover the cubic action expected from the effective field theory of inflation, but with a prediction for the usual Wilson coefficients in terms of the multifield parameters, thus proposing a new interpretation of the bispectrum in this generic $N_\mathrm{field}$ context., Comment: 30 pages, 1 figure

We argue that the large-$N$ master field of the Lorentzian IIB matrix model can give the points and metric of a classical spacetime., 13 pages, v7: published version

The large-$N$ master field of the Lorentzian IIB matrix model can, in principle, give rise to a particular degenerate metric relevant to a regularized big bang. The length parameter of this degenerate metric is then calculated in terms of the IIB-matrix-model length scale., Comment: 14 pages, v6: published version (expanded references in the preprint)

We study some general properties of coupled quantum systems. We consider simple interactions between two copies of identical Hamiltonians such as the SYK model, Pauli spin chains with random magnetic field and harmonic oscillators. Such couplings make the ground states close to the thermofield double states of the uncoupled Hamiltonians. For the coupled SYK model, we push the numerical computation further towards the thermodynamic limit so that an extrapolation in the size of the system is possible. We find good agreement between the extrapolated numerical result and the analytic result in the large-$q$ limit. We also consider the coupled gauged matrix model and vector model, and argue that the deconfinement is associated with the loss of the entanglement, similarly to the previous observation for the coupled SYK model. The understanding of the microscopic mechanism of the confinement/deconfinement transition enables us to estimate the quantum entanglement precisely, and backs up the dual gravity interpretation which relates the deconfinement to the disappearance of the wormhole. Our results demonstrate the importance of the entanglement between the color degrees of freedom in the emergence of the bulk geometry from quantum field theory via holography., 32 pages, 13 figures. v2: published version

We construct the one-dimensional topological sector of $\mathcal N = 6$ ABJ(M) theory and study its relation with the mass-deformed partition function on $S^3$. Supersymmetric localization provides an exact representation of this partition function as a matrix integral, which interpolates between weak and strong coupling regimes. It has been proposed that correlation functions of dimension-one topological operators should be computed through suitable derivatives with respect to the masses, but a precise proof is still lacking. We present non-trivial evidence for this relation by computing the two-point function at twoloop, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product we obtain the two-loop explicit expression for the central charge $c_T$ of ABJ(M) theory. Three- and four-point functions up to one-loop confirm the relation as well. Our result points towards the possibility to localize the one-dimensional topological sector of ABJ(M) and may also be useful in the bootstrap program for 3d SCFTs., 38 pages, 2 figures, 3 tables

We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $\tau$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$ are $\tau$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$ for the $\hbar$-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the $\hbar$-formulation of the KP hierarchy arXiv:1509.04472, arXiv:1512.07172 with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of $\tau$-functions is straightforward and algorithmic., Comment: 26 pages, no figures

Recently, it has been found that JT gravity, which is a two-dimensional theory with bulk action $ -\frac{1}{2}\int {\mathrm d}^2x \sqrt g\phi(R+2)$, is dual to a matrix model, that is, a random ensemble of quantum systems rather than a specific quantum mechanical system. In this article, we argue that a deformation of JT gravity with bulk action $ -\frac{1}{2}\int {\mathrm d}^2x \sqrt g(\phi R+W(\phi))$ is likewise dual to a matrix model. With a specific procedure for defining the path integral of the theory, we determine the density of eigenvalues of the dual matrix model. There is a simple answer if $W(0)=0$, and otherwise a rather complicated answer., Comment: 37 pp

In the matrix model approaches of string/M theories, one starts from a generic symmetry $gl(\infty)$ to reproduce the space-time manifold. In this paper, we consider the generalization in which the space-time manifold emerges from a gauge symmetry algebra which is not necessarily $gl(\infty)$. We focus on the second nontrivial example after the toroidal compactification, the coset space $G/H$, and propose a specific infinite-dimensional symmetry which realizes the geometry. It consists of the gauge-algebra valued functions on the coset and Lorentzian generator pairs associated with the isometry. We show that the $0$-dimensional gauge theory with the mass and Chern-Simons terms gives the gauge theory on the coset with scalar fields associated with $H$., Comment: 29 pages

There are different numerical models, such as the transmission-line matrix model or partially uniform knee model used to predict Schumann radiation. This report introduces a new method build on the...

Motivated by the BPS/CFT correspondence, we explore the similarities between the classical $\beta$-deformed Hermitean matrix model and the $q$-deformed matrix models associated to 3d $\mathcal{N}=2$ supersymmetric gauge theories on $D^2\times_{q}S^1$ and $S_b^3$ by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the $W$-algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results., Comment: 45 pages, 2 figures; v2: typos fixed, comments added to section 4, JHEP version

We derive the analogues of the Harer-Zagier formulas for single- and double-trace correlators in the q-deformed Hermitian Gaussian matrix model. This fully describes single-trace correlators and opens a road to q-deformations of important matrix models properties, such as genus expansion and Wick theorem.

We study the confining/deconfining phase transition in the mass deformed Yang-Mills matrix model which is obtained by the dimensional reduction of the bosonic sector of the four-dimensional maximally supersymmetric Yang-Mills theory compactified on the three sphere, i.e. the bosonic BMN model. The $1/D$ (with $D$ the number of matrices) expansion suggests that the model may have two closely separated transitions. However, using a second order lattice formulation of the model we find that for the small value of the mass parameter, $\mu=2$, those two apparent critical temperatures merge at large $N$, leaving only a single weakly first-order phase transition, in agreement with recent numerical results for $\mu=0$ (the bosonic BFSS model)., Comment: 17 pages and 7 figures; v2: typos fixed, references added, minor editing and comments on double Gaussian approximation

It has been shown recently by Saad, Shenker and Stanford that the genus expansion of a certain matrix integral generates partition functions of Jackiw-Teitelboim (JT) quantum gravity on Riemann surfaces of arbitrary genus with any fixed number of boundaries. We use an extension of this integral for studying gas of baby universes or wormholes in JT gravity. To investigate the gas nonperturbatively we explore the generating functional of baby universes in the matrix model. The simple particular case when the matrix integral includes the exponential potential is discussed in some detail. We argue that there is a phase transition in the gas of baby universes, 29 pages, 12 figures

We study the logarithmic accuracy of angular-ordered parton showers by considering the singular limits of multiple emission matrix elements. This allows us to consider different choices for the evolution variable and propose a new choice which has both the correct logarithmic behaviour and improved performance away from the singular regions. In particular the description of e$^{+}$e$^{−}$ event shapes in the non-logarithmic region is significantly improved. We study the logarithmic accuracy of angular-ordered parton showers by considering the singular limits of multiple emission matrix elements. This allows us to consider different choices for the evolution variable and propose a new choice which has both the correct logarithmic behaviour and improved performance away from the singular regions. In particular the description of e$^{+}$e$^{−}$ event shapes in the non-logarithmic region is significantly improved. We study the logarithmic accuracy of angular-ordered parton showers by considering the singular limits of multiple emission matrix elements. This allows us to consider different choices for the evolution variable and propose a new choice which has both the correct logarithmic behaviour and improved performance away from the singular regions. In particular the description of $e^+e^-$ event shapes in the non-logarithmic region is significantly improved.

In a recent paper, Balthazar, Rodriguez and Yin found some remarkable agreement between the results of c=1 matrix model and D-instanton corrections in two dimensional string theory. Their analysis left undetermined two constants in the string theory computation which had to be fixed by comparing the results with the matrix model results. One of these constants is affected by possible renormalization of the D-instanton action that needs to be computed separately. In this paper we fix the other constant by reformulating the world-sheet analysis in the language of string field theory., Comment: LaTeX file, 15 pages; v2: references added

By using the matrix-model representation, we show that correlation numbers of boundary changing operators (BCO) in $(2,2p+1)$ minimal Liouville gravity satisfy some identities, which we call the null identities. These identities enable us to express the correlation numbers of BCO in terms of those of boundary preserving operators. We also discuss a physical implication of the null identities as the manifestation of the boundary interaction., 8 pages

A U-Net is trained to recover acoustic interference striations (AISs) from distorted ones. A random mode-coupling matrix model is introduced to generate a large number of training data quickly, which are used to train the U-Net. The performance of AIS recovery of the U-Net is tested in range-dependent waveguides with nonlinear internal waves (NLIWs). Although the random mode-coupling matrix model is not an accurate physical model, the test results show that the U-Net successfully recovers AISs under different signal-to-noise ratios (SNRs) and different amplitudes and widths of NLIWs for different shapes.

The canonical tensor model, which is a tensor model in the Hamilton formalism, can be straightforwardly quantized and has an exactly solved physical state. The state is expressed by a wave function with a generalized form of the Airy function. The simplest observable on the state can be expressed by a matrix model which contains non-pairwise index contractions. This matrix model has the same form as the one that appears when the replica trick is applied to the spherical $p$-spin model for spin glasses, but our case has different ranges of variables and parameters from the spin glass case. We analyze the matrix model analytically and numerically. We show some evidences for the presence of a continuous phase transition at the location required by a consistency condition of the canonical tensor model. We also show that there are dimensional transitions of configurations around the transition region. Implications of the results to the canonical tensor model are discussed., Comment: 9 pages, 6 figures; Contribution to the Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019), 31 August - 25 September 2019, Corfu, Greece

The circular Wilson loop in the two-node quiver CFT is computed at large-N and strong 't Hooft coupling by solving the localization matrix model., Comment: 30 pages, 6 figures; v2: misprints corrected

In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings., Comment: LaTeX, 14 pages

I review some recent progresses in counting the number of microstates of AdS supersymmetric black holes in dimension equal or greater than four using holography. The counting is obtained by applying localization and matrix model techniques to the dual field theory. I cover in details the case of dyonic AdS$$_4$$4black holes, corresponding to a twisted compactification of the dual field theory, and I discuss the state of the art for rotating AdS$$_5$$5black holes.

The q-Painleve equation, satisfied by the Fourier transform of the q-Virasoro conformal blocks at c=1, is interpreted as a reformulation of the string equation and two other Virasoro constraints in the $5d$ Dotsenko-Fateev matrix model., 6 pages

We apply the renormalized coupling constants and Virasoro constraints to derive the Itzykson-Zuber Ansatz on the form of the free energy in 2D topological gravity. We also treat the 1D topological gravity and the Hermitian one-matrix models in the same fashion. Some uniform behaviors are discovered in this approach.

In ABJ(M) theory, we propose a matrix model for the exact evaluation of BPS Wilson loops on a latitude circular contour, so providing a new weak-strong interpolation tool. Intriguingly, the matrix model turns out to be a particular case of that computing torus knot invariants in $U(N_1|N_2)$ Chern-Simons theory. At weak coupling we check our proposal against a three-loop computation, performed for generic framing, winding number and representation. The matrix model is amenable of a Fermi gas formulation, which we use to systematically compute the strong coupling and genus expansions. For the fermionic Wilson loop the leading planar behavior agrees with a previous string theory prediction. For the bosonic operator our result provides a clue for finding the corresponding string dual configuration. Our matrix model is consistent with recent proposals for computing Bremsstrahlung functions exactly in terms of latitude Wilson loops. As a by-product, we extend the conjecture for the exact $B^{\theta}_{1/6}$ Bremsstrahlung function to generic representations and test it with a four-loop perturbative computation. Finally, we propose an exact prediction for $B_{1/2}$ at unequal gauge group ranks., Comment: 73 pages; v2: several improvements, JHEP published version

We elaborate the Chern-Simons (CS) matrix models at large $N$. The saddle point equations of these matrix models have a curious structure which cannot be seen in the ordinary one matrix models. Thanks to this structure, an infinite number of multi-cut solutions exist in the CS matrix models. Particularly we exactly derive the two-cut solutions at finite 't\,Hooft coupling in the pure CS matrix model. In the ABJM matrix model, we argue that some of multi-cut solutions might be interpreted as a condensation of the D2-brane instantons., 25 pages, 9 figures; v2: published version, minor changes

We study the finite temperature properties of a recently introduced string bit model designed to capture some features of the emergent string in the tensionless limit. The model consists of a pair of bosonic and fermionic bit operators transforming in the adjoint representation of the color group SU(N). Color confinement is not achieved as a dynamical effect, but instead is enforced by an explicit singlet projection. At large N and finite temperature, the model has a non trivial thermodynamics. In particular, there is a Hagedorn type transition at a finite temperature $T=T_H$ where the string degrees of freedom are liberated and the free energy gets a large contribution $\sim N^{2}$ that plays the role of an order parameter. For $T>T_H$, the low temperature phase becomes unstable. In the new phase, the thermodynamically favoured configurations are characterized by a non-trivial gapped density of the SU(N) angles associated with the singlet projection. We present an accurate algorithm for the determination of the density profile at $N=\infty$. In particular, we determine the gap endpoint at generic temperature and analytical expansions valid near the Hagedorn transition as well as at high temperature. The leading order corrections are characterized by non-trivial exponents that are determined analytically and compared with explicit numerical calculations., 15 pages, 8 pdf figures

Ward identities in the most general "network matrix model" can be described in terms of the Ding-Iohara-Miki algebras (DIM). This confirms an expectation that such algebras and their various limits/reductions are the relevant substitutes/deformations of the Virasoro/W-algebra for (q, t) and (q_1, q_2, q_3) deformed network matrix models. Exhaustive for these purposes should be the Pagoda triple-affine elliptic DIM, which corresponds to networks associated with 6d gauge theories with adjoint matter (double elliptic systems). We provide some details on elliptic qq-characters., 20 pages, 2 figures

The gravity side of the gauge/gravity duality predicts the existence of small black holes with negative specific heat. A free theory of strings has a Hagedorn behavior, but it does not lead to negative specific heat. To understand such states one needs to consider a theory of interacting strings. In the dual gauge theory, the string interactions are related to non-planar diagrams. In this paper the simplest gauge matrix model of two free matrices, that has Hagedorn behavior is analyzed in detail. A simple double trace deformation of the Hamiltonian, proportional to square of the free Hamiltonian square with a negative sign that mimics a gravitational attraction is enough to produce states with negative specific heat perturbatively and one can still compute the equation of state relating the entropy and the energy. A more general argument based on non-planar interactions that are random and that grow faster in strength than the energy suggests that states with negative specific heat appear generically., 20 pages. v2: added references, v3: added references

We study a mass-deformed N=4 version of the Banks-Fischler-Shenker-Susskind (BFSS) matrix model with three matrices and gauge group SU(2). This model has zero Witten index. Despite this, we give numerical evidence for the existence of four supersymmetric ground states, two bosonic and two fermionic, in the limit where the mass deformation is tuned to zero.

We show in details that the all-orders genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the $H_1$ Argyres-Douglas theory coupled to the $\Omega$ background. In the self-dual limit we use the Painlev\'e/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painlev\'e II without taking any double scaling limit or adding any external fields. We decode such solution within the context of trans-series. Finally in the Nekrasov-Shatashvili limit we connect the $H_1$ and the $H_0$ Argyres-Douglas theories to the quantum mechanical models with cubic and double well potentials., Comment: 33 pages, 3 figures, minor corrections, some references added

The IKKT model is proposed as a non-perturbative formulation of superstring theory. We propose a Dirac operator on the noncommutative torus,which is consistent with the IKKT model, based on noncommutative geometry. Next, we consider zero-mode equations of the Dirac operator with magnetic fluxes. We find that zero-mode solutions have the chirality and the generation structures similar to the commutative case. Moreover, we compute Yukawa couplings of chiral matter fields., 32 pages, no figure, v3 published version