Yang-Mills Matrix Models Research Articles

Covariant 4-dimensional fuzzy spheres, matrix models and higher spin

We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as -equivariant projections of quantized coadjoint orbits of . We show that they arise as solutions in Yang–Mills matrix models, which naturally leads to higher-spin gauge theories on S4. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions , which is expected to entail 2 + 1 generations.

Yang–Mills matrix mechanics and quantum phases

The [Formula: see text] Yang–Mills matrix model coupled to fundamental fermions is studied in the adiabatic limit, and quantum critical behavior is seen at special corners of the gauge field configuration space. The quantum scalar potential for the gauge field induced by the fermions diverges at the corners, and is intimately related to points of enhanced degeneracy of the fermionic Hamiltonian. This in turn leads to superselection sectors in the Hilbert space of the gauge field, the ground states in different sectors being orthogonal to each other. The [Formula: see text] Yang–Mills matrix model coupled to two Weyl fermions has three quantum phases. When coupled to a massless Dirac fermion, the number of quantum phases is four. One of these phases is the color-spin locked phase. This paper is an extended version of the lectures given by the second author (SV) at the International Workshop on Quantum Physics: Foundations and Applications, Bangalore, in February 2016, and is based on [1].

Quantum phases of Yang-Mills matrix model coupled to fundamental fermions

By investigating the SU(2) Yang-Mills matrix model coupled to fundamental fermions in the adiabatic limit, we demonstrate quantum critical behaviour at special corners of the gauge field configuration space. The quantum scalar potential for the gauge field induced by the fermions diverges at the corners and is intimately related to points of enhanced degeneracy of the fermionic Hamiltonian. This in turn leads to superselection sectors in the Hilbert space of the gauge field, the ground states in different sectors being orthogonal to each other. As a consequence of our analysis, we show that 2-color quantum chromodynamics coupled to two Weyl fermions has three quantum phases. When coupled to a massless Dirac fermion, the number of quantum phases is four. One of these phases is the color-spin locked phase.

Emergent fuzzy geometry and fuzzy physics in four dimensions

A detailed Monte Carlo calculation of the phase diagram of bosonic mass-deformed IKKT Yang–Mills matrix models in three and six dimensions with quartic mass deformations is given. Background emergent fuzzy geometries in two and four dimensions are observed with a fluctuation given by a noncommutative U(1) gauge theory very weakly coupled to normal scalar fields. The geometry, which is determined dynamically, is given by the fuzzy spheres SN2 and SN2×SN2 respectively. The three and six matrix models are effectively in the same universality class. For example, in two dimensions the geometry is completely stable, whereas in four dimensions the geometry is stable only in the limit M⟶∞, where M is the mass of the normal fluctuations. The behaviors of the eigenvalue distribution in the two theories are also different. We also sketch how we can obtain a stable fuzzy four-sphere SN2×SN2 in the large N limit for all values of M as well as models of topology change in which the transition between spheres of different dimensions is observed. The stable fuzzy spheres in two and four dimensions act precisely as regulators which is the original goal of fuzzy geometry and fuzzy physics. Fuzzy physics and fuzzy field theory on these spaces are briefly discussed.

Remarks on the eigenvalues distributions of D≤4 Yang–Mills matrix models

The phenomenon of emergent fuzzy geometry and noncommutative gauge theory from Yang–Mills matrix models is briefly reviewed. In particular, the eigenvalue distributions of Yang–Mills matrix models in lower dimensions in the commuting (matrix or Yang–Mills) phase of these models are discussed.

New algorithm and phase diagram of noncommutative $ \varPhi $ 4 on the fuzzy sphere

We propose a new algorithm for simulating noncommutative phi-four theory on the fuzzy sphere based on, i) coupling the scalar field to a U(1) gauge field, in such a way that in the commutative limit N\longrightarrow \infty, the two modes decouple and we are left with pure scalar phi-four on the sphere, and ii) diagonalizing the scalar field by means of a U(N) unitary matrix, and then integrating out the unitary group from the partition function. The number of degrees of freedom in the scalar sector reduces, therefore, from N^2 to the N eigenvalues of the scalar field, whereas the dynamics of the U(1) gauge field, is given by D=3 Yang-Mills matrix model with a Myers term. As an application, the phase diagram, including the triple point, of noncommutative phi-four theory on the fuzzy sphere, is reconstructed with small values of N up to N=10, and large numbers of statistics.

The curvature of branes, currents and gravity in matrix models

The curvature of brane solutions in Yang-Mills matrix models is expressed in terms of conserved currents associated with global symmetries of the model. This implies a relation between the Ricci tensor and the energy-momentum tensor due to the basic matrix model action, without invoking an Einstein-Hilbert term. The coupling is governed by the extrinsic curvature of the brane embedding, which arises naturally for compactified brane solutions. The effective gravity on the brane is thereby related to the compactification moduli, and protected from quantum corrections due to the relation with global symmetries.

Gravity and compactified branes in matrix models

A mechanism for emergent gravity on brane solutions in Yang-Mills matrix models is exhibited. Newtonian gravity and a partial relation between the Einstein tensor and the energy-momentum tensor can arise from the basic matrix model action, without invoking an Einstein-Hilbert-type term. The key requirements are compactified extra dimensions with extrinsic curvature M^4 x K \subset R^D and split noncommutativity, with a Poisson tensor \theta^{ab} linking the compact with the noncompact directions. The moduli of the compactification provide the dominant degrees of freedom for gravity, which are transmitted to the 4 noncompact directions via the Poisson tensor. The effective Newton constant is determined by the scale of noncommutativity and the compactification. This gravity theory is well suited for quantization, and argued to be perturbatively finite for the IKKT model. Since no compactification of the target space is needed, it might provide a way to avoid the landscape problem in string theory.

IMPACT OF SUPERSYMMETRY ON EMERGENT GEOMETRY IN YANG–MILLS MATRIX MODELS II

We present a study of D = 4 supersymmetric Yang–Mills matrix models with SO(3) mass terms based on the Monte Carlo method. In the bosonic models we show the existence of an exotic first-/second-order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first-order due to the discontinuity in the action whereas it looks second-order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N. We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a nonpolynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this nonpolynomial law.

Intersecting branes and a standard model realization in matrix models

We consider intersecting brane solutions of the type IIB matrix model. It is shown that fermionic zero-modes arise on such backgrounds, localized at the brane intersections. They lead to chiral fermions in four dimensions under certain conditions. Such configurations reproduce many of the welcome features in similar string-theoretic constructions. Therefore they can be used to construct semi-realistic particle physics models in the framework of Yang-Mills matrix models. In particular, we present a brane configuration which realizes the correct chiral spectrum of the standard model in the matrix model. Furthermore, the stability of intersecting branes is discussed by analyzing the 1-loop effective action. It is shown that intersecting branes may form a bound state for certain flux configurations. The four-dimensional geometry of the branes is generic, and determined by the (emergent) gravity sector of the matrix model.

Heat kernel expansion and induced action for the matrix model Dirac operator

We compute the quantum effective action induced by integrating out fermions in Yang-Mills matrix models on a 4-dimensional background, expanded in powers of a gauge-invariant UV cutoff. The resulting action is recast into the form of generalized matrix models, manifestly preserving the SO(D) symmetry of the bare action. This provides non-commutative (NC) analogs of the Seeley-de Witt coefficients for the emergent gravity which arises on NC branes, such as curvature terms. From the gauge theory point of view, this provides strong evidence that the non-commutative \( \mathcal{N} = 4 \) SYM has a hidden SO(10) symmetry even at the quantum level, which is spontaneously broken by the space-time background. The geometrical view proves to be very powerful, and allows to predict nontrivial loop computations in the gauge theory.

Curvature and gravity actions for matrix models

We show how gravitational actions, in particular the Einstein–Hilbert action, can be obtained from additional terms in Yang–Mills matrix models. This is consistent with recent results on induced gravitational actions in these matrix models, realizing spacetime as four-dimensional brane solutions. It opens up the possibility for a controlled non-perturbative description of gravity through simple matrix models, with interesting perspectives for the problem of vacuum energy. The relation with UV/IR mixing and non-commutative gauge theory is discussed.

Noncommutative gauge theory and symmetry breaking in matrix models

We show how the fields and particles of the standard model can be naturally realized in noncommutative gauge theory. Starting with a Yang-Mills matrix model in more than 4 dimensions, a SU(n) gauge theory on a Moyal-Weyl space arises with all matter and fields in the adjoint of the gauge group. We show how this gauge symmetry can be broken spontaneously down to SU(3)_c x SU(2)_L x U(1)_Q (resp. SU(3)_c x U(1)_Q), which couples appropriately to all fields in the standard model. An additional U(1)_B gauge group arises which is anomalous at low energies, while the trace-U(1) sector is understood in terms of emergent gravity. A number of additional fields arise which we assume to be massive, in a pattern that is reminiscent of supersymmetry. The symmetry breaking might arise via spontaneously generated fuzzy spheres, in which case the mechanism is similar to brane constructions in string theory.

Fermions and noncommutative emergent gravity II: curved branes in extra dimensions

We study fermions coupled to Yang-Mills matrix models from the point of view of emergent gravity. The matrix model Dirac operator provides an appropriate coupling for fermions to the effective gravitational metric for general branes with nontrivial embedding, albeit with a non-standard spin connection. This generalizes previous results for 4-dimensional matrix models. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action of the effective metric, as well as additional terms which couple the Poisson tensor to the Riemann tensor, and a dilaton-like term.

On the Newtonian limit of emergent NC gravity and long-distance corrections

We show how Newtonian gravity emerges on 4-dimensional non-commutative spacetime branes in Yang-Mills matrix models. Large matter clusters such as galaxies are embedded in large-scale harmonic deformations of the space-time brane, which screen gravity for long distances. On shorter scales, the local matter distribution reproduces Newtonian gravity via local deformations of the brane and its metric. The harmonic ``gravity bag'' acts as a halo with effective positive energy density. This leads in particular to a significant enhancement of the orbital velocities around galaxies at large distances compared with the Newtonian case, before dropping to zero as the geometry merges with a Milne-like cosmology. Besides these ``harmonic'' solutions, there is another class of solutions which is more similar to Einstein gravity. Thus the IKKT model provides an accessible candidate for a quantum theory of gravity.

On the shape of a D-brane bound state and its topology change

As is well known, coordinates of D-branes are described by NxN matrices. From generic non-commuting matrices, it is difficult to extract physics, for example, the shape of the distribution of positions of D-branes. To overcome this problem, we generalize and elaborate on a simple prescription, first introduced by Hotta, Nishimura and Tsuchiya, which determines the most appropriate gauge to make the separation between diagonal components (D-brane positions) and off-diagonal components. This prescription makes it possible to extract the distribution of D-branes directly from matrices. We verify the power of it by applying it to Monte-Carlo simulations for various lower dimensional Yang-Mills matrix models. In particular, we detect the topology change of the D-brane bound state for a phase transition of a matrix model; the existence of this phase transition is expected from the gauge/gravity duality, and the pattern of the topology change is strikingly similar to the counterpart in the gravity side, the black hole/black string transition. We also propose a criterion, based on the behavior of the off-diagonal components, which determines when our prescription gives a sensible definition of D-brane positions. We provide numerical evidence that our criterion is satisfied for the typical distance between D-branes. For a supersymmetric model, positions of D-branes can be defined even at a shorter distance scale. The behavior of off-diagonal elements found in this analysis gives some support for previous studies of D-brane bound states.

Covariant field equations, gauge fields and conservation laws from Yang-Mills matrix models

The effective geometry and the gravitational coupling of nonabelian gauge and scalar fields on generic NC branes in Yang-Mills matrix models is determined. Covariant field equations are derived from the basic matrix equations of motions, known as Yang-Mills algebra. Remarkably, the equations of motion for the Poisson structure and for the nonabelian gauge fields follow from a matrix Noether theorem, and are therefore protected from quantum corrections. This provides a transparent derivation and generalization of the effective action governing the SU(n) gauge fields obtained in [1], including the would-be topological term. In particular, the IKKT matrix model is capable of describing 4-dimensional NC space-times with a general effective metric. Metric deformations of flat Moyal-Weyl space are briefly discussed.

Emergent gravity and noncommutative branes from Yang–Mills matrix models

The framework of emergent gravity arising from Yang–Mills matrix models is developed further, for general noncommutative branes embedded in R D . The effective metric on the brane turns out to have a universal form reminiscent of the open string metric, depending on the dynamical Poisson structure and the embedding metric in R D . A covariant form of the tree-level equations of motion is derived, and the Newtonian limit is discussed. This points to the necessity of branes in higher dimensions. The quantization is discussed qualitatively, which singles out the IKKT model as a prime candidate for a quantum theory of gravity coupled to matter. The Planck scale is then identified with the scale of N = 4 SUSY breaking. A mechanism for avoiding the cosmological constant problem is exhibited.

Fermions and emergent noncommutative gravity

Fermions coupled to Yang-Mills matrix models are studied from the point of view of emergent gravity. We show that the simple matrix model action provides an appropriate coupling for fermions to gravity, albeit with a non-standard spin connection. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action and a dilaton-like term, at least for on-shell geometries. This explains and precisely reproduces the UV/IR mixing for fermions in noncommutative gauge theory, extending recent results for scalar fields. It also explains why some UV/IR mixing remains even in supersymmetric models, except in the N=4 case.

Schwinger–Dyson operator of Yang–Mills matrix models with ghosts and derivations of the graded shuffle algebra

We consider large-N multi-matrix models whose action closely mimics that of Yang–Mills theory, including gauge-fixing and ghost terms. We show that the factorized Schwinger–Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G(ξ), are quadratic equations in concatenation of correlations. The Schwinger–Dyson operator is built from the left annihilation operator, which does not satisfy the Leibnitz rule with respect to concatenation. So the loop equations are not differential equations. We show that left annihilation is a derivation of the graded shuffle product of gluon and ghost correlations. The shuffle product is the point-wise product of Wilson loops, expressed in terms of correlations. So in the limit where concatenation is approximated by shuffle products, the loop equations become differential equations. Remarkably, the Schwinger–Dyson operator as a whole is also a derivation of the graded shuffle product. This allows us to turn the loop equations into linear equations for the shuffle reciprocal, which might serve as a starting point for an approximation scheme.