Type IIB Matrix Model Research Articles

Progress in the numerical studies of the type IIB matrix model

The type IIB matrix model, also known as the IKKT model, has been proposed as a promising candidate for a non-perturbative formulation of superstring theory. Based on this proposal, various attempts have been made to explain how our four-dimensional space-time can emerge dynamically from superstring theory. In this article, we review the progress in numerical studies on the type IIB matrix model. We particularly focus on the most recent results for the Euclidean and Lorentzian versions, which are obtained using the complex Langevin method to overcome the sign problem. We also review the earlier results obtained using conventional Monte Carlo methods and clarify the relationship among different calculations.

Replicas, averaging and factorization in the IIB matrix model

We study the partition functions of multiple replicas (copies) of D-brane configurations in the type IIB (IKKT) matrix model. We consider the quenched regime, where small fluctuations of the matrices are superimposed onto the slow (quenched) dynamics of the background, so the partition function is an ensemble average over the background. Interacting D-branes always factorize in a simple way. On the other hand, the non-interacting BPS configurations may or may not factorize depending on the number of replicas, and their factorization mechanism is more involved as the corresponding saddle-point solutions (half-wormholes) break the replica symmetry. We argue that the simple factorization mechanism of interacting branes is actually more interesting as it carries the specific signatures of quantum gravity, which are absent from disordered field theories like the SYK model.

The emergence of expanding space–time and intersecting D-branes from classical solutions in the Lorentzian type IIB matrix model

Abstract The type IIB matrix model is a promising candidate for a nonperturbative formulation of superstring theory. As such, it is expected to explain the origin of space–time and matter at the same time. This has been partially demonstrated by the previous Monte Carlo studies on the Lorentzian version of the model, which suggested the emergence of (3+1)-dimensional expanding space–time. Here we investigate the same model by solving numerically the classical equation of motion, which is expected to be valid at late times since the action becomes large due to the expansion of space. Many solutions are obtained by the gradient descent method starting from random matrix configurations, assuming a quasi-direct-product structure for the (3+1)-dimensions and the extra 6 dimensions. We find that these solutions generally admit the emergence of expanding space–time and a block-diagonal structure in the extra dimensions, the latter being important for the emergence of intersecting D-branes. For solutions corresponding to D-branes with appropriate dimensionality, the Dirac operator is shown to acquire a zero mode in the limit of infinite matrix size.

On the structure of the emergent 3D expanding space in the Lorentzian type IIB matrix model

Abstract The emergence of (3+1)D expanding space-time in the Lorentzian type IIB matrix model is an intriguing phenomenon that has been observed in Monte Carlo studies of this model. In particular, this may be taken as support for the conjecture that the model is a nonperturbative formulation of superstring theory in (9+1) dimensions. In this paper we investigate the space-time structure of the matrices generated by simulating this model and its simplified versions, and find that the expanding part of the space is described essentially by the Pauli matrices. We argue that this is due to an approximation used in the simulation to avoid the sign problem, which actually amounts to replacing ${e}^{iS_{\rm b}}$ by ${e}^{\beta S_{\rm b}}$ ($\beta>0$) in the partition function, where $S_{\rm b}$ is the bosonic part of the action. We also discuss the possibility of obtaining a regular space-time with the (3+1)D expanding behavior in the original model with the correct ${e}^{iS_{\rm b}}$ factor.

Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model

The Lorentzian type IIB matrix model has been studied as a promising candidate for a nonperturbative formulation of superstring theory. In particular, the emergence of (3+1)D expanding space-time was observed by Monte Carlo studies of this model. It has been found recently, however, that the matrix configurations generated by the simulation is singular in that the submatrices representing the expanding 3D space have only two large eigenvalues associated with the Pauli matrices. This problem has been attributed to the approximation used to avoid the sign problem in simulating the model. Here we investigate the model using the complex Langevin method to overcome the sign problem instead of using the approximation. Our results indicate a clear departure from the Pauli-matrix structure, while the (3+1)D expanding behavior is kept intact.

Complex Langevin analysis of the spontaneous symmetry breaking in dimensionally reduced super Yang-Mills models

In recent years the complex Langevin method (CLM) has proven a powerful method in studying statistical systems which suffer from the sign problem. Here we show that it can also be applied to an important problem concerning why we live in four-dimensional spacetime. Our target system is the type IIB matrix model, which is conjectured to be a nonperturbative definition of type IIB superstring theory in ten dimensions. The fermion determinant of the model becomes complex upon Euclideanization, which causes a severe sign problem in its Monte Carlo studies. It is speculated that the phase of the fermion determinant actually induces the spontaneous breaking of the SO(10) rotational symmetry, which has direct consequences on the aforementioned question. In this paper, we apply the CLM to the 6D version of the type IIB matrix model and show clear evidence that the SO(6) symmetry is broken down to SO(3). Our results are consistent with those obtained previously by the Gaussian expansion method.

A type IIB matrix model via non-Abelian T-dualities

We construct a new solution of type IIB supergravity with eight supercharges by applying non-Abelian T-duality to the AdS5 × S5 solution, along the SU(2) isometries in both the internal and Lorentz symmetries. The study of quantised charges and D-brane embeddings suggests a configuration of D1 and D3’-branes that polarise into concentric, spherical, D3 and D5-branes due to the Myers dielectric effect. We find that our solution is dual to a IIB matrix model with fuzzy sphere vacua characterised by partitions of the number of D1 and D3’-branes. We also study the solution obtained via (Abelian) T-dualities along the S1 fiber angles inside the S3’s of the AdS5 × S5 solution. In addition, we point out a precise relation between NATD and ATD solutions by considering a double scaling limit in the NATD solutions. In the particular case of the solutions obtained by a single (non-)Abelian T-duality inside the Lorentz symmetries of the AdS5 × S5 solution, we demonstrate that the double scaling limit provides a supergravity realisation of the relations between the vacua of the half-BPS theories studied by Lin and Maldacena.

A new method for probing the late-time dynamics in the Lorentzian type IIB matrix model

The type IIB matrix model has been investigated as a possible nonperturbative formulation of superstring theory. In particular, it was found by Monte Carlo simulation of the Lorentzian version that the 9-dimensional rotational symmetry of the spatial matrices is broken spontaneously to the 3-dimensional one after some "critical time". In this paper we develop a new simulation method based on the effective theory for the submatrices corresponding to the late time. Using this method, one can obtain the results for $N\times N$ matrices by simulating matrices typically of the size $O(\sqrt{N})$. We confirm the validity of this method and demonstrate its usefulness in simplified models.

Universality and the dynamical space-time dimensionality in the Lorentzian type IIB matrix model

The type IIB matrix model is one of the most promising candidates for a nonperturbative formulation of superstring theory. In particular, its Lorentzian version was shown to exhibit an interesting real-time dynamics such as the spontaneous breaking of the 9-dimensional rotational symmetry to the 3-dimensional one. This result, however, was obtained after regularizing the original matrix integration by introducing “infrared” cutoffs on the quadratic moments of the Hermitian matrices. In this paper, we generalize the form of the cutoffs in such a way that it involves an arbitrary power (2p) of the matrices. By performing Monte Carlo simulation of a simplified model, we find that the results become independent of p and hence universal for p ≳ 1.3. For p as large as 2.0, however, we find that large-N scaling behaviors do not show up, and we cannot take a sensible large-N limit. Thus we find that there is a certain range of p in which a universal large-N limit can be taken. Within this range of p, the dynamical space-time dimensionality turns out to be (3 + 1), while for p = 2.0, where we cannot take a sensible large-N limit, we observe a (5+1)d structure.

The complex Langevin analysis of spontaneous symmetry breaking induced by complex fermion determinant

In many interesting physical systems, the determinant which appears from integrating out fermions becomes complex, and its phase plays a crucial role in the determination of the vacuum. An example of this is QCD at low temperature and high density, where various exotic fermion condensates are conjectured to form. Another example is the Euclidean version of the type IIB matrix model for 10d superstring theory, where spontaneous breaking of the SO(10) rotational symmetry down to SO(4) is expected to occur. When one applies the complex Langevin method to these systems, one encounters the singular-drift problem associated with the appearance of nearly zero eigenvalues of the Dirac operator. Here we propose to avoid this problem by deforming the action with a fermion bilinear term. The results for the original system are obtained by extrapolations with respect to the deformation parameter. We demonstrate the power of this approach by applying it to a simple matrix model, in which spontaneous symmetry breaking from SO(4) to SO(2) is expected to occur due to the phase of the complex fermion determinant. Unlike previous work based on a reweighting-type method, we are able to determine the true vacuum by calculating the order parameters, which agree with the prediction by the Gaussian expansion method.

Power-law expansion of the Universe from the bosonic Lorentzian type IIB matrix model

Recent studies on the Lorentzian version of the type IIB matrix model show that (3+1)D expanding universe emerges dynamically from (9+1)D space-time predicted by superstring theory. Here we study a bosonic matrix model obtained by omitting the fermionic matrices. With the adopted simplification and the usage of a large-scale parallel computer, we are able to perform Monte Carlo calculations with matrix size up to $N=512$, which is twenty times larger than that used previously for the studies of the original model. When the matrix size is larger than some critical value $N_{\rm c}\simeq 110$, we find that (3+1)D expanding universe emerges dynamically with a clear large-$N$ scaling property. Furthermore, at sufficiently late times, we observe a power-law behavior $t^{1/2}$ of the spatial extent with respect to time $t$, which is reminiscent of the expanding behavior of the Friedmann-Robertson-Walker universe in the radiation dominated era. We discuss possible implications of this result on the original model including fermionic matrices.

Dynamical phase space from anSO(d,d)matrix model

It is shown that a matrix model with SO(d,d) global symmetry is derived from a generalized Yang-Mills theory on the standard Courant algebroid. This model keeps all the positive features of the well-studied type IIB matrix model, and it has many additional welcome properties. We show that it does not only capture the dynamics of spacetime, but it should be associated with the dynamics of phase space. This is supported by a large set of classical solutions of its equations of motion, which corresponds to phase spaces of noncommutative curved manifolds and points to a new mechanism of emergent gravity. The model possesses an additional symmetry that exchanges positions and momenta, in analogy to quantum mechanics. It is argued that the emergence of phase space in the model is an essential feature for the investigation of the precise relation of matrix models to string theory and quantum gravity.

Realizing the Standard Model particles in the type IIB matrix model

AbstractWe discuss how three generations of the Standard Model particles are realized in the type IIB matrix model.

Recent developments in the type IIB matrix model

AbstractWe review recent developments in the type IIB matrix model, which was conjectured to be a nonperturbative formulation of superstring theory. In the first part we review the recent results for the Euclidean model, which suggest that SO(10) symmetry is spontaneously broken. In the second part we review the recent results for the Lorentzian model. In particular, we discuss Monte Carlo results, which suggest that (3+1)‐dimensional expanding universe emerges dynamically. We also discuss some results suggesting the emergence of exponential expansion and the power‐law expansion at later times. The behaviors at much later times are studied by the classical equation of motion. We discuss a solution representing 3d expanding space, which suggests a possible solution to the cosmological constant problem.

Realizing three generations of the Standard Model fermions in the type IIB matrix model

We discuss how the Standard Model particles appear from the type IIB matrix model, which is considered to be a nonperturbative formulation of superstring theory. In particular, we are concerned with a constructive definition of the theory, in which we start with finite-N matrices and take the large-N limit afterwards. In that case, it was pointed out recently that realizing chiral fermions in the model is more difficult than it had been thought from formal arguments at N=infinity and that introduction of a matrix version of the warp factor is necessary. Based on this new insight, we show that two generations of the Standard Model fermions can be realized by considering a rather generic configuration of fuzzy S^2 and fuzzy S^2 * S^2 in the extra dimensions. We also show that three generations can be obtained by squashing one of the S^2's that appear in the configuration. Chiral fermions appear at the intersections of the fuzzy manifolds with nontrivial Yukawa couplings to the Higgs field, which can be calculated from the overlap of their wave functions.

Realizing chiral fermions in the type IIB matrix model at finite N

We discuss how chiral fermions can appear in the type IIB matrix model, which is considered to be a nonperturbative formulation of superstring theory. In particular, we are concerned with a constructive definition of the theory, in which we start with a finite-N configuration and take the large-N limit later on. We point out that there exists a certain necessary condition which the structure of the extra dimensions should satisfy. As an example, we consider a previous proposal using intersecting branes and show that chiral fermions can indeed be realized in four dimensions by introducing a matrix counterpart of warped space-time. This is remarkable in view of the well-known difficulty in realizing chiral fermions in lattice gauge theory.

(3+1)-DIMENSIONAL EXPANDING UNIVERSE FROM A LORENTZIAN MATRIX MODEL FOR SUPERSTRING THEORY IN (9+1)-DIMENSIONS

We study the Lorentzian version of the type IIB matrix model as a nonperturbative formulation of superstring theory in (9+1)-dimensions. Monte Carlo results show that not only space but also time emerges dynamically in this model. Furthermore, the real-time dynamics extracted from the matrices turns out to be remarkable: 3 out of 9 spatial directions start to expand at some critical time. This can be interpreted as the birth of our Universe.

Lie-algebraic solutions of the type IIB matrix model

A systematic search for Lie algebra solutions of the type IIB matrix model is performed. Our survey is based on the classification of all Lie algebras for dimensions up to five and of all nilpotent Lie algebras of dimension six. It is shown that Lie-type solutions of the equations of motion of the type IIB matrix model exist and they correspond to certain nilpotent and solvable Lie algebras. Their representation in terms of Hermitian matrices is discussed in detail. These algebras give rise to certain non-commutative spaces for which the corresponding star-products are provided. Finally the issue of constructing quantized compact nilmanifolds and solvmanifolds based on the above algebras is addressed.

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.

String coupling and interactions in type IIB matrix model

String coupling and interactions in type IIB matrix model