In our project we study superstring theory from first principles using the IKKT matrix model. This is totally different from the previous investigations of superstring theory. The conventional formulation of superstring theory is based on perturbation theory, which assumes a particular background as a starting point of the perturbative expansion. In fact, there are extremely many backgrounds that are consistent within perturbation theory, and therefore one cannot really predict anything in that way. In the IKKT matrix model, on the other hand, the background itself is determined dynamically. In this project, we aim at large-scale computations of the IKKT model in collaboration with experts from lattice QCD. Our primary goal is to understand the cosmic inflation from a microscopic point of view, and to compare our results against the observational data of the cosmic microwave background. We also apply similar ideas to investigate the interior structure of blackholes.
Another important subject of our project is to extend the lattice formulation of QCD to the finite density regime. This is known to be extremely difficult due to the so-called sign problem, which occurs when one tries to include the vacuum polarization effects of sea quarks at finite baryon number density. As a promising approach to this problem, the complex Langevin method has attracted a lot of attention in the past five years. We develop this method further by using new techniques such as the generalized cooling, and aim at investigating interesting parameter regions of the QCD phase diagram.
While the above two subjects may look somewhat disconnected, they are actually closely related from methodological points of view, and various interplays between them are to be expected. A common spirit in the two subjects is that calculations based on first principles should be crucial in opening up new frontiers of particle physics. Below we describe each subject in slightly more detail.
The model we investigate in this project is the IKKT matrix model, which was proposed as a nonperturbative definition of superstring theory by Ishibashi, Kawai, Kitazawa and Tsuchiya (Nucl.Phys. B498 (1997) 467-491, arXiv:hep-th/9612115, KEK-TH-503) in 1996. In fact, the idea that matrix models can be used to formulate string theory in a way that goes beyond the perturbation theory is an old one, which dates back to 70s. In 1974, it was noticed by 't Hooft that string worldsheets appear naturally from matrix models. Extending this old idea, a simple matrix model was shown to describe a bosonic string theory in low dimensions in a nonperturbative manner in the early 90s. The IKKT matrix model may be viewed as an extension of such a formulation to superstring theory in ten dimensions. Most importantly, this theory is considered to be a consistent theory of quantum gravity, and as such, the IKKT matrix model has the desired feature that the background space-time itself is determined dynamically. Therefore, even the space-time dimensionality can be predicted by investigating this model.
This issue of predicting the space-time dimensionality has been addressed by many researchers. For a long time, the Euclidean version of the IKKT model was investigated for this purpose. In this case, the space-time is assumed to be Euclidean, and there is actually no distinction between space and time. While such a Euclidean space-time is commonly used in formulating quantum field theory in a nonperturbative manner, it is known to be subtle whether it can be used in formulating quantum gravity. Indeed it was found that the background space-time that appears from the Euclidean IKKT model has only three dimensions instead of four dimensions, and the extent of the three dimensions is only five or six times larger than that of the remaining seven dimensions. (Jun Nishimura, Toshiyuki Okubo, Fumihiko Sugino (JHEP 1110 (2011) 135, arXiv:1108.1293 [hep-th], KEK-TH-1483, OIQP-11-06)
For these reasons, the Lorentzian version of the IKKT model was investigated in 2011 by Sang-Woo Kim, Jun Nishimura and Asato Tsuchiya (Phys.Rev.Lett. 108 (2012) 011601, arXiv:1108.1540 [hep-th], KEK-TH-1484, OU-HET-720-2011). In this case, space and time are distinguishable notions unlike in the Euclidean model, and in fact, one can extract the time evolution of the space from the matrix configurations generated by simulating the Lorentzian model. Surprisingly, it was found that (3+1)-dimensional expanding universe appears dynamically in this way. This is highly nontrivial because the obtained result implies that the 9d rotational symmetry of the model is spontaneously broken down to 3d rotational symmetry. In other words, the matrix model formulation of superstring theory seems to predict the space-time dimensionality of our universe correctly.
Since then, the expanding behavior of the universe has been studied in various simplified versions of this model. From the results, it is speculated for the Lorentzian IKKT matrix model that the exponential expansion analogous to the cosmic inflation occurs at early times, and then the expansion behavior changes into a power-law analogous to that of the Friedmann-Robertson-Walker universe in the radiation dominated era. One of the goals of this project is to examine this speculation for the Lorentzian IKKT model without any simplifications. We also aim at measuring the correlation functions to see whether the expected power spectrum of the density fluctuations can be reproduced.
Lattice formulation of quantum chromodynamics (QCD) has been extremely successful in investigating various phenomena associated with the strong interaction between quarks and gluons. For instance, the basic properties of the QCD vacuum such as the confinement of quarks and the spontaneous breakdown of chiral symmetry were understood by first principle calculations based on lattice QCD. On the other hand, the properties of QCD at finite baryon number density are still poorly understood. While there are various interesting speculations based on simplified models and crude approximations, investigations from first principles conducted so far are limited to the low density region.
The technical problem that hampers the lattice QCD studies at finite density is the so-called sign problem. This problem occurs because the integrand of the path integral which defines the theory becomes complex, and tremendous cancellation occurs due to the fluctuating phase of the integrand. This should be compared with the situation at zero density, where the integrand is real non-negative and the idea of importance sampling can be readily applied. The complex Langevin method is a promising approach to evade this problem by complexifying the dynamical variables and solving the Langevin equation, which describes a fictitious time evolution. Here it is important that various functions of the dynamical variables are extended in a holomorphic manner. In 2011, the condition for the equivalence to the original path integral formulation was discussed, which attracted a lot of attention in the lattice QCD community, and in 2014 the method has been successfully applied to finite density QCD at high temperature. Whether it can be applied to the interesting low temperature regime with reasonably light quarks is an important open question, which is being pursued by many research groups in the world.
In our project, we perform the complex Langevin simulation of finite density QCD. In one of our previous publications (Keitaro Nagata, Jun Nishimura, Shinji Shimasaki, arXiv:1604.07717[hep-lat], KEK-TH-1854, KEK-CP-322), we proposed a new technique called the generalized gauge cooling to circumvent a technical problem that occurs in the parameter region we are interested in. This technique was tested in a simplified model, and the result looks very promising. Developing the complex Langevin method further in this way, we aim at investigating the interesting parameter region of finite density QCD.
Last Update on May 12, 2016
