Membrane configurations in the Banks-Fischler-Shenker-Susskind matrix model are unstable due to the existence of flat directions in the potential and the decay process can be seen as a realization of chaotic scattering. In this note, we compute the lifetime of a membrane in a reduced model. The resulting lifetime exhibits scaling laws with respect to energy, coupling constant and a cut-off scale. We numerically evaluate the scaling exponents, which cannot be fixed by the dimensional analysis. Finally, some applications of the results are discussed.

Japan

Osamu Fukushima

Tomohiro Shigemura

Kentaroh Yoshida

Scaling law for membrane lifetime

We study the origin of turbulence on the string worldsheet with boundaries laid in anti de Sitter (AdS) spacetime. While the classical motion of a single closed string in AdS is integrable, it has recently been recognized that weak turbulence arises in the case of an open string suspended from the AdS boundary. In the open string case, it is necessary to impose boundary conditions on the worldsheet boundaries. We classify which boundary conditions preserve integrability. Based on this classification, we anticipate that turbulence may occur on the string worldsheet if integrability is not guaranteed by the boundary conditions. Numerical investigations of the classical open-string dynamics support that turbulence occurs when the boundary conditions are not integrable.

Takaaki Ishii

Keiju Murata

Boundary driven turbulence on string worldsheet

We revisit classical string motion in a near pp-wave limit of AdS5 × S5. It is known that the Toda lattice models are integrable. But if the exponential potential is truncated at finite order, then the system may become non-integrable. In particular, when the exponential potential in a three-particle periodic Toda chain is truncated at the third order of the dynamical variables, the resulting system becomes a well-known non-integrable system, Henon-Heiles model. The same thing may happen in a near pp-wave limit of AdS5 × S5, on which the classical string motion becomes chaotic.

Shodai Kushiro

Chaotic string motion in a near pp-wave limit

Chaotic scattering is a manifestation of transient chaos realized by the scattering with non-integrable potential. When the initial position is taken in the potential, a particle initially exhibits chaotic motion, but escapes outside after a certain period of time. The time to stay inside the potential can be seen as lifetime and this escape process may be regarded as a kind of instability. The process of this type exists in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model in which the potential has flat directions. We discuss this chaotic instability by reducing the system with an ansatz to a simple dynamical system and present the associated fractal structure. We also show the singular behavior of the time delay function and compute the fractal dimension. This chaotic instability is the basic mechanism by which membranes are unstable, which is also common to supermembranes at quantum level.

Chaotic instability in the BFSS matrix model

We derive non-abelian Toda field theories (NATFTs) from a 4d Chern-Simons (CS) theory with two order defects by employing a certain asymptotic boundary condition. The 4d CS theory is characterized by a meromorphic 1-form ω. We adopt ω with two simple poles and no zeros, and each of the order defects is located at each pole. As a result, an anisotropy parameter β 2 can be identified with the distance between the two defects. As examples, we can derive the (complex) sine-Gordon model and the Liouville theory.

Taiwan

Jun-ichi Sakamoto

Non-Abelian Toda field theories from a 4D Chern-Simons theory

Recently, a variety of deformed T 1,1 manifolds, with which 2D non-linear sigma models (NLSMs) are classically integrable, have been presented by Arutyunov, Bassi and Lacroix (ABL) [46]. We refer to the NLSMs with the integrable deformed T 1,1 as the ABL model for brevity. Motivated by this progress, we consider deriving the ABL model from a 4D Chern-Simons (CS) theory with a meromorphic one-form with four double poles and six simple zeros. We specify boundary conditions in the CS theory that give rise to the ABL model and derive the sigma-model background with target-space metric and anti-symmetric two-form. Finally, we present two simple examples 1) an anisotropic T 1,1 model and 2) a G/H λ-model. The latter one can be seen as a one-parameter deformation of the Guadagnini-Martellini-Mintchev model.

Integrable deformed T 1,1 sigma models from 4D Chern-Simons theory

We study a <math><mi>T</mi><mover><mrow><mi>T</mi></mrow><mrow><mo>¯</mo></mrow></mover></math>-deformed <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> vector model, which is classically equivalent to the Nambu-Goto action with static gauge. The thermal free energy density can be computed exactly by using the Burgers equation as a special property of <math><mi>T</mi><mover><mrow><mi>T</mi></mrow><mrow><mo>¯</mo></mrow></mover></math>-deformation. The resulting expression is valid for an arbitrary value of N. One may consider a large N limit while preserving this expression. We try to derive this result in the field-theoretical approach directly by employing the large N limit. As a result, the leading contribution coincides with the exact one. That is, the <math><mn>1</mn><mo>/</mo><mi>N</mi></math> corrections are canceled out through a non-trivial mechanism.

Junichi Haruna

Katsuta Sakai

Burgers equation vs. large Limit in -deformed ( ) Vector Model

Recently, Arutyunov, Bassi and Lacroix have shown that 2D non-linear sigma model with a deformed T 1,1 background is classically integrable [arXiv:2010.05573 [hep-th]]. This background includes a Kalb-Ramond two-form with a critical value. Then the sigma model has been conjectured to be non-integrable when the two-form is off critical. We confirm this conjecure by explicitly presenting classical chaos. With a winding string ansatz, the system is reduced to a dynamical system described by a set of ordinary differential equations. Then we find classical chaos, which indicates non-integrability, by numerically computing Poincaré sections and Lyapunov spectra for some initial conditions.

Chaotic string dynamics in deformed T 1,1

We derive the Faddeev-Reshetikhin (FR) model from a four-dimensional Chern-Simons theory with two order surface defects by following the work by Costello and Yamazaki [arXiv:1908.02289]. Then we present a trigonometric deformation of the FR model by employing a boundary condition with an R-operator of Drinfeld-Jimbo type. This is a generalization of the work by Delduc, Lacroix, Magro and Vicedo [arXiv:1909.13824] from the disorder surface defect case to the order one.

Faddeev-Reshetikhin model from a 4D Chern-Simons theory

We present homogeneous Yang-Baxter deformations of the AdS5 ×S5 supercoset sigma model as boundary conditions of a 4D Chern-Simons theory. We first generalize the procedure for the 2D principal chiral model developed by Delduc et al. [5] so as to reproduce the 2D symmetric coset sigma model, and specify boundary conditions governing homogeneous Yang-Baxter deformations. Then the conditions are applicable for the AdS5 ×S5 supercoset sigma model case as well. In addition, homogeneous bi-Yang-Baxter deformation is also discussed.