In our previous work [1], we introduced to an arbitrary Markov chain Monte Carlo algorithm a distance between configurations. This measures the difficulty of transition from one configuration to the other, and enables us to investigate the relaxation of probability distribution from a geometrical point of view. In this paper, we investigate the global geometry of a stochastic system whose equilibrium distribution is highly multimodal with a large number of degenerate vacua. We show that, when the simulated tempering algorithm is implemented to such a system, the extended configuration space has an asymptotically Euclidean anti-de Sitter (AdS) geometry. We further show that this knowledge of geometry enables us to optimize the tempering parameter in a simple, geometrical way.

Japan

Masafumi Fukuma

Nobuyuki Matsumoto

Naoya Umeda

Emergence of AdS geometry in the simulated tempering algorithm

For a given Markov chain Monte Carlo algorithm we introduce a distance between two configurations that quantifies the difficulty of transition from one configuration to the other configuration. We argue that the distance takes a universal form for the class of algorithms which generate local moves in the configuration space. We explicitly calculate the distance for the Langevin algorithm, and show that it certainly has desired and expected properties as distance. We further show that the distance for a multimodal distribution gets dramatically reduced from a large value by the introduction of a tempering method. We also argue that, when the original distribution is highly multimodal with large number of degenerate vacua, an anti-de Sitter-like geometry naturally emerges in the extended configuration space.

Distance between configurations in Markov chain Monte Carlo simulations

The algorithm based on integration over Lefschetz thimbles is a promising method to resolve the sign problem for complex actions. However, this algorithm often meets a difficulty in actual Monte Carlo calculations because the configuration space is not easily explored due to the infinitely high potential barriers between different thimbles. In this paper, we propose to use the flow time of the antiholomorphic gradient flow as an auxiliary variable for the highly multimodal distribution. To illustrate this, we implement the parallel tempering method by taking the flow time as a tempering parameter. In this algorithm, we can take the maximum flow time to be sufficiently large such that the sign problem disappears there, and two separate modes are connected through configurations at small flow times. To exemplify that this algorithm does work, we investigate the (0 + 1)-dimensional massive Thirring model at finite density and show that our algorithm correctly reproduces the analytic results for large flow times such as T = 2.

Parallel tempering algorithm for integration over Lefschetz thimbles

Triangle–hinge models [M. Fukuma, S. Sugishita, and N. Umeda, J. High Energy Phys. 1507 , 088 (2015)] are introduced to describe worldvolume dynamics of membranes. The Feynman diagrams consist of triangles glued together along hinges and can be restricted to tetrahedral decompositions in a large- limit. In this paper, after clarifying that all the tetrahedra resulting in the original models are orientable, we define a version of triangle–hinge models that can describe the dynamics of unoriented membranes. By regarding each triangle as representing a propagation of an open membrane of disk topology, we introduce a local worldvolume parity transformation which inverts the orientation of a triangle, and define unoriented triangle–hinge models by gauging the transformation. Unlike two-dimensional cases, this local transformation generally relates a manifold to a nonmanifold, but still is a well-defined manipulation among tetrahedral decompositions. We further show that matter fields can be introduced in the same way as in the original oriented models. In particular, the models will describe unoriented membranes in a target spacetime by taking matter fields to be the target space coordinates.

Sotaro Sugishita

Triangle–hinge models for unoriented membranes

The worldvolume theory of membranes is mathematically equivalent to 3D quantum gravity coupled to matter fields corresponding to the target space coordinates of an embedded membrane. In a recent paper [M. Fukuma et al., J. High Energy Phys. 1507 , 088 (2015) [arXiv:1503.08812 [hep-th]]], a new class of models that generate 3D random volumes was introduced, where the Boltzmann weight of each configuration is given by the product of values assigned to the triangles and the hinges. These triangle–hinge models describe 3D pure gravity and are characterized by semisimple associative algebras. In this paper, we introduce matter degrees of freedom to the models by coloring simplices in such a way that they have local interactions. This is achieved simply by extending the associative algebras of the original triangle–hinge models, and the profile of the matter field is specified by the set of colors and the form of the interactions. The dynamics of a membrane in -dimensional spacetime can then be described by taking the set of colors to be . By taking another set of colors, we can also realize 3D quantum gravity coupled to various spin models such as the Ising model. 3D colored tensor models can also be realized as triangle–hinge models by coloring tetrahedra, triangles, and edges at one time.

Matter fields in triangle–hinge models

We propose a class of models which generate three-dimensional random volumes, where each configuration consists of triangles glued together along multiple hinges. The models have matrices as the dynamical variables and are characterized by semisimple associative algebras A $$ \mathcal{A} $$ . Although most of the diagrams represent configurations which are not manifolds, we show that the set of possible diagrams can be drastically reduced such that only (and all of the) three-dimensional manifolds with tetrahedral decompositions appear, by introducing a color structure and taking an appropriate large N limit. We examine the analytic properties when A $$ \mathcal{A} $$ is a matrix ring or a group ring, and show that the models with matrix ring have a novel strong-weak duality which interchanges the roles of triangles and hinges. We also give a brief comment on the relationship of our models with the colored tensor models.