Wedge dislocations, three-dimensional gravity, and the Riemann-Hilbert problem

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.

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The N-soliton solutions to the Hirota and Maxwell–Bloch equation via the Riemann–Hilbert approach

International Journal of Modern Physics B ◽

10.1142/s0217979221501538 ◽

2021 ◽

pp. 2150153

Author(s):

Jian Li ◽

Tiecheng Xia ◽

Hanyu Wei

Keyword(s):

Physical Meaning ◽

Hilbert Problem ◽

Three Dimensional ◽

Soliton Solutions ◽

Bloch Equation ◽

Lax Pair ◽

Integrable Equation ◽

Motion Trajectory ◽

The Matrix ◽

Riemann Hilbert Problem

In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

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A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

Mathematics ◽

10.3390/math9070776 ◽

2021 ◽

Vol 9 (7) ◽

pp. 776

Author(s):

Osamu Suzuki ◽

Zhidong Zhang

Keyword(s):

Ising Model ◽

External Field ◽

Hilbert Problem ◽

Three Dimensional ◽

Mathematical Structure ◽

Many Body ◽

Non Local ◽

Knot Space ◽

Riemann Hilbert Problem ◽

3D Ising Model

A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry.

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On the Modular Operator of Mutli-component Regions in Chiral CFT

Communications in Mathematical Physics ◽

10.1007/s00220-021-04054-6 ◽

2021 ◽

Author(s):

Stefan Hollands

Keyword(s):

Field Theory ◽

Integral Equation ◽

Hilbert Problem ◽

Matrix Elements ◽

New Approach ◽

Conformal Field ◽

The Matrix ◽

Kms Condition ◽

Modular Operator ◽

Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

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Free partition functions and an averaged holographic duality

Journal of High Energy Physics ◽

10.1007/jhep01(2021)130 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Nima Afkhami-Jeddi ◽

Henry Cohn ◽

Thomas Hartman ◽

Amirhossein Tajdini

Keyword(s):

Partition Function ◽

Spectral Gap ◽

Central Charge ◽

Three Dimensional ◽

Two Dimensions ◽

Three Dimensions ◽

Partition Functions ◽

General Constraints ◽

Dimensional Gravity ◽

Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

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