Tensor renormalization group and the volume independence in 2D U(N) and SU(N) gauge theories

The tensor renormalization group method is a promising approach to lattice field theories, which is free from the sign problem unlike standard Monte Carlo methods. One of the remaining issues is the application to gauge theories, which is so far limited to U(1) and SU(2) gauge groups. In the case of higher rank, it becomes highly nontrivial to restrict the number of representations in the character expansion to be used in constructing the fundamental tensor. We propose a practical strategy to accomplish this and demonstrate it in 2D U(N) and SU(N) gauge theories, which are exactly solvable. Using this strategy, we obtain the singular-value spectrum of the fundamental tensor, which turns out to have a definite profile in the large-N limit. For the U(N) case, in particular, we show that the large-N behavior of the singular-value spectrum changes qualitatively at the critical coupling of the Gross-Witten-Wadia phase transition. As an interesting consequence, we find a new type of volume independence in the large-N limit of the 2D U(N) gauge theory with the θ term in the strong coupling phase, which goes beyond the Eguchi-Kawai reduction.

The Dаrbоuх Theory and Geodesics in the Metric Space with Torsion

The geometry of n Yn space is generated congruently together by the metric tensor and the torsion tensor. In the presented article has been obtained an analog of the Dаrbоuх theory in the n Yn space, also studied the deduction of the equation of the geodesic lines on the hypersurface that embedded in such spaces, showed that in the n Yn space the structure of the curvature tensor has special features and for curvature tensor obtained Ricci - Jacobi identity. We establish that the equations of the geodesics have additional summands, which are caused by the presence of torsion in the space. In n Yn space, the variation of the length of the geodesic lines is proportional to the product of metric and torsion tensors gijSjpk. We have introduced the second fundamental tensor παβ for the hypersurface n Yn-1 and established its structure, which is fundamentally different from the case of the Riemannian spaces with zero torsion. Furthermore, the results on the structure of the curvature tensor have been obtained.

Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald.

On a Riemannian manifold with a circulant structure whose third power is the identity

It is studied a 3-dimensional Riemannian manifold equipped with a tensor structure of type (1,1), whose third power is the identity. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant. On such a manifold a fundamental tensor by the metric and by the covariant derivative of the circulant structure is defined. An important characteristic identity for this tensor is obtained. It is established that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity. A Lie group as a manifold of the considered type is constructed and some of its geometrical characteristics are found.

The Ether Theory Unifying the Relativistic Gravito-Electromagnetism Including also the Gravitons and the Gravitational Waves

In previous publications, we showed that Maxwell’s equations are an approximation to those of General Relativity when V<<c, where V is the velocity of the particle submitted to the electromagnetic field. This was demonstrated by showing that the Lienard-Wiechert potential four-vector A_u created by an electric charge is the equivalent of the gravitational four-vector G_u created by a massive neutral point when V<<c. In the present paper, we generalize these results for V non-restricted to be small. To this purpose, we show first that the exact Lagrange-Einstein function of an electric charge q submitted to the field due an immobile charge q_0 is of the same form as that of a particle of mass m submitted to the field created by an immobile particle of mass m_0. Maxwell’s electrostatics is then generalized as a case of the Einstein’s general relativity. In particular, it appears that an immobile q_0 creates also an electromagnetic horizon that behaves like a Schwarzschild horizon. Then, there exist ether gravitational waves constituted by gravitons in the same way as the electromagnetic waves are constituted by photons. Now, since A_u and G_u, are equivalent, and as we show, G_u produces the approximation, for V<<c, of g_u4 created by m_0 mobile, where the g_uv are the components of Einstein’s fundamental tensor, it follows that A_u+u_u produces the approximation, for V<<c, of Bet_u4 , where the Bet_uv created by m_0 and by q_0, generalize the g_uv.

Derivation of Field Equations in Space with the Geometric Structure Generated by Metric and Torsion

This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsion tensors. We also study the geometry of the space generated jointly and agreed on by the metric tensor and the torsion tensor. We showed that in such space the structure of the curvature tensor has special features and for this tensor we obtained analog Ricci-Jacobi identity and evaluated the gap that occurs at the transition from the original to the image and vice versa, in the case of infinitely small contours. We have researched the geodesic lines equation. We introduce the tensor παβ which is similar to the second fundamental tensor of hypersurfaces Yn-1, but the structure of this tensor is substantially different from the case of Riemannian spaces with zero torsion. Then we obtained formulas which characterize the change of vectors in accompanying basis relative to this basis itself. Taking into considerations our results about the structure of such space we derived from the variation principle the general field equations (electromagnetic and gravitational).