Institute for Theoretical and Experimental Physics

The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY–PT polynomials for any $$N$$ in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can easily be implemented as a computer program. Bipartite links form a rather large family, including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY–PT polynomials using our developed generalized Goeritz method.

Матрица Гeрица является альтернативой скобке Кауффмана и, кроме того, позволяет быстрее вычислять многочлен Джонса при некотором минимальном выборе шахматной раскраски, построенной по зацеплению. Вводится модификация метода Гeрица, обобщающая матрицу Гeрица для вычисления полиномов ХОМФЛИ-ПТ при любых $N$ в частном случае бипартитных зацеплений. Метод сводится к чисто алгебраическим операциям над матрицами, поэтому его легко можно реализовать в виде компьютерной программы. Бипартитные зацепления образуют довольно большое семейство, включающее особый класс зацеплений Монтезиноса, построенных из так называемых рациональных тэнглов. Продемонстрировано, как получить бипартитную диаграмму таких зацеплений и вычислить соответствующие полиномы ХОМФЛИ-ПТ с помощью разработанного обобщенного метода Гeрица.

In this paper we provide a detailed account of our calculation, briefly reported in arXiv:2209.09263, of a two-particle form factor of the lowest components of the stress-tensor multiplet in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 sYM theory on its Coulomb branch, which is interpreted as an off-shell kinematical regime. We demonstrate that up to three-loop order, both its infrared-divergent as well as finite parts do exponentiate in the Sudakov regime, with the coefficient accompanying the double logarithm being determined by the octagon anomalous dimension Γoct. We also observe that up to this order in ’t Hooft coupling the logarithm of the Sudakov form factor is identical to twice the logarithm of the null octagon, which was introduced within the context of integrability-based computation of four point correlators with infinitely large R-charges. The null octagon is known in a closed form for all values of the ’t Hooft coupling constant and kinematical parameters. We conjecture that the relation between the former and the off-shell Sudakov form factor holds to all loop orders.

We investigate partition functions of the circular-quiver supersymmetric Chern-Simons theory which corresponds to the q-deformed Painlevé VI equation. From the partition functions with the lowest rank vanishing, where the circular quiver reduces to a linear one, we find 40 bilinear relations. The bilinear relations extend naturally to higher ranks if we regard these partition functions as those in the lowest order of the grand canonical partition functions in the fugacity. Furthermore, we show that these bilinear relations are a powerful tool to determine some unknown partition functions. We also elaborate the relation with some previous works on q-Painlevé equations.

Определен дайсоновский диффузионный процесс на гладком замкнутом криволинейном контуре на плоскости и получено уравнение Фоккера-Планка для плотности вероятности. Показано, что его стационарное решение является больцмановским весом для логарифмического газа, сосредоточенного на контуре.

Recently, systems with strong coupling between electronic and vibrational degrees of freedom attract a great attention. In this work, we consider the transient dynamics of the system consisting of strongly coupled vibron and exciton driven by external monochromatic field. We show that under coherent pumping, polarization of exciton exhibits complex quantum dynamics which can be divided into three stages. At the first stage, exciton oscillations at its eigenfrequency relax due to the transition to set of shifted Fock states of vibrons. We demonstrate that these shifted Fock states play the role of an effective reservoir for the excited exciton state. The time of relaxation to this reservoir depends on exciton-vibron coupling. At the second stage, excitation, transferred to the reservoir of the vibronic shifted states at the first stage, returns into electronic degrees of freedom and revival of oscillations at exciton eigenfrequency appears. Thus, the dynamics of molecular polarization exhibit collapses and revivals. At the final stage, these collapses and revivals dissipate and polarization exhibits Rayleigh response at the frequency of the external field. Discovered collapses and revivals manifest in radiation spectrum as multiple splitting of the spectral line near the exciton transition frequency.

We have seen in the previous chapter that the twisted reduced models reproduce planar graphs of the d-dimensional quantum field theories as N → ∞. However, the twisted reduced models make sense order by order in 1/N 2 . For the continuum twisted reduced models, the topological expansion goes in the parameter det (B µν ). At finite B µν , the twisted reduced models are mapped [CDS98, AII00] into quantum field theories on noncommutative space characterized by a (dimensional) parameter of noncommutativity θ µν = B −1 µν . The noncommutative gauge field is no longer matrix-valued as in Yang-Mills theory but noncommutativity of matrices in the reduced models is transformed into noncommutativity of coordinates in the noncommutative gauge theory. The planar limit of ordinary Yang-Mills theory is reproduced at large noncommutativity parameter θ µν → ∞, while ordinary quantum electrodynamics is reproduced as θ µν → 0. Noncommutative gauge theories possess a number of remarkable properties. The noncommutative extension of Maxwell's theory is interacting and asymptotically free. The group of noncommutative gauge symmetry is very large and incorporates space-time symmetries, in particular, translation, Lorentz transformation, parity reflection. This restricts a set of observables in noncommutative gauge theory which are built out of both closed and open Wilson loops. At rational values of a (dimensionless) noncommutativity parameter, noncommutative gauge theories on a torus are equivalent to ordinary Yang-Mills theories on a smaller torus with twisted boundary conditions representing the 't Hooft flux. We begin this chapter by mapping the twisted reduced models into noncommutative theories. Then we discuss some properties of noncommutative scalar and gauge theories including their lattice regularization.

There is an elegant alternative to the quenched Eguchi-Kawai model, described in the previous chapter, which also preserves the U (1) d symmetry. It was proposed by GO83b] on the basis of a twisting reduction prescription. The corresponding lattice version of the reduced model lives on a hypercube with twisted boundary conditions. The twisted reduced model for a scalar field was constructed by Eguchi and Nakayama [EN83]. The twisted reduced models reveal interesting mathematical structures associated with representations of the Heisenberg commutation relation (in the continuum) or its finite-dimensional approximation by unitary matrices (on the lattice). In contrast to the quenched reduced models which describe only planar graphs, the twisted reduced models make sense order by order in 1/N and even at finite N . In this case they are associated with gauge theories on noncommutative space, whose limit of large noncommutativity is given by planar graphs thereby reproducing a d-dimensional Yang-Mills theory at large N . We begin this chapter with a description of the twisted reduced models first on the lattice and then in the continuum and show how they describe planar graphs of a d-dimensional theory. We start by working on a lattice to make the results rigorous and then repeat them for the continuum. The twisting reduction prescription is a version of the general reduction prescription described in Sect. 14.1. We again perform the unitary transformation (14.4), where the matrices D(x) are now expressed via a 351

This book introduces the quantum theory of gauge fields, emphasising four non-perturbative methods which have important applications: path integrals, lattice gauge theories, the 1/N expansion, and reduced matrix models. Written as a textbook, it assumes a knowledge of quantum mechanics and elements of perturbation theory, while many relevant concepts are introduced at a basic level in the first half of the book. The second half comprehensively covers large-N Yang–Mills theory. The book uses an approach to gauge theories based on path-dependent phase factors known as Wilson loops, and contains problems with detailed solutions to aid understanding. Suitable for advanced graduate courses in quantum field theory, the book will also be of interest to researchers in high energy theory and condensed matter physics as a survey of recent developments in gauge theory. Originally published in 2002, this title has been reissued as an Open Access publication on Cambridge Core.