Link polynomial calculus and the AENV conjecture

Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of [1] that the large representation limit (the same as considered in the knot volume conjecture) of this quantity matches the prediction from mirror symmetry consideration. We also provide, using the evolution method, the HOMFLY polynomial in two arbitrary symmetric representations for an arbitrary member of the one-parametric family of 2-component 3-strand links, which includes the Hopf and Whitehead links

Introduction to Khovanov homologies. III. A new and simple tensor-algebra construction of Khovanov–Rozansky invariants

We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>1</mn></math> was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">SL</mi></mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> to <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">SL</mi></mrow><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> and reproduces Khovanov–Rozansky (KR) polynomials, simultaneously for all values of N. No matrix factorization and related tedious calculations are needed in such approach, which can therefore become not only conceptually, but also practically useful

Knot invariants from Virasoro related representation and pretzel knots

We remind the method to calculate colored Jones polynomials for the plat representations of knot diagrams from the knowledge of modular transformation (monodromies) of Virasoro conformal blocks with insertions of degenerate fields. As an illustration we use a rich family of pretzel knots, lying on a surface of arbitrary genus g , which was recently analyzed by the evolution method. Further generalizations can be to generic Virasoro modular transformations, provided by integral kernels, which can lead to the Hikami invariants

Towards effective topological field theory for knots

Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when “fingers” and “propagators” are substituting R -matrices in arbitrary closed braids with m -strands. Original version of [25] corresponds to the case m=2 , and our generalization sheds additional light on the structure of those mysterious formulas. Explicit expressions are now combined from Racah matrices of the type R⊗R⊗R¯⟶R¯ and mixing matrices in the sectors R⊗3⟶Q . Further extension is provided by composition rules, allowing to glue two blocks, connected by an m -strand braid (they generalize the product formula for ordinary composite knots with m=1 )

Towards ℛ-matrix construction of Khovanov-Rozansky polynomials I. Primary T -deformation of HOMFLY

We elaborate on the simple alternative [1] to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL( N ). Construction consists of two steps: with every link diagram with m vertices one associates an m -dimensional hypercube with certain q -graded vector spaces, associated to its 2 m vertices. A generating function for q -dimensions of these spaces is what we suggest to call the primary T -deformation of HOMFLY polynovmial — because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R -matrices, what brings the story completely inside the ordinary Chern-Simons theory. The second step is a certain minimization of residues of this new polynomial with respect to T + 1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube — this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of associated complex, just in the original Khovanov’s construction at N = 2. This second step is still somewhat sophisticated — though incomparably simpler than its conventional matrix-factorization counterpart. In this paper we concentrate on the first step, and provide just a mnemonic treatment of the second step. Still, this is enough to demonstrate that all the currently known examples of KR polynomials in the fundamental representation can be easily reproduced in this new approach. As additional bonus we get a simple description of the DGR relation between KR polynomials and superpolynomials and demonstrate that the difference between reduced and unreduced cases, which looks essential at KR level, practically disappears after transition to superpolynomials. However, a careful derivation of all these results from cohomologies of cut-and-join morphisms remains for further studies

Colored HOMFLY polynomials for the pretzel knots and links

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [ r ] for a huge family of (generalized) pretzel links, which are made from g + 1 two strand braids, parallel or antiparallel, and depend on g + 1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g + 1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU q ( N ) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R

Predictions on the transverse momentum spectra for charged particle production at LHC-energies from a two component model

Transverse momentum spectra, d2σ/(dηdpT2) , of charged hadron production in pp -collisions are considered in terms of a recently introduced two component model. The shapes of the particle distributions vary as a function of the c.m.s. energy in the collision and the measured pseudorapidity interval. As a result the pseudorapidity of a secondary hadron in the moving proton rest frame is shown to be a universal parameter describing the shape of the spectra in pp -collisions. In order to extract predictions on the double-differential cross sections d2σ/(dηdpT2) of hadron production for future LHC-measurements the different sets of available experimental data have been used in this study

Spectrum of quantum transfer matrices via classical many-body systems

In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous $gl$ n -invariant XXX spin chain on N sites with twisted boundary conditions can be found in terms of velocities of particles in the rational N -body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all N particles and the other one is an N -dimensional Lagrangian submanifold obtained by fixing levels of N classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the $gl$ n Gaudin model with N marked points (on the quantum side) and the Calogero-Moser system with N particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed

Cosmic Ray Energy Measurement with EAS Cherenkov Light: Experiment QUEST and CORSIKA Simulation

A new method of a primary cosmic particle energy measurement with the extensive air shower (EAS) technique has been developed by exploiting: a) the joint analysis of the shower size, obtained by the EAS-TOP array, and of the EAS Cherenkov light lateral distribution (LDF), obtained by the QUEST array, and b) simulations based on the CORSIKA code. The method is based on the strict correlation between the size/energy ratio and the steepness of the Cherenkov light lateral distribution and has been compared with a "classical" one based on the Cherenkov light flux at a fixed distance (175 m) from the EAS core. The independence of the energy measurement both on the mass of primary particle and the hadronic interaction model used for the analysis is shown. Based on this approach the experimental integral intensity of cosmic rays flux with energy more than 3*10^15 eV is obtained with good systematic and statistical accuracy

The specificity of searches for W ′, Z ′ and γ ′ coming from extra dimensions

We discuss the specificity of searches for hypothetical W ′, Z ′ and γ ′ bosons at hadron colliders in single top quark and μ + ν μ production and Drell-Yan processes assuming these particles to be the Kaluza-Klein excitations of the gauge bosons of the Standard Model. In this case any process mediated by W is also mediated by the whole KK tower of its excitations, whereas to the processes mediated by Z and γ there is not only a contribution from their KK towers, but also from that of the graviton. The contributions of the towers above W ′, Z ′ and γ ′ and above the first excitation of the graviton are included with the help of effective four-fermion Lagrangians. We compute the cross-sections of these processes taking into account the contributions of the Standard Model gauge bosons, of their first KK modes and of the corresponding KK towers and discuss the impact of the interference between them. For pp-collisions at the LHC with the center of mass energy 14 TeV we found specific changes of the distribution tails due to the interference effects. Such a modification of distribution tails is characteristic for the processes mediated by particles coming from extra dimensions and should always be taken into account when looking for them