Knots,BPS States,and Algebraic Curves

We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.     

Torus Knots and the Topological Vertex

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S ³. The key role is played by the \({SL(2, \mathbb{Z})}\) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Mariño. Compared to the recent proposal by Aganagic and Vafa for knots on S ³, we demonstrate that the disk amplitude of the A-brane associated with any knot is sufficient to reconstruct the entire B-model spectral curve. Finally, the construction of toric Lagrangian A-branes is generalized to other local toric Calabi–Yau geometries, which paves the road to study knots in other three-manifolds such as lens spaces.     

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models, we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group \({\mathcal{U}_{q}su(N)}\). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation {2, 1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or—in the limit to classical groups—to determine color factors for Yang Mills amplitudes.     

String Theory and the Kauffman Polynomial

We propose a new, precise integrality conjecture for the colored Kauffman polynomial of knots and links inspired by large N dualities and the structure of topological string theory on orientifolds. According to this conjecture, the natural knot invariant in an unoriented theory involves both the colored Kauffman polynomial and the colored HOMFLY polynomial for composite representations, i.e. it involves the full HOMFLY skein of the annulus. The conjecture sheds new light on the relationship between the Kauffman and the HOMFLY polynomials, and it implies for example Rudolph’s theorem. We provide various non-trivial tests of the conjecture and we sketch the string theory arguments that lead to it.     

Torus Knot Polynomials and Susy Wilson Loops

We give, using an explicit expression obtained in (Jones V, Ann Math 126:335, 1987), a basic hypergeometric representation of the HOMFLY polynomial of (n, m) torus knots, and present a number of equivalent expressions, all related by Heine’s transformations. Using this result, the \({ (m, n) \leftrightarrow (n, m)}\) symmetry and the leading polynomial at large N are explicit. We show the latter to be the Wilson loop of 2d Yang–Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang–Mills theory, which is known to give averages of Wilson loops in \({\mathcal{N}}\) = 4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones–Rosso representation in terms of q-harmonic oscillators.     

Towards matrix model representation of HOMFLY polynomials

We investigate possibilities of generalizing the TBEM (Tierz, Brini-Eynard-Mariño) eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern-Simons mixture of the Gaussian potential, typical for Hermitian models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the N = 2 case. However, beyond the torus family, both the measure and its lifting to larger N contain non-trivial corrections in ? = logq. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz τ-function approach to knot theory, but this remains a subject for future investigation.     

Wilson lines in Chern-Simons theory and link invariants

The vacuum expectation values of Wilson line operators W(L) in the Chern-Simons theory are computed to second order to perturbation theory. The meaning of the framing procedure for knots is analyzed in the context of the Chern-simons field theory. The relation between W(L) and the link invariant polynomials is discussed. We derive an explicit analytic expression for the second coefficient of the Alexander-Conway polynomial, which is related to the Arf- and Casson-invariant. We present also some new relations between the HOMFLY coefficients.     

On eleven-dimensional supergravity and Chern-Simons theory

We probe in some depth into the structure of eleven-dimensional, osp(32|1)-based Chern-Simons supergravity, as put forward by Troncoso and Zanelli (TZ) in 1997. We find that the TZ Lagrangian may be cast as a polynomial in 1/l, where l is a length, and compute explicitly the first three dominant terms. The term proportional to 1/l⁹ turns out to be essentially the Lagrangian of the standard 1978 supergravity theory of Cremmer, Julia and Scherk, thus establishing a previously unknown relation between the two theories. The computation is nontrivial because, when written in a sufficiently explicit way, the TZ Lagrangian has roughly one thousand non-explicitly Lorentz-covariant terms. Specially designed algebraic techniques are used to accomplish the results.     

Geometry and analysis on isospectral sets. I. Riemannian geometry,asymptotic case

A short introduction to the analytical and algebraic aspects of integrable systems is given. We consider the Riemannian geometry of the isospectral set belonging to the Dirichlet problem −y′' + q(x)y = λy, y(0) = y(1) = 0, where q is a square integrable function of the real Hilbert space L²_ℝ([0,1]). We derive the metric and the connection for the isospectral set, which is an infinite dimensional real analytic submanifold of LL²_ℝ([0,1 ]), in the case of large eigenvalues. The curvature in the asymptotic case is then derived and it is proved that the connection and the curvature are well defined if we take their coefficients in the discrete Sobolev spaces. We further give the explicit formulae for the parallel transport and a sufficiency condition is derived such that a curve on the isospectral set is a geodesic.     

Non-minimally coupled quintom model inspired by string theory

In this Letter we consider a quintom model of dark energy with a single scalar field T given by a Lagrangian which inspired by tachyonic Lagrangian in string theory. We consider non-minimal coupling of tachyon field to the scalar curvature, then we obtain the equation of state (EoS), and the condition required for the model parameters when ω crosses over −1.     

Multiplicity-free Quantum 6j-Symbols for {U_q(\mathfrak{sl}_N)}

We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for ${U_q(\mathfrak{sl}_N)}$ . The expression is a natural generalization of the quantum 6j-symbols for ${U_q(\mathfrak{sl}_2)}$ obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.     

The Casimir Effect from the Point of View of Algebraic Quantum Field Theory

We consider a region of Minkowski spacetime bounded either by one or by two parallel, infinitely extended plates orthogonal to a spatial direction and a real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize these two systems within the algebraic approach to quantum field theory using the so-called functional formalism. As a first step we construct a suitable unital ?-algebra of observables whose generating functionals are characterized by a labelling space which is at the same time optimal and separating and fulfils the F-locality property. Subsequently we give a definition for these systems of Hadamard states and we investigate explicit examples. In the case of a single plate, it turns out that one can build algebraic states via a pull-back of those on the whole Minkowski spacetime, moreover inheriting from them the Hadamard property. When we consider instead two plates, algebraic states can be put in correspondence with those on flat spacetime via the so-called method of images, which we translate to the algebraic setting. For a massless scalar field we show that this procedure works perfectly for a large class of quasi-free states including the Poincaré vacuum and KMS states. Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the extended algebras, built in globally hyperbolic subregions can be collected in a global counterpart only after a suitable deformation which is expressed locally in terms of a *-isomorphism. As a last step, we construct explicitly the two-point function and the regularized energy density, showing, moreover, that the outcome is consistent with the standard results of the Casimir effect.     

Bethe ansatz for the XXX-S chain with non-diagonal open boundaries

We consider the algebraic Bethe ansatz solution of the integrable and isotropic XXX-S Heisenberg chain with non-diagonal open boundaries. We show that the corresponding K-matrices are similar to diagonal matrices with the help of suitable transformations independent of the spectral parameter. When the boundary parameters satisfy certain constraints we are able to formulate the diagonalization of the associated double-row transfer matrix by means of the quantum inverse scattering method. This allows us to derive explicit expressions for the eigenvalues and the corresponding Bethe ansatz equations. We also present evidences that the eigenvectors can be build up in terms of multiparticle states for arbitrary S.     

Non-zero masses of π, K, η and κ mesons and algebraic realization of SU(3) × SU(3) chiral symmetry

An algebraic realization of SU(3) × SU(3) symmetry, with linearization on the SU(2) × Y subgroup taking into account masses of π, K, η mesons in virtual states is investigated. This is achieved by explicit breaking of the symmetry. The chosen form of the breaking term in the Lagrangian enables us to estimate the parameter c appearing in the squared mass operator. The obtained value of c(c ≈ ?1.17) is very close to that predicted by Gell-Mann, Oakes and Renner.     

On the geometry of Lagrangian mechanics with non-holonomic constraints

As is well known, Lagrangian mechanics have been entirely geometrized in terms of symplectic geometry. On the other hand, the geometrization of non-holonomic mechanics has been less developed. However, due to the interest aroused by non-holonomic geometry, many papers have been devoted to this subject. In this article we generalize the construction of a connection whose geodesics are the trajectories of a system, obtained by Vershik and Feddeef in the case where the Lagrangian is quadratic and the constraints are linear on the velocities. Using the algebraic formalism of the connections theory introduced by the first author, we carry out the construction in the general case of an arbitrary mechanical system (i.e. of a manifold with a convex Lagrangian not necessarily homogeneous) with ideal non-holonomic constraints. Moreover, we prove something stronger than the result of Vershik and Feddeev: our connection has not only the above-mentioned property for the geodesics, but it preserves the Hamiltonian by parallel transport. This connection is then a generalization of the Levi-Civita connection for the Riemannian manifolds for which the metric (i.e. the kinetic energy) is preserved by parallel transport.     

Phases of renormalized lattice gauge theories with fermions

Starting from the formulation of gauge theories on a lattice we derive renormalization group transformation of the Migdal-Kadanoff type in the presence of fermions. We consider the effect of the fermion vacuum polarization on the gauge Lagrangian but we neglect fermion mass renormalization. We work out the weak coupling and strong coupling expansion in the same framework. Asymptotic freedom is recovered for the non-Abelian case provided the number of fermion multiplets is lower than a critical number. Fixed points are determined both for the U(1) and SU(2) case. We determine the renormalized trajectories and the phases of the theory.     

Lagrangian relative equilibria for a gyrostat in the three-body problem

In this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order k. After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order k, and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order k for k ≥1 is independent of the order of truncation of the potential, if the gyrostat S ₀ is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies.      

Quantum Racah matrices and 3-strand braids in irreps R with |<Emphasis Type="Italic">R</Emphasis>| = 4

We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R = [2, 2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6 × 6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate eigenvalues and is evaluated by the highest weight method. Together with the much harder calculation for R = [3, 1] and with the new method to extract exclusive matrices J and \(\overline J\) from the inclusive ones, this completes the story of Racah matrices for |R| ≤ 4 and allows one to calculate and investigate the corresponding colored HOMFLY polynomials for arbitrary 3-strand and arborescent knots.     

On Vlasov–Manev Equations. I: Foundations, Properties, and Nonglobal Existence

We consider the classical stellar dynamic (Vlasov) equation with a so-called Manev correction (based on a pair potential γ/r + ε/r ²). For the pure Manev potential γ = 0 we discuss both the continuous case and the N-body problem and show that global solutions will not exist if the initial energy is negative. Certain global solutions can be constructed from local ones by a transformation which is peculiar for the ε/r ² law. Moreover, scaling arguments are used to show that Boltzmann collision terms are meaningful in conjunction with Manev force terms. In an appendix, a formal justification of the Manev correction based on the quasirelativistic Lagrangian formalism for the motion of a particle in a central force field is given.