119673307
Generalized Jack polynomials and the AGT relations for the SU(3) group
We find generalized Jack polynomials for the SU(3) group and verify that their Selberg averages for several first levels are given by Nekrasov functions. To compute the averages, we derive recurrence relations for the $\mathfrak{s}\mathfrak{l}_3 $ Selberg integrals.
119577066
HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
A bstractExplicit answer is given for the HOMFLY polynomial of the figure eight knot 41 in arbitrary symmetric representation R = [p]. It generalizes the old answers for p = 1 and 2 and the recently derived results for p = 3, 4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the identity for the “special” polynomials (they define the leading asymptotics of HOMFLY at q = 1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation (“non-commutative $ \mathcal{A} $-polynomial”) in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R = [1p], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.
119162645
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 ħ = 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.
119623409
On possible existence of HOMFLY polynomials for virtual knots
Abstract Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N = 2 to arbitrary N of the Kauffman–Khovanov calculus of cycles in resolved diagrams can be straightforwardly applied to non-planar case. In simple examples we demonstrate that this construction preserves topological invariance – thus implying the existence of HOMFLY extension of cabled Jones polynomials for virtual knots and links.
204011318
Knot polynomials in the first non-symmetric representation
Abstract We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure-8 and some other 3-strand knots in representation [21] . This is the first result for non-torus knots beyond (anti)symmetric representations, and its evaluation is far more complicated. We provide a whole variety of different arguments, allowing one to guess the answer for the figure-8 knot, which can be also partly used in more complicated situations. Finally we report the result of exact calculation for figure-8 and some other 3-strand knots based on the previously developed sophisticated technique of multi-strand calculations. We also discuss a formula for the superpolynomial in representation [21] for the figure-8 knot, which heavily relies on the conjectural form of superpolynomial expansion nearby the special polynomial point. Generalizations and details will be presented elsewhere.
119155537
Evolution method and HOMFLY polynomials for virtual knots
Following the suggestion of Alexei Morozov, Andrey Morozov and Anton Morozov, Phys. Lett. B737, 48 (2014), arXiv:1407.6319, to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel two-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.
204007506
Proving AGT conjecture as HS duality: Extension to five dimensions
Abstract We extend the proof from Mironov et al. (2011) [25] , which interprets the AGT relation as the Hubbard–Stratonovich duality relation to the case of 5 d gauge theories. This involves an additional q -deformation. Not surprisingly, the extension turns out to be straightforward: it is enough to substitute all relevant numbers by q -numbers in all the formulas, Dotsenko–Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β = 1 , i.e. for q = t in MacDonaldʼs notation. For β ≠ 1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams.
119172776
Genus expansion of HOMFLY polynomials
In the planar limit of the’ t Hooft expansion, the Wilson-loop vacuum average in the three-dimensional Chern-Simons theory (in other words, the HOMFLY polynomial) depends very simply on the representation (Young diagram), HR(A|q)|q=1 = (σ1(A)|R|. As a result, the (knot-dependent) Ooguri-Vafa partition function $$\sum\nolimits_R {H_{R\chi R} \left\{ {\bar pk} \right\}}$$ becomes a trivial τ -function of the Kadomtsev-Petviashvili hierarchy. We study higher-genus corrections to this formula for HR in the form of an expansion in powers of z = q − q−1. The expansion coefficients are expressed in terms of the eigenvalues of cut-and-join operators, i.e., symmetric group characters. Moreover, the z-expansion is naturally written in a product form. The representation in terms of cut-and-join operators relates to the Hurwitz theory and its sophisticated integrability. The obtained relations describe the form of the genus expansion for the HOMFLY polynomials, which for the corresponding matrix model is usually given using Virasoro-like constraints and the topological recursion. The genus expansion differs from the better-studied weak-coupling expansion at a finite number N of colors, which is described in terms of Vassiliev invariants and the Kontsevich integral.
119611129
Differential hierarchy and additional grading of knot polynomials
Colored knot polynomials have a special Z-expansion in certain combinations of differentials, which depend on the representation. The expansion coefficients are functions of three variables A, q, and t and can be regarded as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of the alternative character expansion. These new variables decompose especially simply when the representation is embedded into a product of fundamental representations. The recently proposed fourth grading is seemingly a simple redefinition of these new coordinates, elegant, but in no way distinguished. If this is so, then it does not provide any new independent knot invariants, but it can instead be regarded as one more piece of evidence in support of a hidden differential hierarchy (Z-expansion) structure behind the knot polynomials.
15381199
On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri–Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivial exponential tau-function. In fact, in the double scaling Kashaev limit (the one associated with the volume conjecture) dominant in the genus expansion are terms associated with the symmetric representations and with the integrability preserving Casimir operators, though we stop one step from converting this fact into a clear statement about the OVPF behavior in the vicinity of q=1. Instead, we explain that the genus expansion provides a hierarchical decomposition of the Hurwitz tau-function, similar to the Takasaki–Takebe expansion of the KP tau-functions. This analogy can be helpful to develop a substitute for the universal Grassmannian description in the Hurwitz tau-functions.
15661590
The H1 forward proton spectrometer at HERA
Abstract The forward proton spectrometer is part of the H1 detector at the HERA collider. Protons with energies above 500 GeV and polar angles below 1 mrad can be detected by this spectrometer. The main detector components are scintillating fiber detectors read out by position-sensitive photo-multipliers. These detectors are housed in the so-called Roman Pots which allow them to be moved close to the circulating proton beam. Four Roman Pot stations are located at distances between 60 and 90 m from the interaction point.