119237278
Linearized Lorentz-Violating Gravity and Discriminant Locus in the Moduli Space of Mass Terms
We analyze the pattern of normal modes in linearized Lorentz-violating massive gravity over the 5-dimensional moduli space of mass terms. Ghost-free theories arise at bifurcation points when the ghosts get out of the spectrum of propagating particles due to vanishing of the coecient in front of ! 2 in the propagator. Similarly, the van Dam-Veltman-Zakharov (DVZ) discontinuities in the Newton law arise at another type of bifurcations, when the coecient vanishes in front of ~ 2 . When the Lorentz invariance is broken, these two kinds of bifurcations get independent and one can easily nd a ghost-free model without the DVZ discontinuity in the moduli space, at least, in the quadratic (linearized) approximation.
119174057
Link polynomial calculus and the AENV conjecture
A bstractUsing 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.
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.
119133466
Spectral dualities in XXZ spin chains and five dimensional gauge theories
A bstractMotivated by recent progress in the study of supersymmetric gauge theories we propose a very compact formulation of spectral duality between XXZ spin chains. The action of the quantum duality is given by the Fourier transform in the spectral parameter. We investigate the duality in various limits and, in particular, prove it for q → 1, i.e. when it reduces to the XXX/Gaudin duality. We also show that the universal difference operators are given by the normal ordering of the classical spectral curves.
204005491
Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
Abstract 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 N − 1 = 1 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 SL ( 2 ) to SL ( N ) 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.
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.
15734591
Universal Mandelbrot Set as a Model of Phase Transition Theory
The study of Mandelbrot Sets (MS) is a promising new approach to the phase transition theory. We suggest two improvements which drastically simplify the construction of MS. They could be used to modify the existing computer programs so that they start building MS properly not only for the simplest families. This allows us to add one more parameter to the base function of MS and demonstrate that this is not enough to make the phase diagram connected
118782522
Non-conformal limit of AGT relation from the 1-point torus conformal block
Given a 4d N=2 SUSY gauge theory, one can construct the Seiberg-Witten prepotentional, which involves a sum over instantons. Integrals over instanton moduli spaces require regularisation. For UV-finite theories the AGT conjecture favours particular, Nekrasov's way of regularization. It implies that Nekrasov's partition function equals conformal blocks in 2d theories with W_{N_c} chiral algebra. For $N_c=2$ and one adjoint multiplet it coincides with a torus 1-point Virasoro conformal block. We check the AGT relation between conformal dimension and adjoint multiplet's mass in this case and investigate the limit of the conformal block, which corresponds to the large mass limit of the 4d theory e.i. the asymptotically free 4d N=2 supersymmetric Yang-Mills theory. Though technically more involved, the limit is the same as in the case of fundamental multiplets, and this provides one more non-trivial check of AGT conjecture.
119312897
Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
We continue the program of systematic study of extended HOMFLY polynomials, suggested in [A. Mironov, A. Morozov and And. Morozov, arXiv:1112.5754] and [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654]. Extended polynomials depend on infinitely many time-variables, are close relatives of integrable τ-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(∞). Being restricted to "the topological locus" in the space of time-variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m = 3, when the braid is parametrized by a sequence of integers (a1, b1, a2, b2, …) and for the first nonfundamental representation R = [2]. Instead of calculating the mixing matrices directly, as suggested [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654], we deduce them from comparison with the known answers for torus and composite knots. A simple reflection symmetry converts the answer for the symmetric representation [2] into that for the antisymmetric one [1, 1]. The result applies, in particular, to the figure eight knot 41, and was further extended to superpolynomials in arbitrary symmetric and antisymmetric representations in H. Itoyama, A. Mironov, A. Morozov and And. Morozov, arXiv:1203.5978.
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.
121203905
Check of AGT relation for conformal blocks on sphere
Abstract The AGT conjecture identifying conformal blocks with the Nekrasov functions is investigated for the spherical conformal blocks with more than 4 external legs. The diagram technique which arises in conformal block calculation involves propagators and vertices. We evaluated vertices with two Virasoro algebra descendants and explicitly checked the AGT relation up to the third order of the expansion for the 5-point and 6-point conformal blocks on sphere confirming all the predictions of arXiv:0906.3219 relevant in this situation. We propose that U ( 1 ) -factor can be extracted from the matrix elements of the free field vertex operators. We studied the n -point case, and found out that our results confirm the AGT conjecture up to the third order expansions.