117639345
Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation
An extension of the two-dimensional (2d) Alday-Gaiotto-Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two S-dual supersymmetric Yang-Mills (SYM) models to the 3d Chern-Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the “conservation laws” for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov-Shatashvili limit of the 5d SYM theory.
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.
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.
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.
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.
119190491
Resolving puzzles of massive gravity with and without violation of Lorentz symmetry
We perform a systematic study of various versions of massive gravity with and without violations of the Lorentz symmetry in arbitrary dimension. These theories are well known to possess very unusual properties, unfamiliar from studies of gauge and Lorentz invariant models. These peculiarities are caused by the mixing of familiar transverse fields with the revived longitudinal and pure gauge (Stueckelberg) fields and are all seen already in the quadratic approximation. They are all associated with non-trivial dispersion laws, which easily allow superluminal propagation, ghosts, tachyons and essential irrationalities. Moreover, the coefficients in front of emerging modes are small, which makes the theories essentially non-perturbative within a large Vainshtein radius. Attempts to get rid of unwanted degrees of freedom by giving them infinite masses lead to the DVZ discontinuities in the parameter (moduli) space, caused by non-permutability of different limits. Also, the condition mgh = ∞ can not be preserved already in non-trivial gravitational backgrounds and is unstable under any other perturbations of the linearized gravity. At the same time, an a priori healthy model of massive gravity in the quadratic approximation definitely exists: it is provided by any mass level of the Kaluza–Klein tower. It bypasses the problems because the gravity field is mixed with other fields, and this explains why such mixing helps in other models. At the same time, this can imply that the really healthy massive gravity can still require an infinite number of extra fields beyond the quadratic approximation.
204005968
EIGENVALUE HYPOTHESIS FOR RACAH MATRICES AND HOMFLY POLYNOMIALS FOR 3-STRAND KNOTS IN ANY SYMMETRIC AND ANTISYMMETRIC REPRESENTATIONS
Character expansion expresses extended HOMFLY polynomials through traces of products of finite-dimensional - and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding -matrices. Even a weaker (and, perhaps, more reliable) version of this conjecture is sufficient to explicitly calculate HOMFLY polynomials for all the 3-strand braids in arbitrary (anti)symmetric representations. We list the examples of so obtained polynomials for R = [3] and R = [4], and they are in accordance with the known answers for torus and figure-eight knots, as well as for the colored special and Jones polynomials. This provides an indirect evidence in support of our conjecture.
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.
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.
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.
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