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.
119163121
Integrability in non-perturbative QFT
Exact non-perturbative partition functions of coupling constants and external fields exhibit huge hidden symmetry, reflecting the possibility to change integration variables in the functional integral. In many cases this implies also some non-linear relations between correlation functions, typical for the tau-functions of integrable systems. To a variety of old examples, from matrix models to Seiberg-Witten theory and AdS/CFT correspondence, now adds the Chern-Simons theory of knot invariants. Some knot polynomials are already shown to combine into tau-functions, the search for entire set of relations is still in progress. It is already known, that generic knot polynomials fit into the set of Hurwitz partition functions – and this provides one more stimulus for studying this increasingly important class of deformations of the ordinary KP/Toda τ -functions.
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.
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.
118994915
Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials
Original proofs of the AGT relations with the help of the Hubbard–Stratanovich duality of the modified Dotsenko–Fateev matrix model did not work for β ≠ 1, because Nekrasov functions were not properly reproduced by Selberg–Kadell integrals of Jack polynomials. We demonstrate that if the generalized Jack polynomials, depending on the N-ples of Young diagrams from the very beginning, are used instead of the N-linear combinations of ordinary Jacks, this resolves the problem. Such polynomials naturally arise as special elements in the equivariant cohomologies of the GL(N)-instanton moduli spaces, and this also establishes connection to alternative ABBFLT approach to the AGT relations, studying the action of chiral algebras on the instanton moduli spaces. In this paper, we describe a complete proof of AGT in the simple case of GL(2) (N = 2) Yang–Mills theory, i.e., the 4-point spherical conformal block of the Virasoro algebra.
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.
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
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.
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.
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.