119303988
Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities
Abstract We claim that some non-trivial theta-function identities at higher genus can stand behind the Poisson commutativity of the Hamiltonians of elliptic integrable systems, which were introduced in [1] , [2] and are made from the theta-functions on Jacobians of the Seiberg–Witten curves. For the case of three-particle systems the genus-2 identities are found and presented in the Letter. The connection with the Macdonald identities is established. The genus-2 theta-function identities provide the direct way to construct the Poisson structure in terms of the coordinates on the Jacobian of the spectral curve and the elements of its period matrix. The Lax representations for the two-particle systems are also obtained.
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.
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.
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.
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.
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.
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
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.
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.