Elliptic Hypergeometry of Supersymmetric Dualities

We give a full list of known ${\mathcal{N}=1}$ supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for SU(N), SP(2N) and G ₂ gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related R?melsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author.     

Polynomial Invariant of Knots and Links from Two-Parameter Quantum Groups

A new link polynomial is obtained by introducinga two-parameter Hecke algebra associated with thetwo-parameter quantum groups.     

Elliptic Quantum Groups and Ruijsenaars Models

We construct symmetric and exterior powers of the vector representation of the elliptic quantum groupsE _Τ,η(slN). The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators.     

Real Forms of Quantum Orthogonal Groups,q-Lorentz Groups in any Dimension

We review known real forms of the quantum orthogonal groups SO _q (N). New *-conjugations are then introduced and we contruct all real forms of quantum orthogonal groups. We thus give an RTT formulation of the *-conjugations on SO _q (N) that is complementary to the U _q (g) *-structure classification of Twietmeyer. In particular, we easily find and describe the real forms SO _q (N-1,1) for any value of N. Quantum subspaces of the q-Minkowski space are analyzed.     

Vortices in He II,current algebras and quantum knots

A canonical quantization scheme is developed for vertices in superfluid He II, using Dirac's technique for constrained hamiltonian systems. Quantization introduces in the theory in natural way the structure of the infinite Lie algebra of incompressible flows. We argue that all the topological invariants of the vortex, considered as a knot, can be regarded as observables of the system. Finally unitary representations of measure preserving flows on R³ and current algebra are discussed.     

N = 1 Dualities for Exceptional Gauge Groups and Quantum Global Symmetries

We discuss our attempts to generalize the known examples of dualities in N = 1 supersymmetric gauge theories to exceptional gauge groups. We derive some dual pairs from known examples connected to exceptional groups and find an interesting phenomenon: sometimes the full global symmetry is “hidden” on the magnetic side. It is not realized as a symmetry on the fundamental fields in the Lagrangian. Rather, it emerges as a symmetry of the quantum theory. We then focus on an approach based on self-dual models. We construct duals for some very special matter content of E₇, E₆ and F₄. Again we find that the full global symmetry is not realized on the fundamental fields.     

Supersymmetric BPHZ renormalization: (II). Supersymmetric extension of the pure Yang-Mills model

We generalize the supersymmetry way of performing the BPHZ renormalization in introduced in paper (I) to cases including massless fields. This is used to prove the Slavnov identity in the supersymmetric extension of the pure Yang-Mills theory.     

A Model for Spacetime II. The Emergence of Higher Dimensions and Field Theory/Strings Dualities

We show that in 4-spacetime modified at very short distances due to the weakening of classical logic, the higher dimensions emerge. We analyse the case of some smooth topoi, and the case of some class of pointless topoi. The pointless topoi raise the dimensionality due to the forcing adding “string” objects and thus replacing classical points in spacetime. Turning to strings would be something fundamental and connected with set theoretical forcing. The field theory/strings dualities originate at the set theoretical level of the theories. It is argued that this fundamental level can help solving some difficulties of the physical dualities.     

Knots,BPS States,and Algebraic Curves

We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.     

Thermodynamic system far from equilibrium. II. Supersymmetric method for the φ4-model

We have used second-order perturbation theory to derive expressions for the supersymmetric components of the self-energy parts of a superfield propagator.Sumskii Physico-Technical Institute, Scientific-Industrial Enterprise of the RETO. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 39–42, August, 1994.     

Supersymmetric dual string theory: (II). Vertices and trees

Vertex operators for the emission of massless open-string states are constructed using the light-cone gauge formalism introduced in the first paper of this series. The vertices for vector and spinor emission transform into one another under supersymmetry in the same way as the states themselves. This enables one to construct amplitudes and deduce the relationships among them implied by supersymmetry. Various four-particle tree graphs are explicitly investigated and shown to satisfy the duality requirement.     

Chaotic Quantum Vortices in He II: Thermodynamic Equilibrium and Turbulence

The use of superfluid helium as a refrigerant in cryogenic systems is governed by the presence of a chaotic tangle of quantum filaments in the superfluid component of helium. Therefore, to describe any hydrodynamic phenomena (in particular, heat transfer) in quantumliquids containing vortex tangles, it is necessary to have information on their structure and statistics. The paper discusses two possible statistical configurations of chaotic vortices: the thermodynamic equilibrium and the highly nonequilibrium turbulent state, as well as the transition between them. Basing on the Langevin approach, we discuss the mechanism of establishment of thermodynamic equilibrium for a chaotic set of quantum vortex filaments. The corresponding Fokker–Planck equation for the probability density functional has a solution in the form of the Gibbs distribution. Basing on the above analysis, we discuss the possible causes and mechanisms of violation of thermodynamic equilibrium and transition to the turbulent regime.     

The (N=1) Supersymmetric Sine-Gordon model in two dimensions. II

In this second paper the technical part of the results about the SupersymmetricN=1 massless Sine-Gordon field theory, at finite (space) volume, is given. The proof that the theory exists and is analytic in the coupling constant and that, at finite (space) volume, its Witten index is 1, is, therefore, completed.     

Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

We consider CMV matrices with Verblunsky coefficients determined in an appropriate way by the Fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics. In our first application we estimate the spreading rates of quantum walks on the line with time-independent coins following the Fibonacci sequence. The estimates we obtain are explicit in terms of the parameters of the system. In our second application, we establish a connection between the classical nearest neighbor Ising model on the one-dimensional lattice in the complex magnetic field regime, and CMV operators. In particular, given a sequence of nearest-neighbor interaction couplings, we construct a sequence of Verblunsky coefficients, such that the support of the Lee-Yang zeros of the partition function for the Ising model in the thermodynamic limit coincides with the essential spectrum of the CMV matrix with the constructed Verblunsky coefficients. Under certain technical conditions, we also show that the zeros distribution measure coincides with the density of states measure for the CMV matrix.     

Large N Duality, Lagrangian Cycles, and Algebraic Knots

We consider knot invariants in the context of large N transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicitly constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.