Symmetric Products,Permutation Orbifolds and Discrete Torsion

Symmetric product orbifolds, i.e. permutation orbifolds of the full symmetric group S _n are considered by applying the general techniques of permutation orbifolds. Generating functions for various quantities, e.g. the torus partition functions and the Klein-bottle amplitudes are presented, as well as a simple expression for the discrete torsion coefficients.     

Affine Yang-Mills theory and affine self-dual Chern-Simons Solitons

A four-dimensional affine Yang-Mills theory, i.e. Yang—Mills gauge theory with values in an affine Kac-Moody algebra, is constructed which can give rise to a spontaneous breakdown of the affine symmetry. The affine self-dual Yang-Mills equation (which is a special kind of affine YM theory) in four dimensions is dimensionally reduced to the affine self-dual Chem-Simons equation in two dimensions. The latter is shown to have soliton solutions which satisfy the conformal affine Toda equations.K. C. Wong Research Award Winner; address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

Affine Orbifolds and Rational Conformal Field Theory Extensions of W 1+∞

Chiral orbifold models are defined as gauge field theories with a finite gauge group Γ. We start with a conformal current algebra associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Γ of inner automorphisms or (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra of local observables invariant under Γ. A set of positive energy modules is constructed whose characters span, under some assumptions on Γ, a finite dimensional unitary representation of . We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of W _1+∞ that appear to provide a bridge between two approaches to the quantum Hall effect. Received: 5 December 1996 / Accepted: 1 April 1997     

Orbifolds and Topological Defects

We study orbifolds of two-dimensional topological field theories using defects. If the TFT arises as the twist of a superconformal field theory, we recover results on the Neveu–Schwarz and Ramond sectors of the orbifold theory, as well as bulk-boundary correlators from a novel, universal perspective. This entails a structure somewhat weaker than ordinary TFT, which however still describes a sector of the underlying conformal theory. The case of B-twisted Landau–Ginzburg models is discussed in detail, where we compute charge vectors and superpotential terms for B-type branes. Our construction also works in the absence of supersymmetry and for generalised “orbifolds” that need not arise from symmetry groups. In general, this involves a natural appearance of Hochschild (co)homology in a 2-categorical setting, in which among other things we provide simple presentations of Serre functors and a further generalisation of the Cardy condition.     

Quantum Orbifolds

This is a study of orbifold-quotients of quantum groups (quantum orbifolds \({\Theta } \rightrightarrows G_{q}\)). These structures have been studied extensively in the case of the quantum S U ₂ group. A generalized theory of quantum orbifolds over compact simple and simply connected quantum groups is developed. Associated with a quantum orbifold there is an invariant subalgebra and a crossed product algebra. For each spin quantum orbifold, there is a unitary equivalence class of Dirac spectral triples over the invariant subalgebra, and for each effective spin quantum orbifold associated with a finite group action, there is a unitary equivalence class of Dirac spectral triples over the crossed product algebra. A Hopf-equivariant Fredholm index problem is studied as an application.     

Gerbes over Orbifolds and Twisted K-Theory

In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H³(X) via Chern-Weil theory. For an arbitrary gerbe , a twisting K_orb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold.The first author was partially supported by the National Science Foundation and Conacyt-México     

Topological Open Strings on Orbifolds

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3} orbifold point of local \mathbb P²{{\mathbb P}^2}, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov–Witten invariants of \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3}, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.     

Self-Induced Transparency Solitons and Dissipative Solitons in Microwave Electronic Systems

Radiophysics and Quantum Electronics - Solitons are a well-studied subject in many domains of nonlinear physics, including hydrodynamics, plasma physics, optics, etc. However, it has been...     

Mirror Pairs of Quintic Orbifolds

JETP Letters - Two constructions of mirror pairs of Calabi-Yau manifolds are compared by example of quintic orbifolds $$\mathcal{Q}$$ . The first, Berglund—Hubsch—Krawitz, construction...     

Solitons and supertranslations

We construct the quantum theory for a two-dimensional supersymmetric theory around a supertranslated soliton solution. This is done by using Dirac's method for constrained systems. The fermionic parameter of supertranslation is treated as a collective co-ordinate. The path-integral expression is also obtained.     

Orbifolds,Hopf algebras,and the Moonshine

We describe the modular properties and fusion rules of holomorphic orbifold models by Hopf algebraic techniques, using the representation theory of the orbifold quantum group. We apply this theory to the construction of generalized Thompson series, and discuss its connections with Moonshine.     

Solitons and Vortices in Plasmas

The various types of nonlinear wave solutions in plasmas are reviewed. First, the generation, pro- pagation, and stability of solitary waves are demonstrated for some simple examples. Then, relevant two-and three-dimensional models for Langniuir solitons are proposed and investigated. The collapse as an effective dissipation mechanism in plasmas is discussed in detail. Finally, simple models of two-dimensional vortex motion and their consequences are considered.     

Solitons and vortices in lasers

We give in this article the mathematical background for pattern formation in nonlinear active resonators, elucidating the relation of optics with other fields of physics, and demonstrate experimentally the existence, properties, and dynamics of: (i) vortices in lasers, (ii) bright spatial solitons in lasers with saturable absorber, and (iii) spatial solitons in degenerate parametric mixing. All these structures are by definition bistable so that they are potentially useful for parallel optical information processing. Received: 8 June 1998     

Correlation Functions for MN/SN Orbifolds

We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M ^N /S _N , where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is “universal” in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non-vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of N – this is a manifestation of the stringy exclusion principle. Received: 20 July 2000 / Accepted: 17 December 2000     

Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold ${\mathcal{X}}$ and that of its toric crepant resolution Y coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Ruan’s original CRC (Gromov-Witten theory of spin curves and orbifolds, contemp math, Amer. Math. Soc., Providence, RI, pp 117–126, 2006). We prove the open CRC for the weighted projective spaces ${\mathcal{X} = \mathbb{P}(1,\ldots,1, n)}$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.     

Acousto-Optic Solitons in Fibers

It is shown that the deceleration of light pulses down to the velocity of a sound value can be realized in a case of unidirectional parametric interaction of two electromagnetic waves with an acoustic one in the regime of forming three wave acousto-optic solitons. This nonlinear acousto-optic interaction can be realized in long distance systems like fibers. As the result of such an interaction, two types of acousto-optic envelope solitons can propagate in fibers. Modulation of the amplitude of the electromagnetic pump wave can control the soliton velocity.