Ascertaining the values of sigma x,sigma y,and sigma z of a polarization qubit

In the 1987 spin-retrodiction puzzle of Vaidman, Aharonov, and Albert one is challenged to ascertain the values of sigma(x), sigma(y), and sigma(z) of a spin-1/2 particle by utilizing entanglement. We report the experimental realization of a quantum-optical version in which the outcome of an intermediate polarization projection is inferred by exploiting single-photon two-qubit quantum gates. The experimental success probability is consistently above the 90.2% threshold of the optimal one-qubit strategy, with an average success probability of 95.6%.     

Quadratic pseudopotentials for Gl(N, ) principal sigma models

Matrix pseudopotentials for the Gl(n, ) principal sigma model in two-dimensional Minkowski space are derived through the method of closed differential ideals. The structure of the pseudopotential equations is chosen to contain only quadratic nonlinearities of the matrix Riccati type. A set of further constraints is deduced which leads to inner Bäcklund transformations depending on two complex parameters. Through a geometrical interpretation of such matrix Riccati equations and the associated constraints, a linearization procedure is derived and the equations are reduced to those of Zakharov, Mikhaïlov and Shabat [7–10]. An inductive procedure is applied to explicitly solve an iterated sequence of such Bäcklund transformations through a purely algebraic transformation of solutions to the first step. The resulting nonlinear superposition formula is used to demonstrate a permutability theorem, and to analyse the appearance of singularities.fr|Nous déduisons les pseudo-potentiels matriciels pour le modèle sigma principal à valeurs dans Gl(N, ), défini sur l'espace de Minkowski bi-dimensionnel, en utilisant la méthode des idéaux différentiels fermés. On restreint la structure des équations des pseudopotentiels de façon à ce que n'apparaissent que des non-linéarités quadratiques du type Riccati matriciel. Nous déduisons un ensemble de contraintes supplémentaires menant à des transformations de Bäcklund internes dépendantes de deux paramètres complexes. Grâce à l'interprétation géométrique de ces équations de Riccati matricielles et des contraintes associées, nous déduisons un procédé de linéarisation et les équations se réduisent à celles de Zakharov, Mikhailov et Shabat [7–10]. Un procédé inductif est utilisé pour résoudre explicitement une succession de transformations de Bäcklund par une transformation purement algébrique des solutions de la première ètape. La formule de superposition non-linéaire qui en résulte permet de démontrer un théorème de permutabilité et d'analyser l'apparition de singularités.     

Supersymmetric nonlinear sigma model in AdS5

We construct the supersymmetric nonlinear sigma model in a fixed AdS₅ background. We use component fields and find that the complex bosons must be the coordinates of a hyper-Kähler manifold that admits a Killing vector satisfying an inhomogeneous tri-holomorphic condition. We propose boundary conditions that map the on-shell bulk hypermultiplets into off-shell chiral multiplets on 3-branes that foliate the bulk. The supersymmetric AdS₅ isometries reduce to superconformal transformations on the brane fields.     

Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries

Chern–Simons (CS) gauge theories in three dimensions and the Poisson sigma model (PSM) in two dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the literature are not globally well defined for non-parallelizable Poisson manifolds and propose a covariant definition of the off-shell gauge symmetries as left action of some finite-dimensional Lie algebroid.

Our approach allows us to avoid complications arising in the infinite-dimensional super-geometry of the BV- and AKSZ-formalism. This preprint is a starting point in a series of papers meant to introduce Yang–Mills type gauge theories of Lie algebroids, which include the standard YM theory, gerbes, and the PSM.     

Topological sigma models

A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surface to an arbitrary almost complex manifoldM. It possesses a fermionic BRST-like symmetry, conserved for arbitrary, and obeyingQ ²=0. In a suitable version, the quantum ground states are the 1+1 dimensional Floer groups. The correlation functions of the BRST-invariant operators are invariants (depending only on the homotopy type of the almost complex structure ofM) similar to those that have entered in recent work of Gromov on symplectic geometry. The model can be coupled to dynamical gravitational or gauge fields while preserving the fermionic symmetry; some observations by Atiyah suggest that the latter coupling may be related to the Jones polynomial of knot theory. From the point of view of string theory, the main novelty of this type of sigma model is that the graviton vertex operator is a BRST commutator. Thus, models of this type may correspond to a realization at the level of string theory of an unbroken phase of quantum gravity.On leave from Department of Physics, Princeton University. Supported in part by NSF Grants No. 80-19754, 86-16129, 86-20266     

On the CP topological sigma model and the Toda lattice hierarchy

We propose, in bihamiltonian formalism, a version of the Toda lattice hierarchy that is satisfied by the two point correlation functions of the CP¹ topological sigma model at genus one approximation, and we also show that this bihamiltonian hierarchy is compatible with the Virasoro constraints of Eguchi–Hori–Xiong up to genus two approximation.     

Euclidean pseudoduality and boundary conditions in sigma models

We discuss pseudoduality transformations in two-dimensional conformally invariant classical sigma models, and extend our analysis to a given boundaries of world-sheet, which gives rise to an appropriate framework for the discussion of the pseudoduality between D-branes. We perform analysis using the Euclidean spacetime and show that structures on the target space can be transformed into pseudodual manifold identically. This map requires that torsions and curvatures related to individual spaces are the same when connections are Riemannian. Boundary pseudoduality imposes locality condition.     

Current conservation and parallelism in nonlinear sigma models

We examine the consequences of a restrictive current conservation condition, which figures in the bosonization of certain free fermion theories, when applied to a general bosonic nonlinear sigma model in two dimensions. The condition is shown to lead directly to the existence of parallelizing Killing vector fields on the internal manifold. The current algebra is checked to have the expected Kac-Moody form.