This paper demostrates a method for analysing almost CR geometries (H, J), by uniquely defining a partially integrable structure (H,K) from the same data. Thus two almost CR geometries (H,J) and (H ′, J ′) are equivalent if and and only if they generate isomorphic induced partially integrable CR geometries (H,K) and (H ′, K′), and the set of CR morphisms between these spaces contains an element...

The Lagrangian density in the problem is L = − 1 4 (F µν) 2 + (D µ φ) † D µ φ − m 2 φ † φ − λ 6 (φ † φ) 2 , (1) where φ(x) is a complex scalar field in four spacetime dimensions (except for part (h)), and we are interested in the theory when m 2 = −µ 2 < 0. The covariant derivative includes the vector potential D µ φ = (∂ µ + ieA µ)φ. (2) I don't know why they chose this funny normalization for...

We study nonsingular branched coverings of a homogeneous space X. There is a vector bundle associated with such a covering which was conjectured by O. Debarre to be ample when the Picard number of X is one. We prove this conjecture, which implies Barth-Lefschetz type theorems, for lagrangian grassmannians, and for quadrics up to dimension six. We propose a conjectural extension to homogeneous s...

Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed curve F in the complex plane, we introduce the notion of the co...

This article explains how to construct immersed Lagrangian submanifolds in C2 that are asymptotic at large distance from the origin to a given braid in the 3–sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in the vector bundle O(−1)⊕ O(−1) over the Riemann sphere which are asymptotic a...

Based on recent developments, in this letter we study the one parameter deformation of 2+1 dimensional gauge theories with scale invariance and N = 8 supersymmetry, which is expected to be the field theory living on a stack of M2 branes. The deformed gauge theory is defined by a Lagrangian and is based on an infinite set of novel 3-algebras constructed by relaxing the assumption that the invari...

We develop a simple model for coupling the pseudo-Goldstone bosons of QCD Or's, K's, -q) to quark fields. This nonlocal model incorporates a momentum dependent dynamical quark mass ,~ (p) in a manner consistent with global chiral symmetry. Explicit current quark masses and gauge fields are also introduced. All low energy parameters of the chiral lagrangian are expressed in terms of ~(p). In par...

A discussion of the number of degrees of freedom, and their dynamical properties, in higher derivative gravitational theories is presented. It is shown that non-vanishing (Cmnpq) 2 terms arise in N=1, D=4 superstring Lagrangians due to one-loop radiative corrections with light field internal lines. 1 Bosonic Gravitation The usual Einstein theory of gravitation involves a symmetric tensor gμν wh...

It has recently been argued that soft-collinear effective theory for processes involving both soft and collinear partons contains a new soft-collinear mode, which can communicate between the soft and collinear sectors of the theory. The formalism incorporating the corresponding fields into the effective Lagrangian is extended to include external current and four-quark operators relevant to weak...