Global supersymmetry on curved spaces in various dimensions

We propose methods towards a systematic determination of d  -dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity point of view. In particular, by using appropriate gauged supergravities in various dimensions we show that supersymmetry can be defined in conformally flat spaces, such as non-compact hyperboloids Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and compact spheres Sⁿ${\mathbb{S}}^n$ or – by turning on appropriate Wilson lines corresponding to R-symmetry vector fields – on S¹×Sⁿ${\mathbb{S}}^1\times {\mathbb{S}}^n$, with n<6$n<6$. By group theory arguments we show that Euclidean field theories with rigid supersymmetry cannot be consistently defined on round spheres S^d${\mathbb{S}}^d$ if d>5$d>5$ (despite the existence of Killing spinors). We also show that distorted spheres and certain orbifolds are also allowed by the group theory classification.     

Conformal field theories in six-dimensional twistor space

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space–time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the six-dimensional case in which twistor space is the 6-quadric Q$Q$ in CP⁷${\mathbb{C}\mathbb{P}}^7$ with a view to applications to the self-dual (0,2)$(0,2)$-theory. We show how spinor-helicity momentum eigenstates have canonically defined distributional representatives on twistor space (a story that we extend to arbitrary dimension). These yield an elementary proof of the surjectivity of the Penrose transform. We give a direct construction of the twistor transform between the two different representations of massless fields on twistor space (H²$H^2$ and H³$H^3$) in which the H³$H^3$s arise as obstructions to extending the H²$H^2$s off Q$Q$ into CP⁷${\mathbb{C}\mathbb{P}}^7$.     

Force free projective motions of the sphere

Suppose that the sphere Sⁿ$S^n$ has initially a homogeneous distribution of mass and let G$G$ be the Lie group of orientation preserving projective diffeomorphisms of Sⁿ$S^n$. A projective motion of the sphere, that is, a smooth curve in G$G$, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of Sⁿ$S^n$ and, more generally, examples of subgroups H$H$ of G$G$ such that a force free motion initially tangent to H$H$ remains in H$H$ for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H=S_On+1$H=S{O}_{n+1}$). The main tool is a Riemannian metric on G$G$, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.     

Biminimal properly immersed submanifolds in the Euclidean spaces

We consider a complete nonnegative biminimal   submanifold M$M$ (that is, a complete biminimal submanifold with λ≥0$\lambda \ge 0$) in a Euclidean space E^N${\mathbb{E}}^N$. Assume that the immersion is proper  , that is, the preimage of every compact set in E^N${\mathbb{E}}^N$ is also compact in M$M$. Then, we prove that M$M$ is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0$\lambda <0$, we construct examples of biminimal submanifolds and curves.     

Minimal hypersurfaces in a nearly Kaehler 6-sphere

In this paper we show that for a compact minimal hypersurface M$M$ of constant scalar curvature in the unit sphere S⁶$S^6$ with the shape operator A$A$ satisfying ‖A‖²>5${\Vert A\Vert }^2>5$, there exists an eigenvalue λ>10$\lambda >10$ of the Laplace operator of the hypersurface M$M$ such that ‖A‖²=λ−5${\Vert A\Vert }^2=\lambda -5$. This gives the next discrete value of ‖A‖²${\Vert A\Vert }^2$ greater than 0 and 5.     

Pre-quantization of the moduli space of flat G-bundles over a surface

For a simply connected, compact, simple Lie group G$G$, the moduli space of flat G$G$-bundles over a closed surface Σ$\Sigma$ is known to be pre-quantizable at integer levels. For non-simply connected G$G$, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction–namely a certain cohomology class in H³(G²;Z)$H^3(G^2;\mathbb{Z})$–that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G$G$.     

On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold P$P$, there are two approaches to its geometric quantization: one involves a circle bundle Q$Q$ over P$P$ endowed with a Jacobi (or Jacobi–Dirac) structure; the other one involves a circle bundle with a (pre)contact groupoid structure over the (pre)symplectic groupoid of P$P$. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of P$P$ is obtained from the Lie groupoid of Q$Q$ via an S¹$S^1$ reduction that preserves both the Lie groupoid and the geometric structures.     

Entanglement entropy and variational methods: Interacting scalar fields

We develop a variational approximation to the entanglement entropy for scalar ?⁴${?}^4$ theory in 1+1$1+1$, 2+1$2+1$, and 3+1$3+1$ dimensions, and then examine the entanglement entropy as a function of the coupling. We find that in 1+1$1+1$ and 2+1$2+1$ dimensions, the entanglement entropy of ?⁴${?}^4$ theory as a function of coupling is monotonically decreasing and convex. While ?⁴${?}^4$ theory with positive bare coupling in 3+1$3+1$ dimensions is thought to lead to a trivial free theory, we analyze a version of ?⁴${?}^4$ with infinitesimal negative bare coupling, an asymptotically free theory known as precarious  ?⁴${?}^4$ theory, and explore the monotonicity and convexity of its entanglement entropy as a function of coupling. Within the variational approximation, the stability of precarious ?⁴${?}^4$ theory is related to the sign of the first and second derivatives of the entanglement entropy with respect to the coupling.     

Dworkin’s argument revisited: Point processes,dynamics, diffraction,and correlations

The setting is an ergodic dynamical system (X,μ)$(X,\mu )$ whose points are themselves uniformly discrete point sets Λ$\Lambda$ in some space R^d${\mathbb{R}}^d$ and whose group action is that of translation of these point sets by the vectors of R^d${\mathbb{R}}^d$. Steven Dworkin’s argument relates the diffraction of the typical point sets comprising X$X$ to the dynamical spectrum of X$X$. In this paper we look more deeply at this relationship, particularly in the context of point processes.     

On the length of stabilograms: A study performed with detrended fluctuation analysis

The effects associated to the length of stabilograms, a measure of the time dependence of the center of pressure of an individual standing up, are analyzed. The fractal characteristics of 27 signals with a length of 2¹⁴$2^{14}$ points, each one corresponding to a different individual, are studied by using the Detrended Fluctuation Analysis technique. The properties of the complete signals are compared to those of various subsignals extracted from them. No differences have been found between the characteristic exponents found for x$x$ and y$y$ signals. The relation between the exponents of the position and velocity signals is accomplished by the 2¹⁴$2^{14}$ point signals, while subsignals with up to 2¹²$2^{12}$ points do not verify it. Using artificial signals with 2¹⁴$2^{14}$ points, generated for α$\alpha$ values given, it has been demonstrated that the exponents obtained from these signals take values larger than expected for α<0.3$\alpha <0.3$, while the exponents of the accumulated series are smaller than expected for 0.7<α$0.7<\alpha$. For CoP trajectories this indicates that DFA-1 provides feasible exponents for the short τ$\tau$-end region of the velocity signal and the large τ$\tau$-end region of the accumulated (position) one. It has been found that the characteristic exponents vary along the series. A slightly larger persistence is found in the last part of the signal for large frequencies in the x$x$ direction.     

Non-Abelian fusion rules from an Abelian system

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n$n$ dimensional vector space which we call H_n$H_n$. The Z_p${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus H_n$H_n$ can be thought of as a C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n$n$ and p$p$, though we believe this feature holds for all n>p$n>p$. We will see that non-Abelian anyons of the quantum double of C(S₃)$\mathbb{C}\left(S_3\right)$ are obtained as part of the vertex excitations of the model with n=6$n=6$ and p=3$p=3$. Ising anyons are obtained in the model with n=4$n=4$ and p=2$p=2$. The n=3$n=3$ and p=2$p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z_p${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.     

Resolutions of non-regular Ricci-flat Kähler cones

We present explicit constructions of complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki–Einstein manifolds. The metrics are constructed from a complete Kähler–Einstein manifold (V,_gV)$(V,g_V)$ of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kähler metrics on the total spaces of (i) holomorphic C²/_Zp${\mathbb{C}}^2/{\mathbb{Z}}_p$ orbifold fibrations over V$V$, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP¹${\mathbb{W}\mathbb{C}\mathbb{P}}^1$, with generic fibres being the canonical complex cone over V$V$, and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kähler metrics on the total spaces of (a) rank two holomorphic vector bundles over V$V$, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base V$V$. When V=CP¹$V={\mathbb{C}\mathbb{P}}^1$ our results give Ricci-flat Kähler orbifold metrics on various toric partial resolutions of the cone over the Sasaki–Einstein manifolds Y^p,q$Y^{p,q}$.     

Curvatures,volumes and norms of derivatives for curves in Riemannian manifolds

The well-known formulas express the curvature and the torsion of a curve in R³${\mathbb{R}}^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in arbitrary Riemannian manifolds. Our motivation comes from physics. It follows that regular curves in Rⁿ${\mathbb{R}}^n$ are determined up to isometry by the norms of their n$n$ consecutive derivatives. We extend this fact to two-point homogeneous spaces.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

Singular self-dual Zollfrei metrics and twistor correspondence

We construct examples of singular self-dual Zollfrei metrics explicitly, by patching a pair of Petean’s self-dual split-signature metrics. We prove that there is a natural one-to-one correspondence between these singular metrics and a certain set of embeddings of RP³$\mathbb{R}{\mathbb{P}}^3$ to CP³$\mathbb{C}{\mathbb{P}}^3$ which has one singular point. This embedding corresponds to an odd function on R$\mathbb{R}$ that is rapidly decreasing and pure imaginary valued. The one-to-one correspondence is explicitly given by using the Radon transform.