Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

We derive the general form for a three-dimensional scale-invariant field theory with $\mathcal{N}=6$ supersymmetry, $SU(4)$ R-symmetry and a $U(1)$ global symmetry. The results can be written in terms of a 3-algebra in which the triple product is not antisymmetric. For a specific choice of 3-algebra we obtain the $\mathcal{N}=6$ theories that have been recently proposed as models for M2-branes in an ${\mathbb{R}}^{8}/{\mathbb{Z}}_{k}$ orbifold background.

Three-algebras and N=6 Chern-Simons gauge theories

We study the recent construction of maximally supersymmetric field theory Lagrangians in three spacetime dimensions that are based on algebras with a triple product. Assuming that the algebra has a positive definite metric compatible with the triple product, we prove that the only non-trivial examples are either the well known case based on a four dimensional algebra or direct sums thereof.

Constraining Maximally Supersymmetric Membrane Actions

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This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connec- tions with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a G-structure as unifying principles. The General Holonomy Principle bridges over to parallel objects, thus moti- vating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenbock formulas for Dirac operators associ- ated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory. They also provide the link to Kostant's cubic Dirac operator.

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The Srní lectures on non-integrable geometries with torsion

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/2002/0006/0003/AJM-2002-0006-0003-a005.pdf

Nearly Kähler geometry and Riemannian foliations

We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quaternionic Kaehler manifolds and 6-dimensional nearly Kaehler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kaehler structure. Finally, we obtain a classification result for the homogenous case, reducing a conjecture of Wolf and Gray to its 6-dimensional form.

Nearly Kaehler geometry and Riemannian foliations

We consider complete nearly-Kahler manifolds with a canonicalHermitian connection. We prove some metric properties of strict nearly-Kahler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly-Kahler manifold to be a twistor space over a quaternionic-Kahler manifold is given. This enables us to give classification results in 10-dimensions.

On Nearly-Kähler Geometry

We study the skew-symmetric prolongation of a Lie subalgebra $\g \subseteq \mathfrak{so}(n)$, in other words the intersection $\Lambda^3 \cap (\Lambda^1 \otimes \g)$We compute this space in full generality Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean version of a conjecture posed in \cite{ofarill} concerning a class of Pl\"ucker-type embeddings We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension Prolongations of Lie algebras can also be used to finish the classification, started in \cite{datri}, of manifolds admitting Killing frames, or equivalently flat connections with 3-form torsion Next we study specific properties of invariant 4-forms of a given metric representation and apply these considerations to classify the holonomy representation of metric connections with vectorial torsion, that is with torsion contained in $\Lambda^1 \subseteq \Lambda^1 \otimes \Lambda^2$

Prolongations of Lie algebras and applications

We study the space of nearly Kahler structures on compact 6-dimensional manifolds. In particular, we prove that the space of infinitesimal deformations of a strictly nearly Kahler structure (with scalar curvature scal) modulo the group of diffeomorphisms, is isomorphic to the space of primitive co-closed (1,1)-eigenforms of the Laplace operator for the eigenvalue 2scal/5.

https://msp.org/pjm/2008/235-1/pjm-v235-n1-p05-p.pdf

Paul-Andi Nagy

Papers

Nearly Kähler geometry and Riemannian foliations

Nearly Kaehler geometry and Riemannian foliations

On Nearly-Kähler Geometry

Prolongations of Lie algebras and applications

Deformations of Nearly Kahler Structures