Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti-commuting for real hypersurfaces in complex two-plane Grassmannians $G_2\left({\mathbb{C}}^{m+2}\right)$ and prove a complete classification theorem, which gives a shrinking Ricci soliton with potential Reeb flow on Hopf real hypersurfaces and a tube over a totally real totally geodesic $\mathbb{Q}{P}^n$, $m=2n$ in $G_2\left({\mathbb{C}}^{m+2}\right)$.     

Monopoles on the Bryant–Salamon G2-manifolds

$G_2$-monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant–Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the ${\mathbb{S}}^4$ and ${\mathbb{C}\mathbb{P}}^2$. These are the first nontrivial examples of $G_2$-monopoles.Associated with each monopole there is a parameter $m\in {\mathbb{R}}^{+}$, known as the mass of the monopole. We prove that under a symmetry assumption, for each given $m\in {\mathbb{R}}^{+}$ there is a unique monopole with mass $m$. We also find explicit irreducible $G_2$-instantons on ${\Lambda }_{-}^2\left({\mathbb{S}}^4\right)$ and on ${\Lambda }_{-}^2\left({\mathbb{C}\mathbb{P}}^2\right)$.The third Bryant–Salamon $G_2$-metric lives on the spinor bundle over the $3$-sphere. In this case we produce a vanishing theorem for monopoles.     

Liberations and twists of real and complex spheres

We study the 10 noncommutative spheres obtained by liberating, twisting, and liberating +twisting the real and complex spheres $S_{\mathbb{R}}^{N-1},S_{\mathbb{C}}^{N-1}$. At the axiomatic level, we show that, under very strong axioms, these 10 spheres are the only ones. Our main results concern the computation of the quantum isometry groups of these 10 spheres, taken in an affine real/complex sense. We formulate as well a proposal for an extended formalism, comprising 18 spheres.     

Chern–Simons formulation of three-dimensional gravity with torsion and nonmetricity

We consider various models of three-dimensional gravity with torsion or nonmetricity (metric affine gravity), and show that they can be written as Chern–Simons theories with suitable gauge groups. Using the groups $\mathrm{ISO}(2,1)$, $\mathrm{SL}(2,\mathbb{C})$ and $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$, and the fact that they admit two independent coupling constants, we obtain the Mielke–Baekler model for zero, positive and negative effective cosmological constant respectively. Choosing $\mathrm{SO}(3,2)$ as the gauge group, one gets a generalization of conformal gravity that has zero torsion and only the trace part of the nonmetricity. This characterizes a Weyl structure. Finally, we present a new topological model of metric affine gravity in three dimensions arising from an $\mathrm{SL}(4,\mathbb{R})$ Chern–Simons theory.     

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

We enhance the action of higher abelian gauge theory associated to a gerbe on an M5-brane with an action of a torus ${\mathbb{T}}^n(n\ge 2)$, by a noncommutative ${\mathbb{T}}^n$-deformation of the M5-brane. The ingredients of the noncommutative action and equations of motion include the deformed Hodge duality, deformed wedge product, and the noncommutative integral over the noncommutative space obtained by strict deformation quantization. As an application we then introduce a variant model with an enhanced action in which we show that the corresponding partition function is a modular form, which is a purely noncommutative geometry phenomenon since the usual theory only has a ${\mathbb{Z}}_2$-symmetry. In particular, S-duality in this 6-dimensional higher abelian gauge theory model is shown to be, in this sense, on par with the usual 4-dimensional case.     

The geometry of Π-invertible sheaves

Using the fact that $\Pi$-invertible sheaves can be interpreted as locally free sheaves of modules for the super skew field $\mathbb{D}$, we give a new construction of the $\Pi$-projective superspace ${\mathbb{P}}_{\Pi ,B}^n$ over affine $k$ superschemes $B$, $k$ an algebraically closed field. We characterize morphisms into ${\mathbb{P}}_{\Pi ,B}^n$ and give a new interpretation of the composition of $\Pi$-invertible sheaves in terms of the algebra of $\mathbb{D}$.