Non-degeneracy of cohomological traces for general Landau-Ginzburg models

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models.Comment: 29 page

On the slow roll expansion of one-field cosmological models

We study the infrared scale expansion of single field cosmological models using the Hamilton-Jacobi formalism, showing that its specialization at unit scale parameter recovers the slow roll expansion. In particular, we show that the latter coincides with a Laurent expansion of the Hamilton-Jacobi function in powers of the Planck mass, whose terms are controlled by certain recursively-defined polynomials. This allows us to give an explicit recursion procedure for constructing all higher order terms of the slow roll expansion. We also discuss the corresponding effective potential and the action of the universal similarity group.Comment: 52 pages, no figure

Cosmological flows on hyperbolic surfaces

We outline the geometric formulation of cosmological flows for FLRW models with scalar matter as well as certain aspects which arise in their study with methods originating from the geometric theory of dynamical systems. We briefly summarize certain results of numerical analysis which we carried out when the scalar manifold of the model is a hyperbolic surface of infinite area.Comment: 10 pages, conference proceeding

Singular foliations for M-theory compactification

We use the theory of singular foliations to study ${\cal N}=1$ compactifications of eleven-dimensional supergravity on eight-manifolds $M$ down to $\mathrm{AdS}_3$ spaces, allowing for the possibility that the internal part $\xi$ of the supersymmetry generator is chiral on some locus ${\cal W}$ which does not coincide with $M$. We show that the complement $M\setminus {\cal W}$ must be a dense open subset of $M$ and that $M$ admits a singular foliation ${\bar {\cal F}}$ endowed with a longitudinal $G_2$ structure and defined by a closed one-form $\boldsymbol{\omega}$, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet ${\cal W}$. When $\boldsymbol{\omega}$ is a Morse form, the chiral locus is a finite set of points, consisting of isolated zero-dimensional leaves and of conical singularities of seven-dimensional leaves. In that case, we describe the topology of ${\bar {\cal F}}$ using results from Novikov theory. We also show how this description fits in with previous formulas which were extracted by exploiting the $\mathrm{Spin}(7)_\pm$ structures which exist on the complement of ${\cal W}$.Comment: 66 pages, 6 tables, 4 figures; v2: added discussion of limit $kappa=0

Geometric algebra and M-theory compactifications

We show how supersymmetry conditions for flux compactifications of supergravity and string theory can be described in terms of a flat subalgebra of the Kahler-Atiyah algebra of the compactification space, a description which has wide-ranging applications. As a motivating example, we consider the most general M-theory compactifications on eight-manifolds down to AdS3 spaces which preserve N=2 supersymmetry in 3 dimensions. We also give a brief sketch of the lift of such equations to the cone over the compactification space and of the geometric algebra approach to `constrained generalized Killing pinors', which forms the technical and conceptual core of our investigation.Comment: 8 pages, no figures; conference proceeding

Internal circle uplifts, transversality and stratified G-structures

We study stratified G-structures in ${\cal N}=2$ compactifications of M-theory on eight-manifolds $M$ using the uplift to the auxiliary nine-manifold ${\hat M}=M\times S^1$. We show that the cosmooth generalized distribution ${\hat {\cal D}}$ on ${\hat M}$ which arises in this formalism may have pointwise transverse or non-transverse intersection with the pull-back of the tangent bundle of $M$, a fact which is responsible for the subtle relation between the spinor stabilizers arising on $M$ and ${\hat M}$ and for the complicated stratified G-structure on $M$ which we uncovered in previous work. We give a direct explanation of the latter in terms of the former and relate explicitly the defining forms of the $\mathrm{SU}(2)$ structure which exists on the generic locus ${\cal U}$ of $M$ to the defining forms of the $\mathrm{SU}(3)$ structure which exists on an open subset ${\hat {\cal U}}$ of ${\hat M}$, thus providing a dictionary between the eight- and nine-dimensional formalisms.Comment: 24 page