arXiv research

Study of ${\cal M}$-theory dual of thermal QCD-like theories at intermediate coupling.

Conjectures on universal structures in algebraic geometry enumerative invariants.

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measu…

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 $ξ$ of the supersymmetry generator is chiral on some locus ${\cal W}$ which does not coincide wi…

For any principal bundle $P$, one can consider the subspace of the space of connections on its tangent bundle $TP$ given by the tangent bundle $T{\cal A}$ of the space of connections ${\cal A}$ on $P$. The tangent gauge group acts freely on $T{\cal A}$. Appropriate BRST operators are introduced for quantum field theori…

Alternative algebraic characterization of 3D cobordisms category.

The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D = G_0(z) \subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous holomorphic vector bundle $\mathbb E \to D$. The topic here is the double fibration tr…

Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices $\bar S$, which control non-associativity of the representation product in a peculiar channel $R\otimes \bar R \otimes R \longrightarrow R$. These $\bar S$ are simultaneously symmetric and orthogo…

Study on a new type of manifolds that generalize almost C-manifolds.

We summarize the foliation approach to ${\cal N}=1$ compactifications of eleven-dimensional supergravity on eight-manifolds $M$ down to $\mathrm{AdS}_3$ spaces for the case when the internal part $ξ$ of the supersymmetry generator is chiral on some proper subset ${\cal W}$ of $M$. In this case, a topological no-go theo…

In this short article we review how the classical theory of principal fibre bundles (PFB) transcribes in an algebraic formalism. In this dual formulation, a PFB is given by a right co-module algebra ${\cal P}$ over a Hopf algebra ${\cal H}$ with a mapping $Δ_R:{\cal P}\to{\cal P}\otimes{\cal H}$. In our case ${\cal P}$…

The large $k$ asymptotics (perturbation series) for integrals of the form $\int_{\cal F}μe^{i k S}$, where $μ$ is a smooth top form and $S$ is a smooth function on a manifold ${\cal F}$, both of which are invariant under the action of a symmetry group ${\cal G}$, may be computed using the stationary phase approximation…

This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf $\Cal{T}_M$, satisfying an integrability condition. $F$--manifolds and compatible fl…

The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle ${\cal P}$. The structure group G=SO(3) is the vorticity group, and the bundle ${\cal P}=GL_+(3, R})$ is the connected component of the general linear group. The base manifold is the space of positive-defi…

Study curvature properties of w.a. S-manifolds with new conditions.

The classical duality theory of Kantorovich and Kellerer for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice $\cal{X}$ with a order unit. The primal problem is given as the supremum o…

We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\cal R}_i = U_i{\cal R}_1U_i^{-1}…

New structures defined for studying contact foliations and their geometry.

Definition of $({\cal T}_{a},{\cal T}_{b})$-pseudosymmetric semi-Riemannian manifold is given. $({\cal T}_{a},{\cal T}_{b})$-pseudosy mmetric $(N(k),ξ)$-semi-Riemannian manifolds are classified. Some results for ${\cal T}_{a}$-pseudosymmetric $(N(k),ξ)$-semi-Riemannian manifolds are obtained. $({\cal T}_{a},{\cal T}_{b…

This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $¶^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge th…

The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.

We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and show th…

Level-rank duality relates the observables of two different Chern-Simons theories in which the roles of the Chern-Simons level and the rank of the gauge group are exchanged. In this note, we explore the consequences of this duality in the realm of topological string theory. We show that this duality induces a number of…

We construct ${\cal N}=2$ supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. It turns out that for every fixed point one can allocate either instanton or anti-instanton contributions to the partition function, and that this is compatible with supersymmetry. The equi…

This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed ${\cal N}=2$ supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a …

The aim of this paper is to present the main geometrical objects on the dual 1-jet bundle $J^{1*}(\cal{T},M)$ (this is the polymomentum phase space of the De Donder-Weyl covariant Hamiltonian formulation of field theory) that characterize our approach of multi-time Hamilton geometry. In this direction, we firstly intro…

Starting from a Lie algebroid ${\cal A}$ over a space V we lift its action to the canonical transformations on the principle affine bundle ${\cal R}$ over the cotangent bundle $T^*V$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is the Hamiltonian algebroid ${\cal A}^H$ over $…

We introduce the class $Σ_k(d)$ of $k$-stellated (combinatorial) spheres of dimension $d$ ($0 \leq k \leq d + 1$) and compare and contrast it with the class ${\cal S}_k(d)$ ($0 \leq k \leq d$) of $k$-stacked homology $d$-spheres. We have $Σ_1(d) = {\cal S}_1(d)$, and $Σ_k(d) \subseteq {\cal S}_k(d)$ for $d \geq 2k - 1$…

A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyp…

$(N(k),ξ)$-semi-Riemannian manifolds are defined. Examples and properties of $(N(k),ξ)$-semi-Riemannian manifolds are given. Some relations involving ${\cal T}_{a}$-curvature tensor in $(N(k),ξ)$-semi-Riemannian manifolds are proved. $ξ$-${\cal T}_{a}$-flat $(N(k),ξ)$-semi-Riemannian manifolds are defined. It is proved…

We study the Poincaré disk ${\cal D}=\{a\in {\cal A}: \|a\|<1\}$ of a C$^*$-algebra ${\cal A}$ from a projective point of view: ${\cal D}$ is regarded as an open subset of the projective line $\mathbb{P}_1{\cal A}$, the space of complemented rank one submodules of ${\cal A}^2$. We introduce the concept of cross ratio o…

Given two étale groupoids $\Cal G$ and $\Cal G'$, we consider the set of pointed morphisms from $\Cal G$ to $\Cal G'$. Under suitable hypothesis we introduce on this set a structure of Banach manifold which can be considered as the space of objects of an étale groupoid whose space of orbits is the space of morphisms fr…

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration $T=({\cal L}\to {\cal X}\to {\bf C})$, canonically determines a weak geodesic ra…

We study weakened $f$-structures on manifolds, generalizing classical results.

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…

We study the propagation of bosonic strings in singular target space-times. For describing this, we assume this target space to be the quotient of a smooth manifold $M$ by a singular foliation ${\cal F}$ on it. Using the technical tool of a gauge theory, we propose a smooth functional for this scenario, such that the p…

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must…

New bounds found for minimum tests needed in group testing.

The requirement of ${\cal N}=1$ supersymmetry for M-theory backgrounds of the form of a warped product ${\cal M}\times_{w}X$, where $X$ is an eight-manifold and ${\cal M}$ is three-dimensional Minkowski or AdS space, implies the existence of a nowhere-vanishing Majorana spinor $ξ$ on $X$. $ξ$ lifts to a nowhere-vanishi…

Analyzes the differential expansion of knot polynomials, focusing on its applicability and modifications.

We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, which are Mukai dual for zeta=0 and infinity, amd mirror partners…

The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometrie…

Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $\{upu^*: u\in U_2({\cal H})\},$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with t…

Let ${\cal F}_λ$ be the space of tensor densities on ${\bf R}^n$ of degree $λ$ (or, equivalently, of conformal densities of degree $-λn$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\cal F}_λ\otimes{\cal F}_μ$ to~${\cal F}_ν$. The…

Given a Riemannian manifold $N^n$ and ${\cal Z}\in \mathfrak{X}(N)$, an isometric immersion $f\colon M^m\to N^n$ is said to have the \emph{constant ratio property with respect to ${\cal Z}$} either if the tangent component ${\cal Z}^T_f$ of ${\cal Z}$ vanishes identically or if ${\cal Z}^T_f$ vanishes nowhere and the r…

The purpose of this paper is to prove that each of the following conditions is equivalent to that the foliation ${\cal F}$ is riemannian: 1) the lifted foliation ${\cal F}^{r}$ on the $r$-transverse bundle $ν^{r}{\cal F}$ is riemannian for an $r\geq 1$; 2) the foliation ${\cal F}_{0}^{r}$ on a slashed $ν_{\ast}^{r}{\ca…

We consider manifolds equipped with a foliation $\cal F$ of codimension $4q$, and an almost quaternionic structure $Q$ on the transversal bundle of ${\cal F}$. After discussing conditions of projectability and integrability of $Q$, we study the transversal twistor space $Z{\cal F}$ which, by definition, consists of the…

This note revisits recent results regarding the geometry and moduli of solutions of the heterotic string on manifolds $Y$ with a $G_2$ structure. In particular, such heterotic $G_2$ systems can be rephrased in terms of a differential $\check {\cal D}$ acting on a complex $\checkΩ^*(Y , {\cal Q})$, where ${\cal Q}=T^*Y\…