arXiv research

In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative de…

New approach simplifies topological T-duality for torus bundles.

Extends T-duality to non-principal torus actions with elliptic tangent bundles.

A (smooth) dynamical system with transformation group $\mathbb{T}^n$ is a triple $(A,\mathbb{T}^n,α)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\mathbb{T}^n$ and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of $\mathbb{T}^n$ on $A$. In this…

We construct a new example of an A-manifold, i.e. a Riemannian manifold with a cyclic-parallel Ricci tensor, which can be viewed as a generalization of the Einstein condition. The underlying manifold for our construction is a principal torus bundle over Kähler-Einstein manifold a with fibre a torus of arbitrary dimensi…

Generalized complex structures on certain torus bundles are explored.

The main result we give in this brief note relates, under suitable hypotheses, the φ-null Osserman, the null Osserman and the classical Osserman conditions to each other, via semi-Riemannian submersions as projection maps of principal torus bundles arising from a Lorentzian S-manifold.

In this paper we construct Cech cohomology groups that form a Gysin-type long exact sequence for principal torus bundles. This sequence is modeled on a de Rham cohomology sequence published in earlier work by Bouwknegt, Hannabuss and Mathai, which was developed to compute the global properties of T-duality in the prese…

We construct new examples of manifolds with cyclic-parallel Ricci tensor, so called A-manifolds, on a r-torus bundle over a product of almost Hodge A-manifolds.

Study heterotic G2-system on 2-step nilmanifolds with torus bundles.

New metrics found on non-Kähler Calabi-Yau manifolds.

In this note we define a lifting of a local torus action modeled on the standard representation (we call it a local torus action for simplicity) to a principal torus bundle, and show that there is an obstruction class for the existence of liftings in the first cohomology of the fundamental group of the orbit space with…

We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficien…

Ancient solutions to Ricci flow on torus bundles have additional symmetries.

Study rigidifies torus bundles under first Betti number constraints.

A principal toric bundle $M$ is a complex manifold equipped with a free holomorphic action of a compact complex torus $T$. Such a manifold is fibered over $M/T$, with fiber $T$. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety $X\subset M$ o…

The paper classifies manifolds with free torus actions and positive Ricci curvature.

We study the small eigenvalues of the Hodge Laplacian on collaping torus bundles with bounded curvature. In the first part of this dissertation, we consider examples of bundles on S^1 and T^2 with homogeneous structure. In the second part, we give a lower bound of the first non-zero eigenvalue of the 1-form Laplacian o…

Logarithmic connections on principal bundles over normal varieties are studied.

This paper establishes a correspondence between principal bundles and invariant connections.

Ancient Ricci flow solutions on bundles discovered.

New solutions found for $G_2$ system using K3 orbifolds.

Constructs complex structures on bundles, leading to new non-Kähler manifolds.

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor…

The paper provides conditions for factoring equivariant spectral triples in unbounded KK-theory.

We describe principal 3-bundles with adjusted connections using Lie algebras and groupoids.

Let $G$ be a connected reductive complex affine algebraic group and $K\subset G$ a maximal compact subgroup. Let $M$ be a compact complex torus equipped with a flat Kähler structure and $(E_G ,θ)$ a polystable Higgs $G$-bundle on $M$. Take any $C^\infty$ reduction of structure group $E_K \subset E_G$ to the subgroup $K…

Study of $t$-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.

This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifo…

We give a precise formulation of T-duality for Ramond-Ramond fields. This gives a canonical isomorphism between the "geometrically invariant" subgroups of the twisted differential K-theory of certain principal torus bundles. Our result combines topological T-duality with the Buscher rules found in physics.

Study spin structures on compact homogeneous pseudo-Riemannian manifolds.

The relation between nilmanifolds with left-invariant complex structure and iterated principal holomorphic torus bundles is clarified and we give criteria under which deformations in the large are again of such type. As an application we obtain a fairly complete picture in complex dimension three.

We present a new infinite class of near-horizon geometries of degenerate horizons, satisfying Einstein's equations for all odd dimensions greater than five. The symmetry and topology of these solutions is compatible with those of black holes. The simplest examples give horizons of spatial topology S^3xS^2 or the non-tr…

The paper studies self-covers of manifolds and their algebraic structures.

Study on LCK metrics and their existence criteria on complex manifolds.

Motivated by some questions in the path integral approach to (topological) gauge theories, we are led to address the following question: given a smooth map from a manifold $M$ to a compact group $G$, is it possible to smoothly `diagonalize' it, i.e.~conjugate it into a map to a maximal torus $T$ of $G$? We analyze the …

Equivariant T-duality connects bundles with twists.

Study of harmonic maps into principal bundles with applications to magnetic interactions.

Global Double Field Theory is a higher-dimensional generalization of Kaluza-Klein theory.

We prove that the Halperin-Carlsson conjecture holds for any free (Z_2)^m action on a compact manifold whose orbit space is a small cover. In addition, we show that if the total space of a principal (Z_2)^m bundle over a small cover is connected, it must be equivalent to a partial quotient of the corresponding real mom…

Study of semi-principal bundles using group actions and wreath products.

We introduce an elementary way of constructing principal (Z_2)^m-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard (Z_2)^m-actions on closed manifolds for all m>0, and then give a general way to construct all such (Z_2)^m-actions from the orbit space. Some related t…

Introduces generalized principal bundles and connections, linking them to standard gauge theories.

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi-Yau manifolds. For example we prove that given any real-analytic one parameter family of Riemannian metrics $g_t$ on a 3-dimensional manifold $Y$ with volume form independent of $t$ and with a real-analytic family of nowhere vanishin…

Develops n-tuple principal bundles from Lie group actions.

The paper classifies Toda equations for noncompact symmetric spaces and their solutions.

Defines double principal bundles with applications in geometry.