arXiv research

No Hantzsche-Wendt manifolds over 3D admit spin^c structures.

Simplified proof of spin^c structures using twistor spaces.

New cobordism invariants derived from BPS q-series.

We show that a homotopy equivalence between manifolds induces a correspondence between their spin^c-structures, even in the presence of 2-torsion. This is proved by generalizing spin^c-structures to Poincare complexes. A procedure is given for explicitly computing the correspondence under reasonable hypotheses.

We characterize certain CR structures of arbitrary codimension (different from 3, 4 and 5) on Riemannian Spin$^c$ manifolds by the existence of a Spin$^c$ structure carrying a strictly partially pure spinor field. Furthermore, we study the geometry of Riemannian Spin$^c$ manifolds carrying a strictly partially pure spi…

We define spin-c prequantization of a symplectic manifold to be a spin-c structure and a connection which are compatible with the symplectic form. We describe the cutting of an S^1-equivariant spin-c prequantization. The cutting process involves a choice of a spin-c prequantization for the complex plane. We prove that …

It is well known that spinors on oriented Riemannian manifolds cannot be defined as sections of a vector bundle associated with the frame bundle. For this reason spin and spin^c structures are often introduced. In this paper we prove that spin^c structures have a universal property among all other structures that enabl…

The Seiberg-Witten equations are lifted to Kaluza-Klein 5-manifolds.

The study finds hyperbolic manifolds without spin^c structures in dimensions 5 and above.

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of…

The study explores psc-metrics on non-spin manifolds with pin^\pm or spin^c structures.

A G-equivariant spin^c structure on a manifold gives rise to a virtual representation of the group G, called the spin^c quantization of the manifold. We present a cutting construction for S^1-equivariant spin^c manifolds, and show that the quantization of the original manifold is isomorphic to the direct sum of the qua…

Study spin structures on Kac-Moody symmetric spaces.

Two Spin^c structures characterize hypersurfaces in product spaces.

Develops spinorial description of CR structures of arbitrary codimension.

We define a `Higgs field' for a four-dimensional spin$^c$-manifold to be a smooth section of its positive half-spinor bundle, transverse to the zero section, and defined only up to a positive functional factor. This is intended to be a generalization of almost complex structures on real four-manifolds, each of which ma…

Quantum invariant constructed for sutured 3-manifolds using Hopf superalgebra.

This paper gives a combinatorial description of spin and spin^c-structures on triangulated PL-manifolds of arbitrary dimension. These formulations of spin and spin^c-structures are established primarily for the purpose of aiding in computations. The novelty of the approach is we rely heavily on the naturality of binary…

Alternative proof and description of orientations for instanton moduli spaces.

New real invariants for 3-manifolds and links.

Simply connected 3-dimensional homogeneous manifolds $E(κ, τ)$, with 4-dimensional isometry group, have a canonical Spin$^c$ structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $E(κ, τ)$. As…

Study String$^c$ structures using algebraic topology and generalize Witten genera.

We construct examples of four dimensional manifolds with Spin$^c$-structures, whose moduli spaces of solutions to the Seiberg-Witten equations, represent a non-trivial bordism class of positive dimension, i.e. the Spin$^c$-structures are not induced by almost complex structures. As an application, we show the existence…

The paper classifies hypersurfaces in Spin$^c$ manifolds that satisfy a specific inequality.

We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus $g \geq 1$ times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-man…

Proves almost flat spin^c manifolds bound compact manifolds.

We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin-c quantization. We prove the analog of Kodaira vanishing for the Spin-c Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semicl…

Proves functoriality in geometric quantization for non-compact spin-c manifolds.

In this paper, we extend the study of generalized Killing spinors on Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin$^c$ Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing that the dimension of t…

Study shows non-vanishing Stiefel-Whitney classes and absence of spin^C structures in certain hyperbolic manifolds.

New rigidity theorems for spin^c manifolds using modular invariance.

The paper describes spectra of operators on rational homogeneous varieties.

Sharp inequality for complex projective space's systoles under scalar curvature bounds.

Study pinching of small Dirac eigenvalues on spin^c manifolds.

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fund…

Let $M$ a compact connected orientable 4-manifold. We study the space $Ξ$ of $Spin^c$-structures of fixed fundamental class, as an infinite dimensional principal bundle on the manifold of riemannian metrics on $M$. In order to study perturbations of the metric in Seiberg-Witten equations, we study the transversality of…

In this note, we look at estimates for the scalar curvature k of a Riemannian manifold M which are related to spin^c Dirac operators: We show that one may not enlarge a Kaehler metric with positive Ricci curvature without making k smaller somewhere on M. We also give explicit upper bounds for min(k) for arbitrary Riema…

The study classifies spin$^c$ manifolds with positive generalized scalar curvature.

Killing spinors help classify surfaces in complex projective plane.

The study explores dimensions for connected sums of almost complex manifolds and extends results to rational homology spheres.

Ozsvath and Szabo construct a spectral sequence with E_2 term Λ^*(H^1(Y;Z))\otimes Z[U,U^{-1}] converging to HF^\infty(Y,s) for a torsion Spin^c structure s. They conjecture that the differentials are completely determined by the integral triple cup product form via a proposed formula. In this paper, we prove that HF^\…

Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure σon M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister-Turaev torsion of (M,σ), while the other one can be defined using the intersectio…

We use Heegaard decompositions and the theta divisor on a Riemannian surface to define a three-manifold invariant for rational homology three-spheres. This invariant is defined on the set of $Spin^c$ structures ${\hat θ}\colon Spin^c(Y)\longrightarrow Q.$ In the first part of the paper, we give the definition of th…

Classifies manifolds with specific spinors and constructs parallel spinors.

Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.

In this paper, we write down Seiberg-Witten equations on contact metric manifolds of dimension 5. Any contact metric manifold has a spin^c structure. For Dirac equation we use Dirac type operators associated to the generalized Tanaka-Webster connection on spin^c spinor bundle of a contact metric manifold. For curvature…

On $Spin^c$ manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a $Spin^c$ manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and po…

The Yamabe Invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using spin^c Dirac o…