arXiv research

Real vector bundles are determined by their Dirac indices on specific spin manifolds.

Computes conditions for real spinor bundles on pseudo-Riemannian manifolds.

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…

New spin frame transformations affect Dirac equations without preserving metric structures.

We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin …

Formula for relative Chern character number on spin manifolds.

We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural only in this extended sense. We classify fibre bundles with this property, assu…

The paper classifies vector bundles over spin 5-manifolds and introduces a splitting invariant.

New Spin(7) manifolds created from Calabi-Yau bundles.

A new spin structure is constructed for a bundle of harmonic forms.

First non-trivial examples of deformed Spin(7)-instantons constructed.

We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds which involves a general study on convergence of Riemannian manifolds with a pr…

Paper finds first hyperbolic 4-manifold with harmonic spinors.

Unified Higgs bundle vacua from M-theory on Spin(7) spaces.

Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined.

Extended spinor connections associated with composite spin-tensorial bundles are considered. Commutation relationships for covariant and multivariate differentiations and corresponding curvature spin-tensors are derived.

We demonstrate that it is conceptually and computationally favorable to regard spin-weighted spherical harmonics as vector valued functions on the total space $SO(3)$ of the Hopf bundle, satisfying a covariance condition with respect to the gauge group $U(1)$ of this bundle. A key role is played by the invariant connec…

New formulas derived for anomaly cancellation using modular forms and E8 bundles.

Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.

Eta invariant computed for circle bundles over Fano manifolds.

The paper extends Nahm transforms to Spin(7) structures on tori, finding obstructions.

We find all m-spin structures on Klein surfaces of genus larger than one. An m-spin structure on a Riemann surface P is a complex line bundle on P whose m-th tensor power is the cotangent bundle of P. A Klein surface can be described by a pair (P,tau), where P is a Riemann surface and tau is an anti-holomorphic involut…

We give a method of decomposing bundle-valued polynomials compatible with the action of the Lie group $Spin(n)$, where important tools are $Spin(n)$-equivariant operators and their spectral decompositions. In particular, the top irreducible component is realized as an intersection of kernels of these operators.

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

Introduces spin structures and orientations for gauge-theoretic moduli spaces.

Functional for Spin(7) forms defined on compact manifolds.

Let $M_{n}\stackrel{\mathbb R P^1}\to M_{n-1}\stackrel{\mathbb R P^1}\to\ldots\stackrel{\mathbb R P^1}\to M_{1}\stackrel{\mathbb R P^1}\to M_0 = \{ \bullet\}$ be a sequence of real projective bundles such that $M_i\to M_{i-1}$, $i=1,2,\ldots,n$, is a projective bundle of a Whitney sum of a real line bundle $L_{i-1}$…

Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure …

The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a $Spin(9)$-reduction of the frame …

The study examines conditions for high-dimensional Lorentzian cobordisms with specific structure groups.

In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role i…

A Dirac structure on a vector bundle V is a maximal isotropic subbundle E of the direct sum of V with its dual. We show how to associate to any Dirac structure a Dixmier-Douady bundle A, that is, a Z/2Z-graded bundle of C*-algebras with typical fiber the compact operators on a Hilbert space. The construction has good f…

We present a generalization of the Clifford action for other representations spaces of $Spin(n)$, which is called the Clifford homomorphism. Their properties extend to the ones for the higher spin Dirac operators on spin manifolds. In particular, we have general Bochner identities for them, and an eigenvalue estimate o…

Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.

We study the asymptotic of the spectrum of the \spin Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin-Uribe, which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin.

Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.

We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor c…

We show that the Atiyah-Patodi-Singer reduced $η$-invariant of the twisted Dirac operator on a closed $4m-1$ dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight $2m$ up to an integral $q$-series. We prove this resu…

Short proof given for a complex formula in spin manifold geometry.

Study ${ m Spin}(7)$-dDT connections on manifolds with ${ m Spin}(7)$-structures.

Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.

Formulas for spectra of higher spin operators on sphere subbundles.

The paper studies quotient spaces of Spin(7) manifolds by S^1 actions and their G2 structures.

Paper extends gauge theory results to homology tori, preserving spin structure obstructions.

The paper extends Strichartz's conjecture to spinor bundles over real hyperbolic spaces.

Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.

We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over th…

Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.