arXiv research

Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.

Constructs spin hyperbolic surfaces with a spectral gap for Dirac operator.

Method constrains spectral gaps of hyperbolic spin surfaces using identities and semidefinite programming.

Study asymptotics of Selberg zeta function on spin moduli space.

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 …

New theory maps spin structures on surfaces, generating key elements.

We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with sel…

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…

No spin structures found in a hyperbolic 4D space.

We construct stationary flat three-dimensional Lorentzian manifolds with singularities that are obtained from Euclidean surfaces with cone singularities and closed one-forms on these surfaces. In the application to (2+1)-gravity, these spacetimes correspond to models containing massive particles with spin. We analyse t…

Paper computes G-index for specific hyperbolic manifolds.

In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtl…

Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.

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

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

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

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfie…

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…

This paper connects spinors to horospheres in hyperbolic space.

Constructs chiral rational homology spheres with hyperbolic groups.

Study Dirac operator on cusped hyperbolic manifolds, finding spectrum properties.

We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces …

This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold $(M,g)$ of real dimension $4n+2$ which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. The f…

We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of $\mathbb{Z}_r$-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state sum construction of two-dimensional topolog…

Geometric correspondence between spinors and horospheres in hyperbolic space.

Automorphisms of surfaces extendable over 4-sphere with invariant spin structures.

We prove the rigidity of positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds or under special asymptotics.

We study Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy and show the so-called generalized energy identity in the case that the domain converges to a spin surface with only Neveu-Schwarz type nodes. We find condition that is both necessary and sufficient for the $W^{1,2} \times L^{4}$…

We prove that any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action. Our proof is based on the representation of the space of m-spin structures on a Riemann surface …

We investigate the action of the automorphism group of a closed Riemann surface on its set of theta characteristics (or spin structures). We give criteria for when an automorphism fixes all spin structures, or when it fixes just one. The case of hyperelliptic curves and of the Klein quartic are discussed in detail.

We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…

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

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or…

New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.

Researchers compute spin structures on hyperelliptic curves using braid groups.

New quantum integrals discovered for a spin chain model.

The abstract theorem is extended to higher genus surfaces.

Study eigenvalues of Dirac operator on surfaces, proving existence and deriving inequalities.

Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.

New spinor fields reveal local or global geometric properties of manifolds.

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…

Extends spinor-horosphere correspondence to higher dimensions and new spinor types.

We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these metrics has a simple description in terms of Teichmüller theory. In the hyperbolic s…

Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold Σare r-fold fibrewise connected coverings (also called r-th roots) of its unit tangent bundle STΣ. We investigate such structures on hyperbolic orbifolds. The conditions on r for such structures to exist are given. The action of the diffeomo…

In a previous paper, we showed how certain orientations of the edges of a graph G embedded in a closed oriented surface S can be understood as discrete spin structures on S. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on G. In the pres…

The mass of asymptotically hyperbolic ends and manifolds is analyzed.

We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to products of hyperbolic space with a compact manifold. As conclusion we show the Yamabe inequality for some noncompact manifolds which are important to understand the behaviour of Yamabe invariants under surgeries.

Let $M$ be an oriented closed 4-manifold and $\cL$ be a $spin^c$ structure on $M$. In this paper we prove that under a suitable condition the Seiberg-Witten moduli space has a canonical spin structure and its spin bordism class is an invariant for $M$. We show that the invariant for $M=#_{j=1}^l M_j$ is not zero, where…