Study of higher spin bundles on Klein surfaces, linking to singularity theory.
problem Understanding the structure of spaces of higher spin bundles on Klein surfaces.
method Analysis of topological invariants and quotient spaces of Euclidean spaces.
result Connected components of the space of higher spin bundles are homeomorphic to quotients of Euclidean spaces.
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
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…
Formulas for spectra of higher spin operators on sphere subbundles.
problem Finding spectra of higher spin operators on specific subbundles of spinor-valued tensors.
method Explicit formulas derived for spectra in both even and odd dimensions.
result Spectra formulas for higher spin operators and their squares.
We study the higher spin Dirac operators on 3-dimensional manifolds and show that there exist two Laplace type operators for each associated bundle. Furthermore, we give lower bound estimations for the first eigenvalues of these Laplace type operators.
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.
Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.
problem Defining and rigorously studying higher differential geometric objects like stringor bundles.
method Developed a framework of 2-Hilbert bundles, including an associated bundle construction.
result Proves the Stolz-Teichner stringor bundle is canonically isomorphic to the 2-Hilbert bundle.
The paper studies rational higher tangential structures from Lie groups, revealing systematic effects of variations.
problem Understanding higher tangential structures at the rational level and their connections to geometry.
method Defining and studying Spin-Fivebrane and Spin-Ninebrane structures, analyzing their spaces and gauge groups.
result Variations of higher structures systematically involve lower ones, with interesting effects revealed.
New Spin(7) manifolds created from Calabi-Yau bundles.
problem Creating complete non-compact Spin(7) manifolds.
method Building on T2-bundles over AC Calabi-Yau manifolds. result First known examples of complete toric Spin(7) manifolds.
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…
New geometric obstructions found for positive scalar curvature on fiber bundles over aspherical manifolds.
problem Obstructing positive scalar curvature on fiber bundles over aspherical manifolds.
method Using higher indices of the spinor Dirac operator and a new variant of the multi-partitioned manifold index theorem.
result Rosenberg index is an obstruction to positive scalar curvature on M if N↪MwoheadrightarrowB is a fiber bundle of spin manifolds with B aspherical and π1(B) of finite asymptotic dimension. Constructs small bundle gerbes and proves index theorems for manifolds.
problem Constructing and analyzing bundle gerbes on manifolds.
method Defines and constructs small bundle gerbes, uses pseudodifferential and semiclassical smoothing operators, proves index theorems.
result Proves the Atiyah-Singer type theorem for small bundle gerbes, showing their relation to twisted K-theory.
Real vector bundles are determined by their Dirac indices on specific spin manifolds.
problem Determining real vector bundles using Dirac indices.
method Mapping spin or spinh manifolds into a compact smooth manifold and using Dirac indices. result Real vector bundles are uniquely determined by their Dirac indices on prescribed manifolds.
Constructs Spin(7) manifolds from self-dual Einstein 4-orbifolds.
problem Creating complete non-compact Spin(7) manifolds.
method Analytic construction using adiabatic limits of Spin(7) metrics on Seifert bundles over G2 orbifolds.
result Produces complete non-compact Spin(7) manifolds with arbitrary second Betti numbers and infinitely many families of ALC metrics.
We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their …
The paper decomposes and analyzes the higher spin Laplace operator.
problem Understanding the properties and solutions of the higher spin Laplace operator.
method Decomposition into Rrita-Schwinger operators, proving conformal invariance, establishing integral formulas.
result Established a Borel-Pompeiu type formula and a Green type integral formula for the higher spin Laplace operator.
We establish the cancellation of the first 2j terms in the diagonal asymptotic expansion of the restriction to the (0,2j)-forms of the Bergman kernel associated to the spinc Dirac operator on high tensor powers of a positive line bundle twisted by a (non necessarily holomorphic) complex vector bundle, over a c…
The paper classifies higher order fermionic and bosonic operators in spin spaces.
problem Understanding higher order conformally invariant differential operators.
method Using higher spin theory in Clifford analysis, constructing operators as generalizations of the Euclidean Dirac operator.
result These operators act on functions in homogeneous harmonic or monogenic polynomial spaces, and are classified as fermionic or bosonic based on spin.
Motivated by the description of N=1 M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional manifold equipped with a torsion-free structure. As a n…
Geometrically computes cohomotopy groups for higher-dimensional manifolds.
problem Computing cohomotopy groups for higher-dimensional manifolds.
method Geometric arguments and introduction of new manifold types.
result Refined Euler class and obstruction to non-vanishing sections.
Computes conditions for real spinor bundles on pseudo-Riemannian manifolds.
problem Obtaining Dirac operators on real spinor bundles of complex type.
method Using Lipschitz structures and Karoubi Stiefel-Whitney classes, reformulating the problem in terms of Spin^o_α structures.
result Explicit construction and characterization of real spinor bundles of irreducible complex type.
Study pinching of small Dirac eigenvalues on spin^c manifolds.
problem Pinching small Dirac eigenvalues on spin^c manifolds.
method Classification of spin^c and spin manifolds with nontrivial Killing spinors, introducing convergence for spin^c manifolds involving principal S^1-bundles.
result Derive pinching results for small Dirac eigenvalues.
What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and …
We establish the cancellation of the first |2j-q| terms in the diagonal asymptotic expansion of the restriction to the (0, 2j)-forms of the Bergman kernel associated to the modified spin^c Dirac operator on high tensor powers of a line bundle with mixed curvature twisted by a (non necessarily holomorphic) complex vecto…
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…
Explains conformal maps and holomorphic structures for surfaces in higher-dimensional Euclidean spaces.
problem Explains conformal maps and holomorphic structures for surfaces in higher-dimensional Euclidean spaces.
method Uses Clifford algebra and spin transforms to explain conformal maps and holomorphic structures.
result Calculates the degree of the spinor bundle associated with a conformal immersion and defines analogues of polar and bipolar surfaces.
The paper extends higher-order operators in higher spin spaces.
problem Generalizing higher-order operators in higher spin spaces.
method Constructing 3rd order fermionic and 4th order bosonic operators in higher spin spaces.
result Fundamental solutions and intertwining operators of 3rd order fermionic and 4th order bosonic operators are presented.
We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements of infinite order in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (o…
Let M be a closed spin manifold of dimension at least three with a fixed topological spin structure. For any Riemannian metric, we can construct the associated Dirac operator. The spectrum of this Dirac operator depends on the metric of course. In 2005, Dahl conjectured that M can be given a metric, for which a finite …
New spin frame transformations affect Dirac equations without preserving metric structures.
problem Extending spin structures to spin manifolds with fixed signature.
method Defining spin frames and studying their effects on connections and Dirac equations.
result New transformations affect Dirac equations more generally than usual spin transformations.
Study 3d N=1 vacua from M-theory compactification on Spin(7) space.
problem Quantum corrections in 3d N=1 vacua from M-theory compactification.
method Use Higgs bundles to analyze 3d N=1 vacua and track corrections.
result Topological anomalies are robust and calculable in 3d effective field theory.
This work takes place over a conformally flat spin manifold (M,g). We prove existence and uniqueness of the conformally equivariant quantization valued in spinor differential operators, and provide an explicit formula for it when restricted to first order operators. The Poisson algebra of symbols is realized as a space…
Formula for relative Chern character number on spin manifolds.
problem Computing relative Chern characteristic numbers on complex vector bundles.
method Formula derivation for homomorphisms between vector bundles over odd-dimensional spheres.
result Formula for index of twisted Dirac operators 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…
In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structure…
The paper classifies vector bundles over spin 5-manifolds and introduces a splitting invariant.
problem Classifying vector bundles over spin 5-manifolds and understanding their cohomotopy.
method Study quaternionic line bundles and their correspondence to elements in the first cohomotopy group.
result Provides a bordism theoretic splitting map for the short exact sequence of cohomotopy groups.
Study on decay rates of higher derivatives for nonlinear Dirac equations.
problem Estimating decay rates of higher derivatives of solutions to nonlinear Dirac equations.
method Similar to Li and Zang's method, focusing on 'good' spin null form.
result Obtained decay rates of higher derivatives of solutions.
A new spin structure is constructed for a bundle of harmonic forms.
problem Constructing a spin structure for a bundle of harmonic forms.
method Using families Seiberg-Witten equations to construct a lifting O(1)-gerbe.
result A canonical spin structure is constructed for the bundle TBX⊕π∗H+(X). First non-trivial examples of deformed Spin(7)-instantons constructed.
problem Constructing deformed Spin(7)-instantons and connections.
method Constructing on cotangent bundles of CP2 and cones over 3-Sasakian 7-manifolds. result First non-trivial examples of deformed Spin(7)-instantons.
The paper constructs arbitrary order conformally invariant operators in higher spin spaces.
problem Classifying and constructing conformally invariant differential operators in higher spin spaces.
method Explicit expressions and convolution type operators, intertwining operators, and representation theory.
result Explicit expressions and properties of conformally invariant differential operators in higher spin spaces.
The paper calculates determinants for Laplacians on spinor bundles over surfaces with flat metrics.
problem Calculating determinants for Laplacians on spinor bundles over surfaces with flat metrics.
method Explicit expressions for determinants of self-adjoint extensions of Laplacians using Bergman tau-function and theta-constants.
result An explicit expression for the determinant of the Szegö extension and comparison formulas for different extensions.
We study 4-dimensional higher-derivative conformal higher spin (CHS) fields generalising Weyl graviton and conformal gravitino. They appear, in particular, as "induced" theories in the AdS/CFT context. We consider their partition function on curved Einstein-space backgrounds like (A)dS or sphere and Ricci-flat spaces. …
The paper classifies obstructions to real Clifford module bundles and their relation to spin structures.
problem Topological obstructions to real Clifford module bundles over pseudo-Riemannian manifolds.
method Analyzes real Lipschitz structures and their relation to spin structures.
result Classifies real Lipschitz structures in all dimensions and signatures.
Paper finds first hyperbolic 4-manifold with harmonic spinors.
problem Finding harmonic spinors on hyperbolic 4-manifolds.
method Using G-spin theorem and explicit calculations for spin hyperbolic manifolds.
result First example of a closed hyperbolic 4-manifold with harmonic spinors.
The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geome…
Unified Higgs bundle vacua from M-theory on Spin(7) spaces.
problem Unifying Higgs bundle vacua from different string compactifications.
method Developed formalism for M-theory on local Spin(7) spaces and constructed explicit solutions.
result Unified 3D effective field theory from 4D M- and F-theory vacua.
Study higher order fermionic and bosonic operators on cylinders and Hopf manifolds.
problem Understanding higher order higher spin operators on specific manifolds.
method Analysis of higher order operators on cylinders and Hopf manifolds.
result Construction of kernels for these operators on the studied manifolds.
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.