Spin Manifolds Research Articles

Killing superalgebras for lorentzian five-manifolds

We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an mathfrak{sp} (1)-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric theories on curved backgrounds. We define the Killing superalgebra of bosonic backgrounds and show that their existence is implied by the field equations. The maximally supersymmetric backgrounds are characterised and their Killing superalgebras are explicitly described as filtered deformations of the Poincaré superalgebra.

Parallel spinors on Lorentzian Weyl spaces

Holonomy algebras of Lorentzian Weyl spin manifolds with weighted parallel spinors are found. For Lorentzian Weyl manifolds admitting recurrent null vector fields are introduced special local coordinates similar to Kundt and Walker ones. Using that, the local form of all Lorentzian Weyl spin manifolds with weighted parallel spinors is given. The Einstein-Weyl equation for the obtained Weyl structures is analyzed and examples of Einstein-Weyl spaces with weighted parallel spinors are given.

The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.

A spinorial proof of the rigidity of the Riemannian Schwarzschild manifold

We revisit and generalize a recent result of Cederbaum (Cederbaum in preparation; Cederbaum 2017 Oberwolfach Report) concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems (Bunting and Masood-ul-Alam 1987 Gen. Relativ. Gravit. 19 147; Gibbons, Ida and Shiromizu 2002 Phys. Rev. Lett. 89 041101; Hwang 1998 Geom. Dedicata 71 5) as well as the more recent uniqueness theorems for photon spheres (Cederbaum 2015 Contemp. Math. 667 86; Cederbaum and Galloway 2017 Commun. Anal. Geom. 25 303; Cederbaum and Fehrenbach personal communication).

A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent

Let (M,g,σ) be a compact Riemannian spin manifold of dimension m≥2, let S(M) denote the spinor bundle on M, and let D be the Atiyah-Singer Dirac operator acting on spinors ψ:M→S(M). We study the existence of solutions of the nonlinear Dirac equation with critical exponent(NLD)Dψ=λψ+f(|ψ|)ψ+|ψ|2m−1ψ where λ∈R and f(|ψ|)ψ is a subcritical nonlinearity in the sense that f(s)=o(s2m−1) as s→∞. A model nonlinearity is f(s)=αsp−2 with 2<p<2mm−1, α∈R. In particular we study the nonlinear Dirac equation(BND)Dψ=λψ+|ψ|2m−1ψ,λ∈R. This equation is a spinorial analogue of the Brezis-Nirenberg problem. As corollary of our main results we obtain the existence of least energy solutions (λ,ψ) of (BND) for every λ>0, even if λ is an eigenvalue of D. For some classes of nonlinearities f we also obtain solutions of (NLD) for every λ∈R, except for non-positive eigenvalues. If m≢3 (mod 4) we obtain solutions of (BND) for every λ∈R, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it.The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD).

Skew Killing spinors in four dimensions

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor psi is a spinor that satisfies the equation nabla _Xpsi =AXcdot psi with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product {mathbb {R}}times N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to {mathbb {S}}^2times {mathbb {R}}^2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.

Complete noncompact Spin(7) manifolds from self-dual Einstein 4–orbifolds

We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperk\"ahler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete non-compact Spin(7) manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7) metrics on the same smooth 8-manifold.

An invariant related to the existence of conformally compact Einstein fillings

We define an invariant for compact spin manifolds X X of dimension 4 k 4k equipped with a metric h h of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of h h to be the conformal infinity of a conformally compact Einstein metric on X X .

An analytical approach to the equivariant index and Witten genus on spin manifolds

This work is divided in two cases. In the first case, we consider a spin manifold M as the set of fixed points of an S1-action on a spin manifold X, and in the second case we consider the spin manifold M as the set of fixed points of an S1-action on the loop space of M. For each case, we build on M a vector bundle, a connection and a set of bundle endomorphisms. These objects are used to build global operators on M which define an analytical index in each case. In the first case, the analytical index is equal to the topological equivariant Atiyah–Singer index, and in the second case the analytical index is equal to a topological expression where the Witten genus appears.

Spin[formula omitted] and further generalisations of spin

The question of which manifolds are spin or spinc has a simple and complete answer. In this paper we address the same question for spinh manifolds, which are less studied but have appeared in geometry and physics in recent decades. We determine that the first obstruction to being spinh is the fifth integral Stiefel–Whitney class W5. Moreover, we show that every orientable manifold of dimension 7 and lower is spinh, and that there are orientable manifolds which are not spinh in all higher dimensions. We are then led to consider an infinite sequence of generalised spin structures. In doing so, we show that there is no integer k such that every manifold embeds in a spin manifold with codimension k.

Functoriality for higher rho invariants of elliptic operators

Let N be a closed spin manifold with positive scalar curvature and DN the Dirac operator on N. Let M1 and M2 be two Galois covers of N such that M2 is a quotient of M1. Then the quotient map from M1 to M2 naturally induces maps between the geometric C⁎-algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the maximal higher rho invariants of the lifts of DN to M1 and M2 behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.

Gravity, Robertson–Walker metric and the Wodzicki residue

In this paper, for lower dimensional spin manifolds with boundary, we prove the Kastler–Kalau–Walze type theorems for the Dirac operator associated with conformal Robertson–Walker metric. We find the gravitational action for manifolds with boundaries, which is associated with asymmetric operators, and yields the boundary term.

Fiber Floer cohomology and conormal stops

Let $S$ be a closed orientable spin manifold. Let $K \subset S$ be a submanifold and denote its complement by $M_K$. In this paper we prove that there exists an isomorphism between partially wrapped Floer cochains of a cotangent fiber stopped by the unit conormal $\varLambda_K$ and chains of a Morse theoretic model of the based loop space of $M_K$, which intertwines the $A_\infty$-structure with the Pontryagin product. As an application, we restrict to codimension 2 spheres $K \subset S^n$ where $n = 5$ or $n\geq 7$. Then we show that there is a family of knots $K$ so that the partially wrapped Floer cohomology of a cotangent fiber is related to the Alexander invariant of $K$. A consequence of this relation is that the link $\varLambda_K \cup \varLambda_x$ is not Legendrian isotopic to $\varLambda_{\mathrm{unknot}} \cup \varLambda_x$ where $x\in M_K$.

Transfer maps in generalized group homology via submanifolds

Let N \subset M be a submanifold embedding of spin manifolds of some codimension k \geq 1 . A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that M does not admit a metric of positive scalar curvature if k = 2 and the Dirac operator of N has non-trivial index, provided that suitable geometric conditions on N \subset M are satisfied. In the cases k=1 and k=2 , Zeidler and Kubota, respectively, established more systematic results: There exists a transfer \text{KO}_\ast(\text{C}^{\ast} \pi_1 M)\to \text{KO}_{\ast - k}(\text{C}^\ast \pi_1 N) which maps the index class of M to the index class of N . The main goal of this article is to construct analogous transfer maps E_\ast(\text{B}\pi_1M) \to E_{\ast-k}(\text{B}\pi_1N) for different generalized homology theories E and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer E_\ast(M) \to E_{\ast-k}(N) induced by the inclusion N \subset M for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for k \leq 3 . For k \leq 2 , we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

Positive Scalar Curvature due to the Cokernel of the Classifying Map

This paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold of dimension $\ge 5$ which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics over $M$ up to bordism in terms of the corank of the canonical map $KO_*(M)\to KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for $\pi_1(M)$.

Width, Largeness and Index Theory

In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands M×[−1,1], and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on M×R. We show that there is a more general geometric statement underlying both of them implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on M×R if the scalar curvature is positive in some neighborhood. We study (A hat-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties and width.

Overlap-Driven Splitting of Triplet Pairs in Singlet Fission.

We analyze correlated-triplet-pair (TT) singlet-fission intermediates toward two-triplet separation (T...T) using spin-state-averaged density matrix renormalization group electronic-structure calculations. Specifically, we compare the triplet-triplet exchange (J) for tetracene dimers, bipentacene, a subunit of the benzodithiophene-thiophene dioxide polymer, and a carotenoid (neurosporene). Exchange-split energy gaps of J and 3J separate a singlet from a triplet and a singlet from a quintet, respectively. We draw two new insights: (a) the canonical tetracene singlet-fission unit cell supports precisely three low-lying TT intermediates with order-of-magnitude differences in J, and (b) the separable TT intermediate in carotenoids emanates from a pair of excitations to the second triplet state. Therefore, unlike with tetracenes, carotenoid fission requires above-gap excitations. In all cases, the distinguishability of the molecular triplets-that is, the extent of orbital overlap-determines the splitting within the spin manifold of TT states. Consequently, J represents a spectroscopic observable that distnguishes the resemblance between TT intermediates and the T...T product.

Spin Quintet in a Silicon Double Quantum Dot: Spin Blockade and Relaxation

Spins in gate-defined silicon quantum dots are promising candidates for implementing large-scale quantum computing. To read the spin state of these qubits, the mechanism that has provided the highest fidelity is spin-to-charge conversion via singlet-triplet spin blockade, which can be detected in-situ using gate-based dispersive sensing. In systems with a complex energy spectrum, like silicon quantum dots, accurately identifying when singlet-triplet blockade occurs is hence of major importance for scalable qubit readout. In this work, we present a description of spin blockade physics in a tunnel-coupled silicon double quantum dot defined in the corners of a split-gate transistor. Using gate-based magnetospectroscopy, we report successive steps of spin blockade and spin blockade lifting involving spin states with total spin angular momentum up to $S=3$. More particularly, we report the formation of a hybridized spin quintet state and show triplet-quintet and quintet-septet spin blockade. This enables studies of the quintet relaxation dynamics from which we find $T_1 \sim 4 ~\mu s$. Finally, we develop a quantum capacitance model that can be applied generally to reconstruct the energy spectrum of a double quantum dot including the spin-dependent tunnel couplings and the energy splitting between different spin manifolds. Our results open for the possibility of using Si CMOS quantum dots as a tuneable platform for studying high-spin systems.

Non-negative versus positive scalar curvature

In this note, we look at the difference, or rather the absence of a difference, between the space of metrics of positive scalar curvature and metrics of non-negative scalar curvature. The main tool to analyze the former on a spin manifold is the spectral theory of the Dirac operator and refinements thereof. This can be used, for example, to distinguish between path components in the space of positive scalar curvature metrics. Despite the fact that non-negative scalar curvature a priori does not have the same spectral implications as positive scalar curvature, we show that all invariants based on the Dirac operator extend over the bigger space. Under mild conditions we show that the inclusion of the space of metrics of positive scalar curvature into that of non-negative scalar curvature is a weak homotopy equivalence.

A long neck principle for Riemannian spin manifolds with positive scalar curvature

We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if {{,mathrm{scal},}}(X)ge n(n-1) and there is a nonzero degree map into the sphere f:Xrightarrow S^n which is strictly area decreasing, then the distance between the support of text {d}f and the boundary of X is at most pi /n. This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if {{,mathrm{scal},}}(X)>sigma >0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of partial X is at most pi sqrt{(n-1)/(nsigma )}. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to Ntimes [-1,1], with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if {{,mathrm{scal},}}(V)ge sigma >0, then the distance between the boundary components of V is at most 2pi sqrt{(n-1)/(nsigma )}. This last constant is sharp by an argument due to Gromov.