Riemannian Spin Manifold Research Articles

A Chern-Simons transgression formula for supersymmetric path integrals on spin manifolds

Earlier results show that the N=1/2 supersymmetric path integral Jg on a closed even dimensional Riemannian spin manifold (X,g) can be constructed in a mathematically rigorous way via Chen differential forms and techniques from noncommutative geometry, if one considers Jg as a current on the loop space LX, that is, as a linear form on differential forms on LX. This construction admits a Duistermaat-Heckman localization formula. In this note, fixing a topological spin structure on X, we prove that any smooth family g•=(gt)t∈[0,1] of Riemannian metrics on X canonically induces a Chern-Simons current Cg• which fits into a transgression formula for the supersymmetric path integral. In particular, this result entails that the supersymmetric path integral induces a differential topological invariant on X, which essentially stems from the Aˆ-genus of X.

The J-twist DJ of the Dirac operator and the Kastler–Kalau–Walze type theorem for six-dimensional manifolds with boundary

In [S. Liu and Y. Wang, A Kastler–Kalau–Walze type theorem for the J-twist DJ of the Dirac operator, preprint (2022), arXiv:2203.10467], the authors proved a Kastler–Kalau–Walze type theorem for the [Formula: see text]-twist [Formula: see text] of the Dirac operator on [Formula: see text]-dimensional and [Formula: see text]-dimensional almost product Riemannian spin manifold with boundary. In this paper, we develop the Kastler–Kalau–Walze type theorem for the [Formula: see text]-twist [Formula: see text] of the Dirac operator on a [Formula: see text]-dimensional almost product Riemannian spin manifold with boundary.

A Kastler–Kalau–Walze Type Theorem for the J-Twist D_{J} of the Dirac Operator

In this paper, we give a Lichnerowicz type formula for the J-twist D_{J} of the Dirac operator. And we prove a Kastler–Kalau–Walze type theorem for the J-twist D_{J} of the Dirac operator on three-dimensional and four-dimensional almost product Riemannian spin manifold with boundary.

A short proof of the localization formula for the loop space Chern character of spin manifolds

In this note, we give a short proof of the localization formula for the loop space Chern character of a compact Riemannian spin manifold M , using the rescaled spinor bundle on the tangent groupoid associated to M .

The mass of an asymptotically hyperbolic end and distance estimates

Let (M, g) be a completely connected n-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies Rg ≥ −n(n − 1), and let E⊂M be an asymptotically hyperbolic end. We prove that the mass functional of the end E is timelike future-directed or zero. Moreover, it vanishes if and only if (M, g) is isometrically diffeomorphic to the hyperbolic space. We also consider the mass of an asymptotically hyperbolic manifold with a compact boundary, and we prove that the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by a function defined by using distance estimates. For applications, the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by −(n − 1) or the scalar curvature satisfies Rg ≥ (−1 + κ)n(n − 1) for any positive constant κ less than one.

First Boundary Dirac Eigenvalue and Boundary Capacity Potential

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface \(\Sigma \) bounding a noncompact domain in a spin asymptotically flat manifold \((M^n,g)\) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of \(\Sigma \) in \((M^n,g)\) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest.

Feynman–Kac formula for perturbations of order le 1, and noncommutative geometry

Let Q be a differential operator of order le 1 on a complex metric vector bundle mathscr {E}rightarrow mathscr {M} with metric connection nabla over a possibly noncompact Riemannian manifold mathscr {M}. Under very mild regularity assumptions on Q that guarantee that nabla ^{dagger }nabla /2+Q canonically induces a holomorphic semigroup mathrm {e}^{-zH^{nabla }_{Q}} in Gamma _{L^2}(mathscr {M},mathscr {E}) (where z runs through a complex sector which contains [0,infty )), we prove an explicit Feynman–Kac type formula for mathrm {e}^{-tH^{nabla }_{Q}}, t>0, generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact mathscr {M}’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form TrV~∫0te-sHV∇Pe-(t-s)HV∇ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathrm {Tr}\\left( \\widetilde{V}\\int ^t_0\\mathrm {e}^{-sH^{\ abla }_{V}}P\\mathrm {e}^{-(t-s)H^{\ abla }_{V}}\\mathrm {d}s\\right) , \\end{aligned}$$\\end{document}where V,widetilde{V} are of zeroth order and P is of order le 1. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.

Curvature effect in the spinorial Yamabe problem on product manifolds

Let $(M_1,\textit{g}^{(1)})$, $(M_2,\textit{g}^{(2)})$ be closed Riemannian spin manifolds. We study the existence of solutions of the spinorial Yamabe problem on the product $M_1\times M_2$ equipped with a family of metrics $\varepsilon^{-2}\textit{g}^{(1)}\oplus\textit{g}^{(2)}$, $\varepsilon>0$. Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor $M_1$, and which exhibit a spike layer as $\varepsilon\to0$. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $(M_1,\textit{g}^{(1)})$.

The spinor and tensor fields with higher spin on spaces of constant curvature

In this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin j+1/2 over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power j+1 and understand how the spinor fields with spin j+1/2 are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge–de Rham decomposition on spaces of constant curvature.

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.

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.

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah–Singer Dirac operator / D B in L 2 depends Riesz continuously on L ∞ perturbations of local boundary conditions B . The Lipschitz bound for the map B → / D B ( 1 + / D B 2 ) − 1 2 depends on Lipschitz smoothness and ellipticity of B and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

Comparison between two differential graded algebras in noncommutative geometry

Starting with a spectral triple, one can associate two canonical differential graded algebras (DGA) defined by Connes (Noncommutative geometry (1994) Academic Press Inc., San Diego) and Frohlich et al. (Comm. Math. Phys. 203(1) (1999) 119–184). For the classical spectral triples associated with compact Riemannian spin manifolds, both these DGAs coincide with the de-Rham DGA. Therefore, both are candidates for the noncommutative space of differential forms. Here we compare these two DGAs in a very precise sense.

The Kernel of the Rarita–Schwinger Operator on Riemannian Spin Manifolds

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion K\"ahler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperk\"ahler, $G_2$ and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields.

Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac equations with such boundary conditions. We also apply these estimates to a new nonlinear elliptic-parabolic problem, the Dirac-harmonic heat flow on Riemannian spin manifolds. This problem is motivated by the supersymmetric nonlinear \sigma -model and combines a harmonic heat flow type equation with a Dirac equation that depends nonlinearly on the flow.

Large scale index of multi-partitioned manifolds

Let M be a complete n-dimensional Riemannian spin manifold, partitioned by q two-sided hypersurfaces which have a compact transverse intersection N and which in addition satisfy a certain coarse transversality condition. Let E be a Hermitean bundle on M with connection. We define a coarse multi-partitioned index of the spin Dirac operator on M twisted by E . Our main result is the computation of this multi-partitioned index as the Fredholm index of the Dirac operator on the compact manifold N , twisted by the restriction of E to N . We establish the following main application: if the scalar curvature of M is bounded below by a positive constant everywhere (or even if this happens only on one of the quadrants defined by the partitioning hypersurfaces) then the multi-partitioned index vanishes. Consequently, the multi-partitioned index is an obstruction to uniformly positive scalar curvature on M . The proof of the multi-partitioned index theorem proceeds in two steps: first we establish a strong new localization property of the multi-partitioned index which is the main novelty of this note. This we establish even when we twist with an arbitrary Hilbert A -module bundle E (for an auxiliary C^* -algebra A ). This allows to reduce to the case where M is the product of N with Euclidean space of dimension q . For this special case, standard methods for the explicit calculation of the index in this product situation can be adapted to obtain the result.

Spacetimes in Noncommutative Geometry: a definition and some examples

We quickly review the notion of spectral triples - the noncommutative generalization of Riemannian spin manifolds devised by Alain Connes - and propose extensions of it to the Lorentzian and anti-Lorentzian signatures, which we call “spectral spacetimes”. We motivate this definition and give several examples.

Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.

A New $$\frac{1}{2}$$ 1 2 -Ricci Type Formula on the Spinor Bundle and Applications

Consider a Riemannian spin manifold $$(M^{n}, g)$$ $$(n\ge 3)$$ endowed with a non-trivial 3-form $$T\in \Lambda ^{3}T^{*}M$$ , such that $$\nabla ^{c}T=0$$ , where $$\nabla ^{c}:=\nabla ^{g}+\frac{1}{2}T$$ is the metric connection with skew-torsion T. In this note we introduce a generalized $$\frac{1}{2}$$ -Ricci type formula for the spinorial action of the Ricci endomorphism $${{\mathrm{Ric}}}^{s}(X)$$ , induced by the one-parameter family of metric connections $$\nabla ^{s}:=\nabla ^{g}+2sT$$ . This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schrodinger–Lichnerowicz formula related to the square of the Dirac operator $$D^{s}$$ , induced by $$\nabla ^{s}$$ , under the condition $$\nabla ^{c}T=0$$ . In the same case, we provide integrability conditions for $$\nabla ^{s}$$ -parallel spinors, $$\nabla ^{c}$$ -parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly Kahler manifolds and nearly parallel $$\hbox {G}_2$$ -manifolds, in dimensions 5, 6 and 7, respectively.