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Register a new userLinear perturbations of metrics with holonomy Spin(7)
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We apply the method of linear perturbations to the case of Spin(7)-structures, showing that the only nontrivial perturbations are those determined by a rank one nilpotent matrix. We consider linear perturbations of the Bryant-Salamon metric on the spin bundle over $S^4$ that retain invariance under the action of Sp(2), showing that the metrics obtained in this way are isometric.
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The total space of the spinor bundle on the four dimensional sphere S^4 is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang-Mills instanton on this eight dimensional gravitational instanton. This is a higher dimensional generalization of (anti-)self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang-Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution, when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.
We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $Amapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Delta^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.
We formulate and study the isometric flow of $mathrm{Spin}(7)$-structures on compact $8$-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of $H$-structures. We then specialise to $H=mathrm{Spin}(7)subsetmathrm{SO}(8)$, obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an $varepsilon$-regularity theorem. Moreover, we prove Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-$mathrm{I}$ singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric $mathrm{Spin}(7)$-structures, based on squares of spinors, which may be of independent interest.
In this paper we study ${rm Spin}(7)$-instantons on asymptotically conical ${rm Spin}(7)$-orbifolds (and manifolds) obtained by filling in certain squashed $3$-Sasakian $7$-manifolds. We construct a $1$-parameter family of explicit ${rm Spin}(7)$-instantons. Taking the parameter to infinity, the family (a) bubbles off an ASD connection in directions transverse to a certain Cayley submanifold $Z$, (b) away from $Z$ smoothly converges to a limit ${rm Spin}(7)$-instanton that extends across $Z$ onto a topologically distinct bundle, (c) satisfies an energy conservation law for the instantons and the bubbles concentrated on $Z$, and (d) determines a Fueter section, in the sense of Donaldson and Segal, Haydys and Walpuski.
This paper generalizes Bismuts equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe along any closed oriented surface, as well as the derivative of 2-holonomy, which in the case of a torus fits into a sequence of higher holonomies and their differentials.
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