arXiv research

Survey of G2-holonomy manifold constructions.

The paper studies the geometry of flat manifolds with controlled holonomy.

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge.

We find G2-manifolds with specific asymptotic properties.

The paper extends Nahm transforms to Spin(7) structures on tori, finding obstructions.

This is the sequel to "Asymptotically Locally Euclidean metrics with holonomy SU(m)", math.AG/9905041. Let G be a subgroup of U(m), and X a resolution of C^m/G. We define a special class of Kahler metrics g on X called Quasi Asymptotically Locally Euclidean (QALE) metrics. These satisfy a complicated asymptotic conditi…

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU…

We construct examples of asymptotically cylindrical Riemannian 8-manifolds with holonomy group Spin(7). To our knowledge, these are the first such examples. The construction uses an extension to asymptotically cylindrical setting of Joyce's existence result for torsion-free Spin(7)-structures. One source of examples ar…

We construct examples of exponentially asymptotically cylindrical Riemannian 7-manifolds with holonomy group equal to G_2. To our knowledge, these are the first such examples. We also obtain exponentially asymptotically cylindrical coassociative calibrated submanifolds. Finally, we apply our results to show that one of…

The paper studies the asymptotic behavior of twisted Alexander polynomials for hyperbolic knots and manifolds, linking them to volume.

We give a construction of $G_2$ and $Spin(7)$ instantons on exceptional holonomy manifolds constructed by Bryant and Salamon, by using an ansatz of spherical symmetry coming from the manifolds being the total spaces of rank-4 vector bundles. In the $G_2$ case, we show that, in the asymptotically conical model, the conn…

Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.

Let G be a nontrivial finite subgroup of U(m) acting freely on C^m - 0. Then C^m/G has an isolated quotient singularity at 0. Let X be a resolution of C^m/G, and g a Kahler metric on X. We say that g is Asymptotically Locally Euclidean (ALE) if it is asymptotic in a certain way to the Euclidean metric on C^m/G. In this…

Sharp decay estimate for Yang-Mills-Higgs fields near singular points.

The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.

The study reveals a persistent bias in the distribution of holonomy on compact hyperbolic 3-manifolds.

Constructs new coassociative fibrations for G2 manifolds.

Random square-tiled surfaces have normal genus distribution and cover all integer vectors.

New transitions found in Spin(7) holonomy metrics related to a dynamical system.

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 …

Labourie and the author independently showed that a convex real projective structure on an oriented surface of genus at least 2 is equivalent to a conformal structure plus a holomorphic cubic differential U. We analyze the behavior of the real-projective structure as the conformal structure is fixed and the cubic diffe…

This article proves a uniform exponential decay estimate for Seiberg-Witten equations on non-compact 4-manifolds with exact symplectic ends of bounded geometry. This is an extension of the analysis for asymptotically flat almost Kähler (AFAK) structures by Kronheimer and Mrowka. As an application, we construct an invar…

We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.

M-theory on compact eight-manifolds with $\mathrm{Spin}(7)$-holonomy is a framework for geometric engineering of 3d $\mathcal{N}=1$ gauge theories coupled to gravity. We propose a new construction of such $\mathrm{Spin}(7)$-manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four…

Let G be one of the Ricci-flat holonomy groups SU(n), Sp(n), Spin(7) or G_2, and M a compact manifold of dimension 2n, 4n, 8 or 7, respectively. We prove that the natural map from the moduli space of torsion-free G-structures on M to the moduli space of Ricci-flat metrics is open, and that the image is a smooth manifol…

In this paper, a metric with G$_2$ holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold.

Study on Chern-Simons theory at generic levels, revealing universal resurgent structure.

A concrete model for a 7-dimensional gauge theory under special holonomy is proposed, within the paradigm outlined by Donaldson and Thomas, over the asymptotically cylindrical G2-manifolds provided by Kovalev's noncompact version of the Calabi conjecture. One obtains a solution to the $G_2$-instanton equation from the …

We classify all cohomogeneity one manifolds with principal orbit Q^111=SU(2)^3/U(1)^2 or M^110=(SU(3) x SU(2))/(SU(2) x U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically …

We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstr…

We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G_2 structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically…

Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindri…

A class of examples of Riemannian metrics with holonomy G_2 on compact 7-manifolds was constructed by the author in arXiv:math.DG/0012189 and later in a joint work with N.-H. Lee in arXiv:0810.0957, using a certain `generalized connected sum' of two asymptotically cylindrical manifolds with holonomy SU(3). We consider,…

We analyze 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit.

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any …

Continuous family of Ricci-flat metrics found on complex projective spaces.

New geometric approach realizes 5D bulk theories with 4D edge modes.

Given a Riemann surface $X = (Σ, J)$ we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form $(0,0,\cdots,tω_n)$. Moreover, we find an associated equivariant map from the universal cover $(\tildeΣ,\tilde{J})…

We investigate Yang--Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel--Hunsicker--Mazzeo. First referring to the codimension 2 …

A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same con…

We study classical spin networks with group SU(2). In the first part, using gaussian integrals, we compute their generating series in the case where the networks are equipped with holonomies; this generalizes Westbury's formula. In the second part, we use an integral formula for the square of the spin network and perfo…

New Spin(7) manifolds created from Calabi-Yau bundles.

We show that any grafting ray in Teichmüller space determined by an arational lamination or a multi-curve is (strongly) asymptotic to a Teichmüller geodesic ray. As a consequence the projection of a generic grafting ray to moduli space is dense. We also show that the set of points in Teichmüller space obtained by integ…

The goal of this paper is the construction of a compact manifold with G$_2$ holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by …

Let $X$ be a closed hyperbolic surface and $λ, η$ be weighted geodesic multicurves which are short on X. We show that the iterated grafting along $λ$ and $η$ is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of $λ$ and $η$. Using this result, we study the h…

Recently, at least 50 million of novel examples of compact $G_2$ holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry for compactifications of Type II superstrings in this context. We focus on $G_2$ …

This is the first nontrivial construction to date of instantons over a compact manifold with holonomy exactly $G_2$. The HYM connections on asymptotically stable bundles over Kovalev's noncompact Calabi-Yau 3-folds, obtained in the first article, are glued compatibly with a twisted connected sum, to produce a $G_2-$ins…

We consider the mapping properties of generalized Laplace-type operators ${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE manifolds considered by Joyce. Our main result gives conditions under which such opera…