arXiv research

New spectral invariants distinguish Joyce orbifolds from other manifolds.

Study constructs associative submanifolds in $G_2$-manifolds from orbifolds.

Article constructs coassociative submanifolds in Joyce's $G_2$-manifolds.

New $Spin(7)$-instantons constructed on Joyce's manifold.

We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces …

We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.

Study on counting flat connections over $G_2$-orbifolds, proving moduli space compact and smooth.

The study of fibrations of the target manifolds of string/M/F-theories has provided many insights to the dualities among these theories or even as a tool to build up dualities since the work of Strominger, Yau, and Zaslow on the Calabi-Yau case. For M-theory compactified on a Joyce manifold $M^7$, the fact that $M^7$ i…

New hyperKähler orbifolds of Kummer type discovered.

Smooth family of G2-instantons over a Kummer construction.

In this article we introduce a method to construct $\rm{G}_2$-instantons on $\rm{G}_2$-manifolds arising from Joyce's generalised Kummer construction. The method is based on gluing ASD instantons over ALE spaces to flat bundles on $\rm{G}_2$-orbifolds of the form $T^7/ Γ$. We use this construction to produce non-trivia…

New compact manifolds with closed G2 structures found.

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…

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…

The paper connects $G_2$-manifolds to Coulomb and Higgs phases of gauge theories.

This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and readers just wanting a general o…

This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algeb…

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…

Defines special Joyce structures for ASK manifolds encoding real HK structures.

Resurgence of Joyce structures gauged to a standard form using gauge transformations.

Computes a $ν$-invariant for Joyce's $G_2$-manifolds.

We explicitly construct the twistor spaces of Joyce metrics with torus action that are not treated in Part I (math.DG/0603242). This finishes a construction of all the twistor spaces of Joyce metrics on the connected sum of four complex projective planes.

Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.

We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki…

Study on holonomy of Obata connection on Joyce hypercomplex manifolds.

We conjecture that the Joyce-Song wall-crossing formula for Donaldson-Thomas invariants arises naturally from an asymptotic expansion in the field theoretic work of Gaiotto, Moore and Neitzke. This would also give a new perspective on how the formulae of Joyce-Song and Kontsevich-Soibelman are related. We check the con…

New geometric Joyce structures on moduli spaces of quadratic differentials.

In 1995 D. Joyce explicitly constructed a series of self-dual metrics with torus action on the connected sums of complex projective planes. In this paper we explicitly construct the twistor spaces of some of Joyce's self-dual metrics. Starting from a fiber space whose fibers are compact singular toric surfaces, we appl…

We prove that any invariant hypercomplex structure on a homogeneous space $M = G/L$ where $G$ is a compact Lie group is obtained via the Joyce's construction, provided that there exists a hyper-Hermitian naturally reductive invariant metric on $M$.

This paper connects complex hyperkähler structures to Donaldson-Thomas invariants.

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups $G_2$ and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum $G_2$ manifolds, mirrors of such Spin(7) manifolds can be fou…

We give a differential-geometric construction of compact manifolds with holonomy $\mathrm{Spin}(7)$ which is based on Joyce's second construction of compact $\mathrm{Spin}(7)$-manifolds in \cite{Joyce00} and Kovalev's gluing construction of $G_2$-manifolds in \cite{Kovalev03}. We also give some examples of compact $\ma…

The article confirms Joyce's examples of G2-holonomy are formal spaces.

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…

We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.

We construct new examples of self-similar solutions and translating solitons for Lagrangian mean curvature flow by extending the method of Joyce, Lee and Tsui. Those examples include examples in which the Lagrangian angle is arbitrarily small as the examples of Joyce, Lee and Tsui.

We provide a simple algebraic construction of the twistor spaces of arbitrary Joyce's self-dual metrics on the 4-manifold H^2 x T^2 that extend smoothly to nCP^2, the connected sum of complex projective planes. Indeed, we explicitly realize projective models of the twistor spaces of arbitrary Joyce metrics on nCP^2 in …

We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-str…

The present paper deals with mirror symmetry aspects of compact ``barely'' $G_2$ manifolds, that is, $G_2$ manifolds of the form (CY$\times S^1)/\mathbb{Z}_2$. We propose that the mirror of any barely $G_2$ manifold is another barely one and which is constructed as a fibration of the \emph{mirror} of the CY base. Also,…

Previously the two of the authors defined a notion of dual Calabi-Yau manifolds in a G_2 manifold, and described a process to obtain them. Here we apply this process to a compact G_2 manifold, constructed by Joyce, and as a result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known to be mirror dual…

Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In the present paper, we discuss the rather complicated extension of Joyce's observation to several different medial quandles and reduced (one-variable) Alexander modules associated with c…

This paper constructs hyper-Kähler metrics and hyper-Lagrangian foliations for a class of complex manifolds.

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…

Improved estimates for $G_2$-structures on a specific manifold.

We revisit our construction of mirror symmetries for compactifications of Type II superstrings on twisted connected sum $G_2$ manifolds. For a given $G_2$ manifold, we discuss evidence for the existence of mirror symmetries of two kinds: one is an autoequivalence for a given Type II superstring on a mirror pair of $G_2…

We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.

In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called…

Alternative proof and description of orientations for instanton moduli spaces.