arXiv research

Proves Weinstein's and Arnold's conjectures using contact instantons.

Develops gluing theory for contact instantons and pseudoholomorphic curves.

Study contact instantons on Sasakian 5-manifolds with Calabi-Yau structures.

We study a natural contact instanton (CI) equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related both to the transverse Hermitian Yang-Mills (tHYM) condition and the G_2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreduci…

Study contact instantons and Legendrian links, proving energy inequalities.

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds -- with or without boundary -- defined in the instanton Floer setting. We prove that our invariant vani…

Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.

Study calculates instanton Floer homology for surgeries on L-space knots.

We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric $5$-manifolds and initiate the study of their structure. In the $K$-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is…

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proven here; and Kronheimer and Mrowka's spe…

Simplified construction of contact triad connection for analysis.

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, a…

Survey article analyzes pseudoholomorphic curves on symplectization via contact instantons.

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invaria…

On a five dimensional simply connected Sasaki-Einstein manifold, one can construct Yang-Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however the contact instanton equations are not elliptic. It turns out that these equa…

Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.

This research studies the smoothness of moduli spaces of self-dual contact instantons on Sasakian manifolds.

In this expository review we discuss various aspects of gauge theory. While the focus is on mathematics, wherever possible we make contact with theoretical high energy physics. Particular emphasis is placed on instantons and monopoles, which admit physical interpretation, and yield interesting and nontrivial mathematic…

New equations connect instantons, spinors, and 3-forms in 6 and 7 dimensions.

Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projecti…

The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.

This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed ${\cal N}=2$ supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a …

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $\varphi$ induced by the Calabi-Yau $3$-orbifold basic geometry. We disti…

We introduce a canonical affine connection on the contact manifold $(Q,ξ)$, which is associated to each contact triad $(Q,λ,J)$ where $λ$ is a contact form and $J:ξ\to ξ$ is an endomorphism with $J^2 = -id$ compatible to $dλ$. We call it the \emph{contact triad connection} of $(Q,λ,J)$ and prove its existence and uniqu…

Constructs non-unital monoidal category of contact manifolds and Legendrian correspondence calculus

The paper defines ASD connections and constructs families over a 5D Heisenberg group.

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with tr…

Classifies gravitational instantons with quadratic volume growth.

Given two hyperkähler manifolds $M$ and $N$ and a quaternionic instanton on their product, a hyperkähler Nahm transform can be defined, which maps quaternionic instantons on $M$ to quaternionic instantons on $N$. This construction includes the case of Nahm transform for periodic instantons on $\bR^4$, the Fourier-Mukai…

Study calculates ring structure in instanton homology for a surface with points.

Survey of recent progress in gravitational instantons

Study shows instantons and monopoles decompose into U(1) components.

We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route…

Study on Yang-Mills equation near instanton-anti-instanton configurations with energy constraints.

Proves exact triangle linking knot instanton Floer homology to surgeries.

The paper proves unique characterization of gravitational instantons with specific volume growth.

We study the deformation theory of $\mathrm{G}_2$-instantons on the 7-sphere, specifically those obtained from instantons on the 4-sphere via the quaternionic Hopf fibration. We find that the pullback of the standard ASD instanton lies in a smooth, complete, 15-dimensional family of $\mathrm{G}_2$-instantons. In genera…

Proves unique ALE instanton with toric Hermitian structure.

Classifies instantons on a specific gravitational instanton and computes partition functions.

We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds, nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifo…

Period maps surjective for certain gravitational instantons.

We consider the Hermitian Yang-Mills (instanton) equations for connections on vector bundles over a 2n-dimensional Kähler manifold X which is a product Y x Z of p- and q-dimensional Riemannian manifold Y and Z with p+q=2n. We show that in the adiabatic limit, when the metric in the Z direction is scaled down, the gauge…

Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.

New homological action on sutured instanton homology defined.

Smooth family of G2-instantons over a Kummer construction.

Proves uniqueness of certain $S^1$-symmetric gravitational instantons.

First non-trivial examples of deformed G_2-instantons, distinguishing nearly parallel G_2-structures.

Instantons on multi-Taub-NUT spaces are mapped to bow representations.