arXiv research

Study deformations of G2-instantons on nearly G2 manifolds.

We formulate the deformation theory for instantons on nearly Kähler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid…

Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.

Proves stability of gravitational instantons, proving operator positivity.

Solve Painleve VI to relate instanton bundles.

Study uses instanton Floer theory to obstruct knot unknotting operations.

Alternative proof and description of orientations for instanton moduli spaces.

Smooth family of G2-instantons over a Kummer construction.

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

Study on deformations of LC Spin(7) instantons simplifies the problem.

Modified Laplacian connects to Yang-Mills instantons on manifolds.

We show that the integral of the first Pontrjagin class is given by an integer and it is identified with instanton number of the U(n) gauge theory on noncommutative ${\bf R^4}$. Here the dimension of the vector space $V$ that appear in the ADHM construction is called Instanton number. The calculation is done in operato…

We develop the deformation theory of instantons on asymptotically conical $G_2$-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the space of deformations to the kernel of a twisted Dirac operator on the $G_2$-manifold and to the eigenvalues of a twisted Dirac op…

This paper upgrades instanton TQFT to infinity-categories for better simplification.

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…

The paper studies gravitational instantons with flat limits and finds elliptic regularity estimates.

Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.

We study finite action anti-self-dual Yang-Mills connections on the multi-Taub-NUT space. We establish the curvature and the harmonic spinors decay rates and compute the index of the associated Dirac operator. This is the first in a series of papers proving the completeness of the bow construction of instantons on mult…

Survey article analyzes pseudoholomorphic curves on symplectization via contact instantons.

This paper develops a local analogue of the ADHM construction, which characterises ASD instantons defined over smooth bounded domains inside Euclidean $\mathbb{R}^4$ diffeomorphic to the 4-ball, in terms of infinite dimensional Hilbert spaces and bounded Hermitian linear operators satisfying an analogue of the ADHM equ…

SU(3) instanton homology counts Tait colorings for webs and foams.

This paper extends Witten's holomorphic Morse inequalities to singular spaces.

Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-…

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…

The Plebański complex is a differential operator that squares to the Laplacian and is composed of two Dirac operators.

The paper constructs instanton complexes on stratified pseudomanifolds.

In dimension 7, we establish a Fredholm theory for a Dirac-type operator associated to a connection with point singularities. There are two applications. $1$. over a closed 7-manifold, under some natural conditions, a $G_{2}-$instanton and its point singularities can still be "seen" when the $G_{2}-$structure is proper…

Study geometric operators on Tian-Yau spaces, finding $L^2$ harmonic forms and asymptotic regularity.

We show how the index formula for manifolds with fibered boundaries can be used to compute the index of the Dirac operator on Taub-NUT space twisted by an anti-self-dual generic instanton connection.

This work concerns the study of certain finite-energy solutions of the anti-self-dual Yang-Mills equations on Euclidean 4-dimensional space which are periodic in two directions, so-called doubly-periodic instantons. We establish a circle of ideas involving equivalent analytical and algebraic-geometric descriptions of t…

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

This is the first step in an attempt at a deformation theory for $G_{2}-$instantons with $1-$dimensional conic singularities. Under a set of model data, the linearization yields a self-adjoint first order elliptic operator $P$ on a certain bundle over $\mathbb{S}^{5}$. As a dimension reduction, the operator $P$ also ar…

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

We explore the nonperturbative aspects of the chiral algebras of N = (0,2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nont…

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…

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

Mutation is an operation on 3-manifolds containing an embedded surface of genus 2. It is defined by cutting along the surface and regluing using the `hyperelliptic' involution, and is known to preserve many 3-manifold invariants. I show that mutation of a homology 3-sphere preserves its (instanton) Floer homology, and …

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

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

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…