Mathematical study of instanton corrected q-map spaces and their isometries.
problem Understanding the isometries of instanton corrected q-map spaces.
method Study of isometries of instanton corrected q-map spaces associated to PSR manifolds.
result Explicit example of instanton corrected q-map space with full SL(2,Z) acting by isometries.
We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special…
The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.
problem Understanding the geometry of the resolved conifold and its associated structures.
method Explicit description of ASK and instanton-corrected HK manifolds, relating them to twistor coordinates and solving Riemann-Hilbert problems.
result The instanton-corrected hyperkähler manifold realizes a smoothing of the semi-flat HK metric associated with the ASK geometry.
Study examines corrections to heterotic geometry on SU(3) manifolds.
problem Analyzing α′2 corrections to heterotic supersymmetry algebra. method Derives integrability condition and pure gauge correction from graviton equation of motion.
result Curvature of tangent bundle connection acquires a (0,2) component, disrupting semi-classical intuition.
Infinitely many M2-instantons affect M-theory on G2-manifolds.
problem Understanding non-perturbative effects in M-theory on G2-manifolds.
method Analyzing superpotential contributions from M2-instantons on TCS G2-manifolds. result Infinitely many M2-instanton corrections to M-theory on G2-manifolds. Study SYZ mirror construction for toric Calabi-Yau manifolds using quasimap theory.
problem Constructing SYZ mirror for toric Calabi-Yau manifolds.
method Use quasimap Floer theory for instanton corrections instead of Lagrangian Floer theory.
result SYZ mirror coincides with physical construction.
New homology q2 from 3D cobordism to integers.
problem Understanding 4-manifolds with boundary and their homology.
method Instanton homology with Z/2 coefficients. result Found a non-linear homomorphism q2 with bounds on 4-manifolds. We show that the four derivative terms in the effective action of three-dimensional N=8 Yang-Mills theory are determined by supersymmetry. These terms receive both perturbative and non-perturbative corrections. Using our technique for constraining the effective action, we are able to determine the exact form of the eig…
Paper corrects mistakes in moduli space orientability for Spin(7)-instantons and coherent sheaves.
problem Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds.
method Uses new theory from Joyce-Upmeier arXiv:2503.20456 to prove corrected versions of main results.
result Corrected versions of main results hold with an extra assumption on H3(X,Z). Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.
problem Analyzing nonlinear elliptic systems associated with contact Hamiltonian trajectories.
method Identifies correct action and energy functionals, develops elliptic regularity theory, and proves asymptotic convergence.
result Established C∞ convergence of perturbed contact instantons under finite energy hypothesis. Study shows 3D shapes can't fit in certain 4D spaces.
problem Proving 3-manifolds can't embed in symplectic 4-manifolds.
method Constructing L-spaces and using Heegaard Floer correction terms, instanton moduli spaces.
result Infinitely many 3-manifolds cannot embed in symplectic 4-manifolds.
We use the twistorial construction of D-instantons in Calabi-Yau compactifications of type II string theory to compute an explicit expression for the metric on the hypermultiplet moduli space affected by these non-perturbative corrections. In this way we obtain an exact quaternion-Kahler metric which is a non-trivial d…
We study the physics of globally consistent four-dimensional N=1 supersymmetric M-theory compactifications on G2 manifolds constructed via twisted connected sum; there are now perhaps fifty million examples of these manifolds. We study a rich example that exhibits U(1)3 gauge symmetry and a spectrum o…
We give geometric explanations and proofs of various mirror symmetry conjectures for Tn-invariant Calabi-Yau manifolds when instanton corrections are absent. This uses fiberwise Fourier transformation together with base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex stru…
The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.
problem Solving the heterotic G2 system on Calabi-Yau 3-orbifolds.
method Adjusting circle bundles and powers of α' to obtain cocalibrated G2 structures.
result Non-trivial solutions to the heterotic G2 system are found.
Study contact instantons and Legendrian links, proving energy inequalities.
problem Estimating Reeb-untangling energy of Legendrian submanifolds.
method Develop contact Hamiltonian geometry, introduce tame contact manifolds, construct moduli spaces, prove convergence results.
result Self Reeb-untangling energy of compact Legendrian submanifolds is greater than period gap.
Study rigid limit of hypermultiplet moduli spaces in string theory.
problem Understanding the rigid limit of hypermultiplet moduli spaces in Calabi-Yau compactifications.
method Performing a hyperkahler quotient of the Swann bundle over the moduli space, induced by a local limit of the Calabi-Yau.
result The resulting hyperkahler manifold is obtained as the target space of a non-linear sigma model for a five-dimensional gauge theory.
Construct quaternionic-Kähler metrics from special Kähler manifolds with specific BPS structure variations.
problem Construct quaternionic-Kähler metrics from special Kähler manifolds with mutually local variations of BPS structures.
method Construct quaternionic-Kähler metrics from a conical special Kähler manifold with a certain type of mutually-local variation of BPS structures. Provide global and local explicit formulas for the quaternionic-Kähler metric.
result Construct quaternionic-Kähler metrics that are positive-definite and deform the 1-loop corrected Ferrara-Sabharval metric.
Recently, Gaiotto, Moore and Neitzke \cite{GMN08} proposed a new construction of hyperkähler metrics. In particular, they gave a new construction of the Ooguri-Vafa metric, in which they came across certain formulas. We interpret those formulas as wall-crossing formulas that appear in the SYZ construction of instanton-…
In this paper, we study hyperkahler metric and practice GMN's construction of hyperkahler metric on focus-focus fibrations. We explicitly compute the action-angel coordinates on the local model of focus-focus fibration, and show its semi-global invariant should be harmonic to admit a compatible holomorphic 2-form. Then…
We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkahler metric of the moduli space of the theory on R^3 x S^1. The wall-crossing formula reduces to th…
Study G2-instantons on 7-sphere from 4-sphere instantons.
problem Deformation theory of G2-instantons on 7-sphere. method Analysis of pullback from 4-sphere instantons and identification of deformation spaces.
result Existence of a smooth, complete family of 15-dimensional G2-instantons. A 'holographic formula' expressing the functional determinant of the scattering operator in an asymptotically locally anti-de Sitter(ALAdS) space has been proposed in terms of a relative functional determinant of the scalar Laplacian in the bulk. It stems from considerations in AdS/CFT correspondence of a quantum corre…
The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details -- in particular the global moduli / instanton / magnetic charge structure -- have remained elusive. Here we deduce from…
Classifies gravitational instantons with quadratic volume growth.
problem Classifying gravitational instantons with specific growth properties.
method Proves a classification theorem for ALG∗ gravitational instantons, determines topology, and proves existence of uniform coordinates. result Proves a relationship between ALG gravitational instantons of different orders.
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.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.
Survey of recent progress in gravitational instantons
problem Classification of hyperkähler and Hermitian gravitational instantons
method Survey of recent progress in gravitational instantons
result Survey of recent progress in gravitational instantons
We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of p…
String geometry theory uniquely determines classical action with T-symmetry.
problem Non-renormalizability and loop corrections in string theory.
method Distinguishes effects of β and ħ parameters, proving no loop corrections.
result No loop corrections in string geometry theory, avoiding non-renormalizability.
Study calculates nilpotency in instanton homology and related framed instanton homology.
problem Understanding instanton homology and framed instanton homology for surfaces times a circle.
method Computes nilpotency degree and uses moduli space of stable rank two holomorphic bundles.
result Framed instanton homology closely related to kernel of u2−64. Abstract relates G2 instantons to Seiberg-Witten monopoles.
problem Understanding the connection between G2 instantons and Seiberg-Witten monopoles.
method Relates Seiberg-Witten monopoles, Fueter sections, and G2 instantons.
result Describes a conjectural relation between G2 instantons and Seiberg-Witten monopoles.
Study shows instantons and monopoles decompose into U(1) components.
problem Understanding the asymptotic structure of instantons and monopoles.
method Analysis of multi-centered Taub-NUT manifolds, calorons, and monopoles on R^3.
result Instantons and monopoles asymptotically decompose into U(1) components.
Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.
problem Understanding Sp(n)-instantons on hyperkahler manifolds with conical singularities.
method Relating Sp(n)-instantons to deformed instantons and studying their properties on hyperkahler manifolds.
result Sp(n)-instantons on hyperkahler manifolds correspond to tri-contact instantons on the 3-Sasakian link.
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.
problem Understanding Yang-Mills connections near instanton-anti-instanton configurations.
method Analyzing the Uhlenbeck limit and bubble configurations, determining obstructions and proving solutions.
result Instantons are the only solutions with energy less than $4π^2 \left( |κ| + 2
ight) + \varepsilon_κ$.
Proves exact triangle linking knot instanton Floer homology to surgeries.
problem Relating knot instanton Floer homology to surgeries.
method Proves an exact triangle using integer coefficients.
result Poincaré Homology Sphere is not an instanton L-space with integer coefficients.
The study reveals the moduli of instantons on product manifolds.
problem Understanding instantons on product manifolds.
method Using a broken gauge transformation and Hermitian Yang-Mills connections.
result Moduli spaces of G2 and Spin(7)−instantons on product manifolds are equivalent to Hermitian Yang-Mills connections. The paper proves unique characterization of gravitational instantons with specific volume growth.
problem Characterizing gravitational instantons with quadratic volume growth.
method Defining a period mapping and proving its surjectivity and openness.
result The periods uniquely characterize ALG∗ and ALG gravitational instantons up to diffeomorphism. Proves unique ALE instanton with toric Hermitian structure.
problem Classify Ricci flat ALE instantons with toric Hermitian non-Kähler structure.
method Direct global analysis of Tod form in Weyl-Papapetrou coordinates, avoiding toric Kähler geometry.
result Eguchi-Hanson instanton is the only smooth, Ricci flat, ALE instanton with toric Hermitian non-Kähler structure.
Classifies instantons on a specific gravitational instanton and computes partition functions.
problem Classifying finite energy harmonic 2-forms and anti-self-dual Yang-Mills instantons.
method Analyzes U(1)-bundles and computes instantons explicitly. result Unique anti-self-dual Yang-Mills instantons exist and are described explicitly.
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…
New instanton homology for webs counts Tait colorings.
problem Counting Tait colorings of complex webs.
method Introducing a local system of coefficients to deform instanton homology.
result Rank of deformed instanton homology equals number of Tait colorings.
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…
Period maps surjective for certain gravitational instantons.
problem Understanding the uniformization of gravitational instantons.
method Uniformization theorems by Chen--Chen, Chen--Viaclovsky, Collins--Jacob--Lin, and Hein--Sun--Viaclovsky--Zhang.
result Period maps surjective for ALH*, ALG, and ALG* gravitational instantons.
Smooth family of G2-instantons over a Kummer construction.
problem Constructing a smooth family of G2-instantons over a Kummer construction.
method Extending gluing construction for G2-instantons to Kummer constructions, using a Z2-action to overcome obstructions.
result Proves the existence of a smooth 1-parameter family of G2-instantons, showing their infinitesimal rigidity and non-flatness.
Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.
problem Classical Euclidean Black Hole Uniqueness conjecture
method Analyzing the Chen-Teo gravitational instanton
result The Chen-Teo instanton is Hermitian and non-Kähler
New homological action on sutured instanton homology defined.
problem Detecting link splitting in knot homology.
method Defining a homological action on sutured instanton Floer homology.
result Instanton knot homology detects link splitting for two-component links.