Searcharxiv⌕ Search

arXiv subjects

Clifford Henry Taubes

Publications and source records attributed to Clifford Henry Taubes.

At least 19 recordsLinked to original sources
arXiv2024

Non-convergent sequences of solutions to the massive Vafa-Witten equations with 'interesting' $\mathbb{Z}/2\mathbb{Z}$ self-dual harmonic 2-form limits

This paper constructs sequences of solutions to the Vafa-Witten equations with non-zero (but small) mass term on the product of a 2-dimensional torus with a Riemann surface of genus greater than 1. These are divergent sequences (modulo principle bundle automorphisms) that converge after renormalization to define an 'interesting' $\mathbb{Z}/2\mathbb{Z}$ harmonic 2-form data set. This data set consists of a non-empty, codimension 2 submanifold, a real line bundle defined on the complement of that submanifold with no extension across it, and a self-dual, harmonic 2-form with values in that line bundle that does not extend over the submanifold. Even so, the norm of this 2-form does extend as a Hölder continuous function with that submanifold being its zero locus.

math.DG↗
arXiv2024

Spectral flow calculations for reducible solutions to the massive Vafa-Witten equations

The Vafa-Witten equations (with or without a mass term) constitute a non-linear, first order system of differential equations on a given oriented, compact, Riemannian 4-manifold. Because these are the variational equations of a functional, the linearized equations at any given solution can be used to define an elliptic, first order, self-adjoint differential operator. The purpose of this article is to give bounds (upper and lower) for the spectral flow between respective versions of this operator that are defined by the elements in diverging sequences of reducible solutions. (The spectral flow is formally the difference between the respective Morse indices of the solutions when they are viewed as critical points of the functional.) In some cases, the absolute value of the spectral flow is bounded along the sequence, whereas in others it diverges. This is a curious state of affairs. In any event, the analysis introduces localization and excision techniques to calculate spectral flow which may be of independent interest.

math.DG↗
arXiv2022

Topological aspects of $\mathbb{Z}/2\mathbb{Z}$ eigenfunctions for the Laplacian on $S^2$

This paper concerns the behavior of the eigenfunctions and eigenvalues of the round sphere's Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of particular interest is how these eigenvalues and eigenvectors change when viewed as functions on the configuration spaces of points.

math.DG↗
arXiv2022

Sequences of Nahm pole solutions to the SU(2) Kapustin-Witten equations

This paper describes the behavior of sequences of solutions to the Kapustin-Witten equations with Nahm pole asymptotics on the product of the half-line with a compact, oriented, Riemannian 3-manifold. These sequences have sub-sequences that either converge to another solution after acting term-wise by an automorphism of the principle bundle, or they converge after renormalization to a (weak) Z/2 harmonic 1-form from the 3-manifold; it is independent of the half-line coordinate in the product structure.

math.DG↗
arXiv2021

The existence of instanton solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$

A non-negative integer labeled set of model solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$ plays a central role in Edward Witten's program to interpret the colored Jones polynomial or a knot in the context of SU(2) gauge theory. This paper explains why there are $\mathbb{R}$-invariant solutions to these equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$ that interpolate between two model solutions as the $(0,\infty)$ parameter increases from 0 to $\infty$ while respecting the $\mathbb{R}^2$ factor asymptotics. The only constraint on the limiting pair of model solutions is this: Letting $m_0$ and $m_\infty$ denote their non-negative integer labels, then $m_0 - m_\infty$ must be a positive, even integer. (As explained in the paper, there is a $\mathbb{C}^{(m_0 -m_\infty - 2)/2}\times \mathbb{C}^*$ moduli space of these interpolation solutions.)

math.DG↗
arXiv2020

Lectures on the linearized Kapustin-Witten equations on $(0,\infty) \times\ $Y

This is a written version of lectures that I would have given myself about aspects of the differential operator that is obtained from the linearized Kapustin-Witten equations on the product of the half-line with a compact, oriented, Riemannian 3-manifold. These lectures concern for the most part certain instances of much more general theorems of R. Mazzeo and E. Witten. There is also a 'lecture series' about the asymptotics of solutions to the same Kapustin-Witten equations as the half-line parameter limits to infinity.

math.DG↗
arXiv2020

Examples of singularity models for $\mathbb{Z}/2$ harmonic 1-forms and spinors in dimension 3

We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a $\mathbb{Z}/2$ harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are $\mathbb{Z}/2$ harmonic 1-forms or spinors on $\mathbb{R}^3$ that are homogeneous with respect to rescaling of $\mathbb{R}^3$ with their zero locus consisting of four or more rays from the origin. The rays point from the origin to the vertices of a centered tetrahedron in one example; and they point from those of a centered octahedron and a centered icosahedron in two others.

math.DG↗
arXiv2017

Tamed to compatible: Symplectic forms via moduli space integration

Fix a compact 4-dimensional manifold with self-dual 2nd Betti number one and with a given symplectic form. This article proves the following: The Frechet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.

math.SG↗
arXiv2017

The behavior of sequences of solutions to the Vafa-Witten equations

The Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. The main theorem in this paper characterizes in part the behavior of sequences of solutions to the Vafa-Witten equations which have no convergent subsequence. The paper proves that a renormalization of a subsequence of the self-dual 2-form components converges on the complement of a closed set with Hausdorff dimension at most 2, with the limit being a harmonic 2-form with values in a real line bundle.

math.DG↗
arXiv2017

Tamed to compatible when b^(2+) = 1 and b^1 = 2

Weiyi Zhang noticed recently a gap in the proof of the main theorem of the authors article "Tamed to compatible: Symplectic forms via moduli space integration" [T] for the case when the symplectic 4-manifold in question has first Betti number 2 (and necessarily self-dual second Betti number 1). This note explains how to fill this gap.

math.SG↗
arXiv2017

Growth of the Higgs field for solutions to the Kapustin-Witten equations on R^4

The Kapustin-Witten equations on R^4 are equations for a pair of connection on the product principle SU(2) bundle and 1-form with values in the product Lie algebra bundle. The 1-form is the Higgs field. A dichotomy is proved to the effect that either the averaged norm of the Higgs field on large radius spheres grows faster than a power of the radius, or its 1-form components everywhere pairwise commute.

math.DG↗
arXiv2016

On the behavior of sequences of solutions to U(1) Seiberg-Witten systems in dimension 4

This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section of the self-dual spinor bundle of a complex Clifford module on the manifold. Examples include the cases where the Clifford module is a direct sum of C2 bundles associated to SpinC structures; and the case of the SU(2) Vafa-Witten equations with an Abelian ansatz.

math.DG↗
arXiv2016

Some 4-manifold geometry from hyperbolic knots in S^3

A 4-manifold is constructed with some curious metric properties; or maybe it is many 4-manifolds masquerading as one, which would explain why it looks curious. Anyway, knots in the 3-sphere with complete finite volume hyperbolic metrics on their complements play a role in this story.

math.DG↗
arXiv2015

PSL(2;C) connections on 3-manifolds with L2 bounds on curvature

Karen Uhlenbeck's compactness theorem for sequences of connections with L2 bounds on curvature applies only to connections on principal bundles with compact structure group. This article states and proves an extension of Uhlenbecks theorem that describes sequences of connections on principal PSL(2;C) bundles over compact three dimensional manifolds.

math.DG↗