arXiv research

New methods for computing a variety of gauge theoretic invariants for homology 3-spheres are developed. These invariants include the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) conne…

This is lecture notes for a course given at the PCMI Summer School "Quantum Field Theory and Manifold Invariants" (July 1 -- July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg--Witten gauge theory. However, I tried to …

This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $¶^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge th…

New gauge theory invariant detects non-smooth isotopy of $P^2$-knots.

Let $(F,J,ω)$ be an almost Kähler manifold, $α$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $ω$ invariant. We introduce gauge theoretical invariants for such triples $(F,α,K)$. The invariants are associated with moduli spaces of solutions of…

New criterion for almost-complex 4-manifolds using polyhedral decompositions.

Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of $S^1$-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which hav…

This article surveys our ongoing project about the relationship between invariants extending the classical Rohlin invariant of homology spheres and those coming from 4-dimensional (Yang-Mills) gauge theory. The main conjecture towards which this project is directed is that the Rohlin invariant and the gauge theoretic i…

The goal of this paper is to give an efficient computation of the 3-point Gromov-Witten invariants of Fano hypersurfaces, starting from the Picard-Fuchs equation. This simplifies and to some extent explains the original computations of Jinzenji. The method involves solving a gauge-theoretic differential equation, and o…

We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits.…

Survey of invariants for knotted 2-spheres in 4-space.

In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fun…

In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the…

Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of $3$-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Fl…

Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed in…

Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.

We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of $S^1 \times S^3$. The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other is essentially the index of the Dirac operator on a non-compact manifold with e…

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss the Casson-type invariant of a 3-manifold with the integral homology of a torus, given by counting projectively flat connections. We show that its mod 2 evaluation is given by the triple cup prod…

This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invarian…

Study gauge theory of real and quaternionic parabolic bundles over real curves.

Novel gauge-theoretic Floer homologies defined from 5d N=2 theory, linking 4, 3, and 2-manifolds.

New gauge-theoretic construction of 4D hyperkähler ALE spaces.

Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey provides an introduction to the categorical language for the construction and extension…

In this article, we consider a gauge-theoretic equation on compact symplectic 6-manifolds, which forms an elliptic system after gauge fixing. This can be thought of as a higher-dimensional analogue of the Seiberg-Witten equation. By using the virtual neighbourhood method by Ruan, we define an integer-valued invariant, …

L-CNNs learn gauge invariant quantities on lattices.

We formulate a more conceptual interpretation of the Cappell-Lee-Miller glueing/splitting theorem using the new language of asymptotic maps and asymptotic exactness. Additionally, we present an asymptotic description of the Mayer-Vietoris sequence naturally associated to the Cech cohomology of the sheaf of local soluti…

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding…

Novel gauge-theoretic Floer homologies defined for 7, 6, and 5-manifolds.

This article investigates a new gauge theoretic approach to Einstein's equations in dimension 4. Whilst aspects of the formalism are already explained in various places in the mathematics and physics literature, our first goal is to give a single coherent account of the theory in purely mathematical language. We then e…

Study extends gauge-theoretic invariant to higher-dimensional Kahler surfaces and calculates homotopy groups.

Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.

We describe the positive cone and the pseudo-effective cone of a non-Kählerian surface. We use these results for two types of applications: - Describe the set $σ(X)$ of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-Kähhlerian surface, and decide whether the assignment $…

We show that the ``classical'' Harder-Narasimhan filtration associated to a non semistable vector bundle $E$ can be viewed as a limit object for the action of the gauge group in the direction of an optimal destabilizing vector. This vector appears as an extremal value of the so called "maximal weight function". We give…

Kronheimer and Mrowka introduced a new knot invariant, called $s^\sharp$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi-positive knots with non-trivial r…

Paper finds exotic diffeomorphisms on specific 4-manifolds.

A principal pair consists of a holomorphic principal $G$-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobaya…

We give a detailed account of the gauge-theoretic approach to Lie applicable surfaces and the resulting transformation theory. In particular, we show that this approach coincides with the classical notion of $Ω$- and $Ω_{0}$-surfaces of Demoulin.

We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest on these, comes from Donaldson and Segal's suggestion \cite{Donaldson2009} that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metri…

Generalizes Bryant's correspondence to gauge-theoretical settings.

The geometry of the universal hyperKaehler implosion for SU(n) is explored. In particular, we show that the universal hyperKaehler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperKaehler implosion.

We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for $b_2=2$, every minimal class VII surface has a cycle of rational curves hence, by a result of Nakamura, is a global deformation of a one parameter family of blown up primary Hopf sur…

We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surfaces of constant negative Gauss curvature and isothermic surfaces. These are purely expository notes written to accompany some lectures I gave in Fukuoka in May 2015.

Study Spin(7) instantons and HYM connections on Stenzel metric.

We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n > 1$ our results indicate that there should not be any non-trivial gauge-theoretical invariants…

The special isothermic surfaces, discovered by Darboux in connection with deformations of quadrics, admit a simple explanation via the gauge-theoretic approach to isothermic surfaces. We find that they fit into a heirarchy of special classes of isothermic surface and extend the theory to arbitrary codimension.

Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds $X$ with $b_2(X) \geq 1$. If $b_2(X)$ is divisible by four and $b_1(X) =1$ a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our first result shows that if such a moduli s…

Study of gauge-theoretic functionals leading to constant scalar curvature almost-Kahler 4-manifolds.

A gauge theoretic description of the Morgan-Shalen compactification of the $\SL$ character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the Hitchin map.