arXiv research

New $\mathcal{C}$-connection characterizes 4D spaces conformal to Einstein spaces.

Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on $Y$ are $non$-$degenerate$. In this article, we prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature. Combining the compactness theorem and a previous…

We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with topologic…

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists …

The article defines conditions for a manifold to be conformal to an Einstein space.

Study $SL(2,\mathbb{C})$ connections on Seifert-fibered spaces using gauge theory.

We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abe…

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construc…

Let $(X,x_0)$ be any one--pointed compact connected Riemann surface of genus $g$, with $g\geq 3$. Fix two mutually coprime integers $r>1$ and $d$. Let ${\mathcal M}_X$ denote the moduli space parametrizing all logarithmic $\text{SL}(r,{\mathbb C})$--connections, singular over $x_0$, on vector bundles over $X$ of degree…

Mid-dimensional $(A,B,A)$ and $(B,B,B)$-branes in the moduli space of flat $G_{\mathbb C}$-connections appearing from finite group actions on compact Riemann surfaces are studied. The geometry and topology of these spaces is then described via the corresponding Higgs bundles and Hitchin fibrations.

The paper connects isomonodromic and isospectral deformations for $\mathfrak{sl}_2(\mathbb{C})$ connections.

Geometrically constructs dilogarithm from Chern-Simons theory.

Researchers compute Floer homotopy types and eta invariants for Seifert 3-manifolds.

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 compa…

The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.

We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furt…

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equati…

Constant mean curvature (CMC) surfaces in space forms can be described by their associated $\mathbb C^*$-family of flat $SL(2,\mathbb C)$-connections $\nabla^λ$. In this paper we consider the asymptotic behavior (for $λ\to0$) of the gauge equivalence classes of $\nabla^λ$ for compact CMC surfaces of genus $g\geq2.$ We …

For a Seifert fibered homology sphere we show that the q-series Z-hat invariant introduced by Gukov, Pei, Putrov and Vafa is a resummation of the Ohtsuki serie. We show that for every even level k there exists a full asymptotic expansion of Z-hat for q tending to a certain k'th root of unity and in particular that the …

A first-order formulation of gravity is developed in which the fundamental fields consist of an SL(2,C) connection and two spinor-valued 1-forms. It is shown that the first term of an expansion of the Einstein-Hilbert action leads to an action for these fields which consists of dynamic L2 inner products of their covari…

Paper describes a new method for character varieties of surface groups.

We propose a method for determining the spins of BPS states supported on line defects in 4d $\mathcal{N}=2$ theories of class S. Via the 2d-4d correspondence, this translates to the construction of quantum holonomies on a punctured Riemann surface $\mathcal{C}$. Our approach combines the technology of spectral networks…

The paper connects harmonic forms to tree maps and character varieties.

We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichmüller TQFT, and we prove it in detail in the case of the knot complements of $4_{1}$ and $5_2$. The conjecture states that the level-$N$ Andersen-Kashaev invariant, $J^{(\mathrm{b},N)}_{M,K}$, is annihilated by the non-homogeneous $\hat{…

Quantum invariants of 3-manifolds linked to splice diagrams.

We review the representation theory of the quantum group $U_εsl_2\mathbb{C}$ at a root of unity $ε$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our analysis relies on the quantum coadjoint action of De Concini-Kac-Procesi, and the theory of Heisenbe…

Wilson lines generate positive Laurent polynomials in decorated triangulations.

Study on spectral points of Inoue surfaces with Tricerri metric.

Study Floer theory of hyperbolic three-manifolds using Dirac spectral flow.

We define Hitchin's moduli space for a principal bundle $P$, whose structure group is a compact semisimple Lie group $K$, over a compact non-orientable Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which identifies Hitchin's moduli space with the moduli space of flat $K^\mathbb{C}$-connections,…

Proposes a graph learning framework for clustering and semi-supervised classification.

We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example,…

Spectral clustering has found extensive use in many areas. Most traditional spectral clustering algorithms work in three separate steps: similarity graph construction; continuous labels learning; discretizing the learned labels by k-means clustering. Such common practice has two potential flaws, which may lead to sever…

The exterior differential system for constant mean curvature (CMC) surfaces in a 3-dimensional space form is an elliptic Monge-Ampere system defined on the unit tangent bundle. We determine the infinite sequence of higher-order symmetries and conservation laws via an enhanced prolongation modelled on a loop algebra val…

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M)…