The index of mesoscopic Wilson--Dirac operators: I

Mikio Furuta; Shinichiroh Matsuo

Home/Archive/arXiv

arXiv · 2611.08266

The index of mesoscopic Wilson--Dirac operators: I

Mikio Furuta·Shinichiroh Matsuo

View original source ↗Read source document ↗Export metadata ↓

Abstract

This is the first of a series of papers which will be devoted to a study of the index of mesoscopic Wilson--Dirac operators on closed Riemannian manifolds. The aim of the series is to discretise the index of the Dirac operator. On flat tori, this is achieved by the Wilson--Dirac operator of lattice gauge theory on a regular lattice. On a general closed Riemannian manifold, however, no regular lattice is available. As an intermediate step between the continuum Dirac operator and its discretisations, we introduce mesoscopic Wilson--Dirac operators on any closed Riemannian manifold, defined by integrating covariant differences over distances comparable to a fixed length scale. In this paper, we define the index of the mesoscopic Wilson--Dirac operators via spectral flow and prove that it agrees with that of the continuum Dirac operator at all sufficiently small scales. The proof rests on an elliptic estimate uniform in the scale and a Rellich-type compactness theorem. In the second paper, we shall pass from mesoscopic operators to discrete Wilson--Dirac operators.

Explore related subjects

math.DG hep-lat math-ph math.KT math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Mikio Furuta, Shinichiroh Matsuo. 2026-11-05. The index of mesoscopic Wilson--Dirac operators: I. https://arxiv.org/abs/2611.08266

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Transformation theorem and monotonicity of the numbers of almost Euclidean factors of geodesic balls

In \cite{HH}, H. Huang-X.Huang introduced the generalized Reifenberg condition which describes the non-increasing property of numbers of almost Euclidean factors of geodesic balls and gave a transformation theorem under this condition. In this note, we study how almost Euclidean factors of concentric geodesic balls interact with harmonic splitting maps under a lower Ricci curvature bound and prove a dimension gap for harmonic functions of almost linear growth and then derive a transformation theorem under a non-decreasing property compared with the non-increasing property above. We also construct complete manifolds with nonnegative Ricci curvature whose balls are uniformly almost $1$-Euclidean at every scale, but for which no single harmonic function can be transformed into a splitting function at all smaller scales which give a negative answer to the existence question for all-scale splitting functions in \cite{HH}.

math.DG↗

arXiv2026

Schwarzian Residue of the Samuelson Obstruction in Tangent Lagrangian 2-Webs

The Samuelson condition, a classical area-ratio condition for planar Lagrangian $2$-webs, is equivalent in local web coordinates $(u,v)$ to $\partial_u\partial_v\log|J_F(u,v)|=0$, where $F$ is the inverse web-coordinate map and $J_F$ is its Jacobian. For the tangent-line family $L_t: y=tx+h(t)$, with $h$ smooth on an open interval, let $Ψ(u,v)$, for $u\neq v$, be the intersection map of $L_u$ and $L_v$, write $J_Ψ$ for its Jacobian, and set $S_h(u,v) := \partial_u\partial_v\log|J_Ψ(u,v)|$ where $J_Ψ(u,v)\neq0$. Near each diagonal point $(t_0,t_0)$ with $h''(t_0)\neq0$, the obstruction admits the decomposition $S_h(u,v) = \frac{1}{(v-u)^2} + R_h(u,v)$, where $R_h$ extends smoothly across the diagonal near $(t_0,t_0)$. The universal pole yields $S_h(u,v)>0$ off the diagonal in a sufficiently small neighborhood of $(t_0,t_0)$, so the tangent web does not satisfy the Samuelson condition there. We call $R_h(t,t)$ the Schwarzian residue and show that $R_h(t,t) = \frac{1}{2}\{σ,t\} = -κ_{\mathrm{aff}}(σ) \left(\frac{dσ}{dt}\right)^2$. Here $σ$ is the equi-affine arclength parameter of the envelope $γ(t)=(-h'(t),h(t)-t h'(t))$, and $\{σ,t\}$ denotes the Schwarzian derivative of $σ$ with respect to $t$. The equi-affine curvature $κ_{\mathrm{aff}}$ is defined by $γ_{σσσ}+κ_{\mathrm{aff}}γ_σ=0$. We also derive the transformation law of the residue under reparametrization of the tangent-line family.

math.DG↗

arXiv2026

Projective Rigidity of $U(k)$-Radial Ricci-Flat Kähler Metrics

We study Kähler immersions of Ricci-flat metrics into finite- or infinite-dimensional complex projective space. Let $L^{\otimes k}\simeq K_X$ over a real-analytic Kähler manifold $X$. Let $ϕ$ be a base potential and $s$ the canonical squared fibre norm on $L^{\oplus k}$. Consider Ricci-flat potentials $ϕ+h(s)$ near the zero section, with $h$ independent of the base point. We prove that local projective inducibility forces the base to be Ricci-flat and the metric to split locally as $X\times\mathbb C^k$. In that case projective inducibility is equivalent to that of the base. The proof combines Calabi's criterion, positivity of the homogeneous fibre-degree blocks, and the singularities of the radial profile. A related obstruction along holomorphic curves excludes every positive multiple of the Taub--NUT metric and Calabi's hyperkähler metrics on $T^{*1,0}\mathbb{CP}^n$. Applications also include the resolved conifold, canonical-bundle metrics in non-Einstein classes, and Cifarelli's rank-two Type-2 metric.

math.DG↗