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Langte Ma

Publications and source records attributed to Langte Ma.

6 recordsLinked to original sources
arXiv2024

Dimension Reduction of Generalized ASD Instantons

We study generalized anti-self-dual instantons defined over Riemannian manifolds equipped with a parallel codimension-$4$ differential form. In particular, for product Riemannian manifolds possessing such a form, we study dimension reduction phenomena, finding a topological criterion for bundles which, when satisfied, allows for a complete characterization of dimension reduction for the corresponding moduli space of generalized ASD instantons. By establishing an integrability result for families of connections, we then deduce explicit descriptions for these moduli spaces, including those of Hermitian Yang--Mills connections, $G_2$-, and $\Spin(7)$-instantons. When one factor in the product is a $4$-manifold, we establish well-behaved compactifications for these moduli spaces.

math.DG↗
arXiv2023

On Counting Flat Connections over $G_2$-Orbifolds

We study the moduli space of $G_2$-instantons on (projectively) flat bundles over torsion-free $G_2$-orbifolds. We prove that the moduli space is compact and smooth at the irreducible locus after adding small and generic holonomy perturbations. Consequently, we define an integer-valued invariant that is invariant under $C^0$-deformation of torsion-free $G_2$-structures. We compute this invariant for some orbifolds that arise in Joyce's construction of compact $G_2$-manifolds

math.DG↗
arXiv2022

Periodic Index Theory and Equivariant Torus Signature

We deduce an index jump formula for first order elliptic complexes over end-periodic manifolds, which generalizes the corresponding result for the DeRham complex. In the case of the anti-self-dual DeRham complex, we define the periodic rho invariant for a class of $4$-manifolds, and identify it with the periodic spectral flow of this complex. As an application, we prove the equivalence (under a mild homological assumption) of two signatures invariants defined by means of Yang-Mills theory and geometric topology respectively for essentially embedded tori in homology $S^1 \times S^3$. We also prove a surgery formula for the singular Furuta-Ohta invariant, which corresponds to a potential exact triangle of singular instanton homology for knots.

math.GT↗
arXiv2020

Surgery and Excision for Furuta-Ohta invariants on Homology $S^1 \times S^3$

We prove a surgery formula and an excision formula for the Furuta-Ohta invariant $\lambda_{FO}$ defined on homology $S^1 \times S^3$, which provides more evidence on its equivalence with the Casson-Seiberg-Witten invariant $\lambda_{SW}$. These formulae are applied to compute $\lambda_{FO}$ of certain families of manifolds obtained as mapping tori under diffeomorphisms of $3$-manifolds. In the course of the proof, we give a complete description of the degree-zero moduli space of ASD instantons on $4$-manifolds of homology $H_*(D^2 \times T^2; \mathbb{Z})$ with a cylindrical end modeled on $[0, \infty) \times T^3$.

math.GT↗
arXiv2020

A Surgery Formula for the Casson-Seiberg-Witten Invariant of Integral Homology $S^1 \times S^3$

We prove a surgery formula of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg-Witten invariants or a manifestation of the surgery exact triangle in $4$-dimensional Seiberg-Witten theory of homology $S^1 \times S^3$. As an application, we compute this invariant for mapping tori of $3$-manifolds under diffeomorphisms of finite order and fixed-point set being a simple closed curve. This computation generalizes the result of Lin-Ruberman-Saveliev.

math.GT↗