arXiv subjects

Shiyu Cao

Publications and source records attributed to Shiyu Cao.

8 recordsLinked to original sources

arXiv2026

Boundary fibers of the holonomy map for complex projective structures

We study the Morgan--Shalen boundary of holonomy representations of complex projective structures on a compact Riemann surface of genus at least two. We prove that every direction defined by a holomorphic quadratic differential with a multiple zero admits distinct projective marked length limits. Together with Dumas's uniqueness theorem for simple zeros, this gives a complete characterization of directions with unique limits. We also establish that no nonzero abelian length function can occur in the holonomy boundary of any such fixed surface, regardless of the limiting quadratic differential. Finally, we characterize all directional boundary fibers by a necessary-and-sufficient analytic condition involving valued representations and the holomorphic jet bundle underlying an oper.

math.DG↗

arXiv2026

Localization of weighted Goldman volume on $\mathrm{SL}(3,\R)$-Hitchin Component

We compute Goldman volume on the moduli space of convex real projective structures on a closed oriented surface of genus at least two, with exponential weight defined by the shifted Blaschke area. For every positive parameter, the integral converges and is a finite sum of inverse powers of that parameter. Its coefficients are intersection numbers involving the bundle of regular cubic differentials on the moduli stack of stable curves. The proof establishes absolute convergence and vanishing signed boundary integrals of the transgression under simultaneous plumbing degeneration. The pushforward of Goldman measure by the shifted Blaschke area therefore has a polynomial density on $[0,\infty)$, and its sublevel volumes are polynomial in the threshold.

math.DG↗

arXiv2026

Asymptotics of the Moduli Spaces of Parabolic $\mathrm{SL}(3,\mathbb C)$-Higgs Bundles

We study the large-scale geometry of the moduli spaces of stable strongly parabolic $\mathrm{SL}(3,\C)$-Higgs bundles on the three-punctured sphere for a family of parabolic weights. We glue the local Toda model metric to a decoupled metric obtained from the spectral line bundle, and perturb the resulting approximate metrics to harmonic metrics adapted to the parabolic structure. We also prove that Hitchin's $L^2$ metric approaches the semi-flat metric exponentially. The metric comparison and the spectral period calculation show that each moduli space is a complete ALG gravitational instanton of type $E_6$. The asymptotic estimates are uniform on compact subintervals of the weight chamber.

math.DG↗

arXiv2026

Stratification of Joint Moduli Spaces of Higgs Bundles

We study the marked analytic moduli space of polystable degree-zero $\mathrm{SL}_r(\C)$-Higgs bundles over Teichmüller space. The connected components of the orbit-type loci defined by the full determinant-one stabilizers form a complex Whitney stratification. The projection to Teichmüller space is a holomorphic submersion on every stratum. Each stratum carries a canonical relative holomorphic symplectic form. These forms determine a vertical holomorphic Poisson bracket on the reduced structure sheaf, whose symplectic leaves are the connected orbit-type strata in the fibers. On each fiber, the Hitchin--Kobayashi correspondence is biholomorphic on every stratum and identifies its hyperkähler structure. The harmonic representatives and vertical hyperkähler tensors vary smoothly along the strata of the total space.

math.AG↗

arXiv2026

An analytic construction of the universal moduli space of $\mathrm{SL}_r(\mathbb C)$-Higgs bundles

We construct the moduli space of marked polystable $\mathrm{SL}_r(\mathbb C)$-Higgs bundles via analytic methods. The resulting space is normal and Hausdorff, and its local models are products of a Teichmüller neighbourhood with a quotient of a quadratic cone. These models preserve the curve parameter and identify the stabilizers. We prove agreement with the complex and unitary gauge-quotient topologies and establish the coarse moduli property for semistable families over reduced analytic bases. The mapping class group and Higgs scaling act holomorphically. Besides, on the stable locus, the moduli spaces that we consider agree with the joint moduli constructed by Collier, Toulisse and Wentworth.

math.AG↗

arXiv2026

Topological quantum color code model on infinite lattice

The color code model is a crucial instance of a Calderbank-Shor-Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local $C^{*}$-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by $\mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2)) \simeq \mathsf{Rep}(D(\mathbb{Z}_2)) \boxtimes \mathsf{Rep}(D(\mathbb{Z}_2))$, which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices. We also establish the existence and uniqueness of the KMS state at every finite inverse temperature and derive an explicit description of the corresponding thermal distribution of stabilizer excitations.

quant-ph↗

arXiv2026

On the Asymptotics of the Volume of Hitchin Moduli Spaces

Let $X$ be a compact Riemann surface of genus $g\geq2$, and let $\M$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $$ \Vol_{g_{L^2}}B_{L^2}(p,R) =\frac{2^{4g-3}π^{9g-9}}{(3g-3)!}\,R^{6g-6}+o(R^{6g-6}) $$ for every $p\in\M$. We also determine the leading asymptotics of Hamiltonian sublevel volumes and exponentially weighted volumes. The proof combines homogeneity of the hyperkähler volume form with metric asymptotics on the regular Hitchin locus. Symplectic reduction and the Prym polarization evaluate the coefficient, which is independently recovered by equivariant localization.

math.DG↗

arXiv2026

On minimally 1-tough $(K_1\cup P_4)$-free graphs

Agraph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases its toughness, where t is a positive real number. It is conjectured that every $(K_1\cup P_4)$-free 1-tough graph is hamiltonian. In this paper, we characterize the structure of minimally 1-tough $(K_1\cup P_4)$-free graphs, and thus show that the above conjecture is true for minimally 1-tough graphs. Furthermore, it is also proved that the Kriesell's conjecture which states that each minimally 1-tough graph has a vertex of degree 2 holds for minimally 1-tough $(K_1\cup P_4)$-free graphs.

math.CO↗