arXiv subjects

Lutian Zhao

Publications and source records attributed to Lutian Zhao.

9 recordsLinked to original sources

arXiv2026

Topological String Blowup Equations via Stable Pairs

The blowup equations of Huang, Sun and Wang are bilinear relations satisfied by the refined topological string partition function of a local Calabi--Yau 3-fold. We prove the blowup equations for the local Hirzebruch surfaces $\operatorname{Tot}_{\mathbb F_\ell}K_{\mathbb F_\ell}$, $0\leq\ell\leq2$, as identities of torus-equivariant symmetrized $K$-theoretic stable pair invariants; for $\ell=2$ the invariants are localized indices. The proof identifies the stable pair vertex sum, after division by the fibre contribution, with the equivariant Euler characteristic of $(\det\mathcal V)^\ell$ on the moduli space of framed rank $2$ sheaves on $\mathbb P^2$. The blowup formulas of Nakajima--Yoshioka for framed sheaves then yield the unity and vanishing equations. For local $\mathbb P^2$ we obtain the blowup equations conditionally on two explicitly stated conjectures. We also state the Huang--Sun--Wang conjecture for general local Calabi--Yau 3-folds in the language of stable pairs, and formulate stable pair conjectures for the local rational elliptic surface involving the $E_8$ lattice.

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arXiv2025

Level structures on parahoric torsors and complete integrability

For a smooth complex algebraic curve $X$ and a reduced effective divisor $D$ on $X$, we introduce a notion of $D$-level structure on parahoric $\mathcal{G}_{\boldsymbol θ}$-torsors over $X$, for any connected complex reductive Lie group $G$. A moduli space of parahoric $\mathcal{G}_{\boldsymbol θ}$-torsors equipped with a $D$-level structure is constructed and we identify a canonical moment map with respect to the action of a level group on this moduli space. This action extends to a Poisson action on the cotangent, thus inducing a Poisson structure on the moduli space of logahoric $\mathcal{G}_{\boldsymbol θ}$-Higgs torsors on $X$. A study of the generic fibers of the parahoric Hitchin fibration of this moduli space identifies them as abelian torsors and introduces new algebraically completely integrable Hamiltonian Hitchin systems in this parahoric setting. We show that this framework generalizes, among other, the integrable system of Beauville and recovers the classical Gaudin model in its simplest form, the space of periodic KP elliptic solitons and the elliptic Calogero--Moser system, thus demonstrating that the logahoric Hitchin integrable system unifies many integrable systems with regular singularities under a single geometric framework.

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arXiv2023

Poisson Structures on Moduli Spaces of Higgs Bundles over Stacky Curves

We demonstrate the construction of Poisson structures via Lie algebroids on moduli spaces of twisted stable Higgs bundles over stacky curves. The construction provides new examples of Poisson structures on such moduli spaces. Special attention is paid at moduli spaces of parabolic Higgs bundles over a root stack.

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arXiv2023

Gopakumar-Vafa Invariants and Macdonald Formula

We use a derived constructible Chow exponential and prove that its coefficients are perverse minimal extensions from reduced cycles. With compatible orientations, we formulate the cohomological PT/GV relation by this exponential. If the relation holds over reduced cycles, its extension to the full Chow varieties is equivalent to the absence of nonreduced strict supports, and it implies the refined and numerical formulas. For local $\Pp^2$ and $0\le n\le d+1$, we identify the stable-pair space with the smooth relative Hilbert scheme and the vanishing-cycle sheaf with its intersection complex. We determine the first reducible summand and give del Pezzo examples. In degree $(2,4)$, we determine the incidence--ribbon union, its canonical PT critical germ, and the attachment triangle, and we formulate the primitive degree-two KKV comparison.

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arXiv2023

Tame parahoric nonabelian Hodge correspondence on curves

The nonabelian Hodge correspondence for vector bundles over noncompact curves is adequately described by implementing a weighted filtration on the objects involved. In order to establish a full correspondence between a Dolbeault and a de Rham space for a general complex reductive group $G$, we introduce torsors given by parahoric group schemes in the sense of Bruhat--Tits. Combined with existing results on the Riemann--Hilbert correspondence for logarithmic parahoric connections, this gives a full nonabelian Hodge correspondence from Higgs bundles to fundamental group representations over a noncompact curve beyond the $\text{GL}_n(\mathbb{C})$-case.

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arXiv2023

Logahoric Higgs Torsors for a Complex Reductive Group

In this article, a logahoric Higgs torsor is defined as a parahoric torsor with a logarithmic Higgs field. For a connected complex reductive group $G$, we introduce a notion of stability for logahoric $\mathcal{G}_{\boldsymbolθ}$-Higgs torsors on a smooth algebraic curve $X$, where $\mathcal{G}_{\boldsymbolθ}$ is a parahoric group scheme on $X$. In the case when the group $G$ is the general linear group ${\rm GL}_n$, we show that the stability condition of a parahoric torsor is equivalent to the stability of a parabolic bundle. A correspondence between semistable logahoric $\mathcal{G}_{\boldsymbolθ}$-Higgs torsors and semistable equivariant logarithmic $G$-Higgs bundles allows us to construct the moduli space explicitly. This moduli space is shown to be equipped with an algebraic Poisson structure.

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arXiv2020

Monodromy of Rank 2 Parabolic Hitchin Systems

We study the monodromy of the Hitchin fibration for moduli spaces of parabolic G-Higgs bundles in the cases when G=SL(2,R), GL(2,R) and PGL(2,R) A calculation of the orbits of the monodromy with Z2-coefficients provides an exact count of the components of the moduli spaces for these groups.

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arXiv2020

The Beauville-Narasimhan-Ramanan correspondence for twisted Higgs $V$-bundles and components of parabolic $\text{Sp}(2n,\mathbb{R})$-Higgs moduli Spaces

We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left( 2n,\mathbb{R} \right)$-Higgs bundles with fixed parabolic structure.

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arXiv2020

Topological invariants of parabolic $G$-Higgs bundles

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{ü}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure.

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