arXiv subjects

Qingsheng Zhang

Publications and source records attributed to Qingsheng Zhang.

11 recordsLinked to original sources

arXiv2026

Structure of higher-genus open-closed Gromov--Witten theory of $\mathcal{O}_{\mathbb{P}^{1}}(p-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-p-1)$

We study the closed and open Gromov--Witten potentials of the toric Calabi--Yau threefold $$ X_p=\operatorname{Tot}\bigl(\mathcal{O}_{\mathbb{P}^{1}}(p-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-p-1)\bigr),\qquad p\geq 2. $$ We prove closed and open mirror symmetry under a nonvanishing condition on the torus weights, relating these potentials to topological recursion on the mirror curves. We establish polynomial structures for both the higher-genus closed potentials and the stable open potentials. We also establish double-scaling limits for topological recursion on the mirror curves. In particular, our results for the closed potentials prove the higher-genus ansatz and the double-scaling conjecture of Caporaso--Griguolo--Mariño--Pasquetti--Seminara.

math.AG↗

arXiv2026

Wall-crossing formula and genus-one Virasoro conjecture for Fano complete intersections

The Virasoro conjecture predicts a set of universal relations among all genera Gromov--Witten invariants of any smooth projective variety. The conjecture is well understood for semisimple theories, but remains largely open in the non-semisimple setting. We prove the genus-one Virasoro conjecture on the ambient state space of smooth Fano complete intersections in projective space. For most of these complete intersections, the big quantum cohomology is nowhere semisimple. We also generalize the wall-crossing formula for quasimap invariants with weighted markings to the equivariant twisted setting, allowing descendant insertions at light markings. Together with genus-one quantum Lefschetz for quasimaps with light markings, this wall-crossing formula provides the key bridge from the Gromov--Witten theory of the complete intersection to the semisimple equivariant twisted theory of the projective space.

math.AG↗

arXiv2026

Generalized Kontsevich model, topological recursion, and $r$-spin theory

By employing polynomial-reduced KP integrability, combined with the string equation, this work establishes explicit relationships between the generalized Kontsevich model, the topological recursion of the spectral curve, and the geometry of moduli spaces of $r$-spin curves. For the generalized Kontsevich model with a polynomial potential, we derive an explicit formulation and provide a proof of these widely expected correspondences. Furthermore, the method is extended to the cases with admissible deformed potentials, where the corresponding geometric theory is a deformed version of $r$-spin theory.

math-ph↗

arXiv2025

Virasoro constraints for topological recursion

This is the second paper in a series on {\it Virasoro constraints for Cohomological Field Theory}. We derive the ancestor Virasoro constraints for the topological recursion (TR) for an arbitrary spectral curve and establish the descendent Virasoro constraints for spectral curves satisfying certain conditions. For higher-genus curves, we further establish the corresponding ancestor and descendent Virasoro constraints for the associated non-perturbative generating series. We present several examples that illustrate the comparison between the descendent Virasoro constraints for TR descendent invariants and the original Virasoro constraints for geometric descendent invariants.

math-ph↗

arXiv2025

A generalization of the Witten conjecture through spectral curve

We propose a generalization of the Witten conjecture, which connects a descendent enumerative theory with a specific reduction of KP integrable hierarchy. Our conjecture is realized by two parts: Part I (Geometry) establishes a correspondence between the geometric descendent potential (apart from ancestors) and the topological recursion of specific spectral curve data $(Σ, x,y)$; Part II (Integrability) claims that the TR descendent potential, defined at the boundary points of the spectral curve (where $dx$ has poles), is a tau-function of a certain reduction of the multi-component KP hierarchy. In this paper, we show the geometric part of the conjecture for any formal descendent theory by using a generalized Laplace transform. Subsequently, we prove the integrability conjecture for the one-boundary cases. As applications, we generalize and prove the $r$KdV integrability of negative $r$-spin theory conjectured by Chidambaram, Garcia-Failde and Giacchetto. We also show the KdV integrability of the total descendent potential associated with the Hurwitz space $M_{1,1}$, whose Frobenius manifold was initially introduced by Dubrovin.

math-ph↗

arXiv2025

Cohomological Field Theory with vacuum and its Virasoro constraints

This is the first part of a series of papers on {\it Virasoro constraints for Cohomological Field Theory (CohFT)}. For a CohFT with vacuum, we introduce the concepts of $S$-calibration and $ν$-calibration. Then, we define the (formal) total descendent potential corresponding to a given calibration. Finally, we introduce an additional structure, namely homogeneity, for both the CohFT and the calibrations. After these preliminary introductions, we propose two crucial conjectures: (1) the ancestor version of the Virasoro conjecture for the homogeneous CohFT with vacuum; and (2) the generalized Virasoro conjecture for the (formal) total descendent potential of a calibrated homogeneous CohFT. We verify the genus-0 part of these conjectures and deduce a simplified form of the genus-1 part of these conjectures for arbitrary CohFTs. Additionally, we prove the full conjectures for semisimple CohFTs. As applications, our results yield the Virasoro constraints for the deformed negative $r$-spin theory. Moreover, by applying the Virasoro constraints, we discover an extension of Grothendieck's dessins d'enfants theory which is widely studied in the literature.

math-ph↗

arXiv2025

BKP-Affine Coordinates and Emergent Geometry of Generalized Brézin-Gross-Witten Tau-Functions

Following Zhou's framework, we consider the emergent geometry of the generalized Brézin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP hierarchy. More precisely, we construct a spectral curve together with its special deformation, and show that the Eynard-Orantin topological recursion on this spectral curve emerges naturally from the Virasoro constraints for the generalized BGW tau-functions. Moreover, we give the explicit expressions for the BKP-affine coordinates of these tau-functions and their generating series. The BKP-affine coordinates and the topological recursion provide two different approaches towards the concrete computations of the connected $n$-point functions. Finally, we show that the quantum spectral curve of type $B$ in the sense of Gukov-Sułkowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.

math-ph↗

arXiv2023

On a new proof of the Okuyama--Sakai conjecture

In [41] Okuyama and Sakai gave a conjectural equality for the higher genus generalized Brézin--Gross--Witten (BGW) free energies. In a recent work [46] we established the Hodge-BGW correspondence on the relationship between certain special cubic Hodge integrals and the generalized BGW correlators, and a proof of the Okuyama--Sakai conjecture was also given ${\it ibid}$. In this paper, we give a new proof of the Okuyama--Sakai conjecture by a further application of the Dubrovin--Zhang theory for the KdV hierarchy.

math-ph↗

arXiv2022

On the Hodge-BGW correspondence

We establish an explicit relationship between the partition function of certain special cubic Hodge integrals and the generalized Brezin--Gross--Witten (BGW) partition function, which we refer to as the Hodge-BGW correspondence. As an application, we obtain an ELSV-like formula for generalized BGW correlators.

math.AG↗

arXiv2019

On Itzykson-Zuber Ansatz

We apply the renormalized coupling constants and Virasoro constraints to derive the Itzykson-Zuber Ansatz on the form of the free energy in topological 2D gravity. We also treat the topological 1D gravity and the Hermitian one-matrix models in the same fashion. Some uniform behaviors are discovered in this approach.

math-ph↗

arXiv2016

On the Dual of the Coulter-Matthews Bent Functions

For any bent function, it is very interesting to determine its dual function because the dual function is also bent in certain cases. For $k$ odd and $\gcd(n, k)=1$, it is known that the Coulter-Matthews bent function $f(x)=Tr(ax^{\frac{3^k+1}{2}})$ is weakly regular bent over $\mathbb{F}_{3^n}$, where $a\in\mathbb{F}_{3^n}^{*}$, and $Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3$ is the trace function. In this paper, we investigate the dual function of $f(x)$, and dig out an universal formula. In particular, for two cases, we determine the formula explicitly: for the case of $n=3t+1$ and $k=2t+1$ with $t\geq 2$, the dual function is given by $$Tr\left(-\frac{x^{3^{2t+1}+3^{t+1}+2}}{a^{3^{2t+1}+3^{t+1}+1}}-\frac{x^{3^{2t}+1}}{a^{-3^{2t}+3^{t}+1}}+\frac{x^{2}}{a^{-3^{2t+1}+3^{t+1}+1}}\right);$$ and for the case of $n=3t+2$ and $k=2t+1$ with $t\geq 2$, the dual function is given by $$Tr\left(-\frac{x^{3^{2t+2}+1}}{a^{3^{2t+2}-3^{t+1}+3}}-\frac{x^{2\cdot3^{2t+1}+3^{t+1}+1}}{a^{3^{2t+2}+3^{t+1}+1}}+\frac{x^2}{a^{-3^{2t+2}+3^{t+1}+3}}\right).$$ As a byproduct, we find two new classes of ternary bent functions with only three terms. Moreover, we also prove that in certain cases $f(x)$ is regular bent.

cs.IT↗