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Chenglang Yang

Publications and source records attributed to Chenglang Yang.

At least 19 recordsLinked to original sources
arXiv2026

Quasi-polynomiality and $N$-point functions of single connected leaky completed Hurwitz numbers

In this paper, we study the structures of single connected $k$-leaky $(r+1)$-completed Hurwitz numbers. We first prove that, for fixed genus, after extracting an explicit product of Pochhammer type factors, the stable connected leaky Hurwitz numbers for partition $\mu$ are polynomials in the quotients $[\mu_i]$ modulo $k+r$. We then give a closed formula for the generating function of single connected leaky Hurwitz numbers, which can be used to study the genus dependence of leaky Hurwitz numbers.

math-ph↗
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↗
arXiv2026

The $n$-Point Function of $t$-Core Partitions and Topological Vertex

In this paper, we study the $n$-point function of $t$-core partitions. The main tool is the topological vertex, originally developed to study the topological string theory for toric Calabi--Yau 3-folds. By virtue of the topological vertex, we introduce the $q$-deformed $n$-point function that generalizes both the ordinary $n$-point function of all integer partitions studied by Bloch--Okounkov and $t$-core partition case treated here. As a consequence, we provide a closed formula for the $n$-point function of $t$-core partitions in terms of theta functions, and prove that the corresponding correlation functions are quasimodular forms.

math-ph↗
arXiv2025

Total partition function with fermionic number fluxes of local toric Calabi--Yau threefold and KP integrability

Aganagic, Dijkgraaf, Klemm, Mari\~{n}o and Vafa \cite{adkmv} predicted that the open string partition function on a smooth toric Calabi--Yau threefold should be a tau-function of multi-component KP hierarchy after considering the contributions from nonzero fermion number fluxes through loops in the toric diagram. In this paper, we prove their prediction in the case of local toric Calabi--Yau threefolds. More precisely, we construct the total partition function of local toric Calabi--Yau threefolds using an operator on the fermionic Fock space which we developed in an earlier work \cite{wyz} to represent the topological vertex, and show that the total partition function is the trace of an operator on the fermionic Fock space. As an application, we prove the KP integrability of the total partition function.

math-ph↗
arXiv2025

The structures of simple Hurwitz numbers and monotone Hurwitz numbers with varying genus

We study the structures of ordinary simple Hurwitz numbers and monotone Hurwitz numbers with varying genus. More precisely, we prove that when the ramification type is fixed and the genus is treated as a variable, the connected monotone Hurwitz number is a linear combination of products of exponentials and polynomials, and the ordinary simple Hurwitz number is a linear combination of exponentials. Using these structural properties, we also derive the large genus asymptotics of these two kinds of Hurwitz numbers. As a result, we prove one conjecture, and disprove another, both proposed by Do, He and Robertson.

math.CO↗
arXiv2025

The bilinear fermionic form for KP and BKP hierarchies

For a tau-function of the KP or BKP hierarchy, we introduce the notion of lifting operator and derive an equation connecting the corresponding fermionic two-point function and fermionic one-point function through the lifting operator. This provides an effective approach to determine the fermionic two-point function of the tau-function from the lifting operator and the fermionic one-point function. As practical applications, we derive concise formulas for the fermionic two-point functions of several models, like the $r$-spin model and the Br{\' e}zin--Gross--Witten model, which respectively serve as examples for KP and BKP tau-functions.

math-ph↗
arXiv2024

Correlation Function of Self-Conjugate Partitions: $q$-Difference Equation and Quasimodularity

In this paper, we study the uniform measure for the self-conjugate partitions. We derive the $q$-difference equation which is satisfied by the $n$-point correlation function related to the uniform measure. As applications, we give explicit formulas for the one-point and two-point functions, and study their quasimodularity. Motivated by this, we also prove the quasimodularity of the general $n$-point function using a combinatorial method. Finally, we derive the limit shape of self-conjugate partitions under the Gibbs uniform measure and compare it to the leading asymptotics of the one-point function.

math-ph↗
arXiv2024

On $\mathbb N$-Coefficient Binomial Polynomiality of Hurwitz Numbers and Generalized Dessin Counting

In this paper, we study a certain type of Hurwitz numbers which count branched covers over the Riemann sphere admitting several branch points with fixed ramification types, one branch point with a fixed number of preimages, and one branch point with an arbitrary ramification type. We prove that the dependence of this kind of Hurwitz numbers on parts of the ramification type over the last point is a polynomial. Moreover, when expanding this polynomial in terms of products of binomial coefficients, we show that the coefficients are always non-negative integers via a pure combinatorial method. Our result generalizes the polynomiality in several models, including the one-part double Hurwitz numbers studied by Goulden-Jackson-Vakil, the one-part double Hurwitz numbers with completed cycles studied by Shadrin-Spitz-Zvonkine, and the generalized dessin counting.

math.CO↗
arXiv2024

Schr\"oder Paths, Their Generalizations and Knot Invariants

We study some kinds of generalizations of Schr\"oder paths below a line with rational slope and derive the $q$-difference equations that are satisfied by their generating functions. As a result, we establish a relation between the generating function of generalized Schr\"oder paths with backwards and the wave function corresponding to colored HOMFLY-PT polynomials of torus knot $T_{1,f}$. We also give a combinatorial proof of a recent result by Sto\v{s}i\'c and Su{\l}kowski, in which the standard generalized Schr\"oder paths are related to the superpolynomial of reduced colored HOMFLY-PT homology of $T_{1,f}$.

math.CO↗
arXiv2024

Hook-Lengths, Symplectic/Orthogonal Contents and Amdeberhan's Conjectures

The symplectic/orthogonal contents of partitions are related to the dimensions of irreducible representations of symplectic/orthogonal groups. In 2012, motivated by Nekrasov--Okounkov's hook-length formula and Stanley's hook-content formula, Amdeberhan proposed several conjectures about infinite product formulas for certain generating functions of hook-lengths and symplectic/orthogonal contents. Some special cases of his conjectures were recently proved by Amdeberhan, Andrews and Ballantine. In this paper, we prove the general cases of Amdeberhan's conjectures.

math.CO↗
arXiv2024

On a Proof of the ADKMV Conjecture -- $3$-KP Integrability of the Topological Vertex

We present a mathematical proof of the Aganagic-Dijkgraaf-Klemm-Mari\~no-Vafa Conjecture proposed in 2006, which states that the generating function of the topological vertex, i.e., the generating function of the open Gromov-Witten invariants of $\mathbb{C}^3$, satisfies the $3$-component KP hierarchy. In our proof we introduce a boson-fermionic field assignment which generalizes the well-known boson-fermion correspondence. The proof also works for the generalization to the framed topological vertex case conjectured by Deng and Zhou. As a consequence, open Gromov-Witten theory of all smooth toric Calabi-Yau threefolds are controlled by the multi-component KP hierarchy.

math-ph↗
arXiv2023

On two families of Nekrasov-Okounkov type formulas

In this paper, we use the vacuum expectation value formula of the topological vertex and its rotation symmetry to derive two families of Nekrasov-Okounkov type formulas. Each family of formulas depends on $2N+1$ parameters for a positive integer $N$.

math-ph↗
arXiv2023

A remark on certain restricted plane partitions and crystal melting model

In this paper, we provide formulas to calculate the partition functions of two types of plane partitions using the crystal melting model introduced by Okounkov, Reshetikhin and Vafa. As applications, we obtain a product formula for the partition function of the plane partitions with a limit boundary. A corollary of this formula is the demonstration of the equivalence between this partition function and the open string amplitude of the double-$\mathbb{P}^1$ model. We also derive a product formula for the partition function of symmetric plane partitions with a limit boundary along the $z$-axis direction.

math-ph↗
arXiv2023

BKP-Affine Coordinates and Emergent Geometry of Generalized Br\'ezin-Gross-Witten Tau-Functions

Following Zhou's framework, we consider the emergent geometry of the generalized Br\'ezin-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-Su{\l}kowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.

math-ph↗
arXiv2022

Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers

Given a tau-function $\tau(t)$ of the BKP hierarchy satisfying $\tau(0)=1$, we discuss the relation between its BKP-affine coordinates on the isotropic Sato Grassmannian and its BKP-wave function. Using this result, we formulate a type of Kac-Schwarz operators for $\tau(t)$ in terms of BKP-affine coordinates. As an example, we compute the affine coordinates of the BKP tau-function for spin single Hurwitz numbers with completed cycles, and find a pair of Kac-Schwarz operators $(P,Q)$ satisfying $[P,Q]=1$. By doing this, we obtain the quantum spectral curve for spin single Hurwitz numbers.

math-ph↗
arXiv2022

Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers

We derive an explicit formula for the connected $(n,m)$-point functions associated to an arbitrary diagonal tau-function $\tau_f(\boldsymbol{t}^+,\boldsymbol{t}^-)$ of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. Then for fixed $\boldsymbol{t}^-$, we compute the KP-affine coordinates of $\tau_f(\boldsymbol{t}^+,\boldsymbol{t}^-)$. As applications, we present a unified approach to compute various types of connected double Hurwitz numbers, including the ordinary double Hurwitz numbers, the double Hurwitz numbers with completed $r$-cycles, and the mixed double Hurwitz numbers. We also apply this method to the computation of the stationary Gromov-Witten invariants of $\mathbb P^1$ relative to two points.

nlin.SI↗
arXiv2022

Action of $W$-type operators on Schur functions and Schur Q-functions

In this paper, we investigate a series of W-type differential operators, which appear naturally in the symmetry algebras of KP and BKP hierarchies. In particular, they include all operators in the W-constraints for tau functions of higher KdV hierarchies which satisfy the string equation. We will give simple uniform formulas for actions of these operators on all ordinary Schur functions and Schur's Q-functions. As applications of such formulas, we will give new simple proofs for Alexandrov's conjecture and Mironov-Morozov's formula, which express the Br\'{e}zin-Gross-Witten and Kontsevich-Witten tau-functions as linear combinations of Q-functions with simple coefficients respectively.

math-ph↗