arXiv subjects

Liuquan Wang

Publications and source records attributed to Liuquan Wang.

At least 19 recordsLinked to original sources

arXiv2026

Proofs of Some Kanade--Russell Mod 12 Conjectures

Kanade and Russell conjectured seventeen Rogers--Ramanujan type identities of modulus 12. Eleven of these identities involving triple sums were proved by Bringmann--Jennings-Shaffer--Mahlburg and by Rosengren. Motivated by these works and using similar methods, we settle all the six remaining conjectures including two triple sum identities labeled $I_{5a}$ and $I_{6a}$ originated from Russell's thesis and four quadruple sum identities labeled 7, 7a, 8 and 8a. Our proof of the triple sum identities combines linear recurrences, $q$-difference equations, and $q$-series summation formulas. For the quadruple sum identities, we represent the sums as contour integrals whose integrands are infinite products and evaluate them by residue calculus. The resulting residues reduce to single sum cubic basic hypergeometric series, and we are able to express them as infinite products.

math.CO↗

arXiv2026

Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities

In 2007, Zagier identified twelve families of candidates for rank-three modular Nahm sums and established the modularity of three of them. The remaining cases were subsequently confirmed by Wang. Zagier's duality observation indicates that the Nahm sums associated with the duals of these examples might still be modular. We confirm that this is indeed true. The modularity of the dual of the sixth example was previously established by Milas and Wang, while that of the dual of the eleventh example was partially established by the present authors. We settle all remaining cases, thereby establishing the modularity of every defined dual in Zagier's rank-three list. We achieve this by proving new Rogers--Ramanujan type identities that express the relevant Nahm sums as finite sums of infinite products. Along the way we discover two new families of rank-three Nahm sums. As applications, we prove some Nahm sum identities conjectured by Cao--Wang, Li and the present authors.

math.NT↗

arXiv2026

Modular Nahm Sums for the Inverse Cartan Matrix of Type $D_r$

For $r\geq 3$ we denote by $\mathcal{C}(D_r)$ the Cartan matrix of type $D_r$. Recently, Sun and Wang conjectured a Rogers--Ramanujan type identity for the Nahm sum associated with $\mathcal{C}(D_r)^{-1}$ and the zero vector. They further conjecture that there exist $r-1$ companion modular Nahm sums associated with nonzero vectors. We partially prove this conjecture by constructing $\lfloor (r+4)/2\rfloor$ modular Nahm sums for $\mathcal{C}(D_r)^{-1}$. To prove their modularity, we utilize the method of Bailey pairs to establish various Rogers--Ramanujan type identities. In particular, we confirm their conjectural identity.

math.NT↗

arXiv2026

Some new modular Nahm sums of ranks 3 and 4

We discover six new families of modular Nahm sums in ranks 3 and 4. Two of them are rank three sums obtained by modifying two of Zagier's rank three examples. Three rank four families are derived by applying the lift-dual operation to the rank three tadpole Nahm sums studied by Milas and Wang, while the other rank four family is found by the constant term method. To prove modularity, we establish Rogers-Ramanujan type identities that express these Nahm sums as infinite products which are modular.

math.NT↗

arXiv2026

Proofs of some conjectures of Okazaki and Smith on line defect half-indices of ${\rm SU}(N)$ Chern-Simons theories

Okazaki and Smith discovered many elegant formulas expressing some matrix integrals as some celebrated $q$-series such as the Rogers--Ramanujan functions or Jacobi theta functions. These integrals arise as Wilson line defect half-indices of 3d $\mathcal{N}=2$ supersymmetric ${\rm SU}(N)$ Chern-Simons theories. We evaluate them by carefully calculating the constant terms of some infinite products. Along the way we use some crucial facts about antisymmetric multivariate formal Laurent series. Consequently, we prove three general conjectures of Okazaki and Smith which provide explicit formulas for half indices of the ${\rm SU}(N)_{-N-k}$ ($k=0,1/2,1$) Chern-Simons theories. During the process, we extend these ${\rm SU}(N)$ formulas to include one additional parameter. Furthermore, we generalize the ${\rm SU}(N)_{-N-1/2}$ and ${\rm SU}(N)_{-N-1}$ conjectures by calculating the corresponding half-indices of Wilson lines of arbitrary charge. As a special instance of our generalizations, we also confirm the ${\rm SU}(3)_{-4}$ conjecture of Okazaki and Smith.

hep-th↗

arXiv2025

Nahm sum identities for Cartan matrices of type $D_k$

Around 2007, Warnaar proved four identities related to Nahm sums associated with twice the inverse of the Cartan matrix of type $D_k$. Three of these had been conjectured by Flohr, Grabow, and Koehn, while special cases of two of the identities were first conjectured in 1993 by Kedem, Klassen, McCoy, and Melzer. Warnaar's proof relies on a multi-sum identity from Andrews' proof of the Andrews-Gordon identities. We give a new proof of all four identities using the theory of Bailey pairs. Furthermore, we establish a parametric generalization of two of the identities and provide two distinct proofs of this generalization.

math.CO↗

arXiv2025

Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples

We find nine new sets of rank four Nahm sums associated with nine different numeric matrices which are likely to be modular. They are discovered by applying the lift-dual operation to some modular rank three Nahm sums in the works of Zagier and the authors. We prove the modularity of four sets of these Nahm sums by establishing Rogers--Ramanujan type identities which express them as modular infinite products. We use various $q$-series techniques including the constant term method and Bailey pairs to prove these identities. Meanwhile, we present some conjectural identities expressing several Nahm sums as modular infinite products.

math.NT↗

arXiv2025

Proofs of Two Conjectural Identities on Partial Nahm Sums

Recently, Wang and Zeng investigated modularity of partial Nahm sums and discovered 14 modular families of such sums. They confirmed modularity for 13 families and proposed a conjecture consisting of two Rogers--Ramanujan type identities for the remaining family. We prove these conjectural identities in two steps. First, employing a transformation formula involving two Bailey pairs, we transform the partial Nahm sums into some specific Hecke-type series. Second, using two distinct approaches, we convert these Hecke-type series to the desired modular infinite products.

math.NT↗

arXiv2025

Modularity of tadpole Nahm sums in ranks 4 and 5

Around 2016, Calinescu, Milas and Penn conjectured that the rank $r$ Nahm sum associated with the $r\times r$ tadpole Cartan matrix is modular, and they provided a proof for $r=2$. The $r=3$ case was recently resolved by Milas and Wang. We prove this conjecture for the next cases $r=4,5$. We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers--Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions.

math.NT↗

arXiv2025

Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$

Mizuno provided 15 examples of generalized rank three Nahm sums with symmetrizer $\mathrm{diag}(1,2,2)$ which are conjecturally modular. Using the theory of Bailey pairs and some $q$-series techniques, we establish a number of triple sum Rogers--Ramanujan type identities. These identities confirm the modularity of all of Mizuno's examples except that two Nahm sums are sums of modular forms of weights $0$ and $1$. We also prove Mizuno's conjectural modular transformation formulas for two vector-valued functions consisting of Nahm sums with symmetrizers $\mathrm{diag}(1,1,2)$ and $\mathrm{diag}(1,2,2)$.

math.NT↗

arXiv2025

Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums

Let $A$ be a $r\times r$ rational nonzero symmetric matrix, $B$ a rational column vector, $C$ a rational scalar. For any integer lattice $L$ and vector $v$ of $\mathbb{Z}^r$, we define Nahm sum on the lattice coset $v+L\in \mathbb{Z}^r/L$: \begin{align*}\label{eq-lattice-sum} f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T} \in v+L} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}. \end{align*} If $L$ is a full rank lattice and a proper subset of $\mathbb{Z}^r$, then we call $f_{A,B,C,v+L}(q)$ a rank $r$ partial Nahm sum. When the rank $r=1$, we find eight modular partial Nahm sums using some known identities. When the rank $r=2$ and $L$ is one of the lattices $\mathbb{Z}(2,0)+\mathbb{Z}(0,1)$, $\mathbb{Z}(1,0)+\mathbb{Z}(0,2)$ or $\mathbb{Z}(2,0)+\mathbb{Z}(0,2)$, we find 14 types of symmetric matrices $A$ such that there exist vectors $B,v$ and scalars $C$ so that the partial Nahm sum $f_{A,B,C,v+L}(q)$ is modular. We establish Rogers--Ramanujan type identities for the corresponding partial Nahm sums which prove their modularity.

math.NT↗

arXiv2025

Counterexamples to Zagier's Duality Conjecture on Nahm Sums

Given any positive integer $r$, Nahm's problem is to determine all $r\times r$ rational positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is modular. Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums which are modular while their duals are not modular. This provides counterexamples to Zagier's duality conjecture.

math.NT↗

arXiv2024

Modularity of Some Nahm Sums as Vector-valued Functions

Zagier observed that modular Nahm sums associated with the same matrix may form a vector-valued modular function on some congruence subgroup. We establish modular transformation formulas for several families of Nahm sums by viewing them as vector-valued functions, and thereby we show that they are indeed modular on the congruence subgroup $Γ_0(N)$ with $N=1,2,3,4$. In particular, we prove two transformation formulas discovered by Mizuno related to the Kanade--Russell mod 9 conjecture and Capparelli's identities. We also establish vector-valued transformation formulas for some theta series. As applications, we give modular transformation formulas for various families of Nahm sums involving those in the Andrews--Gordon identities and Bressoud's identities.

math.NT↗

arXiv2024

Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation

Around 2007, Zagier discovered some rank two and rank three Nahm sums, and their modularity have now all been confirmed. Zagier also observed that the dual of a modular Nahm sum is likely to be modular. This duality observation motivates us to discover some new modular rank three Nahm sums by a lift-dual operation. We first lift Zagier's rank two Nahm sums to rank three and then calculate their dual, and we show that these dual Nahm sums are indeed modular. We achieve this by establishing the corresponding Rogers--Ramanujan type identities, which express these Nahm sums as modular infinite products.

math.NT↗

arXiv2024

Identities on Zagier's rank two examples for Nahm's problem

Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric $r\times r$ matrix, $B$ a vector of length $r$, and $C$ a scalar. Nahm's problem is to describe all such $A,B$ and $C$ with rational entries for which a specific $r$-fold $q$-hypergeometric series (denoted by $f_{A,B,C}(q)$) involving the parameters $A,B,C$ is modular. When the rank $r=2$, Zagier provided eleven sets of examples of $(A,B,C)$ for which $f_{A,B,C}(q)$ is likely to be modular. We present a number of Rogers--Ramanujan type identities involving double sums, which give modular representations for Zagier's rank two examples. Together with several known cases in the literature, we verified ten of Zagier's examples and give conjectural identities for the remaining example.

math.NT↗

arXiv2024

Mizuno's rank three Nahm sums I: identities of index $(1,1,2)$

Mizuno provided 19 examples of generalized rank three Nahm sums with symmetrizer $\mathrm{diag}(1,1,2)$ which are conjecturally modular. We confirm their modularity by establishing Rogers--Ramanujan type identities of index $(1,1,2)$ for these examples. We first reduce these Nahm sums to some double sums or single sums, and then we use known results or apply the theory of Bailey pairs to prove the desired identities. Meanwhile, we generalize some triple sum identities to general multi-sum identities.

math.NT↗

arXiv2024

Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms

It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to $q$-binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the $q$-binomial cases.

math.NT↗

arXiv2023

Proofs of Mizuno's Conjectures on Generalized Rank Two Nahm Sums

Recently, Mizuno studied generalized Nahm sums associated with symmetrizable matrices. He provided 14 sets of candidates of modular Nahm sums in rank two and justified four of them. We prove the modularity for eight other sets of candidates and present conjectural formulas for the remaining two sets of candidates. This is achieved by finding Rogers-Ramanujan type identities associated with these Nahm sums. We also prove Mizuno's conjectural modular transformation formula for a vector-valued function consists of Nahm sums. Meanwhile, we find some new non-modular identities for some other Nahm sums associated with the matrices in Mizuno's candidates.

math.NT↗