18 problems

For $|q|<1$, define $J_m=(q^m;q^m)_\infty$ and $J_{a,m}=(q^a,q^{m-a},q^m;q^m)_\infty$. Let the two dual Nahm sums be the two triple sums displayed below, with denominator factors…

For $h=1,2,3,4$, let $A^{(h)}(q)$, $B^{(h)}(q)$, $C^{(h)}(q)$, and $D^{(h)}(q)$ denote the four Nahm sums introduced in the source, and let $a$, $b$, and $R_4^{(h)}$ be the corresp…

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathcal{T}_r(\mathbf{x}_r;q)$ denote its tadpole Nahm sum. The sum $\mathcal{T}_r(1,\ldots,1;q)$ is the princ…

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathbf{T}_r$ be its Cartan matrix. For $\mathbf{x}_r=(x_1,\ldots,x_r)$, define the tadpole Nahm sum … The sum…

Let $X$ and $Y$ be Dynkin diagrams of ADET type, with Cartan matrices $\mathcal{C}(X)$ and $\mathcal{C}(Y)$, ranks $r(X)$ and $r(Y)$, and Coxeter numbers $h(X)$ and $h(Y)$. Define…

Shi–Wang's conjecture.

The even and odd Nahm sum conjecture.

Cao–Wang's conjecture.

Let $X$ and $Y$ be Dynkin diagrams of type $ABCDEFGT$, with Cartan matrices $C(X)$ and $C(Y)$. Let $D(X)$ be the diagonal matrix whose $j$-th diagonal entry is $\alpha_j^2/\beta^2$…

Let $\mathcal{C}(D_k)$ be the Cartan matrix of the Dynkin diagram of type $D_k$, and let $f_{A,b,C}(q)$ denote the corresponding Nahm sum. Zagier-duality conjecture. For every…

For each triple $(A,B,C)$ listed in Tables 1–4 and Table 5 (the table labels are those used in the source), let $\mathcal{DL}_i(A,B,C)$ be its lift-dual triple and let the associat…

Let $q$ be a formal variable, and for $n\geq0$ let $(q^4;q^4)_n$ and $(q^m;q^m)_\infty$ denote the usual finite and infinite $q$-Pochhammer products; products with several argument…

Mizuno's duality conjecture. If $(A,B,C,D)$ is a modular quadruple, then $(A^\star,B^\star,C^\star,D^\star)$ is also a modular quadruple.

Modular transformation conjecture.

Duality conjecture. If the Nahm sum associated with $(A,b,c,d)$ is modular, then the Nahm sum associated with $(A^*,b^*,c^*,d^*)$ is also modular. The conjecture proposes a general…

General tadpole modularity conjecture. For even $n=2k\geq 2$,

Penn–Calinescu–Wang modularity conjecture. The character $q^a\chi_0(1,\dots,1)$ is modular for some rational number $a$.

Let $q$ be a formal variable, and let $J_{a,b}$ and $J_a$ denote the Jacobi-product notation used in the source. Vlasenko–Zwegers' conjecture. The following identities are conjectu…