30 problems

Let $A$ be a rational symmetric positive-definite $r\times r$ matrix. For $B\in\mathbb{Q}^r$ and $C\in\mathbb{Q}$, define … A triple $(A,B,C)$ is called modular when $f_{A,B,C}(q)$…

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…

Zagier's duality conjecture. The image $\mathcal{D}(A,B,C)$ is still a modular triple.

Modular transformation conjecture.

Let $F^{(1)}(q),\ldots,F^{(4)}(q)$ be the four dual Nahm sums defined from the rank-three quadratic form in the source, with $J_m=(q^m;q^m)_\infty$ and…

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…

Mizuno's duality conjecture. The dual quadruple $(A^*,B^*,C^*,D^*)$ should also be modular. The supplied status evidence reports that Wang discovered counterexamples to this conjec…

Vlasenko–Zwegers conjecture. The two Nahm sums satisfy

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…

For $|q|<1$, define $J_m=(q^m;q^m)_\infty$ and $J_{a,m}=(q^a,q^{m-a},q^m;q^m)_\infty$. Shi–Wang's conjectural identity. … This is the remaining unproved identity among five dual Na…

For $|q|<1$, let $(q;q)_n=\prod_{k=0}^{n-1}(1-q^{k+1})$ and $J_m=(q^m;q^m)_\infty$. Rank-four tadpole identity. The following identity is conjectured: … It is one of the rank-four…

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 $r\geq 3$, let $\mathcal{C}(D_r)$ be the Cartan matrix of type $D_r$, and for a vector $B$ define the associated Nahm sum by … The Neveu–Schwarz and Ramond characters are those…

Let $r\geq 3$, let $\mathcal{C}(D_r)$ be the Cartan matrix of type $D_r$, and let $(q;q)_m=\prod_{j=1}^{m}(1-q^j)$. For $a\in\\{0,1\\}$, consider nonnegative integer vectors…

Shi–Wang's conjecture.

The even and odd Nahm sum conjecture.

Cao–Wang's conjecture.

Fix $r\geq 3$ and consider the pair $(T_1,D_r)$, with $A(T_1,D_r)=C(D_r)^{-1}$. Let $B$ range over vectors of the appropriate dimension, and let the associated generalized Nahm sum…

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…

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 conjecture. The transformed quadruple $(A^\star,b^\star,c^\star,d^\star)$ is also a modular quadruple. This conjecture predicts a symmetry of the modularity conditions gov…

Dual Nahm-sum 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…