13 problems

The $(A_1,D_{2n+1})$ Macdonald-index conjecture. The Macdonald index is

Andrews et al.'s conjecture. The specialization of the Macdonald index at $z=1$ satisfies

Modified KS operator conjecture. The Macdonald index of the class of four-dimensional ${\cal N}=2$ superconformal field theories for which the modified KS operator is defined is

Let $R$ be the coordinate ring of an affine scheme $X=\operatorname{Spec}(R)$, and let $J_\infty(R)$ be its arc space. Let $\operatorname{gr}(J_\infty(R))$ be the associated graded…

Let $(q;q)_m=\prod_{j=1}^{m}(1-q^j)$ and let $\binom{2m}{k}_q$ be the $q$-binomial coefficient. The $(A_1,D_3)$ Macdonald-index conjecture. The Macdonald index of the Argyres–Dougl…

The $(A_1,A_4)$ identity. The displayed identity holds for all $m$. It extends the numerical agreement between the arc-space Hilbert series and the Macdonald index beyond the check…

Let ${\cal V}$ be the strongly finitely generated vertex operator algebra associated with a four-dimensional ${\cal N}=2$ superconformal field theory, and let … be its refined char…

$\mathcal{W}_3$ character–Macdonald index conjecture. The $t$-refined fermionic expressions for the vacuum and next-to-vacuum modules agree with the corresponding Macdonald indices…

Virasoro character–Macdonald index conjecture. The $t$-refined characters of the vacuum and next-to-vacuum modules satisfy

Quasi-particle–Schur operator correspondence. The quasi-particles of statistical mechanics are in one-to-one correspondence with the Schur operators counted by the Schur/Macdonald…

Character–Macdonald index conjecture. The $t$-refined character is equal to the Macdonald index computed from the gauge theory side.

Refined-character conjecture. The fugacity $T$ counts the number of $L$ and $J$ generators used to construct states in the $\widehat{\mathfrak{su}}(2)$ module.

Refined-character conjecture. For an ${\cal N}=2$ superconformal field theory ${\cal T}$ with associated chiral algebra ${\cal A}[{\cal T}]$ and vacuum module ${\cal V}_{\cal T}$,…