12 problems

Let $q$ and $t$ be indeterminates, let $M(h)$ be the Verma module of the deformed Virasoro algebra, and let $w$ be its Whittaker vector. Denote by $\langle w,w\rangle$ its norm wit…

Five-dimensional AGT conjecture. The $K$-theoretic Nekrasov partition function coincides with the norm of the Gaiotto state:

Elementary scalar particle conjecture. The operator $t(\zeta)$ satisfies

Let $Z^{5D}_{\rm inst}$ denote the Nekrasov instanton partition function of the $5D$ $N=1$ pure $\mathrm{SU}(2)$ gauge theory, and let $|G\rangle$ be the corresponding Gaiotto-like…

Let $H_k$ be the operators in the Heisenberg algebra corresponding to the operators $\hat H_k$ defined by … Let $(t\Psi)_k$ denote the mode obtained from the product of the generat…

Let ${\cal F}_{mn}$ and ${\cal F}^{\cal A}_{mn}$ be level-graded spaces, and let the right action of $\Sigma$ define complexes because $\Sigma^2=0$. Consider the cohomology at the…

Maximal-rank conjecture for $W$. For generic values of $r$, these right actions are surjective if $m-n\le -2$ and indegenerate if $m-n\ge -2$; equivalently, the operator $W$ has ma…

Let $Z_{\operatorname{pure}\operatorname{SU}(2)}^{\operatorname{inst} K}(Q,q,t;\Lambda)$ be the instanton part of the five-dimensional, or $K$-theoretic, pure…

Let ${\cal G}_n(s)$ be the nonlocal integrals of motion for the deformed Virasoro algebra at parameter $s$, and let ${\cal G}_n^{DV}$ be the specialized nonlocal integrals defined…

At $s=2$, let ${\cal I}_n^{DV}$ and ${\cal G}_m^{DV}$ be respectively the specialized local and nonlocal integrals of motion for the deformed Virasoro algebra. Mixed specialized co…

At the specialization $s=2$, let ${\cal G}_n^{DV}$ denote the nonlocal integrals of motion for the deformed Virasoro algebra, defined by the multiple contour integrals in the sourc…

Let ${\cal I}_n$ and ${\されcal I}_n^*$ be the local integrals of motion for the deformed Virasoro algebra, defined by contour integrals for generic ${\rm Re}(s)>0$ and…