21 problems

Let $\mu^1$ and $\mu^2$ be partitions, let $Q$ and $q$ be formal variables, and define … Write … Here $\mathcal W_{\mu^1}$ and $\mathcal W_{\mu^2}$ denote the corresponding one-leg…

Let the higher-rank Donaldson–Thomas partition function of $\mathbb{C}^4$ be the partition function whose explicit expression was proposed by Nekrasov and Piazzalunga. The tetrahed…

Consider the pure, non-conformal linear $SU(2)^{\times n}$ quiver system with functions $\tau_0$ and $\tau_1$ satisfying … … Their generalized Kiev ansatz has coefficients…

Let $N,n,p$ be integers with $p\geq N$. Let $\widehat{X}_{\boldsymbol{N}_{\boldsymbol{m}}^{(f)}}^{\,\boldsymbol{r},\boldsymbol{s}}(\mathfrak{q},\boldsymbol{\mathfrak{t}})$ be the…

Finite-instanton conjecture. Only the $\widetilde{\mathfrak{q}}$ and $\widetilde{\mathfrak{q}}^2$ terms in the Schwinger–Dyson equation survive this limit.

First-instanton survival conjecture. This pattern continues at every instanton order, and only the first instanton contribution $k=1$ survives the flat-space limit.

First-instanton survival conjecture. This pattern continues at every instanton order, and only the first instanton contribution $k=1$ survives the flat-space limit.

Higher-instanton cancellation conjecture. The cancellation observed at the tested orders is a generic feature of all higher-instanton contributions.

Let $d$ be the integer appearing in the five-dimensional $\mathcal{N}=1$ gauge-theory blowup equation … The bound on $d$ obtained as a necessary condition for this equation is conj…

The plethystic form conjecture. The instanton partition function can be written as

Simple-pole conjecture. The free energy $\log\mathcal{Z}_{\mathbb{C}^3}^{\mathrm{inst}}(\vec\epsilon,\vec a;\mathsf{q})$ has only simple monomial poles in the equivariant parameter…

Higher-rank Donaldson–Thomas conjecture. The instanton partition function of the $\mathcal{N}=2$ gauge theory on $\mathbb{C}^3$ at a generic point $\vec\epsilon$ of the $\Omega$-de…

Let $r$ and $p$ be parameters, let $U(r)$ denote the gauge group, and let $\mathcal{A}(r,p)$ be the chiral algebra associated with the corresponding coset model. The quotient…

The augmentation curve supplies a canonical form $\Theta(p)$ and a modified kernel $\widetilde{B}(p_1,p_2)$, interpreted respectively as generating functions of disk instanton numb…

The instanton clustering in supersymmetric Yang–Mills theory is assumed to be entirely described by the effective action. Let $\rho(x)$ be the saddle-point density and let $G(x-y)$…

The quantum isomonodromic systems considered here are time-dependent Schrödinger systems obtained by quantizing Hamiltonian systems arising from similarity reductions of the Drinfe…

Instanton counting for an $SU(N)$ superconformal gauge theory can be encoded by conformal blocks of $A_{N-1}$ Toda theory, with external momenta related to matter masses and intern…

Doubly refined instanton-specialization conjecture. The doubly refined partition function of asymptotic ADHM invariants is obtained from the equivariant instanton sum by the altern…

Let $X_p$ be the compactification used in the paper, and let ${\mathcal O}_{{\mathbb P}^1}(-p)$ denote the corresponding local surface. Consider bundles on $X_p$ with fractional to…

Let $p$ be an integer, and consider the local surface ${\mathcal O}_{{\mathbb P}^1}(-p)$ with instantons framed to the trivial connection at infinity. The paper studies the corresp…

Let $Z_{adj}$ be the instanton partition function of the five-dimensional $U(1)$ gauge theory with adjoint matter, expressed as a sum over Young diagrams $Y$. For a box $s\in Y$, l…