48 problems

Let $G$ be a simple Lie group, and let $\widehat{\mathfrak{g}}$ denote the corresponding auxiliary three-dimensional quiver shape. Consider the Coulomb branch of this three-dimensi…

Generic non-jumping-line conjecture. The generic line is not jumping.

Let $A$ be an anti-self-dual connection on a bundle $E\rightarrow T\times\mathbb{C}$, with curvature satisfying $|F_A|\sim O(|w|^{-2})$. A holomorphic structure…

Schwarzschild unit-energy uniqueness conjecture. The positive chirality spin connection is the only unit-energy instanton over the Schwarzschild space.

Let ${\cal I}_{\mathbb{P}^n}(r,c)$ denote the moduli space of rank $r$ instanton sheaves of charge $c$ on $\mathbb{P}^n$. Quasi-projectivity conjecture. In general,…

Let $E$ be a rank $r\geq 2n$ locally-free instanton sheaf of trivial splitting type on $\mathbb{P}^{2n}$ and charge $c$. Lifting conjecture. There is a rank $r$ locally-free instan…

Atiyah–Jones conjecture. The inclusion induces an isomorphism in homology and homotopy through a range that grows with $k$. This conjecture was originally formulated for spheres an…

Let the limit objects arising from bubbling at the boundary be anti-self-dual instantons on the half space. Energy quantization conjecture. There is an energy quantization for thes…

For a positive definite four-manifold $X_q$, let $\mathfrak M_k^\infty(X_q)$ denote the stable instanton moduli space, and let … be the map induced by the stabilization in the inst…

Completeness conjecture. Up to gauge equivalence, every anti-self-dual connection on $\mathfrak{M}^{\rm I}_q$ is one of the connections produced from classical ADHM data.

Finite-action decay conjecture. Every doubly-periodic instanton connection of finite action satisfies

Consider the critical dimension $d=4$ radial Yang–Mills equation … where $\Delta_{(d-2)}=\partial_r^2+\frac{d-3}{r}\partial_r$, and let … be the static instanton profile. Let $t_*$…

Braden–Nekrasov conjecture. The $U(1)$ instanton on noncommutative ${\bf R}^4 \approx {\bf C}^2$ can be transformed, via the specified field redefinition, to a $U(1)$ instanton on…

Braden–Nekrasov conjecture. Instantons constructed on blowups of ${\bf C}^2$ should be related to instantons on noncommutative ${\bf R}^4$.

Regularity conjecture. Performing this singular gauge transformation gives

Let $N$ be a nonnegative integer, let $\#_{N}\overline{\mathbf{CP}}^2$ be the $N$-fold connected sum of oppositely oriented complex projective planes equipped with an appropriate m…

Consider the four-dimensional superconformal Yang–Mills theory discussed in the source, including four-dimensional instantons and insertions of constituents of chiral-algebra compo…

Formula conjecture. The solution of this recurrence is

Let $(M,g_{ab})$ be a Hermitian non-Kähler ALF instanton. Integrability conjecture. The metric $(M,g_{ab})$ is integrable. Mode stability is presented as a step toward this conject…

The ${\mathcal{Y}}$-observable is interpreted as a partition function of an auxiliary four-dimensional theory living on a copy of $\mathbf{R}^{4}$ transverse to the physical spacet…

Let $H^\bullet(\mathcal{V}_{\bullet}(n_1,n_2),Q_{\mathrm{pert}})$ be perturbative cohomology, and let instanton corrections lower $n_1+n_2$ by multiples of $2N$ while raising cohom…

Let $\vec{\mathtt{q}}=(\mathtt{q}_0,\mathtt{q}_1,\mathtt{q}_2,\mathtt{q}_3)$ and $\mathtt{Q}=\mathtt{q}_0\mathtt{q}_1\mathtt{q}_2\mathtt{q}_3$. Define … where…

Let $\vec r=(r_0,r_1)$, set $r=r_0+r_1$ and $\mathtt{Q}=\mathtt{q}_0\mathtt{q}_1$. For $s\in\{0,1\}$, let … where $s(l)$ records the irreducible $\mathbb{Z}_2$-representation…

Higher-rank type-$\vec r$ conjecture. The equivariant instanton partition function of type $\vec r=(r,0,\ldots,0)$ for the cohomological $\mathsf{U}(r)$ gauge theory with massive f…

Let $M$ be a simply connected compact manifold equipped with the supersymmetric gauge-theory data and localization complex described in the source. Let the localization locus consi…