33 problems

Let $X$ be a smooth four-manifold, let $\xi$ be a class defining a wall, and let $\Delta_{\xi,d}^X$ denote the corresponding wall-crossing term for Donaldson invariants. The multip…

Let the $u$-plane be the modular curve parametrizing the vacua of the low-energy effective theory, and let the regularized $u$-plane integral have boundary contributions from the c…

Mariño–Moore's conjecture. There should be a generalisation of Witten's conjecture to these higher-rank invariants, implying in particular that they do not contain new differential…

Let $X$ be a simply connected four-manifold with $b_+(X)>1$. A manifold is of finite type if there is a non-negative integer $r$ such that … for the relevant Donaldson invariant…

Kotschick–Morgan's weak conjecture. The transition formula $\delta_{X,{\mathcal C}_+,{\mathcal C}_-}^w$ is a homotopy invariant of the pair $(X,\xi)$; more precisely, if $\phi$ is…

Let $X$ be a $4$-manifold with $b^+(X)>1$. Finite-type conjecture. Every such $4$-manifold is of finite type. This conjecture concerns the finiteness properties of Donaldson invari…

Spin-c structure conjecture. This $Spin^c$-structure induces a $Spin^c$-structure on the moduli space $M_H^X(c_1,d)$. This would help extend the K-theoretic Donaldson invariants to…

Let $\ell$, $F$, $L_1$, $\mathfrak{s}_0$, $SW(\mathfrak{s}_0\otimes L_1)$, and $Q_X$ denote the level, bundle and line-bundle data, Seiberg–Witten invariant, and intersection form…

Let $X$ be a four-manifold, let $E$ be a rank-two bundle, and let $z\in\mathbb A(X)$. Let $n_a$, $d_a$, $\overline{\operatorname{Red}}(W,E)$, $\overline V(z)$,…

Let $Y=\boldsymbol{\Sigma}\times \mathbb{S}^1$, where $\boldsymbol{\Sigma}$ is a genus-$2$ surface, and let $\gamma\subset Y$ be an embedded curve. The Fukaya–Floer homology…

Let $\bar X_i$ have $b_1=0$ and be of simple type. Suppose that there are embedded surfaces $\Sigma_i\subset\bar X_i$ of genus $g$ and self-intersection zero. Form … Let…

Let $X$ be a four-manifold, let ${\cal M}_k$ be the moduli space of charge-$k$ connections, and let $\nu_p$ be the geometric representative consisting of connections for which…

Let $S$ be a rational surface, let $N,d,r$ be the parameters in the wall-crossing formula, let $K^2$ denote $K_S^2$, and let $Q_i(N,d,r,K^2)$ and $P_i(N,d,r,K^2)$ be the polynomial…

Let $X$ be a $4$-manifold with $b_+(X)=1$ and $b_1(X)=0$, let $\xi\in H^2(X,\mathbb Z)$ satisfy $\xi^2<0$, and let $\lambda_{[X]}(q)$ be the power-series factor associated with the…

Let $X$ be a rational surface, let $[?]$...

Let ${\mathbb{X}}$ be a four-manifold with $b_2^+({\mathbb{X}})=1$. For $\zeta\in\mathsf{H}^2({\mathbb{X}};\mathbb{Z})$ with $(\zeta,\zeta)<0$, let $W_\zeta=\{J:(\zeta,J)=0\}$ be t…

Let $[r-1]=\\{1,\ldots,r-1\\}$, and for $1\leq i<j\leq r-1$ define … and … Let $(X,H)$ be a smooth polarized surface with $H_1(X,\mathbb{Z})=0$ and $h^{2,0}(X)>0$. Let $r>1$ and…

Massive SQCD correlation-function conjecture. The correlation functions on $X$ in the theory with $N_f$ massive fundamental hypermultiplets are conjectured to be

Let $X$ be a compact oriented $4$-manifold with $b_+^2(X)=1$, without assuming that $X$ is simply connected, and consider its Donaldson-theoretic moduli spaces and geometric data a…

Let $S$ be a smooth projective surface with $p_g(S)>0$ and $b_1(S)=0$, and let $L\in \operatorname{Pic}(S)$. Choose $H,c_1,c_2$ so that there are no rank-two strictly Gieseker $H$-…

For integers $(\beta_1,\beta_2,\beta_3,\beta_4)$ and $v\geq0$, let $F_v(\beta_1,\beta_2,\beta_3,\beta_4,y)$ and $G_v(\beta_1,\beta_2,\beta_3,\beta_4,y)$ be the universal functions…

Let $S$ be a smooth projective surface and consider Mochizuki's rank-three descendent formula, whose terms are indexed by Seiberg–Witten basic classes $a_1,a_2,a_3$ satisfying…

Let $M_4$ be a smooth 4-manifold, and let $T[M_4;G]$ be the two-dimensional ${\mathcal N}=(0,2)$ superconformal theory associated to $M_4$ and a root system $G$. For $G=SU(2)$, con…

Let $S$ be a smooth projective surface, let $Q(\alpha)=\int_S\alpha^2$, and let $a_1,\ldots,a_s$ be its Seiberg–Witten basic classes. Define the generating series…

Let $X$ be a four-manifold of ${\rm U}(3)$-simple type, let $\{K_i\}$ be its Kronheimer–Mrowka basic classes, and let $\mathfrak s_i$ be chosen so that the basic class associated t…