28 problems

Dijkgraaf–Moore–Verlinde–Verlinde conjecture.

Let $X$ be a complex manifold equipped with an effective action of a finite group $G$. Write $X/G$ for the quotient, and let $\widehat{Ell}(X/G,;y,q)$ denote the normalized ellipti…

Conjecture 6.1. If $\frac{1}{2}p_1(F)=0$ and the Ricci curvature of $g^F$ along each leaf is positive, then the Witten genus

Modularity conjecture. We conjecture that $\widehat{Ell}({\bf P},q)$ is a modular form. The preceding theorem establishes the stronger specific claim that, for a Gorenstein toric v…

Let $N\ge 2$ and let $t$ be a proper divisor of $N$. Write $I_*^{N,t}$ for the ideal associated with the pair $(N,t)$, and let $\varphi_n$ denote the homomorphism indexed by a divi…

Let $X$ and $Y$ be complex surfaces, let $\Gamma$ be a finite group acting on $Y$, and let $\tau:X\to Y/\Gamma$ be a resolution of singularities. For each $n$, let $\Gamma_n$ denot…

Elliptic genus conjecture for resolutions of surface orbifolds. Then, for every $n$,

Wreath-product orbifold elliptic genus conjecture. The elliptic genus for the wreath product orbifold $Y^n/G_n$ satisfies

Generalized Vafa–Witten conjecture. The analysis relating Yang–Mills theory on $K3$ to generating functions of Euler numbers should generalize so that the generating function of th…

Let $A$ be an abelian surface over $\mathbb{C}$, let $A^{[\ell]}$ be its Hilbert scheme of length-$\ell$ subschemes, and let $A^{\langle\ell-1\rangle}$ be the fiber over $0$ of the…

Let $g$ be a symmetry of a K3 nonlinear sigma model, and let $\phi^g(\mathcal{C},\tau,z)$ denote its $g$-twined elliptic genus. For a four-plane-preserving element of $Co_0$, let…

Let $g\in G_{\Pi^{\natural}}\subset Co_0$ be a four-plane-preserving element, and let $\phi^g(V^{f\natural},\tau,z)$ be the function obtained by tracing $g$ over the Ramond-twisted…

Let $M_1$ be the $24$-dimensional String manifold of Hopkins–Mahowald. For non-negative integers $i$, $j$, and $k$, let $T^i\otimes\wedge^j\otimes S^k$ denote the corresponding ten…

$G_2$-induced generation conjecture. The subring of Dynkin-diagram- and $\Lambda_{cw}(\mathfrak{a}_2)$-shift-symmetric forms appropriate for the numerator $\mathcal{N}$ should be g…

For each positive integer $n$, let $G$ be the symmetry group of the corresponding $H_G^{(n)}$ theory, let $v$ and $x$ be fugacities, let $m_G$ denote the $G$-flavor fugacities, and…

Let $G$ be the global symmetry group of a rank-one $H_G$ theory, let $v$ and $m_G$ denote its fugacities, and let $Q_{\tau}$ be the elliptic-parameter fugacity. Write … Let…

Let $\widetilde{\mathbb{C}/\mathbb{Z}}_{2}$ denote the squashed version of the $\mathbb{Z}_2$ quotient of the complex plane, and let the $SL(2,\mathbb{R})_{k}/U(1)$ coset theory be…

Let ${\mathcal M}$ be the moduli space of $K3$ sigma models, let $N$ range over the 23 Niemeier lattices and the Leech lattice $\Lambda$, and let $\Phi(N)$ denote the corresponding…

Let $a\in\{\operatorname{cob},\operatorname{ell}\}$, let $\underline\beta\in\mathbb Z^4$ satisfy $\beta_1\equiv\beta_2\pmod 2$ and $\beta_3\ge\beta_4-3$, and let…

Let $S$ be a surface satisfying the hypotheses of the corollary in the source, let $H,c_1,c_2$ be chosen so that there are no rank $2$ strictly Gieseker $H$-semistable sheaves with…

Let $S$ be a smooth projective surface with $b_1(S)=0$, $p_g(S)>0$, $K_S\ne0$, and with only Seiberg–Witten basic classes $0$ and $K_S$. Let $H,c_1,c_2$ be chosen so that there are…

Let $\mathbb E(\tau\,|\,\boldsymbol{z})$ be the elliptic genus of a 2d theory, let $E(\boldsymbol{z})$ denote its supersymmetric Casimir energy, and let the anomaly 4-form be the c…

Central-charge conjecture. The GLSM flows to a SCFT with central charge $c=6m$.

Let $n$ be odd, and let $X_W$ and $Y_W$ be the singular Pfaffian-Grassmannian double mirrors constructed from the linear data $W$. Write $Ell(-;y,q)$ for the appropriately defined…

Let $\Phi=D_{1+m/2}$ with $m\neq6$, and let the $\mathbb Z_2$ automorphism exchange the simple roots $f_{m/2}$ and $f_{1+m/2}$. Let $q=e^{2\pi i\tau}$, $y=e^{2\pi iz}$, and let…