84 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…

Kashiwara crystal conjecture. should have the structure of a Kashiwara crystal isomorphic to $B(\lambda)$, the crystal of the integrable highest weight module of the quantized enve…

Quantum Hikita conjecture. For conical theories, the specialized quantum D-modules on Higgs branches and the D-modules of graded traces for Coulomb branches are isomorphic as D-mod…

Let $\mathfrak{M}_0$ be a Nakajima quiver variety resolved by a smooth Nakajima quiver variety $\mathfrak{M}\to\mathfrak{M}_0$. Let $S_{\mathfrak{M}_0}$ be the torus acting on…

Let $\mathcal M_C(\mu,\lambda)$ denote the Coulomb branch of a $3d$ $\mathcal N=4$ gauge theory, equivalently the moduli space of singular monopoles with magnetic charge $\mu$ and…

Let $H^\pm[n]$ denote the Hamiltonians obtained from the quantum determinant construction in the horizontal quantum loop algebra $U_q(\dot{\mathfrak{gl}}_\ell)$. The Hamiltonians a…

Let $\0$ be the nilpotent orbit with Bala–Carter label $E_8(b_6)$, let $\0'=A_4+A_2+A_1$, and let ${\mathcal S}'$ be the Slodowy slice from $\0'$ to $\overline{\0}$. Let…

Let $\mathbf{v}$ be a dimension vector, let $\GL_{\mathbf{v}}$ be the corresponding product of general linear groups, and let $\mathbf{N}_{\mathbf{v}}$ be the associated representa…

Let $G$ be a reductive group, let $\mathbf{N}$ be a representation, and let $\mathcal{M}(G,\mathbf{N})$ and $\mathcal{M}_{\mathsf{Higgs}}(G,\mathbf{N})$ denote the associated Coulo…

Inner-patch vacuum conjecture. If the four-dimensional theory is asymptotically free, the existence of supersymmetric vacua on an inner patch requires that $F_j$ have zeros on that…

Let $Q=({\mathbb I},\Omega)$ be an ADE quiver, let $(V_i;W_i)_{i\in{\mathbb I}}$ be a representation of its framed double quiver, and impose the stated Satake-diagram symmetry and…

Let $\Phi^{-,\mathrm{re}}_\mu$ be the relevant set of negative real roots, let $\langle\alpha,\mu\rangle$ denote the root–coweight pairing, and let $L$ be the unique irreducible ob…

Let $G$ act on ${\bf N}$, let $\nu\colon\mathbb{C}^{\times}\to\mathsf{A}$ define a partial resolution $\widetilde{\mathcal{M}}_{C,\nu}\to\mathcal{M}_C$, and let…

Let $\mathcal{M}^{!}$ be a Coulomb branch whose quantization has a category $\mathcal{O}$ containing a unique irreducible object $L$. Character conjecture. The character of $L$ can…

Let $\widetilde{\mathcal{M}}_H$ be a smooth symplectic Nakajima quiver variety and let a partial resolution of the Coulomb branch be equipped with its torus action. Higgs–Coulomb f…

Let $\mathcal{M}^{!}=\overline{\mathcal{W}}^{\lambda}_{\mu}$ be the 3d mirror dual of a zero-dimensional quiver variety, with unique torus fixed point $p^{!}$, and set…

Quantum K-theoretic Hikita conjecture. This $(\hbar=q)$-specialization should be equal to the $D$-module of graded traces for the quantized K-theoretic Coulomb branch.

Homological Hikita conjecture. There is an isomorphism of graded $\mathbb C[(\mathfrak t_\bv/S_\bv)\times\ff]\otimes\mathbb C[\hbar]$-algebras

K-theoretic Hikita conjecture. There is an isomorphism of algebras over $\mathbb C[(T_{\bf{v}} / S_\bv) \times F] \otimes \mathbb C[q^{\pm 1}]$

Let $\mathfrak{g}$ be a symmetric Kac–Moody Lie algebra, and let $\mathcal M_C(\mathsf G,\mathsf N)$ be the Coulomb branch associated with the data determined by $\lambda$ and…

Lagrangian coincidence conjecture. The Lagrangians constructed in the cited works coincide with those given by fref{IntroThm1}.

Let $S$ be a compact oriented surface of genus one with boundary, and let $\widetilde{G}=G\times F$ and $N$ be the associated group and representation. The quantized Coulomb branch…

Let $U^Z_{q,\mu}(\widehat{\mathfrak{g}})$ be the quotient algebra of $U_{q,\mu}(\widehat{\mathfrak{g}})$ parametrized by a tuple of truncation parameters $Z$. Finkelberg–Tsimbaliuk…

A generalized quiver is a triple $(V,A,K)$, where $V$ is finite, $A:V\times V\to\mathbb{Z}$, and $K:V\to\mathbb{Z}_{>0}$, satisfying the diagonal and symmetrizability conditions st…

Let $Q$ be a good 3d $\mathcal{N}=4$ quiver theory, let $\mathcal{M}_C(Q)$ denote its Coulomb branch, and let $\mathcal{L}$ be the set of decay and fission products of $Q$, ordered…