13 problems

Framed no-exotics conjecture. Framed BPS states form the trivial representation under $SU(2)_R$; consequently, ${\overline{\underline\Omega}}(\mathbb{L},u,\gamma,q)$ is a character…

In a four-dimensional theory with a line defect $L$, let $\mathcal{H}_{L,u,\gamma}^{BPS}$ be the framed BPS Hilbert space at Coulomb-branch point $u$ and charge $\gamma$, and defin…

Let $\langle W_m\rangle_{\mathcal{N},1,0}^{{\rm SU}(3)_{-4}}$ denote the line defect half-index for the $m$-th Wilson line in the ${\rm SU}(3)_{-4}$ Chern–Simons theory with the in…

Let $A_\mathfrak{q}$ be the space on which the line defect Schur indices define a pairing, and let $I_{a,b}(\mathfrak{q})$ denote the Schur index associated with line defects $L_a$…

Let $Q$ be a quiver, let $A$ be its CoHA, and let $\vec f,\vec f'$ be framing vectors. Denote by $\mathrm{sH}(Q;\vec f)$ the corresponding spherical CoHA bimodule, by…

Let $A_u$ be an algebra of BPS states for a SQFT and let $B_{L;u}$ be the associated collection of $A_u$-bimodules for a line defect $L$. Let the map $\operatorname{RG}_u$ assign t…

Topological line defects in the HT twist form a derived monoidal, $\mathbb{Z}$-graded category, and the category $\mathrm{KP}_{G,N}$ is expected to describe the image of half-BPS l…

Let $C$ be an oriented Riemann surface with a pants decomposition determining a weak-coupling Lagrangian duality frame with gauge algebra … For a line defect $L({\cal P},\zeta)$ as…

Stabilization conjecture. For every fixed $k$, the coefficient of $q^k$ in $\operatorname{Tr}M(q)$ stabilizes to a limiting value for sufficiently large $N$.

Generalized commutative-diagram conjecture. The commutative diagram relating line-defect Schur indices to vacuum expectation values at the $U(1)_r$-invariant vacua exists more gene…

Let $C$ be the relevant ultraviolet curve, let $Y$ be a puncture, and let $[1^N]$ denote the full puncture. Write $d_C^{[1^N]}$ and $d_C^Y$ for the corresponding quantities associa…

Let $L$ be a UV line defect, let $\vartheta$ be a central-charge phase, and let $F(L,\vartheta)=\sum_{\gamma\in\Gamma}\overline{\underline{\Omega}}(L,\vartheta,\gamma,q)X_\gamma$ b…

Renormalization-group bijection conjecture. The map $\mathbf{RG}$ is bijective; equivalently, the holomorphic functions arising from line defects can be labelled by the integral po…