42 problems

Let the Witt group classify topological orders in spatial dimension ${\mathsf{d}}=3$. Johnson–Freyd's conjecture. The Witt group of topological orders in ${\mathsf{d}}=3$ is isomor…

In $5$D spacetime, let $F$-, $\Delta$-, and $\Delta^2$-symbols describe fusion, shrinking, and hierarchical shrinking, with the two relevant shrinking-fusion hexagon equations prov…

Let $F$-, $\Delta$-, and $\Delta^2$-symbols implement basis transformations for diagrams involving fusion, shrinking, and hierarchical shrinking processes. The paper verifies that…

In higher-dimensional spacetime, topological orders may contain loop and membrane excitations. Their diagrammatic data include $F$-symbols for fusion and $\Delta$- and $\Delta^2$-s…

Let $Z_N$ denote the cyclic symmetry acting on the model, and let $\mathbf{F}^{(l)}$ be the deformed bulk theory obtained by inserting a $Z_N$ twisted boundary condition, with defo…

Domain-wall decomposition conjecture. Any gapped domain wall of the form

Generalized pull-open conjecture. Any stable gapped domain wall ${}_B\CM_B$ is equivalent to

Let $\mathsf{B}^{n+1}$ be a background phase and let $\mathsf{C}^{n+1}$ be an $(n+1)$D anomaly-free topological order. An MCLP lattice model over $\mathsf{B}^{n+1}$ has a local spi…

A quantum liquid is an anomalous quantum phase described by a pair $(\mathsf{X}_{\mathrm{lqs}},\mathsf{X}_{\mathrm{top}})$, where $\mathsf{X}_{\mathrm{lqs}}$ is its local quantum s…

A super modular tensor category with a fermionic group action determines a $(3+1)$-dimensional topological field theory whose partition functions give anomaly indicators on $4$-man…

Let $K$ be the target space of a continuous $c0$-finite nonlinear $c3$-model, let $\mathbb{G}^{(d)}$ be the higher group associated with its singularity-free generalized symmetry,…

In $(3+1)$-dimensional spacetime, consider bosonic topological orders without emergent fermions. Wang–Levin conjecture. Such topological orders are described by Dijkgraaf–Witten th…

Cobrordism-invariant conjecture. For invertible topological orders in branch-independent bosonic systems, the topological invariant $Z^\text{top}(M)$ is a cobordism invariant.

Branch–frame independence conjecture. Independence of the branch structure of the spacetime complex for lattice models implies independence of the frame structure of the spacetime…

Let $H$ act on the U(1) gauge group by $x\mapsto\sigma(\mathbf{g})\cdot x$, where $\sigma(\mathbf{g})\in\{\pm1\}$, and write $(x,\mathbf{g})\in\mathrm{U}(1)\rtimes H$. Let…

Let $\mathcal{C}$ be a fermionic topological order with anyon quantum dimensions $d_a$, total quantum dimension $\mathcal{D}$, U(1) charges $Q_a$, and modular data $S_{a,b}$. Let…

Let $\mathcal{C}$ be a non-Abelian topological order with anyon quantum dimensions $d_a$, total quantum dimension $\mathcal{D}$, topological spins $\theta_a$, and fractional U(1) c…

Gapless ESB conjecture. The novel concept of ESB can also be defined for gapless systems and might have important implications in fundamental physics, such as the CP violation prob…

Minimal modular extension conjecture. Every slightly degenerate braided fusion 1-category admits a minimal modular extension.

Anomaly-freeness conjecture for 2d fermionic topological orders. All 2d fermionic topological orders are anomaly-free.

Minimal modular extension conjecture. Every slightly degenerate braided fusion category admits a minimal modular extension; equivalently,

Minimal modular extension conjecture. Every slightly degenerate braided fusion category $\mathcal C$ admits a minimal modular extension $\mathcal C'$.

The paper considers $n$d topological orders with gapped or gapless boundaries, together with higher-codimensional gapped or gapless defects on the boundary. Higher-categorical boun…

Let $cal F$ be an arbitrary bosonic $(D+1)$-dimensional topological order. For each $dleq D$, let $cal F[S^d]$ denote its compactification along the $d$-sphere. Compactification co…

Let $\EuScript{C}$ be a unitary modular tensor category (UMTC), and do not fix a central charge. Unitary VOA realization conjecture. Every UMTC $\EuScript{C}$ can be realized as th…