32 problems

Let $G$ be the relevant symmetry group, and consider symmetry protected topological phases in $u$-dimensional quantum spin systems. An SPT phase is a gapped, short-range-entangled…

Let $\d$ be the spatial dimension, let $H$ be an internal symmetry group, and let $\mathcal{S}^{d}(H)$ denote the set of $(d+1)$-dimensional gapped short-range entangled Hamiltonia…

Let a gapped invertible topological phase of matter be a phase with fixed dimension and symmetry type, and let an invertible field theory (IFT) be its low-energy limit. Low-energy…

Cohomological classification conjecture. Such states should be classified by the cohomology group

Completeness conjecture. If two $G$-representations have the same total index, then they belong to the same equivalence class.

Consider $(1+1)$D phases equipped with weak Hopf symmetry, as realized by lattice models whose symmetries are subalgebras of $H \times \hat{H}$ for a weak Hopf algebra $H$ and its…

Anomalous boundary correspondence conjecture. Any two arbitrary boundary phases with the same quantum anomaly can transform into each other by symmetry-allowed operations, possibly…

Cohomological classification conjecture. The topological action is largely classified by $H^{d+1}(BG_s,U(1)_r)$. This proposes a cohomological description of crystalline SPT phases…

Coupled-wire edge-theory conjecture. This theory can be an edge theory of an intrinsic ASPT with exact $\mathbb{Z}_2^f$ and average $\mathbb{Z}_2^T$ symmetry.

A topological twist string is an invertible codimension-2 defect in a $(3+1)$D topological phase. It can be constructed either by a layer construction from a $(2+1)$D topological p…

Let $A_6$ be the alternating group on six letters, and let $\rho_1$ and $\rho_2$ be the automorphisms of the $A_6$ gauge theory discussed in the source. Their composite…

Let $G$ be a finite group, let $g,h\in G$ be a commuting pair, and define $Z_{(g,h)}:=\sigma_{[g]}(h)$, where $\sigma_{[g]}\in\mathcal{H}^1(C_g,\operatorname{U}(1))$ characterizes…

Two-direction translation-invariant SPT classification conjecture. Assuming the stable SPT classes carry the conjectured abelian-group structure, this sequence induces a sequence o…

Stable SPT classification group-structure conjecture. The operation $*$ is well defined, independently of the representatives, and makes $S$ an abelian group.

The three-fermion phase is described by a unitary modular tensor category (UMTC) with four anyons: the trivial anyon and three fermions. Its fusion rules are those of the toric cod…

Let $G$ be a compact gauge group and consider 3D Chern–Simons theory classified by $H^3(BG,U(1))$. For a connected, simply connected simple Lie group, the class is specified by a l…

Real Picard-group classification conjecture.

A three-dimensional topological phase is a topological phase of an interacting quantum system in three spatial dimensions. A symmetry-protected topological (SPT) phase is a short-r…

A local commuting projector Hamiltonian is a Hamiltonian expressed as a sum of mutually commuting local projectors, and its ground state is a trivial product state when it is a ten…

Inverse-phase realizability conjecture. For any bosonic SPT (or invertible) phase, it should be possible to realize the inverse SPT phase in such a system.

A ${\mathbb Z}_2$ symmetry flux is a topological defect associated with the global ${\mathbb Z}_2$ symmetry, and an $E_8$ state is the two-dimensional bosonic invertible phase with…

An LSM-like theorem is a constraint in a quantum spin system, such as the requirement that a system with half-integer total spin per unit cell be gapless, have degenerate ground st…

Lan's gauging conjecture. Gauging 4d fSPTs with finite-group unitary internal symmetry may give a large subclass of all 4d bSETs with finite gauge group, and likely gives a complet…

Let point-group or space-group symmetry act on an interacting fermionic system, and impose the crystalline principle and the relevant spin-statistics relations. Crystalline extensi…

Consider three-dimensional interacting systems and the operation of gauging a global symmetry of an SPT phase. Gauging conjecture. All topological phases in three-dimensional inter…