112 problems

Let $\mathcal{B}$ be an anyon model, let $\mathcal{M}_{\mathcal{B}}$ be the moduli space of gapped systems with intrinsic topological order $\mathcal{B}$, and let…

Let $G$ (or $G_f$) be a spatial symmetry group with given additional symmetry data, and let $G^{\text{eff}}$ (or $G_f^{\text{eff}}$) be the corresponding effective internal symmetr…

Let $h$ denote the helicity of a massless particle and let $C$ denote the Chern number characterizing the topology of its particle bundle. Chern number–helicity conjecture. The rel…

Anyon–superselection sector correspondence conjecture. In these models, anyons should be in one-to-one correspondence with the superselection sectors.

A gapped invertible phase is a phase of matter with an energy gap and an inverse under stacking; the phases may arise from interacting systems, and a loop-spectrum is a spectrum wh…

Let $H_\circ$ be a Hamiltonian whose ground state $G_{\circ,{\bf l}}$ has log-extensive degeneracy, and let $G_{\epsilon,{\bf l}}$ be the strictly less degenerate ground state obta…

Let $Q=\tau+2$ for DFib or $Q=2$ for the toric code, and let $\beta_c$ denote the Potts-model critical inverse temperature. For a net $G$, write $\langle G\rangle_\tau$ for its iso…

Let $\tau=\frac{1+\sqrt{5}}{2}$ and consider the DFib topological phase together with its connection to the $Q=\tau+2$ state Potts model. DFib critical-temperature conjecture. The…

Let $\ket{G}$ be a gapped ground state of a known two- or three-dimensional topological phase. Let $I_{\max}(\ket{G})$ be the maximal amount of information transferable in one roun…

Conjecture that $(-1)^{\int w_2^4}$ is a nontrivial QCA. The response $Y=(-1)^{\int w_2^4}$ is a nontrivial quantum cellular automaton.

Fidkowski's conjecture. $Y$ is a nontrivial QCA if the invertible phase it entangles has a nontrivial partition function on some orientable manifold.

Let $n,m\geq0$ satisfy $d+1=2n+3m$ with $d\geq3$, and let $A_{d-1}\in Z^{d-1}(M_{d+1};\mathbb Z_2)$ and $B_2\in Z^2(M_{d+1};\mathbb Z_2)$ be cocycles. Consider the QCA family whose…

Let $S(X)=\frac{\sigma(X)}{8}\pmod{1}$ be the invertibly normalized response of the seven-dimensional generalized-semion QCA, where $\sigma(X)$ is the signature of a closed oriente…

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…

Consider the invertible phases, including invertible stabilizer phases and quantum cellular automata (QCA), and their classification by a generalized cohomology theory. Kitaev's co…

A topological phase is a phase of matter with topological order, and a commuting projector Hamiltonian is a Hamiltonian expressible in terms of mutually commuting local projector t…

A point quasiparticle is an excitation localized at a point in a three-dimensional topological phase, and an SFC is the class of structures referred to in the source as describing…

Microscopic index conjecture. The index $omega_N$ provides a microscopic definition of the chiral central charge modulo $24$.

Refined invariant conjecture. There exists a $\boldsymbol{Z}$-valued invariant for 2d invertible phases which, under special circumstances involving the emergence of conformal symm…

Field-theoretic classification conjecture. At least a large class of $d$-dimensional topological phases of quantum many-body systems can be classified by field-theoretic methods.

Duals in the unstraightening. The category $\mathbf{Un}_{\mathcal X}(\mathrm{Th})$ is an $(\infty,n)$-category that has duals.

Let $\mathcal H=\ell^2(\mathbb Z^2)$ and let $P$ be the Fermi projection of a gapped Hamiltonian $H$ satisfying exponential locality. Let $\omega_P$ be the associated quasi-free st…

K-theoretic framework conjecture. More broadly, this $K$-theoretic group structure is indeed the correct mathematical setting for the comparison with interacting SPT phases protect…

Let $\mathsf{C}$ and $\mathsf{D}$ be fusion categories and let $_{\mathsf{C}}\mathsf{M}_{\mathsf{D}}$ be a $\mathsf{C}$–$\mathsf{D}$ bimodule category. Write…

A periodic system in two dimensions is described by a Bloch bundle, and a Wannier function is obtained from a Bloch frame by the inverse Bloch transform. For a non-trivial Bloch bu…