1. List of all question
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Problem 1.01.
[Dominic Williamson] For condensable algebras $A_1,A_2, \ldots, A_k \in \mathcal{C}$, sequentially condense in the following order \[ \mathcal{C}^{loc}_{A_1}\; |\mathcal{C}| \; \mathcal{C}^{loc}_{A_2} \;| \mathcal{C}|\; \ldots\; |\mathcal{C}^{loc}_{A_k}\; | \mathcal{C}| \; \mathcal{C}^{loc}_{A_{k-1}}\;|\mathcal{C}| \;\ldots |\mathcal{C}| \;\mathcal{C}^{loc}_{A_1} \] What automorphisms do we get by this process? Can this yield the Floquet code? -
Problem 1.03.
[Zhenghan Wang] Determine Representations of motion groups of links in $S^3$. In particular, study the ones coming from Dijkgraff-Witten $(3+1)$-D TQFTs. -
Problem 1.04.
[Theo Johnson-Freyd] If $\mathcal{Z}(\mathcal{C})\cong \mathcal{Z}(\mathcal{D})$ and $\mathcal{C},\mathcal{D}$ and they both admit a fiber functor, are they Morita equivalent? -
Problem 1.05.
[org.aimpl.user:apteanuj@uchicago.edu] Do generalized spins in $2$-categories need to be rational? This a 2-category version of Vafa’s theorem. -
Problem 1.06.
[Zhenghan Wang] Suppose we have two BFCs $\mathcal{B},\mathcal{C} \subseteq \mathcal{M}$ with $\mathcal{M}$ modular such that $\mathcal{B},\mathcal{C}$ centralize each other. Then what is the Witt class of $\mathcal{M}$? -
Problem 1.07.
[Corey Jones / Zhenghan Wang] What is the correct definition of the $2$-center of categories so that the following statement holds: $\mathcal{Z}(\mathcal{C}) \cong \mathcal{Z}(\mathcal{B})$ if and only if $\mathcal{C}, \mathcal{B}$ are Morita equivalent). -
Problem 1.08.
[David Reutter] What are the braided — categories so that invertibles correspond to cobordisms? -
Problem 1.09.
[Zhenghan Wang] Find a short algebraic definition of a tetracategory (preferably, something less than 10 pages long). Current known definition can be found in ’Spans in 2-categories: A monoidal tricategory’ by Hoffnung. -
Problem 1.11.
[Dominic Williamson] Classify fracton models constructed from $1$-foliated topological defect networks. -
Problem 1.12.
[org.aimpl.user:apteanuj@uchicago.edu] Is there a notion of Symmetry Protected Topological (SPT) phases protected by fusion categorical symmetry? Perhaps, they can be built using the cohomology groups of classifying spaces of the associated Brauer-Picard groupoids. -
Problem 1.13.
[Ryan Thorngren] Compute examples of fusion $2$-categories with fiber functors. -
Work of Kang-Lan-Wen does this for (super) groups.
Problem 1.14.
[Ryan Thorngren] Understand fiber $2$-functors $F$ of a fusion $2$-category and understand boundary anomalies. -
Problem 1.15.
[Zhenghan Wang] Study representations of motion groups of links in $S^3$. For example, study the ones coming from Dijkgraff-Witten $(3+1)$-D TQFTs. -
Problem 1.16.
[Maissam Barkeshli] Define and study $G$-crossed Braided Fusion categories (BFC) with anti-unitary $G$-action. -
Problem 1.17.
[Maissam Barkeshli] In $(3+1)$D there is an $O(5)$ obstruction to the full $G$-crossed theory. Calculate it systematically. This can be rephrased as: calculate the $G$-extension theory of a braided fusion $2$-category. -
Problem 1.18.
[Zhenghan Wang] Find the eigenspace decomposition of the Hilbert space for the golden chain Hamiltonian. Are the eigenspaces $1$-dimensional. -
Problem 1.19.
[Corey Jones] Where to look for counterexamples to the conjecture that: a pseudounitary category is unitary. For a pseudounitary category the quantum dimensions of all objects are positive. -
Problem 1.2.
[David Reutter] Do all fusion categories admit a Hermitian structure (a $\dagger$-structure which is positive) ? -
Problem 1.21.
[Dave Penneys] If a fusion category admits a Hermitian structure, is it unique? -
Problem 1.22.
[Kyle Kawagoe] Given two $(3+1)$D SPTs $\mathcal{A},\mathcal{B}$ with the same group $G$ and an $\mathcal{A}$-$\mathcal{B}$ bimodule category $\mathcal{M}$ representing the boundary between them, construct a $G$-crossed braided fusion category using it. If we gauge both the SPTs, what happens to the domain wall/boundary/bimodule ? -
Problem 1.23.
[Ryan Thorngren, David Reutter] Do QCAs form $\Omega$ spectrum? Are the first four levels the same as the spectrum of modular tensor categories? -
Problem 1.24.
[Nathanan Tantivasadakarn and Zhenghan Wang] Starting with a trivial TQFT, what is the minimum number of abelian extensions that one can do to get reach some given TQFT? -
Problem 1.25.
[Corey Jones] How are QCAs module {finite depth Quantum circuits} related to Witt group of Modular Tensor Categories. -
Problem 1.26.
[Nathanan Tantivasadakarn] Are $(3+1)$D Quantum Cellular Automata (QCA) coming from $(3+1)$D TQFT states beyond cobordisms? -
Problem 1.27.
[Dave Penneys] How to define the skeleton of fusion $2$-categories and their bimodules? What is the generalization of $6j$-symbols? -
Problem 1.29.
[org.aimpl.user:apteanuj@uchicago.edu] If there is a BFC that has order $2$ in the Witt group, does it admit a time-reversal symmetry ? -
Problem 1.3.
[Colleen Delaney] Do fusion $2$-categories exhibit some version of Ocneanu rigidity? -
Problem 1.31.
[Jeongwan Haah] Can we always introduce gapped boundary for the $2$D commuting Hamiltonian? -
Yes! This is known
Problem 1.32.
[Zhenghan Wang] Is factorization of MTCs unique if there are no invertible objects? -
Problem 1.33.
[Kyle Kawagoe] Are there non-trivial links that can be created by only local operators from the vacuum and if so what topology can we get? -
Problem 1.34.
[Anuj Apte] Construct fermionic $2$-categories. In other words, construct enriched fusion $2$-categories. -
Problem 1.36.
[Eric Rowell] Is it possible to detect the trivial Witt Class from the S and T matrices? (In other words, can we tell if a nondegenrate braided fusion category is a Drinfeld center just by looking at the $S$-matrix.) -
Problem 1.37.
[Zhenghan Wang] Given a MTC $\mathcal{C}$ and a condensable algebra $A$, how can one describe the $A$-action on $\mathcal{C}^{loc}_A$? Describe a cohomological theory of it. -
Problem 1.38.
[David Reutter] What is the homotopy type of the space of braided tensor categories? -
Problem 1.39.
[David Reutter] What is the diagrammatic interpretation of Gauss sums for $\mathcal{C}$? -
Problem 1.41.
[Kyle Kawagoe] Understand the topology of the classes of Topological Phases of Matter (TPMs). -
Problem 1.42.
[Noah Snyder and Jacob Bridgeman] Develop extension theory for rings (in place of groups). There is motivation to investigate this problem from the study of Haagerup fusion categories. -
Problem 1.43.
[Laurens Lootens] For Rep($SU(2)$), construct module categories over it. Study its Morita dual and the associated data. -
Problem 1.44.
[Jeongwan Haah] Provide a classification of Lagrangian submodules of a module with an anti-Hermitian form (over a commutative ring).-
Remark. [Jeongwan Haah] This question arises in an attempt to show that every gapped 2+1d Hamiltonian system is described by a unitary modular tensor category, where the meaning of description should be suitably defined. While a general Hamiltonian system might be difficult to treat, a commuting Hamiltonian would be easier to handle. It is a folklore that a commuting Hamiltonian may only realize a Levin-Wen state (Kevin Walker’s conjecture), towards which it seems to be useful to examine a 1+1d boundary. If the boundary can be gapped out, this would mean that the bulk contains a nontrivial boson.
The conjectural existence of a nontrivial boson in any commuting Hamiltonian system in 2+1d would prove the nontriviality of QCA in 3+1d against general locally generated QCA (quantum circuit). As of now, the nontriviality of certain QCA in higher dimensions is only proved against Clifford circuits, and one approach of such nontriviality proof uses the existence of a 1+1d gapped boundary for any 2+1d commuting Pauli Hamiltonian.
Let us give more precise formulation of the problem. Let H be a two-dimensional commuting Hamiltonian on a lattice of uniformly finite dimensional degrees of freedom which satisfies the “local topological order” condition, defined in Bravyi, Hastings, Michalakis 1001.0344. In words, this condition asks that the reduced density matrix of any ground state on any ball-like region must be determined by Hamiltonian terms that touch the region. Colloquially, this means that the Hamiltonian terms are sufficiently constraining the ground state. Now, introduce a spatial 1+1d boundary by dropping terms outside a spatial boundary of the 2+1d system. A gapped boundary means a choice of new local Hermitian operators along the spatial boundary such that the overall Hamiltonian, including both the bulk terms and the new boundary terms, fulfills the local topological order condition. Observe that the local topological order condition has nothing to do with spatial homogeneity; in particular, it can be imposed for a system with boundary.
The boundary terms may in principle be noncommuting, but it is conceivable that we may answer the question within commuting Hamiltonians. Insisting commuting Hamiltonians, the question asks us to investigate the commutant of bulk Hamiltonian terms, where the commutant is taken within the local operator algebra on a narrow strip (1+1d) at the boundary. Note that if the bulk is translation invariant, the boundary Hamiltonian may have to break the translation symmetry, at least weakly. An example is the Wen plaquette model in which the bulk has translation symmetry with a unit cell while any boundary Hamiltonian obeys translation symmetry with twice as large a unit cell.
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Problem 1.46.
[Dave Penneys] Construct a $(p,\mathbf{Z}_p)$ near group fusion category. These are categories generated by two objects $\{\mathbf{1},X\}$ with $X$ invertible and the fusion rule is $X\otimes X= \oplus_{g\in \mathbf{Z}_p} g \oplus pX$ for infinitely many $p$’s. -
Hint: a definition of such a category exists in work of Jones-Penneys-Reutter.
Problem 1.47.
[Yu-An Chen] Interpret $(3+1)$D SETs using $G$-crossed $2$-categories. -
Problem 1.49.
[David Jaklitsch] Provide an intrinsic characterization of exact algebras in tensor categories. -
Problem 1.5.
[Dominic Williamson] Condense out everything below middle dimension excitations in TQFTs. Then, what is left? -
Problem 1.51.
[Cain Edie- Michell] Can you find graph planar algebra embeddings for known near-group categories in small dimensions and spot a pattern? -
Problem 1.53.
[Noah Snyder] Look at Gauss’ proof on positivity of Gauss sums ($\sum_{k=0}^{p-1} e^{2\pi i k^2/p} $). Is there a TQFT interpretation for this positivity?
Cite this as: AimPL: Higher categories and topological order, available at http://aimpl.org/highercattopord.