Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.

Topological field theories and symmetry protected topological phases with fusion category symmetries

Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples (Z, M, i, s, ϕ) where (Z, M, i) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor (Z, M, i). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

Symmetries and strings of adjoint QCD2

We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group SU(N). The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits ∼ 22N non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all N these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, m ≪ gYM, the theory confines and the non-invertible symmetries are softly broken. We use them to compute analytically the k-string tension for N ≤ 5. Our results suggest that the k-string tension, Tk, is Tk ∼ |m| sin(πk/N) for all N. We also consider the dynamics of adjoint QCD deformed by symmetric quartic fermion interactions. These operators are not generated by the RG flow due to the non-invertible symmetries, thus violating the ordinary notion of naturalness. We conjecture partial confinement for the deformed theory by these four-fermion interactions, and prove it for SU(N ≤ 5) gauge theory. Comparing the topological phases at zero and large mass, we find that a massless particle ought to appear on the string for some intermediate nonzero mass, consistent with an emergent supersymmetry at nonzero mass. We also study the possible infrared phases of adjoint QCD allowed by the non-invertible symmetries, which we are able to do exhaustively for small values of N. The paper contains detailed reviews of ideas from fusion category theory that are essential for the results we prove.

Extended TQFTs via generators and relations I: The extended toric code

In his PhD thesis [G. Goosen, Oriented 123-tqfts via string-nets and state-sums, PhD thesis, Stellenbosch University, Stellenbosch (2018)], Goosen combined the string-net and the generators-and-relations formalisms for arbitrary once-extended 3-dimensional topological quantum field theories (TQFTs). In this paper, we work this out in detail for the simplest nontrivial example, where the underlying spherical fusion category is the category of [Formula: see text]-graded vector spaces. This allows us to give an elementary string-net description of the linear maps associated to 3-dimensional bordisms. The string-net formalism also simplifies the description of the mapping class group action in the resulting TQFT. We conclude the paper by performing some example calculations from this viewpoint.

THE CASIMIR NUMBER AND THE DETERMINANT OF A FUSION CATEGORY

Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

On 3-dimensional homotopy quantum field theory III: Comparison of two approaches

Let [Formula: see text] be a discrete group and [Formula: see text] be an additive spherical [Formula: see text]-fusion category. We prove that the state sum 3-dimensional HQFT derived from [Formula: see text] is isomorphic to the surgery 3-dimensional HQFT derived from the [Formula: see text]-center of [Formula: see text].

Braided Extensions of a Pointed Fusion Category with Prime Dimension

We obtain a necessary condition for a strictly weakly integral generalized Tambara–Yamagami fusion category to be braided. Then we use this result to classify braided extensions of a pointed fusion category with prime dimension.

String-net models for nonspherical pivotal fusion categories

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case, the vector spaces agree with the state spaces of the corresponding Turaev–Viro topological quantum field theory. In the present work, some effects of dropping the sphericality condition are investigated. In one example of nonspherical pivotal fusion categories, the string-net space counts the number of [Formula: see text]-spin structures on a surface and carries an isomorphic representation of the mapping class group. Another example concerns the string-net space of a sphere with one marked point labeled by a simple object [Formula: see text] of the Drinfeld center. This space is found to be nonzero iff [Formula: see text] is isomorphic to a nonunit simple object determined by the nonspherical pivotal structure. The last example mirrors the effect of deforming the stress tensor of a two-dimensional conformal field theory, such as in the topological twist of a supersymmetric theory.