A (dummy’s) guide to working with gapped boundaries via (fermion) condensation

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.

Section: Discussionmentioning

confidence: 99%

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

Inamura

2021

“…In most of this paper we have worked with the topological lines in the bosonized theory. This is to avoid technical complications, and also because of lack of literature completing the theory of lines in a fermionic theory, though there are remarkable papers [93,95,[162][163][164][165][166][167][168][169]. 54 Based on this literature, here we give a brief outlook of the topological lines in fermionic theories, and give an explicit example of topological lines in the fermionic SU(3) adjoint QCD.…”

Section: Jhep03(2021)103 G Topological Lines In Fermionic Theorymentioning

confidence: 99%

“…The 54 In particular, most of the literature talks about super-commutative Frobenius algebra object A and the corresponding module category CA, but not about the bimodule category ACB in detail. In [169] constructed the bimoduel category based on [58]. 55 The dimension of a Z2-graded vector space is denoted as p|q, where p is the dimension of the degree-even (bosonic) subspace and q is the dimension of the degree-odd (fermionic) subspace.…”

Section: G1 Topological Lines In Fermionic Theoriesmentioning

confidence: 99%

Symmetries and strings of adjoint QCD2

Komargodski

Ohmori

Roumpedakis

et al. 2021

197

149

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.

“…In this section we demonstrate the existence of anomalies in fermionic TQFTs by looking directly at their Hilbert space. We follow the construction of the Hilbert space of a fermionic TQFT in [61]; see [21,[81][82][83][84][85] for related work. Here we summarize the main ingredients, leaving most details to appendix A.…”

Section: Anomalies In Spin Tqft Hilbert Spacementioning

confidence: 99%

Global anomalies on the Hilbert space

Delmastro

Gaiotto

Gomis

2021

We show that certain global anomalies can be detected in an elementary fashion by analyzing the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory. Distinct anomalous behaviours imprinted in the Hilbert space are identified with the distinct cohomology “layers” that appear in the classification of anomalies in terms of cobordism groups. We illustrate the manifestation of the layers in the Hilbert for a variety of anomalous symmetries and spacetime dimensions, including time-reversal symmetry, and both in systems of fermions and in anomalous topological quantum field theories (TQFTs) in 2 + 1d. We argue that anomalies can imply an exact bose-fermi degeneracy in the Hilbert space, thus revealing a supersymmetric spectrum of states; we provide a sharp characterization of when this phenomenon occurs and give nontrivial examples in various dimensions, including in strongly coupled QFTs. Unraveling the anomalies of TQFTs leads us to develop the construction of the Hilbert spaces, the action of operators and the modular data in spin TQFTs, material that can be read on its own.