Critical Lattice Model for a Haagerup Conformal Field Theory

Abstract

We use the formalism of strange correlators to construct a critical classical lattice model in two dimensions with the Haagerup fusion category H 3 as input data. We present compelling numerical evidence in the form of finite entanglement scaling to support a Haagerup conformal field theory (CFT) with central charge c ¼ 2. Generalized twisted CFT spectra are numerically obtained through exact diagonalization of the transfer matrix, and the conformal towers are separated in the spectra through their identification with the topological sectors. It is further argued that our model can be obtained through an orbifold procedure from a larger lattice model with input ZðH 3 Þ, which is the simplest modular tensor category that does not admit an algebraic construction. This provides a counterexample for the conjecture that all rational CFT can be constructed from standard methods.

Non-invertible symmetries of $$ \mathcal{N} $$ = 4 SYM and twisted compactification

“…By figure 8, the two are seen to be related by PSO(N ) 0,2 = σSO(N ) −,0 . 14 We can derive this result as follows. 13 Note that there is also an operation σ2 corresponding to gauging the Z one-form symmetry and a mixed anomaly between them.…”

“…We will briefly touch upon this possibility later, but will not consider it in detail. 14 Recall that figure 8 is the quotient of the full diagram by C, and thus from this figure alone it is actually unclear whether we should have PSO(N )0,2 = σSO(N )−,0 or PSO(N )0,2 = σ 3 SO(N )−,0. It turns out that the former is correct.…”

“…This leads to the notion of "generalized symmetries", and includes higher-form symmetries (for which the topological operator has codimension greater than one) and non-invertible symmetries (for which the associated topological operators do not possess an inverse, and as such do not form a group). The latter will be especially important for our purposes -they have been studied in two-dimensions in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and more recently in higher dimensions in [18][19][20][21][22][23][24][25][26]. Despite going beyond our traditional notions of symmetry, the topological nature of these operators suggests that they give rise to constraints similar to those provided by ordinary symmetries, including constraints on RG flows [4, 6-8, 21, 26].…”

Numerical Evidence for a Haagerup Conformal Field Theory

“…Before proceeding, we note that almost the same model was considered previously [12], and that its conformality will be announced by another group [13] simultaneously with the present work.…”

Asymptotic density of states in 2d CFTs with non-invertible symmetries

“…Here we use a version H 3 obtained by Grossman and Snyder [51] from a gauging of the original ones found by Haagerup. A two-dimensional CFT having this symmetry was recently found numerically in [52,53]. The category H 3 has six simple objects, 1, a, a 2 , ρ, aρ, a 2 ρ with the fusion rule a 3 = 1, aρ = ρa 2 , and ρ 2 = 1 + ρ + aρ + a 2 ρ; the quantum dimensions are given by dim a = 1 and dim ρ = ( √ 13 + 3)/2.…”