论文标题
孤子对称性超出同型：来自拓扑量子场理论的无可逆转和不可延迟性的可逆性
Solitonic symmetry beyond homotopy: Invertibility from bordism and noninvertibility from topological quantum field theory
论文作者
论文摘要
据信，孤子对称性遵循拓扑孤子的同型组分类。在这里，我们指出一个更复杂的代数结构时，当频谱中的不同维度共存时。我们在具体的量子场理论中揭示了这一现象，即$ 4 $ d $ \ mathbb {c} p^1 $模型。该模型具有两种孤子激发，涡旋和跳跃，根据同型组，将遵循两个$ u（1）$ solitonic对称性。尽管如此，我们通过评估涡旋运营商的跳跃费用来证明Hopfion $ u（1）$对称性的不存在。我们澄清说，保守的跳跃数是由3D旋转拓扑量子场理论（TQFTS）产生的不可固化的对称性。它的可逆零件只是$ \ mathbb {z} _2 $，我们将其识别为旋转的bordism不变。与3D $ \ mathbb {C} p^1 $模型相比，我们的工作提出了对孤子对称性和与拓扑阶段的统一描述。
Solitonic symmetry has been believed to follow the homotopy-group classification of topological solitons. Here, we point out a more sophisticated algebraic structure when solitons of different dimensions coexist in the spectrum. We uncover this phenomenon in a concrete quantum field theory, the $4$d $\mathbb{C}P^1$ model. This model has two kinds of solitonic excitations, vortices and hopfions, which would follow two $U(1)$ solitonic symmetries according to homotopy groups. Nevertheless, we demonstrate the nonexistence of the hopfion $U(1)$ symmetry by evaluating the hopfion charge of vortex operators. We clarify that what conserves hopfion numbers is a non-invertible symmetry generated by 3d spin topological quantum field theories (TQFTs). Its invertible part is just $\mathbb{Z}_2$, which we recognize as a spin bordism invariant. Compared with the 3d $\mathbb{C}P^1$ model, our work suggests a unified description of solitonic symmetries and couplings to topological phases.
