Ran Luo

We show that the ’t Hooft anomaly of a quantum field theory with continuous flavor symmetry can be detected from rearrangements of the topological defect webs implementing the global symmetry in general spacetime dimension, which is concretized in 2D by the of the defect lines. Via dualizing the defects to flat background gauge field configurations, we characterize the ’t Hooft anomaly by various cohomological data of the symmetry group, where the cohomology of Lie groups with discrete topology plays the central role. We find that an extra dimension emerges naturally as a consequence of the mathematical description of the ’t Hooft anomaly in the case of flat gauging.

A bstract: We define the symmetry category in 1+1d for continuous 0-form G -symmetry to be Sky ^τ ( G ), the category of skyscraper sheaves of finite dimensional vector spaces with finite support on the group manifold of G , where τ ∈ H ⁴ ( BG, ℤ) is the anomaly. We propose that the corresponding 2+1d SymTFT is described by the Drinfeld center of Sky ^τ ( G ). We show explicitly the way that τ twists the convolution tensor product of the objects of Sky ^τ ( G ). As a concrete example, we present the S and T -matrices for the simple anyons of the resulting Z ( Sky ^τ ( G )) category for G = U (1), both for the cases without or with anomaly, and discuss the topological boundary conditions as Lagrangian algebra of Z ( Sky ^τ ( U (1))). We also present the definition of Sky ^τ ( G ) and Z ( Sky ^τ ( G )) for the non-abelian case of G = SU (2), as well as the speculated modular data. We point out that in order to have a physically relevant center and Lagrangian algebras, it is necessary to generalize Sky ^τ ( G ) to a larger category, which we argue to be closely related to the category of quasi-coherent sheaves on G _ℂ with convolution tensor product twisted by τ .

We extend the symmetry topological field theory (SymTFT) framework to open quantum systems. Using canonical purification, we embed mixed states into a doubled (2+1)-dimensional topological order and employ the slab construction to study (1+1)-dimensional mixed-state phases through condensable algebras in the doubled SymTFT. Hermiticity and positivity of the density matrix impose additional constraints on allowable anyon condensations, enabling a systematic classification of mixed-state phases—including strong-to-weak symmetry breaking (SWSSB) and average symmetry-protected topological (ASPT) phases. We present examples of mixed-state phase transitions involving SWSSB and show how gauging within the open SymTFT framework reveals connections among different mixed-state phases.

We extend the symmetry topological field theory (SymTFT) framework to open quantum systems. Using canonical purification, we embed mixed states into a doubled (2+1)-dimensional topological order and employ the slab construction to study (1+1)-dimensional mixed-state phases through condensable algebras in the doubled SymTFT. Hermiticity and positivity of the density matrix impose additional constraints on allowable anyon condensations, enabling a systematic classification of mixed-state phases - including strong-to-weak symmetry breaking (SWSSB) and average symmetry-protected topological (ASPT) phases. We present examples of mixed-state phase transitions involving SWSSB and show how gauging within the open SymTFT framework reveals connections among different mixed-state phases.

We explore new aspects of internal fermionic shifting symmetries, present in physical systems such as free Dirac spinors and p -form tensor-spinor fields. We propose a novel procedure to gauge these global symmetries, which also introduces a new Stückelberg mechanism to give a mass to free fermionic fields. Furthermore, we find new magnetic fermionic symmetries in these physical systems whose charged objects are disorder operators. For the case of a 4D Dirac spinor, we discuss a dual description, where the magnetic symmetry acts on the holonomies of a dual 2-form tensor-spinor. Further generalizations such as higher-group-like structures are also discussed. Published by the American Physical Society 2024