Develops local diagnostics for strong symmetries and strong-to-weak symmetry breaking via infinite-volume definitions and local charge coherence, introduces von Neumann symmetries, and derives an LSM-type anomaly constraint for quantum spin chains.
Brylinski,Differentiable cohomology of gauge groups,math/0011069
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abstract
We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilinson lead to differentiable cohomology classes with coefficients in C*. These may be viewed as an enrichment of the Chern-Simons differential forms. By transgression, classes in differentiable cohomology of a Lie group G lead to differentiable cohomology classes for gauge groups Map(M,G). These classes generalize the central extensions of loop groups. We also discuss holomorphic cohomology of complex Lie groups as the natural place to construct secondary classes. We present several conjectures relating the above cohomology classes to the differential forms of Bott-Shulman-Stasheff.
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The symmetry category of a 2D QFT with G-symmetry and anomaly k equals the twisted Hilbert space category Hilb^k(G), whose Drinfeld center is the twisted representation category of the conjugation groupoid C*-algebra, enabling braiding computations in the 3D SymTFT.
A quantum mechanical framework is given for Hilbert and defect spaces of line operators in BF+kCS TQFT, with line operator action realized by convolution kernels and matches to Verlinde and semiclassical Hopf-link data.
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A local description of strong symmetries and strong-to-weak symmetry breaking in quantum many-body systems
Develops local diagnostics for strong symmetries and strong-to-weak symmetry breaking via infinite-volume definitions and local charge coherence, introduces von Neumann symmetries, and derives an LSM-type anomaly constraint for quantum spin chains.