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Straightening Out the Frobenius-Schur Indicator

by Steven H. Simon and Joost K. Slingerland

Submission summary

Authors (as Contributors): Steve Simon
Submission information
Preprint link: scipost_202209_00037v1
Date submitted: 2022-09-19 12:07
Submitted by: Simon, Steve
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


Amplitudes for processes in topological quantum field theory (TQFT) are calculated directly from spacetime diagrams depicting the motion, creation, annihilation, fusion and splitting of any particles involved. One might imagine these amplitudes to be invariant under any deformation of the spacetime diagram, this being almost the meaning of the "topological" in TQFT. However, this is not always the case and we explore this here, paying particular attention to the Frobenius-Schur indicators of particles and vertices. The Frobenius-Schur indicator is a parameter $\kappa_a=\pm 1$ assigned to each self-dual particle $a$ in a TQFT, or more generally in any tensor category. If $\kappa_a$ is negative then straightening out a timelike zig-zag in the worldline of a particle of type $a$ can incur a minus sign and in this case the amplitude associated with the diagram is not invariant under deformation. Negative Frobenius-Schur indicators occur even in some of the simplest TQFTs such as the $SU(2)_1$ Chern-Simons theory, which describes semions. This has caused some confusion about the topological invariance of even such a simple theory to space-time deformations. In this paper, we clarify that, given a TQFT with negative Frobenius-Schur indicators, there are two distinct conventions commonly used to interpret a spacetime diagram as a physical amplitude, only one of which is isotopy invariant --- the non-isotopy invariant interpretation is used more often in the physics literature. We clarify in what sense TQFTs based on Chern-Simons theory with negative Frobenius-Schur indicators are isotopy invariant, and we explain how the Frobenius-Schur indicator is intimately linked with the need to frame world-lines in Chern-Simons theory. Further, in the non-isotopy-invariant interpretation of the diagram algebra we show how a trick of bookkeeping can usually be invoked to push minus signs onto the diagrammatic value of a loop (the ``loop weight''), such that most of the evaluation of a diagram does not incur minus signs from straightening zig-zags, and only at the last step minus signs are added. We explain the conditions required for this to be possible. This bookkeeping trick is particularly useful in the construction of string-net wavefunctions, where it can be interpreted as simply a well-chosen gauge transformation. We then further examine what is required in order for a theory to have full isotopy invariance of planar spacetime diagrams, and discover that, if we have successfully pushed the signs from zig-zags onto the loop weight, the only possible obstruction to this is given by an object related to vertices, known as the ``third Frobenius-Schur indicator''. We finally discuss the extent to which this gives us full isotopy invariance for braided theories.

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In refereeing

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Submission scipost_202209_00037v1 on 19 September 2022

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