SciPost Submission Page
Straightening Out the FrobeniusSchur Indicator
by Steven H. Simon and Joost K. Slingerland
Submission summary
As Contributors:  Steve Simon 
Preprint link:  scipost_202209_00037v1 
Date submitted:  20220919 12:07 
Submitted by:  Simon, Steve 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
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 FrobeniusSchur indicators of particles and vertices. The FrobeniusSchur indicator is a parameter $\kappa_a=\pm 1$ assigned to each selfdual particle $a$ in a TQFT, or more generally in any tensor category. If $\kappa_a$ is negative then straightening out a timelike zigzag 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 FrobeniusSchur indicators occur even in some of the simplest TQFTs such as the $SU(2)_1$ ChernSimons theory, which describes semions. This has caused some confusion about the topological invariance of even such a simple theory to spacetime deformations. In this paper, we clarify that, given a TQFT with negative FrobeniusSchur 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 nonisotopy invariant interpretation is used more often in the physics literature. We clarify in what sense TQFTs based on ChernSimons theory with negative FrobeniusSchur indicators are isotopy invariant, and we explain how the FrobeniusSchur indicator is intimately linked with the need to frame worldlines in ChernSimons theory. Further, in the nonisotopyinvariant 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 zigzags, 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 stringnet wavefunctions, where it can be interpreted as simply a wellchosen 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 zigzags onto the loop weight, the only possible obstruction to this is given by an object related to vertices, known as the ``third FrobeniusSchur indicator''. We finally discuss the extent to which this gives us full isotopy invariance for braided theories.