SciPost Submission Page
Straightening Out the FrobeniusSchur Indicator
by Steven H. Simon and Joost K. Slingerland
Submission summary
Submission information 
Preprint link: 
scipost_202209_00037v1

Date submitted: 
20220919 12:07 
Submitted by: 
Simon, Steve 
Submitted to: 
SciPost Physics 
Ontological classification 
Academic field: 
Physics 
Specialties: 
 Condensed Matter Physics  Theory
 HighEnergy Physics  Theory

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.
Current status:
In refereeing