Straightening Out the Frobenius-Schur Indicator

TL;DR

This paper clarifies the role of the Frobenius-Schur indicator in topological quantum field theories, especially in relation to diagram invariance and framing, resolving confusion about isotopy invariance in theories with negative indicators.

Contribution

It provides a detailed analysis of how Frobenius-Schur indicators affect diagram invariance in TQFTs, introducing methods to manage sign issues and conditions for full isotopy invariance.

Findings

Identifies two conventions for interpreting spacetime diagrams in TQFTs with negative indicators.

Introduces a bookkeeping trick to push minus signs onto loop weights, simplifying diagram evaluation.

Discovers the third Frobenius-Schur indicator as an obstruction to full isotopy invariance.

Abstract

The Frobenius-Schur indicator is a parameter $\kappa_a=\pm 1$ assigned to each self-dual particle $a$ in a TQFT. 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. This has caused some confusion about the topological invariance of even simple theories to space-time deformations. 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. 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. 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.