## SciPost Submission Page

# Why are fractional charges of orientifolds compatible with Dirac quantization?

### by Yuji Tachikawa, Kazuya Yonekura

### Submission summary

As Contributors: | Yuji Tachikawa |

Arxiv Link: | https://arxiv.org/abs/1805.02772v2 |

Date accepted: | 2019-10-10 |

Date submitted: | 2019-10-04 |

Submitted by: | Tachikawa, Yuji |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | High-Energy Physics - Theory |

### Abstract

Orientifold $p$-planes with $p\le4$ have fractional D$p$-charges, and therefore appear inconsistent with Dirac quantization with respect to D$(6-p)$-branes. We explain in detail how this issue is resolved by taking into account the anomaly of the worldvolume fermions using the $\eta$ invariants. We also point out relationships to the classification of interacting fermionic symmetry protected topological phases. In an appendix, we point out that the duality group of type IIB string theory is the pin+ version of the double cover of $GL(2,\mathbb{Z})$.

###### Current status:

### Author comments upon resubmission

### List of changes

- We clarified that we only treat O+ planes. O- planes also induce anomalies of the Maxwell theory on the D-brane worldvolume around them, and are much harder to study.

- Referee 1 suggested that we might improve our brief summary of Gilkey's method to compute the eta invariant. We opted not to follow this advice, as this topic would be covered in more detail in our forthcoming paper (with Chang-Tse Hsieh) on the issue of fractional charges of both O+ and O- planes.

- Referee 2 suggested us to look at four papers related to our Appendix.

As for two papers by Sen hep-th/9603113 and hep-th/9604070, Sen indeed discussed a somewhat related issue, but not exactly on how the SL(2,Z) of IIB needs to be interpreted. There were also some other papers from that period discussing issues similar to what Sen discussed. But they are all cited in Dabholkar's review hep-th/9804208, which we did cite and was actually the input of our discussion in the Appendix, as can be seen above our eq. (A.3). So we consider Sen's two papers already implicitly cited.

As for two papers hep-th/9810153 and hep-th/9810213 on SL(2,Z) anomaly of type IIB theory, we agree that this is an interesting question. But our objective in the Appendix was to clarify the precise group structure of the duality group, and studying its anomaly is something beyond our aim in this paper. We added a brief foonote concening this point.

- We also added other minor clarifications.