Why are fractional charges of orientifolds compatible with Dirac quantization?

Abstract

Orientifold p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> </mml:math> -planes with p\le 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> have fractional D p <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> </mml:math> -charges, and therefore appear inconsistent with Dirac quantization with respect to D (6{-}p) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>6</mml:mn> <mml:mo>−</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> -branes. We explain in detail how this issue is resolved by taking into account the anomaly of the worldvolume fermions using the \eta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>η</mml:mi> </mml:math> 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 ^+ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi/> <mml:mo>+</mml:mo> </mml:msup> </mml:math> version of the double cover of GL(2,Z).