The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this "Hypothesis H", here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman's comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional D-brane charges which do not take irrational values, in any twisted sector. Given that the possibility of irrational D-brane charge has been perceived as a paradox in string theory, we conclude that Hypothesis H serves to resolve this paradox. Concretely, we first explain that the Boardman homomorphism, in the present case, is the map from the Burnside ring to the representation ring of the singularity group given by forming virtual permutation representations. Then we describe an explicit algorithm that computes the image of this comparison map for any finite group. We run this algorithm for binary Platonic groups, hence for finite subgroups of SU(2); and we find explicitly that for the three exceptional subgroups and for the first few cyclic and binary dihedral subgroups the comparison morphism surjects precisely onto the sub-lattice of the real representation ring spanned by the non-irrational characters.

中文翻译：

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将分数 D 膜电荷提升到等变同伦理论

最近假设 K 理论 D 膜电荷对 M 理论的提升会落在同伦上同调理论中。为了进一步验证这个“假设 H”，我们在这里通过 Boardman 的比较同态明确计算了由从等变 K 理论到等变同伦理论的升力的存在所施加的对 ADE 取向奇点处的分数 D 膜电荷的约束。我们检查了相关案例，发现这个条件精确地挑出了那些在任何扭曲扇区中不取非理性值的分数 D 膜电荷。鉴于无理 D 膜电荷的可能性已被视为弦理论中的一个悖论，我们得出结论，假设 H 有助于解决这个悖论。具体来说，我们首先解释博德曼同态，在本例中，是从 Burnside 环到通过形成虚拟置换表示给出的奇点群的表示环的映射。然后我们描述了一个显式算法，该算法为任何有限群计算这个比较图的图像。我们为二元柏拉图群运行这个算法，因此对于 SU(2) 的有限子群；并且我们明确地发现，对于三个例外子群和前几个循环和二元二面体子群，比较态射精确地投射到由非无理字符跨越的实表示环的子格上。我们为二元柏拉图群运行这个算法，因此对于 SU(2) 的有限子群；并且我们明确地发现，对于三个例外子群和前几个循环和二元二面体子群，比较态射精确地投射到由非无理字符跨越的实表示环的子格上。我们为二元柏拉图群运行这个算法，因此对于 SU(2) 的有限子群；并且我们明确地发现，对于三个例外子群和前几个循环和二元二面体子群，比较态射精确地投射到由非无理字符跨越的实表示环的子格上。