Lift of fractional D-brane charge to equivariant Cohomotopy theory

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Abstract

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.

Section snippets

Fractional brane charge quantization in M-theory

The issue of irrational D-brane charge. It is a long-standing conjecture [75, Sec. 5.1] that the charge lattice of fractional D-branes [19] stuck at G-orientifold fixed-point singularities is the G-equivariant K-theory of the singular point, hence the representation ring of G (e.g. [32]). However, it was argued already in [16, 4.5.2] that not all elements of the representation ring can correspond to viable D-brane charges, and a rationale was sought for identifying a sub-lattice of physical

The Burnside ring and K-theory

Finite group actions control orbifold spacetime singularities in string theory. For G a finite group one may consider linear as well as purely combinatorial actions of G on some set (see e.g. [20], [36], [42], [71]). We will be concerned with the relation between these two types of actions (see e.g. [4], [9], [71]).

Basics. Traditionally, for k any field, the linear actions receive more attention as they are the k-linear representations of G, namely the group homomorphisms GAutk(V) from G to

The image of β-Computations

Given a finite group G and its irreducible characters over a given field k of characteristic zero, Theorem 2.37 provides an effective algorithm for identifying the image of A(G)βRk(G) (7) and checking whether β is surjective. We have implemented this algorithm in Python (the code is available at [12]). From this we obtained the results shown in Theorem 3.1. Here the

show that, over the real numbers, β has vanishing cokernel/is surjective onto the ring of integer characters (i.e. onto

Acknowledgments

We thank Tim Dokchitser, James Dolan, James Montaldi and Todd Trimble for discussion. Our algorithm is inspired by the note [72], which in turn goes back to private communication with James Dolan.

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