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Lift of fractional D-brane charge to equivariant Cohomotopy theory. (English) Zbl 1457.81086

Summary: 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 \(\mathrm{SU}(2)\); 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.

MSC:

81T32 Matrix models and tensor models for quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
19L47 Equivariant \(K\)-theory
55N32 Orbifold cohomology
14F42 Motivic cohomology; motivic homotopy theory
19A22 Frobenius induction, Burnside and representation rings
20N02 Sets with a single binary operation (groupoids)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)

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References:

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