Lift of fractional Dbrane charge to equivariant Cohomotopy theory
Abstract
The lift of Ktheoretic Dbrane charge to Mtheory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this "Hypothesis H", here we explicitly compute the constraints on fractional Dbrane charges at ADEorientifold singularities imposed by the existence of lifts from equivariant Ktheory to equivariant Cohomotopy theory, through Boardman's comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional Dbrane charges which do not take irrational values, in any twisted sector. Given that the possibility of irrational Dbrane 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 sublattice of the real representation ring spanned by the nonirrational characters.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.09679
 Bibcode:
 2018arXiv181209679B
 Keywords:

 Mathematics  Representation Theory;
 Mathematical Physics;
 Mathematics  Algebraic Topology;
 Mathematics  Group Theory
 EPrint:
 47 pages, Python code attached as ancillary file