Lift of fractional D-brane charge to equivariant Cohomotopy theory
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 -orientifold fixed-point singularities is the -equivariant K-theory of the singular point, hence the representation ring of (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 a finite group one may consider linear as well as purely combinatorial actions of 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 any field, the linear actions receive more attention as they are the -linear representations of , namely the group homomorphisms from to
The image of -Computations
Given a finite group and its irreducible characters over a given field of characteristic zero, Theorem 2.37 provides an effective algorithm for identifying the image of (7) and checking whether 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, 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.
References (76)
- et al.
Fun with
J. Number Theory
(2009) - et al.
Orbifold resolution by D-branes
Nuclear Phys.
(1997) Congruence relations characterizing the representation ring of the symmetric group
J. Algebr.
(1986)- et al.
Rational sphere valued supercocycles in M-theory and type IIA string theory
J. Geom. Phys.
(2017) D2-brane RR-charge on
Phys. Lett.
(2002)The Burnside algebra of a finite group
J. Combin. Theory
(1967)D-branes in b fields
Nuclear Phys.
(2001)Stable Homotopy and Generalized Homology
(1974)- et al.
Flux stabilization of D-branes
J. High Energy Phys.
(2000) - et al.
Rational representations and permutation representations of finite groups
Math. Ann.
(2016)
Representations and Cohomology Volume 1: Basic Representation Theory of Finite Groups and Associative Algebras
Orbifold boundary states from Cardy’s condition
J. High Energy Phys.
Equivariant homotopy theory lecture notes
HandBook of Categorical Algebra
Sur les groupes des classes de transformations continues
CR Acad. Sci. Paris
Biset Functors for Finite Groups
Gauge enhancement for super M-branes via parameterized stable homotopy theory
Comm. Math. Phys.
On the representation of a group of finite order as a permutation group, and on the composition of permutation groups
Proc. Amer. Math. Soc.
Equivariant stable homotopy and Segal’s burnside ring conjecture
Ann. of Math.
M-theory reconstruction from CFT and the chiral algebra conjecture
J. High Energy Phys.
Triples, fluxes, and strings
Adv. Theor. Math. Phys.
Orientifold Précis
Notes on the theory of representations of finite groups
M-theory (the theory formerly known as strings)
Internat. J. Modern Phys.
The World in Eleven Dimensions: Supergravity, Supermembranes and M-Theory
The WZW term of the M5-brane and differential cohomotopy
J. Math. Phys.
The rational higher structure of M-theory
Fortschr. Phys.
Twisted cohomotopy implies M-theory anomaly cancellation on 8-manifolds
Comm. Math. Phys.
Twisted cohomotopy implies M5 WZ term level quantization
Comm. Math. Phys.
Equivariant version of real and complex connective K-theory
Homology Homotopy Appl.
Real ADE-equivariant (co)homotpy of super M-branes
Comm. Math. Phys.
Which finite groups act freely on spheres?
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On leave from Czech Academy of Science.