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started some bare minimum omn RR-field tadpole cancellation. Currently I am using this just to complement discussion at intersecting D-brane models
added a bunch of references that discuss tadpole cancellation in terms of K-theory charges.
In particular section 4 of
makes explicit the tadpole cancellation condition as equations satisfied by the representation characters corresponding to fractional D-branes.
I started to spell out and discuss some actual mathematical details of tadpole cancellation for the special case of fractional D-branes at orientifold singularities (here), where it turns out to be a neat little thing in elementary representation theory.
It took me a good while to isolate the required general statement from the literature, now referenced to in Marchesano 03.
Beware that in doing so, I am currenly slightly extrapolating: The tadpole cancellation condition as I now state it in the entry is stated in Marchesano 03 (4.9) only for A-type singularities. The formula trivially generalizes to all kinds of singularities, but one should check that in this evident generalization it really still encodes RR-tadpole cancellation. I haven’t yet found any references considering this.
But in either case, the key point is that for a linear representation $V$, read as a fractional D-brane charge, the character value
$M \coloneqq tr_V(e) =dim(V)$at the neutral element plays a different role than those at the non-trivial elements
$Q_{g} \coloneqq tr_V(g) \;\;\;\; g \neq e \,.$The former plays the role of mass, and only the latter should be referred to as RR-charges.
With this finally sorted out, the idea of approaching this via stability conditions becomes viable again…
Thanks!
Right, I was still going through the references that this section 4 is appearently based on, apparently it’s specifically these two
G. Aldazabal, D. Badagnani, Luis Ibáñez, Angel Uranga, Tadpole versus anomaly cancellation in $D=4,6$ compact IIB orientifolds, JHEP 9906:031, 1999 (arXiv:hep-th/9904071)
Gabriele Honecker, Intersecting brane world models from D8-branes on $(T^2 \times T^4\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds, JHEP 0201 (2002) 025 (arXiv:hep-th/0201037)
If that is the case, then (4.9) and (4.15) in
is a real achievement in global clarification
added also the example of tadpole cancellation at a $2 T$-singularity, here
The mass of the generating tadole-free fractional D-brane is… 24.
Interestingly, this still has $\mathbb{Z}$-worth of solutions to the homogeneous tadpole condition, even though (in contrast to the dihedral cases) this system now is overconstrained.
Okay, so the operation of forming minus the sum of the truncated character values recovers the dimension of the rep modulo the order of the group (since it gives the actual dimension on the nontrivial irreps and 1 minus the order of the group on the trivial irrep). This means that the virtual reps in the kernel of the truncated character map must have dimension being a multiple of the order of the group. But by injectivity of the full character map, these must be the multiples of the regular rep. So these are generally the only solutions to that homogeneous tadpole cancellation condition.
maded a bunch of little edits here and elsewhere:
included here graphics for tadpole cancellation from worldsheet perspective
gave O-plane a stub section on O-plane charge and cross-linked with pointer to the literature
gave Riemannian orbifold a graphics illustrating $\mathbb{R}^2 \sslash \mathbb{Z}_2$
added more textbook references
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