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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
added pointer to today’s
I have fixed the statement of the tadpole cancellation condition to match the case of K3-style $\mathbb{T}^4/G^{DE}$-examples actually listed afterwards, which, for orientifold groups $D1:\; \mathbb{Z}_2$ and $D3:\; \mathbb{Z}_4$ is equation (18) in Buchel-Shiu-Tye 99.
The result in this case is, rephrased in maths lingo: The correct element in the $G^{DE}$-equivariant K-theory of the point must be exactly that multiple of the regular representation whose total dimension is 32:
$n \cdot k[G] \in K_G(\ast) \simeq R_G \phantom{AA} \text{such that}\; n \cdot \left\vert G\right\vert = 32 \,.$I still haven’t found literature that would generalize this beyond the case $D3:\; \mathbb{Z}_4$ to binary dihedral groups. But if the statement remains the same for all finite subgroups of $SU(2)$ in the D- and E-series, then, interestingly, there are only the following possible cases (since in all other cases no multiple of the dimension of the regular rep is 32):
Dynkin label | orientifold group $G$ | tadpole cancelling charge in $K_G(\ast)$ |
---|---|---|
D0 | $1$ | $32 \cdot k[1]$ |
D1 | $\mathbb{Z}_2$ | $16 \cdot k[\mathbb{Z}_2/1]$ |
D3 | $\mathbb{Z}_4 = Dic_1$ | $8 \cdot k[Dic_1/1]$ |
D4 | $2 D_4 = Dic_2 = Q_8$ | $4 \cdot k[Dic_2/1]$ |
D6 | $2 D_8 = Dic_4$ | $2 \cdot k[Dic_4/1]$ |
D20 | $2 D_{16} = Dic_8$ | $1 \cdot k[Dic_8/1]$ |
Hm, but then I don’t know at the moment whether all these dicyclic groups in #17 are point groups of 4d crystallographic groups.
It’s easy to see to be the case for $\mathbb{Z}_4$ (and of course for $\mathbb{Z}_2$).
Are $Dic_2$ and/or $Dic_4$ the point groups of any 4d crystallographic groups?
[edit: ah, for $Dic_2 = Q_8$ it should be obviously true, too]
Is this what you’d call a bottom-up approach, bottom-up and top-down model building? You have a theory with anomalies and you look to cancel them by adjustment, “tadpole cancellation via orientifolding”. It sounds like things are added in to achieve a goal, rather than because they have to be there.
Is it possible to go top-down and say there can’t be tadpoles because… (Something like the graph complexes arising from the cohomology of configuration spaces don’t admit tadpoles.)?
It’s another anomaly cancellation issue:
1) From the point of view of perturbative string theory, one tries to construct a consistent 2d SCFT with certain properties by starting somewhere (with some superstring sigma model), and then incrementally imposing further consistency conditions.
2) From the point of view of a putative non-perturbative formulation, one may ask whether these consistency conditions are all just aspects of the condition that the relevant charges be in a certain generalized cohomology theory.
Here with RR-field tadpolce cancellation in toroidal orientifolds, we are looking for a cohomology theory which says that admissible brane charges on a toroidal orienti 5-fold must be of the form: 1) “multiple of regular rep of orienifold group minus number of fixed points times the trivial rep” with 2) total dimension vanishing.
The question is then: Which cohomology theory has cocycles whose component data is just of this form? The answer is going to be: unstable equivariant Cohomotopy, by the equivariant Hopf degree theorem.
Ok, thanks!
By the way, I am still looking for more references for more entries for the table at RR-field tadpole cancellation on toroidal orientifolds – table.
What I’d really like is references computing the tadpole cancellation conditions for the D-brane configurations which correspond to M5-branes in M-theory on S1/G_HW times H/G_ADE, which means: type I’ orientifolds with D4-branes at ADE-singularities and with D8-branes at O8-planes
In Kataoka-Shimojo 01, in a discussion of pure D4-brane orientifolds, the authors actively shy away from diving into the computation involving also D8-branes:
The $\mathbb{Z}_N$ action with even $N$ contains an order 2 element $[ ...]$ Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use of the same dimensional D-branes, we restrict ourselves to the case that the order $N$ of $\mathbb{Z}_N$ is odd. (p. 4)
And indeed, most (all?) of the ensuing activity on pure D4 or pure D8 orientifolds (Honecker et al.) focuses on the odd-order case $\mathbb{Z}_3$.
Does anyone discuss tadpole cancellation for D4+D8 branes on $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \sslash \mathbb{Z}_{2k}$ ??
(It is clear what the result should be, but it would still be good to have a reference for an explicit computation.)
I tend to doubt now that this computation has been published, but if anyone has anything, please drop me a note.
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