Joint rule about loom HOT 5 CLOSED

With the current algorithm for joint creation, if walls S_alpha and S_beta meet, and
-alpha is orthogonal to beta,
-(ij) \in P_alpha and (jk) \in P_beta
There will be a joint between the walls that gives rise to a third wall, right?
(Our theory construction would forbid such a joint instead)

If we know this is correct, why use the current joint algorithm to check our conjecture? already
Isn't this already a difference between the two?

from loom.

Indeed it is, and we see one example of such a joint when we consider D_3. In that case we have to exclude such a joint to get a 2d BPS spectrum that is consistent with a coset model. We will test the 4d BPS spectra of G = SO(6) and compare the result with the BPS quiver analysis result.

Another interesting check will be studying the 2nd fundamental representation of A_3, whose weight structure is the same as the 1st fundamental representation of D_3. Here we can investigate 2d and 4d BPS spectra and compare the result with those from the 1st fundamental rep cover of A_3. This is a good test of our conjectures in many ways. We're heading there.

from loom.

Well, I'm really asking whether it's even worth proceeding with the current joint rule, if we already know it's giving wrong results (spectra that are incompatible with the deformations of coset models).

While it's true that we need to do checks, as you pointed out, there are other ways to make those.
So, I'm wondering if we should proceed right away implementing the joint rule of our note (the 'Lie-algebraic' rule).

from loom.

Ah, I see your point. You're right, I just wanted to have, at the back of my mind, that if the Lie-algebraic rule somehow does not work, then we should consider going back to path concatenation. But definitely we should proceed with our currently conjectured rule for now and see if it gives us consistent results.

from loom.

One more thing. Currently, one of the attributes of S-walls is 'x', which is a list of sheet-coordinates. But, S-walls are really trajectories on C, hence the attribute 'z' already contains their basic geometric info. Moreover, to keep track of which sheets they run on, we already have other ways: the attribute 'local_weight_pairs', for example.
If we want to know the 'x' sheets at a given z, we could simply write a method which gives such x's, instead of keeping such info along the whole wall.
In fact such info seems redundant, especially if we switch to the 'Lie-algebraic rule' for joint formation. Right now we aren't using the x's data anywhere, and again, if we need it at a given point, we can simply generate it via a method of the class.
This way we should save quite some space when saving data files, especially with higher-reps.

from loom.