@Danu Uh. Well. I have a statement "heterotic theory on $T^3$ is equivalent to M-theory on K3"

Now I'm told to apply this "fiberwise", to get that heterotic theory on $W$ is equivalent to M-theory on $X$

I can't help you with udnerstanding how heterotic/M theory come into this, I'm afraid

But I don't really understand how that is supposed to work, i.e. I don't understand what it means to do this "fiber by fiber", as you propose

OK

@EmilioPisanty i'm learning a lot from it and it is very interesting, but beyond that is there any especial reason why it is "awesome"?

So you supposedly have some prescription (a map?!) to go from one manifold, $T^3$, to another, which is a K3 surface.

Does this map only exist when the $T^3$ and K3 are the entire compactifying-manifolds?

@Danu Well, basically one observes that "heterotic on $T^3$" and "M-theory on K3" lead to the same $\mathcal{N}=1$ effective theory.

So they're "dual".

There is no "map" I can see

OK, so that seems to be a duality that depends on the fact that $T^3$ and K3 are the entire compactifying manifolds (CMs), maybe?

@heather @EmilioPisanty was in the same year, at the same Uni, in the same CDT as Mercedes

For the type IIa duality, this is much neater, since I know that you can go from the M-theory to the IIa theory by compactifying on a circle, which gives a kind of map

@Mithrandir24601 true, that

@Danu Yes, so it would seem to me, but everyone just non-chalantly states one can do this "fiberwise"

OK, not so important for now

In any case, this map still makes sense in the case of a product, clearly, because then you can compactify "step by step", i.e. first over the fiber, then over the base

imperial.ac.uk/controlled-quantum-dynamics/people/students/… back in the when

Yes, for a trivial fibration, it's also clear

But since a generic fibration has no "gluing maps" like e.g. a vector bundle would have, I can't see how to glue these local descriptions together. Also I'm not even sure if the fibrations are guaranteed to be locally trivial

probably not

@Mithrandir24601 oh, cool!

they probably want to allow singular fibers, as is the usual case in algebraic geometry

@Danu Yup

@heather And I've met Mercedes (briefly, once) and am doing a PhD on a linear optical quantum chip :) England's a small country...

That seems to rule out the idea of bootstrapping the product case.

@Mithrandir24601 where are you studying?

@EmilioPisanty Bristol Quantum Engineering CDT

@Danu Yup. I've thought a bit about this already before I asked the question ;)

@heather CC @Mithrandir there's another good picture of Mercedes at the end of Terry's inaugural lecture

@Mithrandir24601 cool

and yeah, that'll put you in frequent contact with the Imperial group

though now that Terry's stateside maybe not so much

@ACuriousMind At the start of the paper coauthored by Witten, there are some further comments

First of all, explicit confirmation that one wants to allow singularities

Secondly, they say "taking the $T^3$'s to be small [...]" before doing fiberwise blahblah

@EmilioPisanty Yeah, well, apparently one of my secondary supervisors is in America and there are links with Bristol and the company they've set up (whatever it's called now), so you never know!

(said secondary supervisor is completely unrelated)

heh

fair enough

are you working with Jeremy by any chance?

@EmilioPisanty O'Brien? Nope

My primary supervisor will be Anthony Laing. One of his post-docs and apparently someone from America that I've never met before will be secondary supervisors

There: Yogesh Joglekar

fair enough, don't know either of them

"will be" as in, you're in the MRes year?

we don't get an MRes, but the first year, yeah

cool