Back to AdS/CFT. Which theory represents spacetime accurately? The target metric in string theory (or at least a non-compactified version) plays a pretty traditional role in spacetime.

CFTs are invariant under conformal transformations of the metric.  This means that the thing that might represent ST is the equivalence class of metric conformally equivalent to Minkowski.

Does a spacetime with only conformal structure even represent spacetime? No inertial structure. Geodesics, even, are not invariant under conformal transforms.  Thus the structure is not a candidate for representing phenomenological spacetime.

Giant insects!?  [any notion of being larger or smaller at stake]

Brian Pitts: Are these the same point? If nothing were massive, conformal structure would be all we need.

EK: Sure, but not exactly the same point.

Which version of EM correctly describes spacetime?

They both seem to do a fairly good job.

Which offers a more fundamental picture?

By identifying gauge-related classes of bundles we arrive at the holonomies. Both descriptions describe the same (non-ST) degrees of freedom.

Questions EK doesn’t understand (heading off objections to aforementioned controversial views?)

Q. Is the (e.g) fibre bundle space a *real* *physical* space?

A. It can represent physical d.o.f, sure. But what of it? It’s a mathematical space.

Q: Where does the stuff of the universe *really* *live*?!

A: You are pushing a metaphor too far, give it up. 

We tend to have the view that spacetime and dynamical symmetries should match. (e.g. Earman).  But in GR it is not quite clear to say how this should go.

The roles we care about will be filled by an object if it provides (locally) preferred inertial co-ords. (Brown, more or less).

Thus we can interrogate a theory to see if it has such an object.

What is the arena in which physical reality unfolds? (From Albert)  That is what is at stake.

But EK will be questioning this approach.

1. Which is more fundamental?

Do they have different domains? Cope better with closely related cases?

2. Which objects represent space-time role?

EK: this is to be functionally determined (one of those controversial claims).

Contrast `functional kind’ with `compositional kind.’  We specify functional role and then look for what instantiates that role in the theory.

Take it for granted that AdS/CFT correspondence holds.  What are the realist interpretational options?

1. Bulk theory is reality.

2. Boundary theory is reality

3. Both represent the same reality?

How do we make sense of 3?

Intends to say many controversial things about spacetime. So leave objections til the end.

The debate we will be looking at:  holonomies (Belot, Healey) vs. fibre bundle substantivalism (Arntzenius, Nounou).

Aharonov-Bohm effect used to argue that holonomies about closed paths to be the fundamental physical quantities.

Arntzenius: Fibre bundles *really exist* and their parts.

Obviously dimensionality will be distinct here.

AdS/CFT is a generic name for the class of conjectures about dualities linking higher and lower dimensional theories.

Can we make sense of the idea that these two descriptions are of the *same* reality? Is there not a fact of the matter about how many dimensions there are?

But there questions are not so weird for philosophy, really.  Think of gauge theory arguments; configuration space realists in quantum theory.

Reassures us that this will *really* be non-technical.

Compares self to opinionated weekender in the land of String Theory (along lines of Nick’s earlier metaphor). [Laughter]

We will address a slightly more familiar case: gauge theories on fibre bundles.

Daniele Orti: What about dynamics?  No matter. Pure gravity there is not much to calculate.

Nic: Well the correct CFT dual would have matter states.  But we’re not sure how to get there.

[From the back]

We can look at asymptotic behavior, which we can specify, and that leads to AdS results.

Chris Wuthrich: I don’t quite understand the correspondence we have.  Are you saying that the notions of equivalence we have are not applicable?

Nic: It may be that we’re not at a sufficiently advanced level.  The categorical equivalence fails in other cases.

Another question?

CW: No, I was just saying I didn’t understand. Would you like to comment? [laughter]

Nic: Categorical tools have been applied to holography.

Sam Fletcher: But we can weaken categorical equivalence.  We don’t even know in this case what the categories are in this case to relate.  Would need to be sure of what the categories are.

Jeremy Butterfield: An old man question: long ago, the syntactic view said two theories were equivalent if they were (mtually) definitionally extensible. Halvorson would reject this, since allows labels to stick to structures [poor paraphrase].  But can model isomorphism not be improved so that having the same models i.e. having each other as one’s definitional equivalent?

[Interjection] Clark Glymour has recently responded to Halvorson somewhere along these lines.

Nic: I agree with JNB. But would be hard to write out syntactically.  Category of models is useful as it is native to modern math.  This is a good thing. 

Exits, for taxi awaiting outside.  [Applause]