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I have now what should be a readable pre-version of
This still needs a round of polishing or two. But it should be at least readable.
If anyone is interested, have a look, be critical and try to poke holes into it.
How do we get from the stipulation of the form of a
topological+boundary+defect local field theory
to there being a Lorentzian manifold where we can talk about spacelike separation?
In The Motivic Aspect you have a diagram with and , but then have and later.
the notion of as above is still much more general than the field theories typically of interest in nature and in theory
Could there be a more principled reason for this restriction than merely that they turn out to be what occurs in nature?
I guess the whole construction must be ’natural’ but it sure looks like a Heath Robinson contraption (Americans insert Rube Goldberg).
But less contrived than the usual description, no?
(accidentally double posted)
When all the dust settles I’m sure it will be seen to be God-given.
In fact I’m hopeful that the God-givenness of the whole theory will allow us to see why even number theorists glimpse something gauge-theoretic in their work.
the context of higher differential geometry for the description of bosonic fields and of higher supergeometry (for the descrition of fermion fields).
If particles divide into bosons and fermions, what about branes?
@David - grading by appropriate truncations of the sphere spectrum?
Good morning, just got up.
I’ll reply in more detail just a little later. Just quickly about this here:
I guess the whole construction must be ’natural’ but it sure looks like a Heath Robinson contraption (Americans insert Rube Goldberg).
Okay, thanks for saying this. This means that I failed in the exposition of the simplicity the key idea and need to do something about it.
For the moment, let me just say it more in brief words. The simple three-step is this:
we are in a suitable ambient homotopy theory;
that’s the collection of all its contexts – the choice of any one we call a choice of “prequantum field theory”;
we semi-linearize by dependent sum along a linear 1d representation ;
the resulting slice is a “monoidal topos”;
this essentially canonically maps to the tensor category of modules and linear maps.
finally we consider relations in the original context. The linearization makes them relations in the tensor category. Relations in a tensor category naturally want to turn into functions.
We say: a relation in the prequantum context is a space of field trajectories, and its associated linear function is the corresponding quantum operator.
That’s it.
That should sound non-contrived. If not, then I am still not saying it well. Then I should try to improve further.
But now I first need to hunt for some breakfast.
Hi David,
in between coffee and first talk in the morning, I have started a super-quick note which I hope will be going along the lines that you are hoping to see.
For you and those with tolerance for lightning unpolished notes have a look here.
Have to dash now. More later.
Looking good. Do you have any questions to ask God, as to why He made certain choices?
Have polished a tad more, here.
why He made certain choices?
Which choices are you now thinking of? In a way the whole program here is doing away with the unneceary choices.
Have to run again, more later.
Now I have a minute.
Currently the main open question about choices that I have is “What determines the choices of ring (spectrum) ?”
We know the constraints:
there needs to be a way to twist, hence a map ;
we need to be able to find consistent -orientations on the correspondences of interest.
The second condition is a major constraint and probably cuts down the possibilities immensely. But currently I have litte insight what the systematics of this are, or should be.
I have fine-tuned that section yet a bit more.
Thanks for making me write this!
And now I’ll actually react to the above comments:
How do we get from the stipulation of the form of a
topological+boundary+defect local field theory
to there being a Lorentzian manifold where we can talk about spacelike separation?
Very good question. This is at the heart of the story here: it turns out that this “motivic quantization” encodes the holographic principle, and it is via this relation that non-topological/geometric/physical field theories are indeed encoded, as boundary field theories of the topological bulk theory.
To start with, the central example is that a particle (such as you see around you), with phase space a Poisson manifold, is quantized as the boundary theory of the non-perturbative 2d Poisson-Chern-Simons topological open string whose endpoints are attached to that Poisson manifold.
But everything in sight, bulk and boundary field theory is local, and so I am talking about spacelike separation distance. For the topological theory what matters is just distance as such.
By the way, in case it seems mysterious: the reason why boundary theories pick up non-topological “geometric” data is due to the choice of orientation/measure involved in the pull-push quantization. In the bulk this is highly constrained by functoriality (“consistent orientations”) but on the boundary, there is a free choice.
This has long been known from AdS3-CFT2 and CS-WZW correspondence: the choice of quantization of the CS theory on the boundary is what gives the boundary WZW theory its non-topological geometric (here: conformal) structure.
the notion of Z as above is still much more general than the field theories typically of interest in nature and in theory
Could there be a more principled reason for this restriction than merely that they turn out to be what occurs in nature?
Good question. I don’t know. But this here we had talked about this before: if there is no Lagrangian/action functional, but just the abstract given, then it is pretty unclear how to extract actual physics from this data.
Of course this might just us (or me here) not seeing clearly enough. But that’s the only half-way decent answer that I currently have.
In fact I’m hopeful that the God-givenness of the whole theory will allow us to see why even number theorists glimpse something gauge-theoretic in their work.
Yeah, so the key point of it all here is of course now the “motivic aspect”, which means (best seen at the modern formulation of non-commutative motives and at KK-theory) that one considers a universal linearization of a non-linear situation, subject to some desired conditions. Then the theorems say that these universal linearizations are represented by correspondences that are equipped with cocycles in generalized cohomology. These in turn are integral kernels and under the equivalence relaiton they map to their “path integral”.
This is how from a general criterion that everyone may run into, namely “universal linearization”, one runs into all kinds of QFT-related structures. QFT is – that’s conversely the claim here – the result of universally linearizing, in some sense, infinity-toposes.
@Urs #15
A silly question, but do we ever see and so the twist is just the inclusion? We probably need something more like as some sort of homotopy cohere centre of , but what about in low-dimensional cases?
the twist is just the inclusion?
I guess you are asking if it makes sense to take the “group of phases” to be just itself. Yes. I didn’t quite write it that way because it is also good to have the freedom not to have the identification.
But a place where (up to the relevant truncation degree set by the field theory dimension) apears for KU is the 2-field theory being quantized the type II superstring. Here the B-field is a super line 2-bundle and (as discussed there) the geometric realization of the moduli 2-stack of these is just in the relevant low degrees (even for the relevant “non-connected delooping” that includes the degree twist.)
Ah, I forgot that is an analogue of a commutative ring :-) I wasn’t assuming that was abelian, so yes: my question was “when is ?”
It begs the question as to whether we see in examples that the map giving the twist has certain properties, of which the strongest is that it is an equivalence. For example -truncated or what-have-you (or whatever a sensible notion of ’subgroup’ is in the -setting.
Yes, that is the question, which above in #15 I said I wish I understood more sytstematically. One needs some with some twists such that one can find consistent orientations in twisted -cohomology of the trajectory correspondences such as to have a functorial pull-push.
Whether finding this is a matter of trial and error (as currently in the examples) or whether there is a systematic way to find the suitale linearizations of a given higher prequantum field theory, I don’t know at the moment. I wish I did, though.
I have now concluded the Examples section of the “Expositional summary” section which what is maybe the strongest motivating class of examples for the whole story: the formalization of -WZW models holgraphically dual to -Chern-Simons theory, notably on the 5-brane.
For I noticed in discussion here at GAP XI (and in previous incidents which readers here may or may not remember)that the story to be solved here (Witten 98) is not quite as explicitly appreciated as it might be. So I have added some comments.
Some great slogans around, such as
This is how from a general criterion that everyone may run into, namely “universal linearization”, one runs into all kinds of QFT-related structures. QFT is – that’s conversely the claim here – the result of universally linearizing, in some sense, infinity-toposes
Neatening it up, and assuming you would qualify the last term with ’infinitesimally cohesive’
QFT is the result of universally linearizing infinitesimally cohesive infinity-toposes
What if the ’space of cohesion’ turns out to be vast? Why would Nature content itself with the Supergeometric? What might happen with a Berkovich-ish one, were it to exist?
Perhaps we need a universal infinitesimally cohesive infinity-topos :)
@David C - perhaps this new release will help answer that MO question: Foundations of Rigid Geometry I
So that’s related to another issue which I don’t fully understand yet.
The presence of cohesion makes certain things come out right that are related to quantum observables. Namely slicing over makes the automorphisms of the action functionals be the right quantomprhism/Heisenberg groups with Poisson bracket/Heinebgerg -algebras, and makes actions of groups be Hamiltonian actions, which in turn is the right condition with pull-push quantization to satisfy “quantization commutes with reduction”.
On the other hand, the actual pull-push for quantization is done after applying dependent sum along , hence after forgetting the -connection. This is not a surprise, of course, it is just the familar statement that wavefunctions/quantum states are polarized sections of the prequantum line bundle, not flat sections. In summary, the connection does not affect what the states are just how the quantum observables act.
Anyway, what I am saying is that much of the pull-push will at least be defined also in non-cohesive toposes, even if maybe it may be harder there to find orientations etc. I don’t know, nobody knows this. The vast majority of the literature deals just with pushing down along maps of manifolds, a little bit with Lie groupoids, certainly not along maps of general stacks.
So without cohesion, quantum states are defined, but not quantum observables and their properties.
(That said, I should warn of the following subtlety: the notion of observables in geometric quantization and hence in “motivic” quantization is actually rather different than in deformation quantization).
@David R: That’s just volume 1. An accessible introduction. So their rigid spaces are richer than Berkovich spaces (fig. 6, page 14).
If infinity-toposes are being linearized, shouldn’t we expect tangent infinity-toposes to feature? Hmm, there was a suggestion of parameterized spectra as a means of understanding twitsed cohomology.
there was a suggestion of parameterized spectra as a means of understanding twitsed cohomology.
This is at the heart of what is going on at motivic quantization:
from a prequantum circle -bundle
one obtains the associated prequantum -line bundle
and this is an -(infinity,1)-module bundle, hence a “parameterized ring spectrum” over the moduli stack of fields. The sections of this higher prequantum line bundle – hence the wavefunctions / quantum state are the -twisted -cohomology cocycles of the moduli stack:
Arthur Parzygnat made a video-recording of the talk, see here.
Any interesting questions?
With Cohen I had a very interesting discussion afterwards about how string-topology-operations would fit in as an extended TQFT (that it fits in unextendedly is one of the examples in Joost’s thesis). Via:
which identifies the “space of 2-states” of the string topology-TQFT of a manifold to the Fukaya category of its cotangent bundle, hence with the extended A-model , this is more generally asking how TCFT would fit in.
This question I had talked about a bit with H. Tannaka, who in turn has a “topological” construction of the Fukaya category via cobordisms. One guess was that many of these -categories are categories of modules of -algebra, and as such would very naturally qualify as 2-modules. Maybe this is true for more examples than currently understood when -algebras are considered not just over an ordinary base ring, but over a genuine -ring. That’s one of the questions that were floating around.
I also have a bunch of question myself. Work of Hisham Sati and Westerland strongly suggests that we should be linearizing over the integral Morava K-theory -rings. While I am far from having this under control, with this question I know at least in principle how to proceed and what to check…
Re #28, so are parameterized spectra appearing because they form the tangent -category to ?
Currently they are just appearing by decree, by declaring: “let’s look at the linearization of this in higher linear algebra”.
But eventually it would be nice to exhibit this step more as “god given” by seeing how it implements some universal construction with respect to the tangent cohesion.
Yes, this seems plausible and would be nice. I don’t see it yet in technical detail, though. Need to think about it.
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