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I have started an entry analytic space with material on Berkovich’s non-rigid analytic geometry.
I don’t really know this subject and have been adding material to the entry as I read about it and to the extent that I correctly understood it. Experts are most welcome to help out.
As indicated here, I am motivated by the following: Berkovich’s local contractibility result suggests that the $\infty$-topos of $\infty$-sheaves over the site of $p$-adic analytic spaces might be cohesive.
The idea would be that his result implies (if it does) that the site (category with coverage) of contractible $p$-adic afine spaces is a dense subsite of that of all $p$-adic spaces. Since it should be an infinity-cohesive site that would imply the claim.
But despite looking through Berkovich’s writings for a little bit today, I am still not sure if he just shows that the underlying topological space of a $p$-adic anayltic space is locally contractible, or if one may indeed deduce that they are locally contractible with respect to étale homotopy, as would be needed for the above conclusion.
Is this part of an attempt to survey the range of cohesiveness?
Yes, I am trying to find useful subsites of algebraic spaces that are cohesive.
Perhaps then we might see why there can be a p-adic and adelic physics?
Hmm, that’s the frightening thing about maths. One step away from things you might know a little about, there are whole worlds you knew nothing about, where people have been working away for years. Now I see there’s a type of space in which the unit disk has four types of point and its possible to have a Potential theory and dynamics on the Berkovich projective line.
we might see why there can be a p-adic and adelic physics?
Yes, I was beginning to wonder about that yesterday. I had started to add some references to p-adic number – References – Applications. But I am not feeling well about this, since there is a tremendous amount of wild speculation in there and few concrete results, as far as I can see.
Notice that at the above link that you give, before long the page lists references to the journal “Chaos, Solitons and Fractals”. Deep nonsense territory.
This is not to say that there cannot be something of value here. If locally smooth $p$-adic analytic spaces indeed form a site for a cohesive $\infty$-topos, then the cohesive formalism would systematically spit out sensible $p$-adic analogs of good bits of physics. That would help me understand what’s really going on.
But first we need to clarify that question. I have now emailed Berkovich and asked him if his result implies a certain technical fact from which it would follow (I still can’t see it based on his texts, though it might be implicit in there).
The hope that working with p-adic numbers can remove divergences in QFT is an old hope which drives the p-adic physics for a long while; the main initiator is Vladimirov (the same Vladimir of the theory of generalized functions/distributions), and along with him Volovich and their students (included our Blagrade colleague B. Dragovich listed in the list). The reference list linked above is interesting, including listing recent works of Manin and of Marcolli. As Urs says there are also suspicious items there, to say it mildly, including a reference from C. Carlos, the guy who had serious problems with arXiv elsewhere for posting bad speculation as a science. On the other hand, one should also point out on mathematical side that the idea of rigid analytic geometry for nonarchimedean field does not have its origin in Berkovich, as some of the above comments and some comments in the entry might imply, but in the original works of Tate. Berkovich just gave one of the later/more modern formalisms, based on his discovery of Berkovich spectrum (which should be the entry for that version, while rigid analytic geometry and analytic space should eventually be about all formalisms, including Huber’s etc.). Of course, it would be nice that Frederic Paugam contributes here as he is one of the people around who are competent in the subject. For analytic spaces there are also books by Grauert, Remmert etc. One should also warn that the field of germs of holomorphic functions at zero is also nonarchimedean, what has applications in homological mirror symmetry (works of Kontsevich-Soibelman), what is another relation to physics.
Urs is looking for some homotopical or infinite categorical aspects here. There is an unpublished work on the $\mathbf{B}^1$-homotopy theory, the analogue of Voevodsky’s $\mathbf{A}^1$-theory in rigid analytic context, by some student in France few years ago who is now professor somewhere. I am not quite sure who that was, remotely it might be Joseph Ayoub, but I am not quite remembering (I heard a talk some 6-7 years ago or so). In any case over there certainly some facts similar to ones you might look for must have been thought of, and would be nice to consult the source.
There is an unpublished work on the $\mathbb{B}^1$-homotopy theory, the analogue of Voevodsky’s $\mathbb{A}^1$-theory in rigid analytic context, by some student in France few years ago who is now professor somewhere. I am not quite sure who that was, remotely it might be Joseph Ayoub, but I am not quite remembering (I heard a talk some 6-7 years ago or so).
Ah, interesting. I tried googling around for “analytic geometry homotopy theory” a bit, and looked at Ayoub’s webpage, but so far I haven’t found the kind of discussion you are alluding to. But I’d be most interested in seeing it.
Try to contact him, and if he is not the guy he must know who that is as he came from that Paris circle. I wrote some comments in $n$Community before and could never find anything online.
Correction: there is something related online: pdf
Maybe of no interest, but one place with ’analytic geometry’ and ’homotopy’ on the same page is a course outline which includes
The moduli space of Lubin-Tate formal group laws is a rigid analytic space occurring in the work of Gross and Hopkins, which ties together this sort of geometry with stable homotopy theory.
M.J. Hopkins and B.H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bulletin of the AMS 30 (1994), 76–86
Notes for the course are here.
there is something related online: pdf
Ah, thanks!
This does not seem to help me with my question about cohesiveness, but otherwise it is very good to see this. For the moment I have archived the link in a new stub entry B1-homotopy theory.
Maybe of no interest, but one place with ’analytic geometry’ and ’homotopy’ on the same page is a course outline which includes […]
Thanks, David! That looks useful, I’ll have to look at it. For the moment I have archived the reference here.
Hi Zoran,
coming back to this here:
The hope that working with $p$-adic numbers can remove divergences in QFT is an old hope which drives the $p$-adic physics for a long while;
Do you know about any concrete results that have been obtained? Anything that goes beyond hope and vague speculation? Are there some actual facts about “$p$-adic mathematical physics” that one could point to?
I’ll try to look around myself a bit more. But if you have anything to share, I’d be interested.
How about this here:
in the mathematical literature on $p$-adic analytic geometry it says (I’ll try to dig out a precise reference again) that functions $f : \mathbb{Q}_p \to \mathbb{Q}_p$ are uninteresting, and that the objects to consider are instead maps $\mathbb{A}_{\mathbb{Q}_p} \to \mathbb{A}_{\mathbb{Q}_p}$ (from the Berkovich affine line to itself).
Now, I open a book like
and in the introduction it announces to study the replacement of real functions with maps $\mathbb{Q}_p \to \mathbb{Q}_p$.
Generally, I see little or none mathematical analytic geometry mentioned in these texts on $p$-adic mathematical physics. Is there maybe a communication gap?
Can you work in reverse with cohesive $(\infty, 1)$-toposes, so that if all, or some subset, of the associated structures turn up, you know you must be in the presence of a cohesive $(\infty, 1)$-topos? Are there perhaps a small number of structures whose presence suggests the others?
By the way, there’s an interesting account of the difficulty of Berkovich’s first steps.
13,14
They solve various simple physical systems in p-adic context, how far they got I do not know, but usually these are difficult and exact calculations, so the progress is relatively slow. I do not think that anybody there studies rigid analytic geometry – as usually in mainstream physics one does local computations in just one chart. I would not call that a communication gap. People are busy and can not learn to do effectively everything. You can not expect to have comparable results to ordinary physics where there are thousands of physicist with a direction populated by about 10 enthusiasts, most of whom do other mathematical physics as well.
I added references
to analytic geometry and analytic space.
Maybe one should point out the discovery of $p$-adic Veneziano amplitude for $p$-adic string theory in
See also
Can you say what the result is here? I understand that one can refomrulate the expression for the amplitude in $p$-adic arithmetic. What is the nontrivial result to be proven?
(This is not a rethoric question. I genuinely need to learn this.)
It is not a reformulation. It is a different mathematical expression which is an analogue of Veneziano amplitude in $p$-adic context. I am personally not interested in the subject.
Even doing the integration over p-adics effectively needs hard techniques closely related to motivic integration. This has been subject of much recent research in motivic integration community.
The situation is somewhat similar to the situation with q-beta function of Ramanujan and q-binomial formula etc. which are not reducing to the usual beta function and the usual binomial formula.
I have split off an entry p-adic physics, just in order to collect all these references for the moment.
Good.
re #16:
as usually in mainstream physics one does local computations in just one chart. I would not call that a communication gap.
I am not sure if this is just a matter of globalizing. It seems to me that the basic idea of geometry in mathematical analytic geometry is different from what the texts on “$p$-adic physics” try to connect to:
the latter replaces the real line with $\mathbb{Q}_p$. But the former replaces it with the Berkovich affine line $\mathbb{A}_{\mathbb{Q}_p}$ (“$\mathbb{B}_p^1$”, I guess ). This should be a big difference.
OK. Reason more that they should not be interested. I think nobody claimed in the Vladimirov community to do rigid analytic geometry ever. They do the usual $p$-adic geometry in the sense of $p$-adic analytic manifolds and their examples belong there. On the other hand, I think the mathematical physics-inspired work of Marcolli and Manin (also listed on the link above) does indeed point toward some phenomena in rigid geometry.
they should not be interested
I understand that they are not interested, but I don’t understand why they should not be interested. This seems to be the “communication gap” that I mentioned. These people set out to explore the consequences of replacing in the geometry of physics the real numbers by the $p$-adic numbers, while at the same time there is a huge body of mathematics that does precisely such a replacement, but differently.
I do not think that the idea was to replace the numbers by $p$-adic numbers just for math analogy, but for the fact that in usual $p$-adic coordinates some expressions were finite which are infinite in the other case. I do not recall examples, I have heard few conference talks presenting formulas some of which appear in the calculations in QFT. So they continued this game, starting from the concrete evidence they had. This does not give any concrete evidence however that similar expressions would appear in calculations in the different setup you propose. If you have a striking finiteness result for amplitudes in Berkovich spaces, then the situation will change. This is not matter of communication but matter of lack of evidence as of today.
Okay, so I need to learn that better. In the introduction to that book
for instance, the motivation that is given is: look, there are two kinds of completions of the rationals, and traditonally only one of them is used for the description of physical space. We want to explore what happens when we use the other one.
This is supplemented with the usual speculation about quantum geometry at the Planck scale. Similarly in the other reviews that I have seen.
Well, once somebody is in the game one tries to find the applications everywhere, and then one needs to soften and widen the motivation statement, so I am not surprised that books get into such lowest common denominator. But all of that is not very strong. What is stronger is the initial observation that some infinite expressions in QFT were finite in $p$-adic context. The initial counterargument being what is the $p$ which nature would take and fix ?
The analytic geometry based on the very fine and disconnected topology of non-archimedean field is having problems which needed rigidification. So I have originally viewed entry rigid analytic geometry to be about ALL modern approaches which rectify this problem with various levels of success and in various level of generality: Tate’s rigid analytic spaces, Berkovich analytic spaces, Huber’s adic spaces, Fujiwara-Kato approach via multiple formal models and Zariski Riemann space etc. This lead in past to some misunderstanding, as Urs and Frederic were taking every time I mentioned word “rigid” ot mean the Tate’s approach; though I often hear people saying rigid for the other approaches. On the other hand analytic space is about possibly even non-archimedean versions. So I decided that it is better that we are really split into different levels of generality carefully. So I renamed analytic space into Berkovich space with redirect Berkovich analytic space and creted a new analytic space to lead to both archimedean and non-acrhimedean case. Then I created an entry non-archimedean analytic geometry to have Tate, Berkovich, Huber, global and Fujiwara-Kato in one place while having more details on the pages specially dedicated to each of them. Rigid analytic geometry entry is now predominantly about Tate’s approach only, with agreement with Urs and Frederic.
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