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mirror symmetry (needs more well chosen references, I am runnning out of time and will be busy next few days; there are hundereds of references available so we should choose important and/or well written ones)
Thans, Zoran.
I added some links.
I noticed that symplectic geometry is in a mess. In addition to other problems, there was some strange and rambling paragraph at the end which looks like somebody's informal journal club presentation. I moved it entirely to equivariant localization and elimination of nodes. I found that the source of the text is one cafe discussion.
Urs, the 3 links you put in a comment do not work any more. Reason: you do not have published version but show version now for your publicly available private web. These links you will easily correct.
I still feel the mathematical physics is still badly underrrepresented in nlab. I feel if I write something related to physics it will get much less response and comments from other nlabizants than if I write any math nonsense.
One thing which would be very practical in nlab is to have pages listing all the various actions and couplings which are in standard usage in various falvours of string and M-theory, together with sketch of argument for the shape of corresponding actions. These things are hard to find systematized in the literature, especially from descent mathematical viewpoint. The 1997 two IAS volumes on strings for mathematicians are obsolete as they have almost no D-brane mathematics. Zwiebach has lots of those cases listed, but it is low level, without deeper explanations; still it is the main resource to start. Old books focus on classical cases and sectors.
I still feel the mathematical physics is still badly underrrepresented in nlab. I feel if I write something related to physics it will get much less response and comments from other nlabizants than if I write any math nonsense.
One way to attract people looking into this is to have blog posts on the corresponding topic, with maybe some pointers to the nLab. I can forward a guest post of yours, if you want to write one.
Of course also on the blog the interest in mathematical physics is badly underrepresented. But there are a few regulars who are interested. One would need to lure them to get more involved with the Lab, too.
One way to attract people looking into this is to have blog posts
That is for random people outside of the circle of some 30-40 nlab regulars. I am more disappointed that the 30-40 nlab regulars, do not seem to be that interested in physics contributions. I value nlab work far above the ncafe discussions. You can see from your own posts that those (edit: entries at cafe) which had the deepest technical content (e.g. your post on Bousfield localization) usually got less response than those mild ones where everybody can step in. Thanks, anyway, I could do some guest post occasionally.
I have added two standard references and also linked quantization to a unpolished short post of mine at mathlight on quantization. I do not have enough spare time now to write polished version for cafe now though. Hopefully in few weeks.
Edit: 1. how does one get rid of "Possibly related posts: (automatically generated)" at wordpress ? they link idiotic blogs as possibly related
Good job
triggered by discussion with Zoran and Domenico, I have tried to expand the Idea-section at mirror symmetry. Check.
I basically agree, but I would say “there is a conjectural involution” at the very beginning. also it is not clear to me in which sense the categorical version of mirror symmetry is now pretty much established.
I created a stub for Hodge numbers
True, maybe you could add caveats, as indicated.
A survey of established results as of 2008 I find in
See in particular page 30 and following.
I am scanning the arXiv for recent statements of results. This here seems to have a good deal of such:
Abstract. Katzarkov has proposed a generalization of Kontsevich’s mirror symmetry conjecture, covering some varieties of general type. Seidel has proved a version of this conjecture in the simplest case of the genus two curve. Basing on the paper of Seidel, we prove the conjecture (in the same version) for curves of genus $g\geq 3,$ relating the Fukaya category of a genus $g$ curve to the category of Landau-Ginzburg branes on a certain singular surface. We also prove a kind of reconstruction theorem for hypersurface singularities. Namely, formal type of hypersurface singularity (i.e. a formal power series up to a formal change of variables) can be reconstructed, with some technical assumptions, from its D$(\Z/2)$-G category of Landau-Ginzburg branes. The precise statement is Theorem 1.2.
I know of this paper, Efimov is the most talented undergraduate student at Moscow University, about to graduate with Orlov and to go to graduate school at Harvard with Seidel.
I disagree with the statement that we have really an involution, even not conjecturally. For example, to some Calabi-Yau varieties one associates a rather different mirror which is not a variety but sort of deformation of the corresponding category in noncommutative world, which has no underlying variety in commutative sense. So locally one has another Calabi Yau variety or sometimes a bit different thing. Next there is that business about families and considering this only in the large volume limit. So it is not really a picture of internal involution set-theoretically.
Okay, let’s change the statement about the involution.
We also need to say something about Landau-Ginzburg model, eventually.
Meanwhile, I created a section Complete proofs that is meant to list articles in which (homological) mirror symmetry is completely established on certain (types of) spaces.
Currently this contains the Efimov article and the complete proofs that he cites.
There are some earlier results for abelian varieties and for toric varieties, e.g.
V. Golyshev, V. Lunts, D. Orlov, Mirror symmetry for abelian varieties. J. Algebraic Geom. 10 (2001), no. 3, 433–496, math.AG/9812003
in the Lab entry on Mirror symmetry, at the point here homological mirror symmetry is introduced, I find exposition abit confused: namely, apparently we first say that the TCFT formulation is equivalent to the categorical one, then we say the categorical version is an almost established conjecture, and finally we say that the categorical version is equivalent to the TCFT version. I would rephrase that part as follows:
“This categorical formulation was introduced by Maxim Kontsevich in 1994 under the name homological mirror symmetry. The equivalence of the categorical expression of mirror symmetry to the SCFT formulation has been proven by Maxim Kontsevich and independently by Kevin Costello, who showed how the datum of a topological conformal field theory is equivalent to the datum of a Calabi-Yau A-infinity-category (see TCFT).”
in this I would omit the digression on $(\infty,1)$-categories, not because it is not correct, but because it seems to me to break the sentence, making it harder to follow the argument.
Domenico, feel free to polish. In expanding these paragraphs, I had tried to preserve as much as possible of the previous paragraphs, which maybe led to some repetition. Please, if you have the time, try to make a well-flowing piece out of it.
fine. I’ve now edited it. I’ll try to recover the remark on $A_\infty$-categories as a model for stable $(\infty,1)$-categories at some point, during future revisions of that entry.
Thanks!
so, what is mirror symmetry? it seems to me that, as Zoran points out, the modern use of the term is something more vast than it was at the beginning, which we could nPOV state as: a mirror symmetry is an equivalence between Calabi-Yau $A_\infty$-categories. after this we could add: it derives its name from the following phenomenon, observed for a quantity of Calabi-Yau manifolds: etc. etc.
the one above would at least be a definition, and examples of mirror symmetries in literature would be from this point of view examples of “reamarkable” mirror symmetries, where “remarkable” is something undefined with a matter of personal taste in it (where personal can involve a large part of the community at a certain time), depending on how a priori urelated were the two Calab-Yau categories which turn out to be equivalent in a given example. to make an analogy, one could say that there’s a trivial isomorphism between the group of automorphisms of the set $\{a,b,c\}$ and the group of automorphism of the set $\{1,2,3\}$, but there is a remarkable isomorphism between the the group of automorphisms of the set $\{a,b,c\}$ and the quotient of the free group on two generators $\rho,\sigma$ modulo the subgroup generated by $\{\rho^3,\sigma^2, (\sigma\rho)^2\}$.
I created a stub for Hodge numbers
Thanks, I have edited it slightly.
I notice that we are lacking an entry on complex manifold…
so, what is mirror symmetry? it seems to me that, as Zoran points out, the modern use of the term is something more vast than it was at the beginning, which we could nPOV state as: a mirror symmetry is an equivalence between Calabi-Yau A ∞-categories.
I would think that would not be the right way to say it.
I think the nature of all these “dualities”, T-duality, mirror symmetry, U-duality etc. is that they describe invariances of the sigma-model construction(functor?)
$\sigma-model : geometric background data \to 2dQFT$Here “geometric data” may be quite a bit more sophisticated than just the naive “manifold with extra structure”, it may be some non-commutative geometry etc. but it is still “geometric data” of sorts.
Then what the “dualities” assert is that there are certain systematic operations on the collection of geometric data under which $\sigma-model$ is invariant, up to equivalence.
This is what mirror symmetry is an example of.
A second step then is to notice that parts of the category of 2dQFTs of sorts is equivalent to a category of certain categories, which transports this statement to a statement about an invariant of a construction
$\sigma-model : geometric background data \to CertainCategories$But it is still important, I think, to read it is a statement about such an assignment, not just as a statement that certain categories are equivalent. It is the fact that these categories that are equivalent are induced from geometric data, and from different geometric data, that makes clear in which sense their equivalence is “remarkable” in your sense.
I agree, but what is a geometric background? where does the notion of geometric ends? can we really say that noncommutative geometry is geometry and $A_\infty$-category theory is not? as a matter of personal taste I would restrict mirror symmetry to the switch of A-model and B-model for pairs of Calabi-Yau manifolds, but this is too narrow and has been overcome by the current use of the term mirror symmetry, which now refers to a plethora of different examples. so, even if I’m not satisfied with it, I am unable to see a better definition for what is mirror symmetry than the one I suggested above, and would consider $\sigma$-model dualities as remarkable examples. but I see this is an extreme point of view, and that’s why I didn’t edit Mirror symmetry along these lines.
I agree, but what is a geometric background? where does the notion of geometric ends?
Right, one can in fact take the stance that the SCFT defines the geoemtry. This is the point of view of spectral geometry: a spectral triple is just a 1d SQFT and the idea of spectral triples is really to define geometry as whatever the corresponding 1d SQFT sees it. The case of 2d SCFTs is really a straightforward generalization/categorification of this. (Yan Soibelman is in the process of writing that up for our book…)
So but if we take this point of view, then the statements about dualities vanish. Because then we have adopted a “duality invariant” notion of geometry, where I mean dulity in the strict sense of perturbative string theory. This is a good point of view for many purposes, but it removes the subtlety that “mirror symmetry” is interested in , by definition.
So I think it makes not much sense to talk about mirror symmetry in such a context.
I think what we really need – and we have been discussing this at some length already elsewhere, of course – is a full formalization of the notion “$\sigma$-model”. As you know, I am thinking that there should be one which is such that the input datum is a differential cocycle in an $\infty$-connected $(\infty,1)$-topos. Not clear yet if that works out fully, but somethin like that I expect should count as “geometric input”.
So I think it makes not much sense to talk about mirror symmetry in such a context.
that’s why in my version I didn’t say “mirror symmetry is”, but “a mirror symmetry is”. it would be just a synonym for “equivalence” in the context of Calabi-Yau categories (or of TCFTs, which is the same). even in the duality invariant notion of geometry, I think equivalences would be relevant: we do not only want to say that two “geometries” are equivalent, but also how they are equivalent.
As you know, I am thinking that there should be one which is such that the input datum is a differential cocycle in an $\infty$-connected $(\infty,1)$-topos
and you know I agree with and enjoy a lot this point of view. but I wonder whether forcing to the extreme this generality one would not be able to consider a given Calabi-Yau category as an example. and then involved the sigma-model in this particular case would just be the identity :-) what I mean is that there must be social agreement on what a geometric input is and what is not. so one cannot say whether differential cocycles can encompass all kind of geometric inputs unless there’s agreement on what geometric inputs are. and the only way I can see to comprehend all possible inputs is to include all Calabi-Yau categories.
but now I want to write down a few more specific questions/remarks on “classic” mirror symmetry.. give me a couple of hours to think to what I want to write and I’ll be back :)
Another angle to look at it is in terms of quantization:
“geometric background” in the sense that makes things like mirror symmetry a “remarkable” equivalence in your terms is classical background . In that: it defines a classical sigma-model QFT. The “remarkable” fact then is that two inequivalent classical sigma-models become equivalent after quantization.
That this does not happend for 1dQFT is by the way a famous theorem by Connes, if I recall correctly: there are non-isometric but isospectral Riemannian manifolds, but if one takes on top of the spectrum of the Laplace operator (= Hamiltonian of the quantized 1 d sigma model that they induce) their full spectral triple into account, then that serves to distinguish these and one can “hear the shape of the drum” after all.
I agree fully with 15, it is a bit more complicated to articulate concerns from 16 on.
More about supersymmetry: there is a famous non-renormalization theorem for both twisted models which says that there are sufficiently supersymmetry-related cancellations that there are no quantum corrections from renormalization. This also enables the business of path integral localization which in most examples needs supersymmetry to work. I think, that these localization effects are true for general compact Kahler manifolds, not necessary Calabi-Yau manifolds, and it is also OK for B-model on Calabi-Yau orbifolds, and also in the case of A-model for any closed symplectic manifold.
I agree fully with 15
is that 25?
I agree fully with 15
is that 25?
So you are both saying “A mirror symmetry is a (any) equivalence between two Calabi-Yau $A_\infty$-categories.” ?
Hopefully I can eventually manage to convince you that this is not a good idea… ;-)
By the way: even though I am disagreeing, I am delighted that we are discussing this.
It is kind of weird that this kind of discussion we are having here isn’t present in the literature, resolved or not. I think there is so much confusion in high energy physics and string theory literature due to the fact that such conceptual basics are never properly discussed. There is a huge amount of formalization of concepts that still needs to be done.
There is a huge amount of formalization of concepts that still needs to be done.
we’re here for that! :)
we’re here for that! :)
Yes, indeed. Three of us now. The rest is lost in the landscape, literally. ;-)
Not all the rest of course. For instance Distler-Freed-Moore is all about identifying the right notion of string background.
So you are both saying “A mirror symmetry is a (any) equivalence between two Calabi-Yau A ∞-categories.” ?
I never said that nonsense. E.g. even in the most classical case, one has two equivalences simultaneously: A(X) and B(Y) and A(Y) and B(X).
No, Domenico, I was talking about my full agreement with the quotation long entry 21.
so here is the questio/remak I was promising.. it concerns something very basic, but it seems it has somehow gone lost in the modern take on mirror symmetry: Hodge numbers (for instance they are mentioned only once, and in the introduction, in the recent survey by Ballard mentioned in #15).
at the very beginning of the mirror symmetry story, one had an $n$-dimensional Calabi-Yau manifold, computed its Hodge numbers and organized them into the shape of a square with $h^{0,0}$ as the bottom vertex and $h^{n,n}$ as the top vertex (Hodge diamond). this can be seen as a morphism
$Calabi-Yau manifolds \stackrel{Hodge diamond}{\to} numerology$and it was remarked that for suitable pairs of Calabi-Yau manifolds, the numerologies one obtained were related by a symmetry of the Hodge diamond. the physical interpretation of this is very simple: very roughtly, one has two TCFTs attached to a Calabi-Yau manifold, namely the A-model and the B-model, and the Hodge numbers appear as dimensions of suitable eigenspaces for $\mathfrak{u}(1)\times\mathfrak{u}(1)$-actions.
so the above arrow is refined as
$Calabi-Yau manifolds\stackrel{A/B model}{\to}TCFT \stackrel{Hodge diamond}{\to} numerology$in the categorical approach, this becomes
$Calabi-Yau manifolds\stackrel{A/B model}{\to}Calabi-Yau category \stackrel{Hodge diamond}{\to} numerology$so, what is the arrow $Calabi-Yau category \stackrel{Hodge diamond}{\to} numerology$?
a similar numerological example is Candelas-de la Ossa-Green-Parkes formula (and all the mathematics it generated..). that can be sketched as
$Calabi-Yau manifolds\stackrel{GW potential/ Yukawa coupling}{\to} numerology$so in categorical terms one should have something like
$Calabi-Yau manifolds\stackrel{A/B model}{\to}Calabi-Yau category \stackrel{?}{\to} numerology$where “?” is presumibly Kevin Costello’s GW-potential associated to a TCFT. but in neither of Costello’s papers on TCFTs one can find any occurrence of “Yukawa”. so the question is: where have the dear old basics of mirror symmetry gone in the categorical refomulation? I’m sure Maxim Kontsevich had this extremely clear in his mind when he formulated the homological mirror conjecture, but I’m quite surprised these basics seem to have disappeared from the categorical treatment of mirror symmetry. are we still talking of the same thing? surely yes (at least I hope so), but I’d like to find this written out in more evidence somewhere..
Domenico, it is not only Hodge diamond it is also correspondence between the variation of Hodge structures of type A and of type B. In categorical framework there is a suitable version of a more general “noncommutative Hodge structures” which tell you again more than Hodge diamond. See again Katzarkov, Kontsevich, Pantev arxiv/0806.0107 for recent state of the art.
even in the most classical case, one has two equivalences simultaneously: A(X) and B(Y) and A(Y) and B(X).
but this can be asked only when both models are available for X and Y, and in the modern usage this is not the case: for instance one says that the mirror of the A-model on $\mathbb{P}^n$ is the B-model on $w:(\mathbb{C}^*)^n\to \mathbb{C}$, where $w(z_1,\dots,z_n)=z_1+\cdots+z_n+q/(z_1\cdots z_n)$, where $q\in \mathbb{C}^*$. (this is the first mirror symmetry example in Katzarkov, Kontsevich and Pantev).
it is not only Hodge diamond it is also correspondence between the variation of Hodge structures of type A and of type B
sure. waht I was pointing out is that even something much poorer such as the Hodge diamond seems to be lost in the abstract nonsense of the categorical formulation. for instance in Katzarkov, Kontsevich and Pantev Hodge structures are described in a setting that though noncommutative is still geometric, and there is nothing such as a Hodge structure of a Calabi-Yau category (by teh way there is no occurrence of “Calabi-Yau category” in that paper). so what? I don’t know. it could mean that the real framework for mirror symmetry are not Calabi-Yau categories, but rather “Calabi-Yau categories with a fixed Hodge structure”? by the way, does every Calabi-Yau category admit an Hodge strucure (whatever this means)? clearly “geometric” Calabi-Yau categories do have natural Hodge structures, so this reinforces Urs point of view that one is interested only in Calabi-Yau categories with a geometric origin. as far as concerns me, I’m now in a worse position than a few hours ago, not only I do not now what is mirror symmetry, but neither I know what it is about. :)
#36 is sadly true..
I added that not only F(X) = D(Y) but simultaneously oine requires F(Y)=D(X). Previously just one half of the statement was there.
but this can be asked only when both models are available for X and Y, and in the modern usage this is not the case
It is not easy to show full mirror symmetry.
It is not true that the Hodge diamond is lost in modern formulation; I mean did you ever attend a homological mirror symmetry conference ? There is almost no talk without derived category formulation and without several examples full of Hodge diamond data illustrating various aspects.
“geometric” Calabi-Yau categories do have natural Hodge structures, so this reinforces Urs point of view that one is interested only in Calabi-Yau categories with a geometric origin
One thing is the apperance of Calabi-Yau categories and their relation to TFTs and another (though related) thing is the mirror symmetry. It is not clear what you mean by “geometric origin”. Restricting to commutative varieties is certainly not the proper scape as mirror partner of a variety is often not commutative; families of deformations can be studied using deformation theory and they contain many interesting members which are not geometric in naive sense. Everything is geometric eventually at sufficiently abstract level. It is not good to try to make generalizations with so little experience, this is a huge subject, and making easy souding generalizations leads to easy failures. It is best to go along the program which I proposed few months in nlab without any success: to build in nlab the expositions of separate notions like equivariant localization, path integral localization, Picard-Fuchs equations, variations of Hodge structure, the language of smooth A-infinity categories, Gromov-Witten invariants, Picard-Lefschetz theory, Maslov index, quantum D-module, Floer homology etc.
In Kontsevich’s Homological algebra of mirror symmetry the homological mirror conjecture is only one way, see at page 18, but I agree that for Calabi-Yau manifolds one should ask for both ways.
Hodge structures are described in a setting that though noncommutative is still geometric
Domenico, the complex parameter in the study of monodromy is a complex parameter. This is just one dimension. This is not about the underlying space, but about the business of meromorphic connection. In typical applications in Katzarkov et al. one takes something like cyclic homology of very abstract category and put on top of it such structure.
It is not true that the Hodge diamond is lost in modern formulation
I think that if in a paper from 2008 whose title is “Meet homological mirror symmetry” the word “Hodge” appears once, then one is entitled to say that Hodge diamond is lost.. ;) by that I didn’t mean that who really works in mirror symmetry is not concerned with Hodge structures, but that a basic question like what is mirror symmetry about? seems not to have a clear answer (since I can remember Urs writing somewhere having got no clear answer to What is string theory? I can’t complain too much :) )
Young people learn hi fashionable techniques in particular areas. String theorists of today do not know the papers of classics from 1980s. You picked a paper from a new knowledgeable young specialist so no wonder you face such peculiarieties. Read instad Orlov, Kontsevich, Gross, Seidel, Fukaya…
one takes something like cyclic homology of very abstract category and put on top of it such structure.
and this very abstract category is what has to be thought as some (derived) category of (quasi-coherent) sheves over a noncommutative space, right?
derived) category of (quasi-coherent) sheves over a noncommutative space
something like (derived) noncommutative deformation of a complex projective variety, I guess
Read instad Orlov, Kontsevich, Gross, Seidel, Fukaya…
but I do want to read these! but I need a framework where pieces of the jigsaw puzzle goes in, otherwise I’m lost. can you choose for me a few selected papers to read to get a basic but at the same time neat and complete picture of what is mirror symmetry about?
There is no complete picture. Every picture or formalism captures just some aspects.
I think for the very beginning introduction the recent book might be the best
which I do not have but had browsed it online few pages as googlebooks or whatever. Orlov’s lectures on derived categories and mirror symmetry are very clearly written:
but have very little excursion into physics.
well, actually in Kasputin and Orlov the only appearance of Hodge is in the physics introduction at the beginning of the lecture :)
just joking. now I’ll look more carefully at these references before coming back here. thanks a lot!
If somebody gets a file of Aspinwall et al. above I would like to have it :)
OK, so most of this discussion is waaaaay over my head. However, I spent some time over the last couple of years working with a student on developing a link - primarily grounded in conceptual, i.e. physical, reasoning - between symmetry and entropy. This is based on a certain conceptual interpretation of entropy. We have a very tentative, but rather weak mathematical result, but one that is based on fairly solid physical arguments. So my question to you guys is, at a higher level like what’s being discussed in this thread, could any of you envision a link between symmetry and entropy?
could any of you envision a link between symmetry and entropy?
Sure, it’s not deep: if your system has symmetry, then its entropy is invariant under the symmetry operation.
This has nothing special to do with mirror symmetry, but of course it applies there, too: if you want to compute the entropy of an SCFT and find it too hard, you can equivalently compute the entropy of an equivalent mirror SCFT, if that turns out to be easier. Because they are, well, equivalent.
This is done for instance in this article here:
The authors want to compute entropy of a type IIB SCFT, find that too hard, invoke the mirror symmetry (-conjecture, for their purpose) and instead compute with the mirror dual type IIA SCFT. Or at least argue that there is such a computation.
I added some words, and some references, regarding mirror symmetry beyond the Calabi-Yau case.
@Kevin: thanks for the reference. I actually knew that, but looking back to it after your suggestion has been a good idea: now I more clearly see which is the question I’m interested in. namely, Kontsevich writes on page 18 We expect that the equivalence of derived categories will imply numerical predictions. and this is the statement I’d like to see worked out in detail. any reference? (Costello? Kasputin-Orlov? others?)
@domenico_fiorenza: As far as I know, that statement has not yet been rigorously established in any cases. This is something I’ve been thinking about…
Try looking at Costello’s paper “The Gromov-Witten potential associated to a TCFT”. Also look at Katzarkov-Kontsevich-Pantev.
Edit to: mirror symmetry by Urs Schreiber at 2018-04-01 01:33:24 UTC.
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