**Edit to**: mirror symmetry by Urs Schreiber at 2018-04-01 01:33:24 UTC.

**Author comments**:

adde pointer to textbookby Ibanez-Uranga

]]>@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.

]]>@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?)

I added some words, and some references, regarding mirror symmetry beyond the Calabi-Yau case.

]]>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:

- Aspinwall, Maloney, Simons,
*Black Hole Entropy, Marginal Stability and Mirror Symmetry*(pdf)

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.

]]>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?

]]>If somebody gets a file of Aspinwall et al. above I would like to have it :)

]]>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!

]]>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

- Paul Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Dirichlet branes and mirror symmetry, Amer. Math. Soc. Clay Math. Institute 2009.

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:

- A. N. Kapustin, D. O. Orlov,
*Lectures on mirror symmetry, derived categories, and D-branes*, Uspehi Mat. Nauk**59**(2004), no. 5(359), 101–134; translation in Russian Math. Surveys**59**(2004), no. 5, 907–940, math.AG/0308173

but have very little excursion into physics.

]]>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?

]]>derived) category of (quasi-coherent) sheves over a noncommutative space

something like (derived) noncommutative deformation of a complex projective variety, I guess

]]>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?

]]>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…

]]>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 :) )

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.

]]>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.

]]>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.

]]>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.

]]>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..

]]>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.

]]>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..

]]>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.

]]>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.

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