Talk:Mirror symmetry (string theory)

history of a subject
People who have no clue about the history of a subject should refrain from writing reviews like this one. —Preceding unsigned comment added by 137.138.15.138 (talk • contribs) 13:11, June 29, 2005

proposed renaming
When I typed in "mirror symmetry", this article popped up instead of Reflection symmetry. I propose renaming this article to something like Mirror symmetry (string theory) or Mirror symmetry (manifolds), so that Mirror symmetry can be a redirect to the more elementary topic in Reflection symmetry. A dab link could be added at the top of that article. Alternatively, Mirror symmetry could be made into a dab page. --Jtir (talk) 12:01, 15 August 2008 (UTC)
 * There were three articles related to mirror symmetry, so I made Mirror symmetry a dab page.--Jtir (talk) 18:45, 17 August 2008 (UTC)

Comments

 * I have translated into Japanese about 1 year ago. This new article is rewritten from a very "mathematical" view. So in Japanese version the former version is remained as Appendix with sources that are commented out. I will propose that
 * FIRST, revive the version of july 2013 version as Appendix of this article,
 * Second, be independent this article as, for example, "mirror symmetry (mathematics)".
 * --Enyokoyama (talk) 02:43, 7 September 2013 (UTC)


 * The current version of the article does emphasize the mathematical applications of mirror symmetry, but I'm still in the process of revising, and eventually I intend to add more about mirror symmetry from a physical point of view. The version from July 2013 was mostly not about mirror symmetry per se but another duality between three-dimensional gauge theories. This information is still available in the article 3D mirror symmetry. Polytope24 (talk) 13:44, 7 September 2013 (UTC)


 * I hope your description from a physical point of view. However, even if the old description needs several improvements the following items in the old one could not be deleted.

· Batyrev-Borisov construction.

· electromagnetic duality and t'Hooft-Polyakov monopole.

· mirror symmetry in the sigma model on two-dimensional gauge theory.

· mirror symmetry in three-dimensional gauge theory (not be other argicle but in this article.)


 * They are much important as examples or as applications to string theory. Then I will be revive on this note but not as "Appendix". Please refer.　--Enyokoyama (talk) 14:35, 8 September 2013 (UTC)


 * I'm afraid I don't know what you're trying to say about an appendix. I'm going to keep adding to this article, and I will talk about the most important physical applications. But the most important thing for right now is to cover the BASIC topics. The previous article had essentially zero discussion of enumerative geometry, which is by far the most famous application of mirror symmetry. The old article was neither well sourced nor accessibly written, and other users have complained about this over at the 3D mirror symmetry article.


 * Before adding anything about advanced applications to physics, I'm going to add a section on the SYZ conjecture and improve the citations. Mirror symmetry is an enormous topic, and Wikipedia policy requires that the article be written for the widest possible audience without giving undue weight to any particular topic. Polytope24 (talk) 16:09, 8 September 2013 (UTC)


 * By the way, you are welcome to create subpages on more technical topics and link to them from here. Perhaps that is what you meant by an "appendix". Polytope24 (talk) 16:13, 8 September 2013 (UTC)


 * Thanks! Mr. Polytope24. Firstly, I will only revive the previous version at July 2013 on this note, though they needs some improvements.--Enyokoyama (talk) 22:20, 8 September 2013 (UTC)

Now, I will post the version of "mirror symmetry (string)" at 1st Aug 2013 as a new section, which is the previous version of this from the rather physical view point. I think that this article needs some improvement.--Enyokoyama (talk) 15:12, 10 September 2013 (UTC)

Dear Mr Polytope24! I agree with you completely. The 1st Aug 2013 version has some problematic parts as an article and needs significant improvements. Part of the "Mirror symmetry in 3-dimensional gauge theories" becomes a separate article and I will modify to refer to the separate article. I will never disturb this article.--Enyokoyama (talk) 13:08, 12 September 2013 (UTC)

the 1st Aug 2013 version of this article "mirror symmetry (string theory)"
In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi–Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of string theory. The classical formulation of mirror symmetry relates two Calabi–Yau threefolds M and W whose Hodge numbers h1,1 and h1,2 are swapped; string theory compactified on these two manifolds lead to identical effective field theories.

History
The discovery of mirror symmetry is connected with names such as Lance Dixon, Wolfgang Lerche, Cumrun Vafa, Nicholas Warner, Brian Greene, Ronen Plesser, Philip Candelas, Monika Lynker, Rolf Schimmrigk and others. Andrew Strominger, Shing-Tung Yau, and Eric Zaslow have showed that mirror symmetry is a special example of T-duality: the Calabi–Yau manifold may be written as a fiber bundle whose fiber is a three-dimensional torus. The simultaneous action of T-duality on all three dimensions of this torus is equivalent to mirror symmetry.

Mathematicians became interested in mirror symmetry in 1990, after Candelas-de la Ossa-Green-Parkes gave predictions for numbers of rational curves in a quintic threefold via data coming from variation of Hodge structure on the mirror family. These predictions were mathematically proven a few years later by Alexander Givental and Lian-Liu-Yau.

Applications
Mirror symmetry allowed the physicists to calculate many quantities that seemed virtually incalculable before, by invoking the "mirror" description of a given physical situation, which can be often much easier. Mirror symmetry has also become a tool in mathematics, and although mathematicians have proved theorems based on the physicists' intuition, a full mathematical understanding of the phenomenon of mirror symmetry is still being developed.

Most of the physical examples can be conceptualized by the Batyrev–Borisov mirror construction, which uses the duality of reflexive polytopes and nef partitions. In their construction the mirror partners appear as anticanonically embedded hypersurfaces or certain complete intersections in Fano toric varieties. The Gross–Siebert mirror construction generalizes this to non-embedded cases by looking at degenerating families of Calabi–Yau manifolds. This point of view also includes T-duality. Another mathematical framework is provided by the homological mirror symmetry conjecture.

Generalizations
There are two different, but closely related, string theory statements of mirror symmetry.


 * 1. Type IIA string theory on a Calabi–Yau M is mirror dual to Type IIB on W.
 * 2. Type IIB string theory on a Calabi–Yau M is mirror dual to Type IIA on W.

This follows from the fact that Calabi–Yau hodge numbers satisfy h1,1 $$\geq$$ 1 but h2,1\$$\geq$$ 0. If the Hodge numbers of M are such that h2,1=0 then by definition its mirror dual W is not Calabi–Yau. As a result mirror symmetry allows for the definition of an extended space of compact spaces, which are defined by the W of the above two mirror symmetries.

Mirror symmetry has also been generalized to a duality between supersymmetric gauge theories in various numbers of dimensions. In this generalized context the original mirror symmetry, which relates pairs of toric Calabi–Yau manifolds, relates the moduli spaces of 2-dimensional abelian supersymmetric gauge theories when the sums of the electric charges of the matter are equal to zero.

In all manifestations of mirror symmetry found so far a central role is played by the fact that in a d-dimensional quantum field theory a differential p-form potential admits a dual formulation as a (d-p-2)-form potential. In 4-dimensions this relates the electric and magnetic vector potentials and is called electric–magnetic duality. In 3-dimensions this duality relates a vector and a scalar, which in an abelian gauge theory correspond to a photon and a squark. In 2-dimensions it relates two scalars, but while one carries an electric charge, the dual scalar is an uncharged Fayet-Iliopoulos term. In the process of this duality topological solitons called Abrikosov-Nielsen-Oleson vortices are intercharged with elementary quark fields in the 3-dimensional case and play the role in instantons in the 2-dimensional case.

The derivations of 2-dimensional mirror symmetry and 3-dimensional mirror symmetry are both inspired by Alexander Polyakov's instanton calculation in non-supersymmetric quantum electrodynamics with a scalar Higgs field. In a 1977 article he demonstrated that instanton effects give the photon a mass, where the instanton is a 't Hooft-Polyakov monopole embedded in an ultraviolet nonabelian gauge group.

Mirror symmetry in 2-dimensional gauged sigma models
Mirror symmetries in 2-dimensional sigma models are usually considered in cases with N=(2,2) supersymmetry, which means that the fermionic supersymmetry generators are the four real components of a single Dirac spinor. This is the case which is relevant, for example, to topological string theories and type II superstring theory. Generalizations to N=(2,0) supersymmetry have also appeared.

The matter content of N=(2,2) gauged linear sigma models consists of three kinds of supermultiplet. The gauge bosons occur in vector multiplets, the charged matter occurs in chiral multiplets and the Fayet-Ilipolous (FI) terms of the various abelian gauge symmetries occur in twisted chiral multiplets. Mirror symmetry exchanges chiral and twisted chiral multiplets.

Mirror symmetry, in a class of models of toric varieties with zero first Chern class Calabi–Yau manifolds and positive first Chern class (Fano varieties) was proven by Kentaro Hori and Cumrun Vafa. Their approach is as follows. A sigma model whose target space is a toric variety may be described by an abelian gauge theory with charged chiral multiplets. Mirror symmetry then replaces these charged chiral multiplets with uncharged twisted chiral multiplets whose vacuum expectation values are FI terms. Instantons in the dual theory are now vortices whose action is given by the exponential of the FI term. These vortices each have precisely 2 fermion zeromodes, and so the sole correction to the superpotential is given by a single vortex. The nonperturbative corrections to the dual superpotential may then be found by simply summing the exponentials of the FI terms. Therefore mirror symmetry allows one to find the full nonperturbative solutions to the theory.

In addition to finding many new dualities, this allowed them to demonstrate many dualities that had been conjectured in the literature. For example, beginning with a sigma model whose target space is the 2-sphere they found an exactly solvable Sine-Gordon model. More generally, when the original sigma model's target space is the n-complex dimensional projective space they found that the dual theory is the exactly solvable affine Toda model.

Mirror symmetry in 3-dimensional gauge theories
Please see the article "3D mirror symmetry", which is now separated but originally was this section.

Illustration of Calabi-Yau and its Mirror Dual?
Would it be possible to have an illustration showing a Calabi-Yau and its mirror dual? This could be hugely beneficial for giving readers an understanding of what this is about. I guess the simplest thing would be showing a torus and its mirror dual torus side by side, but maybe it is possible to something more elaborate for higher dimensional CY.TR 16:49, 9 April 2014 (UTC)


 * I would also love to see a picture of dual Calabi-Yau manifolds, but unfortunately I don't have the computer skills to make such a picture myself. The best way to picture a pair of mirror tori would be to draw their fundamental domains, but that's not very exciting. Polytope24 (talk) 23:53, 9 April 2014 (UTC)


 * I'm also not totally sure how this would work in higher dimensions. To make a picture of a Calabi-Yau manifold, you have to choose a particular slice. I'm not sure if there's a natural way of doing this for both the manifold and its mirror… Polytope24 (talk) 00:50, 10 April 2014 (UTC)

Question
The section on the SYZ conjecture seems to suggest that all CY manifolds are toric varieties. Is this true? (It could be, I just cannot remember)TR 16:52, 9 April 2014 (UTC)


 * I think what you mean is that every Calabi-Yau manifold with a mirror can be viewed as a fibration whose fibers are homotopy equivalent to tori. Polytope24 (talk) 23:55, 9 April 2014 (UTC)
 * Well, I think the question is if a Calabi-Yau that is not a toric variety (and I think those exist, otherwise people would not be specifically discussing "toric CY") can still be viewed as "a fibration whose fibers are homotopy equivalent to tori".
 * Or is the construction discussed in the SYZ conjecture section limited to toric CY (as many explicit constructions involving topological string dualities are)?TR 08:13, 10 April 2014 (UTC)


 * I'm afraid the terminology is a little confusing. When people talk about "toric Calabi-Yau varieties", I think they're referring to Calabi-Yau varieties that are realized as subvarieties of a toric variety. There is a famous construction of mirror pairs due to Batyrev where the varieties are realized in this way. (If you're interested, this is all explained in Chapter 7 of the book by Hori et al.)


 * In the original SYZ paper, the authors conjecture that every Calabi-Yau which has a mirror can be realized as a torus fibration. So yes, this presumably applies to any of the toric Calabi-Yau pairs constructed by Batyrev, as well as any other mirror pair. Polytope24 (talk) 16:03, 10 April 2014 (UTC)

Reference fails validation
The article cited as by Candelas et al. (1991) is not at Nuclear Physics B 342 (1): 21–74. The Bibcode refers to an article of the cited title but by Walcher in a different journal, and the DOI to a completely different article by Walcher. Camboxer (talk) 15:17, 9 May 2014 (UTC)


 * The article that the citation refers to is this one. Polytope24 (talk) 15:34, 9 May 2014 (UTC)


 * Should be fixed now. Polytope24 (talk) 15:57, 9 May 2014 (UTC)

In mathematics
The article says, "In mathematics and theoretical physics, mirror symmetry is a relationship between two geometric objects called Calabi–Yau manifolds." I'm not sure a simple "In mathematics" is the best way to start, since in "most" mathematics, from the perspective of an intelligent layman (i.e., lower-level mathematics), the phrase "mirror symmetry" would have the everyday interpretation of being symmetric about an axis (or plane). Perhaps "In advanced mathematics and theoretical physics" might be better? Or, more specifically, "In algebraic geometry and theoretical physics"? - dcljr (talk) 17:52, 9 May 2014 (UTC)
 * I've gone ahead and made the change ("In algebraic geometry and theoretical physics"). - dcljr (talk) 21:31, 12 May 2014 (UTC)

Error?
I do not have expertise in this topic. But I believe this sentence: "From a mathematical point of view, the version of mirror symmetry described above is still only a conjecture, but there is another version of mirror symmetry in the context of topological string theory, a simplified version of string theory introduced by Edward Witten (ref: Witten 1990) which has been rigorously proven by mathematicians. (ref: Givental 1996, 1998; Lian, Liu, Yau 1997, 1999, 2000)" is in error. At the least, I cannot verify it by looking at the two papers of Givental or the three by Lian-Liu-Yau, since they do not even cite Witten's paper. The text by itself (ignoring the given refs) seems to suggest the Witten conjecture in Gromov-Witten theory, which to my understanding is completely distinct from the mirror formulas proved by Givental and Lian-Liu-Yau, which did not originate with Witten. Can someone clarify? Gumshoe2 (talk) 09:49, 8 March 2022 (UTC)


 * I don't think the 1990 paper by Witten is related to string theory (just to counting some curves using two different models of 2d quantum gravity). The papers of Lian-Liu-Yau appear to prove a "mirror conjecture" although it is a bit much to claim that this is "another version of mirror symmetry" since it seems to just be a theorem relating enumerative invariant counts for certain simple classes of Calabi-Yaus. It would be more precise to say "Some early predictions of physical mirror symmetry in the context of topological string theory have been proven, including relating certain counts of enumerative invariants described from algebraic and symplectic perspectives". Mirror symmetry is too vast a subject to make a dangerous statement like some theorems relating GW counts from the 1990s constitute a "different version" rather than a very special case. Tazerenix (talk) 11:02, 8 March 2022 (UTC)