themirrorโsmagicsights: an updateonmirrorsymmetryย ยท orientable special lagrangians admit phase...
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The Mirrorโs Magic Sights: AnUpdate on Mirror Symmetry
Timothy PerutzBut in her web she still delightsTo weave the mirrorโs magic sights
Tennyson, The Lady of Shalott
The Mirror Symmetry MysteryOrigins. The 1980s and โ90s saw an astonishing entangle-ment of research in geometry and mathematical physics.String theorists, developing their candidate for a quantumtheory incorporating gravity, not only drew on state-of-the-art mathematics, but introduced mathematical ideasof great power and prescience: none more so than mirrorsymmetry.
The author is an associate professor of mathematics at the University of Texasat Austin. His work is partially supported by NSF grant CAREER: 1455265.His email address is [email protected].
For permission to reprint this article, please contact:[email protected].
DOI: https://doi.org/10.1090/noti/1854
In a 1989 paper [13], Lerche, Vafa, and Warner studiedthe algebraic structure of 2-dimensional ๐ = 2 supersym-metric conformal field theories (SCFT). I will not define a 2-dimensional ๐ = 2 SCFT, but only note that it is a typeof quantum field theoryโas such, involving operators onHilbert spacesโin which the operators are associated withRiemann surfaces. The authors knew that a CalabiโYaumanifold gives rise to an ๐ = 2 SCFT, the Riemann sur-faces being traced out by the motions and interactions ofclosed strings, i.e., loops, inside the manifold. In such atheory, they wrote,
there are four types of rings arising from the var-ious combinations of chiral and anti-chiral, andleft and right. We will denote these rings by (๐, ๐),(๐, ๐), (๐, ๐), (๐, ๐). ... There is a non-trivial re-lationship between (๐, ๐) and (๐, ๐). ... For su-perconformal models coming from compactifica-tion on Calabi-Yau manifolds, the (๐, ๐) ring be-comes isomorphic to the structure of the cohomol-ogy ring of the manifold in the large radius limit.
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Figure 1. Mirrored Hodge diamonds for mirror pairs of CY 3-folds.
... One possibility might be that [the Poincarรฉ se-ries for the (๐, ๐) ring] is the Poincarรฉ series for (adeformation of) the cohomology ring of anothermanifold. If so, there must be another manifold๏ฟฝฬ๏ฟฝ for which the Betti numbers satisfy๐๐
๐,๐ = ๐๏ฟฝฬ๏ฟฝ๐โ๐,๐.
The possibility tentatively put forward in this passage1 wassoon enunciated with greater precision and certitude, andnamed mirror symmetry [9,19].
Basic explanations: A Kรคhler manifold (๐,๐) is a com-plex manifold ๐, together with a Kรคhler form ๐: a ๐ถโ
real 2-formโi.e., a skew-symmetric bilinear form on thetangent bundle ๐๐โwhich is closed (๐ฝ๐ = 0), invari-ant under the complex structure, and positive on complexlines in ๐๐. Being closed and non-degenerate, a Kรคhlerform is an example of a symplectic form. Complex projec-tive space โ๐ has a unique Kรคhler form (up to a positivescalar factor) that is invariant under the transitive actionof the projective unitary group of โ๐+1; an embedding of๐ into โ๐ determines a Kรคhler form on ๐ by restriction.
ACalabiโYau (CY)manifold (๐,๐,ฮฉ) is a compact Kรคh-ler manifold (๐,๐) endowed with a holomorphic volumeform ฮฉ. In local holomorphic coordinates (๐ง1,โฆ , ๐ง๐),ฮฉ = ๐(๐ง)๐ฝ๐ง1 โง โฏ โง ๐ฝ๐ง๐, where ๐ is holomorphic andnowhere-vanishing. Examples:
โข When ๐ = 1, the only CY manifolds are ellipticcurves โ/๐ ๐บ๐๐๐๐ผ๐พ; one can take ๐ = ๐๐ฝ๐ง โง ๐ฝ ฬ๐งand ฮฉ = ๐ฝ๐ง.
โข CY hypersurfaces ๐ โ โ๐+1, cut out from projec-tive space by a homogeneous polynomial of de-gree ๐+ 2. Elliptic curves arise as cubics in โ2.
โข Complex tori โ๐/๐ ๐บ๐๐๐๐ผ๐พ.The โBetti numbersโ ๐๐
๐,๐ in the quotation are really theHodge numbers, ๐๐
๐,๐ = โ๐,๐(๐) โถ= dimโ ๐ป๐(๐,ฮฉ๐๐):
โ๐,๐ is the vector-space dimension of the ๐th cohomologyof the sheaf ฮฉ๐
๐ of holomorphic ๐-forms. The Betti num-ber ๐๐ = dimโ ๐ป๐(๐;โ), the dimension of the ๐th sin-gular cohomology, is the sum of the โ๐,๐ where ๐ + ๐ =1L. Dixon reportedly also put forward this idea.
๐. The โPoincarรฉ seriesโ ๐(๐ก) of a graded ring is the gen-erating function for the dimensions of its homogeneousparts, so for ๐ปโ(๐;โ) the Poincarรฉ series is the polyno-mial ๐(๐ก) = โ๐๐(๐)๐ก๐.
The term โmirror symmetryโ refers to a literal mirroringof Hodge diamonds expressed by the relation โ๐,๐(๐) =โ๐โ๐,๐(๏ฟฝฬ๏ฟฝ)โthe Hodge diamond is the conventional vi-sualization of the array of Hodge numbers โ๐,๐ (Figure 1).But in retrospect, it seems mistaken to view that as a pri-mary manifestation of mirror symmetry. I prefer to thinkof the term as a metaphor for the reciprocal relationshipof ๐ to ๏ฟฝฬ๏ฟฝโthe mirror of the mirror is the original.
The ๐ = 2 SCFT which, string theorists argue, can beassociated with a CY manifold ๐ is a type of sigma model:it is based on maps ฮฃ โ ๐ where ฮฃ is a Riemann sur-face. There are two topological twists of the sigma modelwhich are 2-dimensional topological field theories, called theA-model and the B-model. Formally they are on an equalfooting, but their physical observables have quite differ-ent geometrical meanings, relating to holomorphic mapsfrom Riemann surfaces to the CY in the A-model, and toperiod integrals of differential forms in the B-model. Astatement of mirror symmetry, arising from string theorybut congenial to mathematicians, is the following:
Mirror symmetry determines an isomorphism of 2-dimensional topological field theories between the A-model of ๐ and the B-model of ๏ฟฝฬ๏ฟฝ, and vice versa.2
Readers familiar with topological field theory will knowthat the state space attached to the circle is a ring: these arethe rings that appear in the quoted passage from [13].
Today, there is an ocean of literature on holomorphicmaps from Riemann surfaces to Kรคhler, or more gener-ally, symplectic manifolds, including GromovโWitten in-variants (the โclosed stringโ part of the story). The theoryof Fukaya categories (the โopen stringโ part) is proceedingrapidly with respect to foundations and the developmentof tools. Laying down completemathematical foundations
2โTopological field theory should be understood in an โextendedโ or โopen-closedโ sense; cf.[5].
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for the A-model topological field theory appears to bewith-in reach. We also have a good formulation of the partsof the B-model where ฮฃ has genus 0, incorporating de-rived categories and variations of Hodge structure, and anemerging understanding of the higher genus part [5,14].Counting curves. It seems that the germinal ideas of mir-ror symmetry elicited littlemore than skeptical shrugs fromgeometers. But in 1991, Candelas, de la Ossa, Greene, andParkes [4] made a prediction which geometers could notignore, for it seemed magical yet the evidence was com-pelling.
Taking the example of a quintic 3-fold ๐ โ โ4, anda mirror consisting of a certain holomorphic 1-parameterfamily ๏ฟฝฬ๏ฟฝ๐ of CY 3-folds (the parameter ๐ varies in a punc-tured disc ฮโ = {๐ โ โ โถ 0 < |๐| < 1}) they studieda facet of SCFT visible in the topologically twisted A- andB-models and expected to match under mirror symmetry:the 3-point Yukawa couplings. For the A-model of ๐, theYukawa coupling was identified as a generating function
๐ธ๐ด(๐) = 5+โโ๐=1
๐๐ ๐3๐๐(1 + ๐๐ +๐2๐ +โฆ), (1)
where ๐๐ counts rational curvesโthe images of holomor-phic maps โ1 โ ๐โmeeting a hyperplane ๐ป โ โ4 withtotal multiplicity ๐. On the B-side, the Yukawa couplingsare period integrals for the family {๏ฟฝฬ๏ฟฝ๐}. Precisely, ๐ธ๐ต(๏ฟฝฬ๏ฟฝ)is the Laurent series expansion of the holomorphic func-tion on ฮโ
๐ โฆ โซ๏ฟฝฬ๏ฟฝ๐
ฮฉฬ โง (๐ ๐๐๐)
3
ฮฉฬ,
where ฮฉฬ is a holomorphic 3-form on the total space of thefamily, defining a volume form ฮฉฬ๐ on each fiber ๏ฟฝฬ๏ฟฝ๐; it hasto be correctly normalized as a function of ๐. Candelas etal. computed that
๐ธ๐ต(๏ฟฝฬ๏ฟฝ) = 5(1 + 55๐ฅ)
1๐ฆ(๐ฅ)2 (๐
๐ฅ๐๐ฅ๐๐)
3
,
where
๐ฆ(๐ฅ) = โ๐โฅ0
(5๐)!(๐! )5 (โ1)๐๐ฅ๐, ๐ฅ(๐) = โ๐+770๐2+โฆ.
The crucial change of coordinates ๐ฅ = ๐ฅ(๐), which theycomputed to all orders, is called the mirror map. Their pre-diction, then, was that
๐ธ๐ด(๐) = ๐ธ๐ต(๏ฟฝฬ๏ฟฝ). (2)
They wrote:
It is gratifying that [assuming (2)] we find that๐1 =2875 which is indeed the number of lines (ra-tional curves of degree one) and ๐2 = 609250which is known to be the number of conics (ratio-nal curves of degree 2).
Mathematicians soon proposed a precise definition for thecoefficients ๐๐ of the series
๐ธ๐ด(๐) = โ๐๐๐๐
(so ๐0 = 5, ๐1 = ๐1, ๐2 = 8๐2 +๐1, etc.).It is rooted in Gromovโs notion of pseudo-holomorphiccurves in symplectic manifolds. One defines๐๐ as a genus-zero GromovโWitten invariant, a homological โcountโ of holo-morphic maps ๐ขโถ โ1 โ ๐ of degree ๐, mapping threespecified points ๐ง๐ โ โ1 (๐ = 0, 1, 2) to ๐ป๐ โฉ๐, where๐ป๐ โ โ4 is a specified hyperplane (Figure 2).
Figure 2. The A-side 3-point Yukawa coupling is a GWinvariant enumerating holomorphic maps ๐ข.
GW invariants do not ultimately depend on the com-plex structure used on๐ used to define them, so any smoothquintic 3-fold will serve. Such maps ๐ข may factor throughbranched coveringsโ1 โ โ1, and there is a qualified sensein which the ๐๐ in (1) count the images, in ๐, of the maps๐ข.Principles. The intense activity inspired by the work ofCandelas et al. made certain principles clear:
โข The A-model of (๐,๐,ฮฉ) concerns the symplecticgeometry of (๐,๐).
GromovโWitten invariantsโsigned, weighted counts ofholomorphic maps from Riemann surfaces into ๐ invokea complex structure on ๐๐, but this should be viewed asan auxiliary choice not affecting the outcome.
โข The mirror to a CY manifold is not a single CYmanifold, but a family of CY manifolds. The B-model concerns the complex analytic geometry ofthis family.3
The next principle is that one cannot expect mirror sym-metry to arise from a single CY manifold ๐, nor from anarbitrary family. Rather,
โข ๐ has a mirror when it undergoes a maximal de-generation to a singular variety, such as the degen-eration of an elliptic curve to three projective lines(a degenerate cubic, Figure 3).4
3And, when these CY manifolds are projective varieties, their complex analytic geometry isinterpretable as algebraic geometry.4A maximal degeneration, parametrized by a small disc in โ, is one with maximally unipo-tent monodromy.
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Figure 3. Plane cubic curves ๐ฅ๐ฆ(๐ฅ + ๐ฆโ 1) = ๐ degeneratingto three lines as ๐ โ 0.
Finally, there is Kontsevichโs eagle-eyed conjecture from1994 [11], today called homological mirror symmetry (HMS),connecting Lagrangian submanifolds of ๐ to coherentsheaves on ๏ฟฝฬ๏ฟฝ. There are โopen stringโ topological field the-ories, governed by categorical structures called ๐ดโ-categories. In the A-model, one has the Fukaya๐ดโ-categoryF(๐,๐) of the symplectic manifold (๐,๐)โits objectsare Lagrangian submanifolds of ๐โand in the B-model,the bounded derived category ๐ท(๏ฟฝฬ๏ฟฝ), whose objects arethose complexes of sheaves โฐโข of ๐ช๏ฟฝฬ๏ฟฝ-modules whose co-homology sheaves โ๐(โฐ) are coherent and of boundeddegree ๐.5
We pause to define two of the terms:Lagrangian submanifolds: A subspace ฮ of a vector space
๐ with a symplectic pairing ๐๐ is called Lagrangian if, foreach ๐ฃ โ ฮ, the linear form ๐๐(๐ฃ, โ ) vanishes preciselyonฮ; this implies dim๐ = 2dimฮ. A submanifold ๐ฟ โ๐ in a symplectic manifold (๐,๐) is one whose tangentspaces ๐๐ฅ๐ฟ are Lagrangian in ๐๐ฅ๐.
Coherence of sheaves: In algebraic geometry, and simi-larly in the rigid analytic geometry we shall discuss later,an algebraic variety ๐ comes with a sheaf of rings ๐ช๐, thestructure sheaf, assigning a commutative ring๐ช๐(๐) to eachopen set ๐ โ ๐. A sheaf of ๐ช๐-modules โฐ assigns an๐ช๐(๐)-module โฐ(๐) to each open ๐. Assuming for sim-plicity that๐ช๐(๐) is a Noetherian ring for small neighbor-hoods ๐ of an arbitrary point ๐ง โ ๐, we say โฐ is coherentif each point of ๐ has an open neighborhood ๐ such that(i) the๐ช๐(๐)-moduleโฐ(๐) is finitely generated; and (ii),for all open sets ๐ โ ๐, the map ๐ช๐(๐)โ๐ช๐(๐) โฐ(๐) โโฐ(๐), ๐ โ ๐ โฆ ๐ โ ๐ |๐, is an isomorphism.
โข HMS: There is a functor F(๐,๐) โ ๐ท(๏ฟฝฬ๏ฟฝ)โmapping Lagrangian submanifolds of ๐ to coher-ent complexes of sheaves on ๏ฟฝฬ๏ฟฝโwhich is, in a cer-tain sense, a categorical equivalence.6
5The derived category should here be treated not as a triangulated category, but its enhance-ment to a differential-graded (hence ๐ดโ) category.6Namely, it induces a quasi-equivalence of the associated ๐ดโ-categories of right modules.It may appear that HMS is incompatible with the notion that the mirror is a family. WhenHMS is formulated more precisely, this apparent disconnect proves illusory.
Kontsevich foresaw that HMS should be an organizing prin-ciple; that it should imply the isomorphism of topologicalfield theories ๐ด(๐) and ๐ต(๏ฟฝฬ๏ฟฝ), and thereby enumerativestatements such as the prediction (2).Verification, explanation. Someofmirror symmetryโs pre-dictionswere soon verified. Candidatemirror partners werefound for many CY manifolds. The Yukawa couplings๐ธ๐ด(๐) were computed for a class of CY manifolds ๐ in-cluding the quintic 3-fold [7] by showing that they satisfythe same differential equations as their B-side counterparts๐ธ๐ต(๏ฟฝฬ๏ฟฝ). Such work bore witness to the mirror symmetryphenomenon, but did not explain it.
Explanations gradually emerged [5,12,17]. The GrossโSiebert program [10] is a systematic and sophisticated con-struction of mirror pairs, for which several of the predic-tions of mirror symmetry have been proven. HMS hasrecently become tractable as basic tools for working withFukaya categories have been developed. We now know [8]that HMS is an indeed an organizing principle, implyingstatements such as (essentially) (2). We know that HMS istrue for (on the A-side) the quintic 3-fold [16], andwe havea prototype for a truly explanatory proof of HMS [1,2].
The Key Questions(a) How do we construct a mirror ๏ฟฝฬ๏ฟฝ to a CY manifold
๐?(b) How can the symplectic geometry of ๐ be read as
analytic geometry of ๏ฟฝฬ๏ฟฝโor vice versa?(c) Why is HMS true?(d) Why is mirror symmetry involutory?โWhy is ๐ the
mirror of its mirror ๏ฟฝฬ๏ฟฝ?
The germ of the answer to (a) and (d) was proposed byStromingerโYauโZaslow (SYZ) in 1996 [17]. The point isto find a smooth, surjective map
๐โถ ๐2๐ โ ๐๐
to a middle-dimensional base ๐ such that the subspaceker๐ท๐ฅ๐ โ ๐๐ฅ๐ is Lagrangian for all regular points ๐ฅ; sothe regular fibers are Lagrangian submanifolds of ๐.
The regular fibers ๐น๐ โถ= ๐โ1(๐) are necessarily tori:each fiber ๐น๐ has the structure of an ๐-dimensional affinevector space๐ดmodulo the action of a lattice ๐ฟ in its vectorspace ๐ of translations. One then obtains the mirror ๏ฟฝฬ๏ฟฝ byreplacing the non-singular fibers of that family by the dualtori ฬ๐น๐ โถ= ๐ป1(๐น๐; โ/โค) โ ๐โ/๐ฟโ, the quotients of thedual vector spaces by the dual lattices. Provided one canfind a way to handle the singular fibers, one obtains in thisway a space ๏ฟฝฬ๏ฟฝ and a map ฬ๐ โถ ๏ฟฝฬ๏ฟฝ โ ๐ with fibers ฬ๐น๐.
A CY manifold ๐ admits an โoptimalโ pair (๐,ฮฉ), onefor which ฮฉ is covariantly constant with respect to theKรคhler metric: this is a famous theorem of S.-T. Yau. ALagrangian ๐ฟ โ ๐ is called special with respect to a CY
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Figure 4. A schematic of ๐โ๐1; the top and bottom are gluedtogether to form (๐โ๐1)/(๐โ๐1)โค. The red lines are thecotangent fibers, which become circles in the quotient.
metric if the imaginary part of ฮฉ vanishes on ฮ๐๐๐ฟ. SYZproposed that the fibers ๐น๐ should be special Lagrangian.Orientable special Lagrangians admit phase functions ๐:there is a non-vanishing section ๐ฃ๐ฟ of ฮ๐๐๐ฟ such thatargฮฉ(๐ฃ๐ฟ)โถ ๐ โ ๐1 admits a continuous logarithm ๐,called a phase function. Lagrangians with phase functionswill suffice for our needs in this article, and special La-grangians will not reappear.
The basic model for the SYZ mirrorโdisregardingspecialnessโis as follows:
Let ๐ be an integral affine ๐-manifold, that is, an๐-manifold with an atlas of charts whose transitionfunctions are affine transformations between open sub-sets of โ๐, of shape ๐ฅ โฆ ๐ด๐ฅ + ๐ with ๐ โ โ๐ and๐ด โ ๐บ๐ฟ๐(โค). The cotangent bundle ๐โ๐ is natu-rally a symplectic manifold, and there is a natural in-teger lattice (๐โ
๐ ๐)โค in each cotangent space ๐โ๐ ๐.
Let ๐ = (๐โ๐)/(๐โโค ๐) be the quotient of ๐โ๐
by fiberwise-translations by lattice-vectors (Figure 4).Then ๐ is symplectic, and is a bundle of Lagrangian๐-tori over ๐. The tangent bundle ๐๐ contains alattice dual to the one in ๐โ๐, and the quotient man-ifold ๏ฟฝฬ๏ฟฝ = ๐๐/๐โค๐ is naturally complex. Then ๐and ๏ฟฝฬ๏ฟฝ are mirrors.
SYZโs idea is at the heart of our current understandingof mirror symmetry, but the version I will outline in thesection on rigid analytic mirrors is purely symplectic ratherthan Riemannian in nature, and, unlike the basic modeljust presented, it makes ๏ฟฝฬ๏ฟฝ a complex 1-parameter family.
Prototypes: Fourier Transforms, Classical andGeometricPontryagin duality. The most basic model for a dualitysuch as mirror symmetry is the passage from a finite-dimensional vector space to its dual. A more instructiveexample is Pontryagin duality. The characters of a locallycompact, abelian topological group ๐บ are the continuoushomomorphisms ๐บ โ ๐ to the circle-group ๐ = โ/โค. InFourier analysis, one takes ๐บ = โ or ๐, so that the respec-tive characters are the maps ๐ฅ โฆ ๐2๐๐๐ก๐ฅ for ๐ก โ โ or ๐ก โ โค.The set ฬ๐บ of characters is again a locally compact topo-logical group, the Pontryagin dual of ๐บ. There is a โuni-versal characterโ, which is the evaluation pairing ๐โถ ฬ๐บ ร๐บ โ ๐, ๐(๐, ๐ฅ) = ๐(๐ฅ). A complex-valued function ๐on ๐บ has a Fourier transform ฬ๐, a function on ฬ๐บ: ฬ๐(๐) =โซ๐บ ๐(๐, ๐ฅ) ๐(๐ฅ)๐๐บ,where๐๐บ is a suitably-normalized left-invariant measure defined on the open sets.
The construction is a duality inasmuch as the evaluation
map ๐พ๐โถ ฬฬ๐บ โ ๐บ is an isomorphism, and ฬฬ๐ = ๐โ๐๐๐โ๐พ๐(where ๐๐๐โถ ๐บ โ ๐บ is inversion).
Mirror symmetry, based on the SYZ idea, is roughlyanalogous to the formation of the Pontryagin dual group,with the Fourier transform a prototype for HMS.
FourierโMukai transforms for K3 surfaces. FourierโMukaitransforms [15] bring us closer to mirror symmetry proper.Consider a simply connected, compact CY complex sur-face (๐,๐,ฮฉ) embedded in a projective space: a projec-tive K3 surface.
Holomorphic vector bundles, or more generally, coher-ent sheaves F, over ๐, have a discrete invariant, the Cherncharacter, which is best packaged as the Mukai vector๐ฃ(F) = ๐ฃ0+๐ฃ2+๐ฃ4 โ ๐ป0(๐; โค)โ๐ป2(๐; โค)โ๐ป4(๐; โค).7There is a moduli space๐๐,๐ฃ, parametrizing isomorphismclasses of โstableโ coherent sheaves F, with fixed Mukai vec-tor ๐ฃ; under assumptions that go unstated here, it is acompact complex manifold, projective, of dimension 2 +(๐ฃ,๐ฃ), where (๐ฃ, ๐ฃ) = โซ๐ (โ2๐ฃ0๐ฃ4 +๐ฃ2
2). In the isotropiccase (๐ฃ, ๐ฃ) = 0, ๐๐,๐ฃ is again a surface, and is again CY.8
In the case that ๐ฃ = 1 โ ๐ป0(๐; โค), one has ๐๐,๐ฃ = ๐,the points of ๐๐,๐ฃ being merely the ideal sheaves for thepoints ๐ โ ๐. But for other choices of Mukai vector, ๐๐,๐ฃis a new K3 surface, and we can recover๐ as amoduli spaceof sheaves of ๐๐,๐ฃ:
๐๐,๐ฃ โ ๐๐(๐,๐ฃ),๐ฃโฒ
for a certainMukai vector๐ฃโฒ for๐๐,๐ฃ. Thus amoduli spaceof geometric objects on a K3 surface gives rise to a new K3surface, in a reciprocal relationship with the original.
7The Mukai vector is ๐ผ๐(F) โง (1 + ๐), where ๐ผ๐ is the Chern character and ๐ is thegenerator for๐ป4(๐; โค).8The holomorphic volume form is the Serre duality pairing on ๐F๐๐,๐ฃ = ๐ค๐๐1๐ช๐ (F,F).
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There is a distinguished sheaf on ๐ ร ๐๐,๐ฃ, the univer-sal sheaf E๐๐๐๐, whose restriction to the slice ๐ร {F} = ๐is isomorphic to F.9 The FourierโMukai transform now in-puts coherent sheaves E on ๐, and outputs (complexes of)coherent sheaves on ๐๐,๐ฃ:
F โฆ ฬF = (๐๐2)โ(E๐๐๐๐ โ๐๐โ1 F).The FourierโMukai transform has a categorical mani-festation, which is strongest when (๐ฃ, ๐ฃ) = 0: it thendefines an equivalence of derived categories of coherentsheaves on ๐ and on๐๐,๐ฃ. This is the model for HMS.
Rigid Analytic MirrorsThe Novikov field and rigid analytic geometry. Fix a field๐น. The vector space ๐นโ of all functions ๐โถ โ โ ๐น has asubspace ฮ๐น of Novikov series: functions ๐ whose supportis discrete and bounded below. One can multiply Novikovseries, by convolution; thus we usually write Novikov se-ries as formal series
๐ =โโ๐=1
๐๐๐๐๐ , ๐๐ โ ๐น,
๐๐ โ โ, ๐1 < ๐2 < โฆ, ๐๐ โ โ.(This series represents the function supported on {๐1,๐2,โฆ} given by ๐๐ โฆ ๐๐.) In this way ฮ๐น becomes a field;the complex Novikov field ฮโ is algebraically closed.
The most important feature of ฮ๐น is that it comes witha complete valuation
๐๐บ๐ (๐) โถ= min supp๐. (3)
A valuation on a field ๐พ is a map ๐๐บ๐ โถ ๐พร โ โ (extendedto ๐พ by setting ๐๐บ๐ (0) = +โ) such that ๐๐บ๐ (๐ฅ + ๐ฆ) โฅmin(๐๐บ๐ (๐ฅ), ๐๐บ๐ (๐ฆ)) and ๐๐บ๐ (๐ฅ๐ฆ) = ๐๐บ๐ (๐ฅ) + ๐๐บ๐ (๐ฆ).There is an associated absolute value, |๐ฅ| = exp(โ๐๐บ๐ (๐ฅ)),and a metric ๐(๐ฅ,๐ฆ) = |๐ฅโ ๐ฆ|. The valuation is completeif ๐-Cauchy sequences converge.
Rigid analytic geometry [18] is a variant of algebraic ge-ometry, applicable over a complete valued field (๐พ, ๐๐บ๐ ):it builds in the internal geometry of the valuation.
In algebraic geometry over a field๐พโwhich, for brevity,we here assume algebraically closedโthe basic objects arepolynomial algebras๐พ[๐ง1,โฆ , ๐ง๐]. Maximal ideals thereincorrespond to points ๐ฅ โ ๐พ๐, as they take the form (๐ง1 โ๐ฅ1,โฆ , ๐ง๐ โ ๐ฅ๐). In rigid analytic geometry, one insteadstudies the Tate algebra ๐๐ = ๐พโจ๐ง1,โฆ , ๐ง๐โฉ, the algebraof power series ๐(๐ง) = โ๐๐ผ๐ง๐ผ, a sum over multi-indices
(๐1,โฆ , ๐๐) โ (โคโฅ0)๐, with ๐๐ผ โ ๐พ and ๐ง๐ผ = โ๐ง๐๐๐ ,
such that |๐๐ผ| โ 0 as โ๐ผโ โ โ, where โ๐ผโ = โ๐ ๐๐. Ifone has a point ๐ฅ = (๐ฅ1,โฆ , ๐ฅ๐) in the โunit polydiskโ๐ป๐ โ ๐พ๐, meaning |๐ฅ๐| โค 1 for all ๐, it defines a max-imal ideal ๐ช๐ฅ = (๐ง1 โ ๐ฅ1,โฆ , ๐ง๐ โ ๐ฅ๐) โ ๐๐: there is
9Mukai develops โquasi-universal sheavesโ in cases where automorphisms preclude a univer-sal sheaf.
an isomorphism ๐๐/๐ช๐ฅ โ ๐พ, given by [๐] โฆ ๐(๐ฅ) =โ๐โฅ0 โโ๐ผโ=๐ ๐๐ผ๐ฅ๐ผ (convergent series). This constructionaccounts for all maximal ideals of ๐๐, and so one thinksof ๐๐ geometrically as the polydisk ๐ป๐.
A quotient ๐ด = ๐พ[๐ง1,โฆ , ๐ง๐]/(๐1,โฆ , ๐๐) determinesa topological space ๐ = ๐ฒ๐๐พ๐ผ๐ด. The points of ๐ are theprime ideals of ๐ด; ๐ has its Zariski topology, in which themaximal ideals are the closed points. One thinks of theclosed points of๐ as the zero-set ๐1(๐ฅ) = โฏ = ๐๐(๐ฅ) = 0inside๐พ๐. There is a๐พ-algebra of โfunctionsโ๐ช๐ on๐, themaps ๐ฅ โฆ ๐(๐ฅ) โ ๐ด/๐ช๐ฅ where ๐ โ ๐ด and ๐ฅ โ ๐ labelsa maximal ideal ๐ช๐ฅ. But actually, ๐ช๐ โ ๐ด.
Likewise, a quotient ๐ด = ๐๐/(๐1,โฆ , ๐๐) determines aspace ๐ = ๐ฒ๐๐ด of maximal ideals, called an affinoid space.As before, it determines ๐ด as its ring of functions ๐ช๐.
Certain subsets ๐ โ ๐ inside an affinoid space ๐ =๐ฒ๐๐ด are called affinoid domains. Take a (suitable) normโ โ โ on ๐ด, and the induced norms โ โ โ๐ฅ on its quotients๐ด/๐ช๐ฅ: โ๐โ๐ฅ = inf{โ๐โ โถ ๐โ๐ โ ๐ช๐ฅ}. Then, for ๐ โ ๐ดand ๐ โ โ, the set ๐(๐, ๐) = {๐ฅ โ ๐ โถ โ๐(๐ฅ)โ๐ฅ โค๐} is an affinoid domain. So too is a finite intersectionโ๐(๐๐, ๐๐).
In algebraic geometry, spectra of ๐พ-algebras can beโgluedโ together to form a global object, a๐พ-scheme, whichis a topological space ๐ equipped with a sheaf ๐ช๐ of ๐พ-algebras, locally the spectrum of a ๐พ-algebra. Tate showedhow affinoid subdomains of affinoid spaces can be gluedtogether to form a global objectโa space ๐ with a sheaf of๐พ-algebras ๐ช๐, which is locally the algebra of functions of anaffinoid domain.
Rigid analytic mirrors. Suppose we have a compact, con-vex polytope ๐ โ โ๐. To this we attach the set
๏ฟฝฬ๏ฟฝ๐ = {๐ฅ โ (ฮรโ )๐ โถ (๐๐บ๐ (๐ฅ1),โฆ , ๐๐บ๐ (๐ฅ๐)) โ ๐}
(Figure 5). This subset is actually an affinoid subdomainof an affinoid space over the Novikov field ฮโ. First, wecan realize the annular domain {๐ฅ โ ฮ๐
โ โถ ๐ โค |๐ฅ๐| โค๐โ1, ๐ = 1,โฆ ,๐} as an affinoid space ๐ด๐
๐ . The polytope๐ is cut out from โ๐ by a finite list of inequalities, eachof shape ๐ โ ๐ฅ โฅ ๐, where ๐ โ โค๐ and ๐ โ โ. And ๏ฟฝฬ๏ฟฝ๐is cut out, inside ๐ด๐
๐ for a suitably small ๐, by inequalities|๐ฅ๐1
1 โฏ๐ฅ๐๐๐ | โค ๐โ๐; this identifies it as an affinoid subdo-main of ๐ด๐
๐ .
XฬP Pval
Figure 5. The values of the coordinates of the affinoid domain๏ฟฝฬ๏ฟฝ๐ form the polytope ๐.
488 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4
Suppose now that one has an ๐-manifold ๐ which isnot merely smooth, but integral affine (cf. โThe Key Ques-tionsโ)โsuch as the base of a fibering of a symplectic man-ifold ๐ by Lagrangian submanifolds {๐น๐}๐โ๐. โTriangu-lateโ๐ by a collection of integral affine polytopes๐๐ผ. Eachof them defines an affinoid domain ๏ฟฝฬ๏ฟฝ๐๐ผ , and these glue to-gether to form a rigid analytic space ๏ฟฝฬ๏ฟฝ overฮโ, which doesnot change when one subdivides the triangulation.
P1 P2
P3P4
P5
Figure 6. Fragment of a triangulation of ๐, as it appears in anintegral affine chart.
The set underlying ๏ฟฝฬ๏ฟฝ is the space of pairs (๐, ๐), where๐ โ ๐ and ๐ โ ๐ป1(๐น๐;๐(ฮ)). Here ๐(ฮ) ={๐ โ ฮร โถ |๐| = 1}: so the mirror is a space ofpairs of a torus-fiber ๐น๐ and a homomorphism fromthe first homology group๐ป1(๐น๐) โ โค๐ to the group ofunit-norm Novikov seriesโmade into a rigid analyticspace.
For example, if ๐ = โ/โค is the circleโthe base ofa Lagrangian fibration on the 2-torus โ2/โค2 viewed as asymplectic manifoldโits affine integral structure is inher-ited from โ, and we can triangulate it by intervals [๐, ๐].The affinoid domain associated with an interval is an โan-nulusโ {๐ง โ ฮร โถ ๐โ๐ โค |๐ง| โค ๐โ๐}, and these gluetogether to form an elliptic curve over ฮ, the Tate curve๐ธ๐๐๐ก๐ = ฮร/๐โค.Pseudo-holomorphic curves. Why should rigid analyticgeometry over the Novikov field have anything whatsoeverto do with symplectic topology? The brief answer is: Gro-mov compactness.
Symplectic topologists probe symplectic manifolds(๐,๐) using pseudo-holomorphic curves: maps ๐ขโถ ฮฃ โ ๐from a Riemann surface ฮฃ to ๐ such that, for some speci-fied complex structure ๐ฝ on ๐๐, the derivative ๐ท๐ข is com-plex linear. Thus, if ๐ is the complex structure on ๐ฮฃ, onehas the โCauchyโRiemann equationโ ๐ฝ โ ๐ท๐ข = ๐ท๐ข โ ๐.In the presence of a Lagrangian submanifold ๐ฟ โ ๐, one
may suppose that ฮฃ has boundary, and impose the bound-ary condition that ๐๐ข (the restriction of ๐ข to the boundary๐ฮฃ) maps ๐ฮฃ to ๐ฟ.
Once one pins down the smooth surface underlying ฮฃ,and the Lagrangian boundary conditions, there is amodulispace โณ of pseudo-holomorphic curves in ๐, which oneshould think of as a smooth manifold. One can also al-low pseudo-holomorphic curves with nodal domains, andfrom these one can construct a larger moduli space โณ.Gromov compactness says that the subspace โณโค๐, wherethe energy ๐ธ(๐ข) = โซฮฃ ๐ขโ๐ is at most ๐, is compact.
One typically imposes conditions on ๐ข so as to cut โณdown to a zero-dimensional manifold ๐. Then the com-pact sub-level sets ๐โค๐ for the energy function ๐ธ are finite.Once one has a recipe for orienting ๐, one can โcountโits points with signs, and the result is a Novikov series,#๐ โถ= โ๐ขโ๐ ๐๐๐๐(๐ข)๐๐ธ(๐ข) โ ฮโ.From Lagrangians to coherent sheaves. Suppose that wehave a compact CY manifold (๐2๐,๐,ฮฉ) and a non-singular fibering ๐โถ ๐2๐ โ ๐๐ by Lagrangiansubmanifoldsโnecessarily toriโwhich admit phase func-tions. Then ๐ acquires an integral affine structure. Sup-pose also that we have identified a section๐โถ ๐ โ ๐ of ๐whose image is Lagrangian; then ๐ = ๐โ๐/(๐โ๐)โค. Aswe discussed in the section on rigid analytic mirrors , wecan use the integral affine structure of ๐ to define a rigidanalyticฮ-space ๏ฟฝฬ๏ฟฝ = ๏ฟฝฬ๏ฟฝ๐๐๐๐๐. This is ourmirror.10 It comeswith a naturalmap ฬ๐ โถ ๏ฟฝฬ๏ฟฝ โ ๐, and the fiber ฬ๐โ1(๐) can beidentified with ๐ป1(๐น๐;๐ฮ), where ๐ฮ = ๐๐บ๐ โ1(0) โ ฮร
is the group of unit-norm Novikov series.Now we come to the โFourier transformโ underlying
HMS, the process by which Lagrangians are converted intocoherent sheaves on the mirror. Suppose ๐ฟ โ ๐ is a com-pact Lagrangian submanifold, equipped with a phase func-tion. One then defines sheaves โ๐(โฐ๐ฟ) of ๐ช๏ฟฝฬ๏ฟฝ-moduleson ๏ฟฝฬ๏ฟฝ: Cover ๐ by integral polytopes ๐๐ผ, and let ๐๐ผ โ ๐๐ผbe a reference point. For each๐ผ, we can perturb ๐ฟ to a newLagrangian ๐ฟ๐ผ such that ๐ฟ๐ผ โฉ ๐น๐ is a transverse intersec-tion for every ๐ โ ๐๐ผ. We define a module โฐ๐ฟ,๐ผ over thering of functions ๐ช๐ผ โถ= ๐ช๏ฟฝฬ๏ฟฝ๐๐ผ
of ๏ฟฝฬ๏ฟฝ๐๐ผ by
โฐ๐ฟ,๐ผ = (๐ช๐ผ)๐ฟ๐ผโฉ๐น๐๐ผ โถthe freemodule on the set of intersection points. Themod-ule โฐ๐ฟ,๐ผ has a grading, defined via phase functions, anda differential ๐ฟโa square-zero endomorphism which in-creases the grading by 1. The construction of ๐ฟ uses familyFloer cohomology. It involves pseudo-holomorphic bigons,discs ฮ โ ๐, with a boundary condition that requires theupper half of ๐ฮ to map to ๐ฟ๐ผ, and the lower half to ๐น๐
10An important and delicate issue is whether there are holomorphic discs in ๐ whoseboundary lies on a fiber of ๐, and if so, how properly to account for them in the construc-tion of the mirror. For present purposes, assume there are none. This assumption is a majorsimplification of what is typically true.
APRIL 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 489
for some ๐ โ ๐๐ผ. For present purposes, we assume an ab-sence of holomorphic discs whose entire boundary lies on๐น๐ or ๐ฟ๐ผ. This is vital; to make things work in generality,one will need to prove their absence rather than assumingit. The fact that ๐ฟ makes sense expresses a compatibilitybetween pseudo-holomorphic curves and rigid analytic ge-ometry [2,6].
We then pass to the cohomology module
๐ปโ(โฐ๐ฟ,๐ผ) = ker๐ฟ/ im๐ฟ.
This is a finitely generated ๐ช๐ผ-module. While patterns ofintersections change under perturbations of Lagrangians,๐ปโ(โฐ๐ฟ,๐ผ) does not depend on the perturbation ๐ฟ โ ๐ฟ๐ผ.One can use that fact to assemble the modules ๐ปโ(โฐ๐ฟ,๐ผ)into a sheaf โโ(โฐ๐ฟ) of ๐ช๏ฟฝฬ๏ฟฝ-modules. Locally, it is thesheaf associated with a finitely generated module over aNoetherian ringโso it is coherent.
The mapping ๐ฟ โฆ โโ(โฐ๐ฟ), sending a Lagrangianto a coherent sheaf on the rigid analytic mirror, is theโFourier transformโ which explains HMS [2].
Mirror Symmetry as an Operation on Holomor-phic FamiliesWe have just seen that the symplectic geometry of fami-lies of Lagrangian submanifolds, fibering ๐, gives rise to arigid analytic mirror ๏ฟฝฬ๏ฟฝ๐๐๐๐๐ over the complex Novikov fieldฮ, and that other Lagrangians in ๐ then produce coher-ent analytic sheaves on ๏ฟฝฬ๏ฟฝ๐๐๐๐๐. But a rigid analytic spaceis not a symplectic manifold, so this cannot be an involu-tory process like Pontryagin duality or the FourierโMukaitransform.
I want to outline, via an example, how the formation ofrigid analytic mirrors should feed into an involutory pro-cedure, not yet fully understood, the construction of themirror partner to a degenerating 1-parameter families of CYmanifolds, whereby the mirror of the mirror is the original.
The first point is that degenerations should give rise to La-grangian torus fibrations. Start with projective spaceโ๐. Thishas a Lagrangian torus fibration โ๐ โ ฮฃ๐, of sorts, whosefibers are โClifford tori,โ the points (๐ง0 โถ โฏ โถ ๐ง๐) withโ|๐ง๐|2 = 1 and |๐ง๐| = ๐๐ (constant) for each ๐. Thebase ฮฃ๐ is a ๐-dimensional simplex. Some of the Cliffordtori, those lying over the boundary of the simplex, are notLagrangian, because they are tori of dimension less than๐.
Now consider the โtotally degenerate CY hypersurfaceโ๐0 = {๐ง0 โฏ๐ง๐ = 0} โ โ๐+1. It is a union of ๐ + 1projective hyperplanes ๐ฅ๐ = 0, and the Lagrangian torusfibrations over these hyperplanes assemble to give a map๐โถ ๐0 โ ๐ to a ๐-dimensional polyhedron formed bygluing the ๐+ 1 simplices along faces (๐ actually just theboundary of a (๐ + 1)-dimensional simplex). The fibers
of ๐ are Lagrangian tori over the interiors of the faces of ๐,and are lower-dimensional tori elsewhere (Figure 7).
X0 = {x0x1x2x3 = 0} ยต
{x0 = 0} {x1 = 0}
Figure 7. The map ๐โถ ๐0 โ ๐ in the case ๐ = 2.
Next, consider the family of CY hypersurfaces
๐๐ก = {(๐ก, ๐ง) โ โรโ๐+1 โถ ๐ก๐น(๐ง) + ๐ง0 โฏ๐ง๐ = 0},
where๐น is a (generic) homogeneous polynomial of degree๐ + 1. Thus ๐1 is a CY manifold, while ๐0 is our singu-lar, totally degenerate CY hypersurface. One can use thesymplectic geometry of the family (with a Kรคhler form in-herited from โ ร โ๐+1) to produce a map ๐โถ ๐1 โ ๐0which is a symplectomorphism over the smooth locus in
๐0. The composite ๐โถ ๐1๐โโ ๐0
๐โโ ๐ is then our candi-date for a Lagrangian torus fibration. Over the interiors ofthe simplices of ๐,๐ has Lagrangian fibers and ๐ is a dif-feomorphism; over a codimension ๐ facet of ๐, the fibersof๐ have dimension ๐โ๐, but those of๐ have dimension๐, so ๐ has fibers of dimension ๐, as we want. However,there is a โbadโ locus ๐ต โ ๐0 where the total space of thefamily is singular, and the mechanism breaks down; that isthe source of singularities in the fibers of ๐ (Figure 8).
Xt X0ฮผ
Pฯ ฮผ(B)
Figure 8. The map ๐โถ ๐๐ก โ ๐ in the case ๐ = 2, showingsome of its fibers in red. The 24 dots on the edges of thetetrahedron ๐ are the images of the singular locus of the totalspace of the family.
490 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4
This example illustrates a mechanism whereby toric de-generations of CY manifoldsโroughly, degenerations to va-rieties each of whose irreducible components is a toricvarietyโshould give rise to Lagrangian torus fibrations.11
The fiber ๐1 comes with a symplectic automorphism๐, the monodromy around the unit circle, whichโin amodel situation, at any rateโpreserves the fibers of ๐, andacts as translation of each of the non-singular fibers. Thisautomorphism corresponds to extra structure on the mir-ror, a line bundle over ๏ฟฝฬ๏ฟฝ๐๐๐๐๐. One expects that this linebundle is ample, and therefore defines an embedding of๏ฟฝฬ๏ฟฝ๐๐๐๐๐ into rigid analytic projective space. Just as in com-plex analytic geometry, the image of an embedding intoprojective space is in fact cut out algebraically bypolynomialsโso the ๏ฟฝฬ๏ฟฝ๐๐๐๐๐ becomes an algebraic scheme๏ฟฝฬ๏ฟฝ๐๐๐ over ฮโ.
Pause for a moment to observe that if we have a family๐๐ก of complex projective varieties, whose defining equa-tions depend holomorphically on ๐ก โ ฮโ (the punctureddisc), we can take the Laurent expansions of these equa-tions to get a family Z over the field โ((๐ก)) of finite-tailedLaurent series, and therefore, by extending scalars, a vari-ety over ฮโ. One can ask whether ๏ฟฝฬ๏ฟฝ๐๐๐ arises in this way,from a family ๏ฟฝฬ๏ฟฝ๐ก of complex projective varieties. This isnot the place to get into the details, but there are geomet-ric reasons to expect that to be true. In this way, we endup with a new family {๏ฟฝฬ๏ฟฝ๐ก} of complex projective CY man-ifolds, mirror to the original family.
While the general picture described here has large gapsstill to be filled, an algebro-geometric analogue of the com-posite process has been fully worked out by GrossโSiebert[10]. Their works centers on a part of the story called wall-crossing that I have not even hinted at.
Example. If one takes a degenerating family of ellipticcurves X โ ฮโ, given as cubic curves in โ2, the genericfiber ๐ is (symplectically) the 2-torus โ2/โค2 and it hasthe Lagrangian fibration given by projection ๐โถ โ2/โค2 โโ/โค. After choosing a section of ๐, one obtains the Tatecurve as rigid analytic mirror, with a degree 1 line bundleover it. Section of powers of this line bundle define anembedding of the Tate curve into โ2(ฮ) as a cubic curve
๐ฆ2 + ๐ฅ๐ฆ = ๐ฅ3 +๐4(๐)๐ฅ + ๐6(๐),
where ๐4 and ๐6 are certain power series in ๐. In particu-lar, this curve is defined over โ((๐)). Since ๐4 and ๐6 areconvergent in the unit disc |๐| < 1, it can also be viewed asa holomorphic family overฮโโthe mirror to the originalfamily.
11This mechanism was first explored by W.-D. Ruan in 1999, but was recently revisited inR. Guadagniโs 2017 University of Texas Ph.D. thesis.
Looking AheadFrom this symplectic geometerโs perspective, the most im-portant task ahead is to fill the gaps in the picture justoutlinedโprecisely how to construct Lagrangian fibrationswith singularities from degenerations, and then, crucially,how to construct their analytic mirrors. The chief difficultyis with Floer theory for singular Lagrangians. The GrossโSiebert program provides an algebro-geometric solution,at the cost of losing the direct connection to symplectictopology and the natural construction of HMS as a Fouriertransform. I hope and suspect that GrossโSiebertโs workwill be precisely linked to symplectic topology, perhapseven in the absence of a full understanding of the singularLagrangians, and that a proof of HMS, valid in vastly moregenerality than we can currently manage, will therebyemerge.
I especially look forward to the weaving together of dif-ferent threads of mirror symmetry, integrating thesymplectic-analytic-algebraic picture with the Riemanniangeometry of special Lagrangians; and the topologicalfield theory of the A- and B-models with rigorous ap-proaches to a quantum field theory on ๏ฟฝฬ๏ฟฝ [3, 14]. In thisaccount I have not even touched on mirror symmetry forFano manifoldsโwhich is just as remarkable as for CYmanifoldsโnor on wall-crossing, applications of mirrorsymmetry in symplectic topology, or connections to theLanglands program. Formathematicians fascinated by hid-den connections, mirror symmetry is a dazzling phenom-enon.
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symmetry. Proceedings of the International Congress ofMathematiciansโSeoul 2014. II, 813โ836, Kyung Moon Sa,Seoul, 2014. MR3728639
[2] Abouzaid M, Homological mirror symmetry without cor-rections. ArXiv:1703.07898
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Topics, Progress in Mathematics, 160 (1998), Birkhรคuser,Boston, MA MR1653024
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Credits
All images are courtesy of Timothy Perutz.
492 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4