Homological Mirror Symmetry and Tropical Geometry

445 Pages·2014·6.67 MB·English

by  Ricardo Castano-Bernard, Fabrizio Catanese, Maxim Kontsevich, Tony Pantev, Yan Soibelman, Ilia Zharkov (eds.)

Lecture Notes of the Unione Matematica Italiana
Ricardo Castano-Bernard
Fabrizio Catanese
Maxim Kontsevich
Tony Pantev
Yan Soibelman
Ilia Zharkov Editors
Homological Mirror
Symmetry and
Tropical Geometry
Lecture Notes of 15
the Unione Matematica Italiana
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TheEditorialPolicycanbefoundatthebackofthevolume.
Ricardo Castano-Bernard Fabrizio Catanese
(cid:2) (cid:2)
Maxim Kontsevich Tony Pantev
(cid:2) (cid:2)
Yan Soibelman Ilia Zharkov
(cid:2)
Editors
Homological Mirror
Symmetry and Tropical
Geometry
123
Editors
RicardoCastano-Bernard FabrizioCatanese
MathematicsDepartment MathematischesInstitut
KansasStateUniversity UniversitätBayreuth
Manhattan,KS,USA Bayreuth,Germany
MaximKontsevich TonyPantev
InstitutdesHautesEtudesScientifiques MathematicsDepartment
Bures-sur-Yvette,France UniversityofPennsylvania
Philadelphia,PA,USA
YanSoibelman
IliaZharkov
DepartmentofMathematics
KansasStateUniversity
Manhattan,KS,USA
ISSN1862-9113
ISBN978-3-319-06513-7 ISBN978-3-319-06514-4(eBook)
DOI10.1007/978-3-319-06514-4
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014950803
MathematicsSubjectClassification(2010):14J33,53D37,14T05,14N35,14D24
(cid:2)c SpringerInternationalPublishingSwitzerland2014
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Introduction
The workshop “Mirror Symmetry and Tropical Geometry” took place in Cetraro,
Italy on July 2–8, 2011. The idea was to bring together mathematicians and
physicistswhoworkedonbothtopicsorinrelatedareas.
HomologicalMirrorSymmetry,hereabbreviatedasHMS,istheareaofmathe-
maticsrevolvingaroundseveralcategoricalequivalencesconnectingsymplecticand
holomorphic(oralgebraic)geometry.ThismathematicalapproachtoMirrorSym-
metrygoesbacktotheworkofMaximKontsevich(1993).Furtherdevelopmentsof
Kontsevich’sprogramwasthesubjectofmanytalksattheworkshop.Thisthemeis
thereforepresentinseveralpapersofthisvolume.
WorksrelatedtoHomologicalMirrorSymmetryincludethepaperonHMSfor
Landau–GinzburgmodelsbyH.Ruddat,thepaperofN.SibillaonHMSforcurves,
the paper of Kontsevich and Y. Soibelman on complex integrable systems, and
the paper by D. Favero, F. Haiden, and L. Katzarkov on the phantom categories
which appear in HMS. The variety of methodsrangingfrom homologicalalgebra
todelicatequestionsofsymplectictopologyandalgebraicgeometryillustratesthe
complexityofthesubject.
The second topic of the workshop was Tropical Geometry. Roughly speak-
ing, Tropical Geometry studies piecewise-linear objects which appear as certain
“degenerations”ofthecorrespondingalgebro-geometricobjects.Therelationshipof
TropicalGeometrywithMirrorSymmetrygoesbacktotheworkbyKontsevichand
Y. Soibelman (2000)where methodsof non-archimedeangeometry (in particular,
tropical curves) were used for the purposes of Homological Mirror Symmetry.
Combined with the subsequent work of Mikhalkin on a “tropical” approach to
Gromov–Witten theory and with the work of Gross, Siebert, and several others,
TropicalGeometryhasbecomeausefultoolforpeopleworkinginMirrorSymme-
try.
On the other hand, “tropical” analogs of many notions of classical symplectic
and algebraic geometry are interesting and nontrivial objects by themselves. The
paperbyG.MikhalkinandI.Zharkov,whichisdevotedtothetropicalanalogofthe
intermediateJacobian,isa goodillustrationofthisstatement. Methodsoftropical
v
vi Introduction
geometryarealsousedinthepaperbyKontsevichandY.Soibelmandevotedtothe
studyofDonaldson–Thomasinvariantsandcorrespondingwall-crossingformulas.
The volume also contains several papers which are related to the main topics
of the workshop in an indirect way. For example, the paper by S. Guillermou
and P. Schapira is devoted to the application of the microlocal theory of sheaves
developed by the second author jointly with M. Kashiwara to the displaceability
problem in symplectic topology. It should be compared with attempts of several
mathematicians to describe the Fukaya category (one of the main objects on the
“symplectic” side of Homological Mirror Symmetry) in terms of constructible
sheavesandcorrespondingdg-categories.
Two papers are devoted to various aspects of the moduli stacks of bundles. In
the paper by O. Ben-Bassat and E. Gasparim the stack of vector bundles on a
formalneighborhoodofarationalcurveinasurfaceisstudied.InthepaperbyA.
Soibelmanthe“verygood”propertyintroducedbyBeilinsonandDrinfeldintheir
work on the Geometric LanglandsProgram is generalized to the case of arbitrary
parabolic bundles on a curve and then applied to the additive Deligne–Simpson
problem.
A. Neitzke gives a nice review of his joint work with D. Gaiotto and G.
Mooreon the constructionof hyperkählermetrics. Theirapproachisbased onthe
thermodynamicalBetheAnsatz-typeintegralequationproposedbythem,aswellas
onthe“Kontsevich–Soibelmanwall-crossingformulas”.Therearemanyinteresting
andnontrivialanalogiesbetweenthepaperbyNeitzkeandthepaperbyKontsevich
andY.Soibelmaninthisvolume.
S.GukovandP.Sulkowskiproposeawaytoquantizespectralcurves.Thenthey
discuss the relationship of arising “quantum spectral curves” with the topological
recursionofEynard–OrantinaswellaswithothertopicssuchasA-polynomialsof
knots.
The paper by M. Kapranov, O. Schiffmann, and E. Vasserot is devoted to the
Hallalgebraofthe“compactifiedSpec.Z/”interpretedasacurve.The“categoryof
vectorbundles”onsuchanobjectisdescribedinArakelovterms,asthecategoryof
metrizedlattices.The(spherical)Hallalgebraofthiscategoryisashufflealgebra,
similar to Hall algebras of the corresponding categories for “usual” curves. The
relationsinthealgebraaredescribedintermsforthe(full)zeta-function.
Webelievethatthepresentvolumerepresentsarathercompleteupdateaboutthe
stateoftheartinthefield,andwehopethatitshallbecomeanimportantreference
for graduate students and researchers who want to enter this exciting new field.
Papersin thisvolumerepresenta tiny portionof the varietyof topicsdiscussed at
the workshop.In order to give to the reader an idea aboutthe latter we finish the
IntroductionwiththelistoftalkspresentedattheCetraroworkshop.
Introduction vii
Acknowledgement ofSupport
ThisprojectwaspartiallysupportedbytheNSFFocusResearchGroupawardDMS-
0854989“MirrorSymmetryandTropicalGeometry”.

Lectures
(cid:129) Mina Aganagic(Berkeley):KnotHomologyfromRefined Chern–SimonsThe-
ory.
(cid:129) Fedor Bogomolov (NYU): On rationality of the fields of invariants of linear
actionsforconnectednonsemisimplealgebraicgroups(basedonmyjointwork
withChristianBoehningandHans-ChristianGrafvonBohtmer).
(cid:129) Fabrizio Catanese (Bayreuth):Special Galois coveringsand the singular set of
themodulispaceofcurves
(cid:129) Alexander Efimov (Steklov Institute): Cohomological Hall algebra and Kac’s
conjecture
(cid:129) Vladimir Fock (Strasbourg): Integrable systems, dimers and cluster varieties.
(JointworkwithAMarshakov)
(cid:129) KenjiFukaya(Kyoto):HomologicalMirrorsymmetryoftoricmanifolds
(cid:129) AlexanderGoncharov(Yale):Dimersandclusterintegrability
(cid:129) MarkGross(UCSD):Examplesofstablelogmapsandtropicalgeometry
(cid:129) Sergei Gukov (Caltech): From hyperholomorphic sheaves to quantum group
invariantsviaLanglandsduality
(cid:129) IliaItenberg(Strasbourg):Topologyofrealtropicalhypersurfaces
(cid:129) MikhailKapranov(Yale):ArithmeticHallalgebras
(cid:129) LudmilKatzarkov(MiamiandVienna):Degenerationsandwallcrossings
(cid:129) Viatcheslav Kharlamov (Strasbourg): Anti-symplectic involutions on rational
symplectic4-manifolds
(cid:129) MaximKontsevich(IHES):Integrablesystemsandcanonicalbases
(cid:129) AndreiLosev(ITEP):Homotopicalbeta-functionoftheinstantonicsigma-model
andbosonicstringEinsteinequationonschemes
(cid:129) DiegoMatessi(UofMilan):Conifoldtransitionsandtropicalgeometry
(cid:129) David Morrison (UCSB): Mirror symmetry and non-complete-intersection
Calabi–Yaumanifolds
(cid:129) AndyNeitzke(UTAustin):A2d-4dwall-crossingformula
(cid:129) NikitaNekrasov(IHES):SurpriseswithfourdimensionalND2gaugetheories
(cid:129) DimitriOrlov(Steklov):Mirrorsymmetry,B-branesandstrangeArnoldduality
(cid:129) TonyPantev(UPenn):MirrorsymmetryandmixedHodgestructures
ix