Singularities, Mirror Symmetry, and the Gauged Linear Sigma Model (Contemporary Mathematics)

218 Pages·2021·18.231 MB·English

by  Tyler J. Jarvis, Nathan Priddis, Jarvis (editor), Tyler J. (editor), Priddis (editor), Nathan (editor)

763
Singularities, Mirror Symmetry,
and the Gauged Linear Sigma
Model
Summer School
Crossing the Walls in Enumerative Geometry
May 21 - June 1, 2018
Snowbird, Utah
Tyler J. Jarvis
Nathan Priddis
Editors
Singularities, Mirror Symmetry,
and the Gauged Linear Sigma
Model
Summer School
Crossing the Walls in Enumerative Geometry
May 21 - June 1, 2018
Snowbird, Utah
Tyler J. Jarvis
Nathan Priddis
Editors

763
Singularities, Mirror Symmetry,
and the Gauged Linear Sigma
Model
Summer School
Crossing the Walls in Enumerative Geometry
May 21 - June 1, 2018
Snowbird, Utah
Tyler J. Jarvis
Nathan Priddis
Editors
EDITORIAL COMMITTEE
Dennis DeTurck, Managing Editor
Michael Loss Kailash Misra Catherine Yan
2020 Mathematics Subject Classification. Primary 14J33, 14N35, 14J32, 14B05, 14D23,
14H10; Secondary 53D37, 53D45, 32Q25, 32S05, 32S30.
Library of Congress Cataloging-in-Publication Data
LibraryofCongressCataloging-in-PublicationData
Names: Jarvis, Tyler Jamison, editor. | Priddis, Nathan C., 1983- editor. | Summer School on
CrossingtheWallsinEnumerativeGeometry(2018: Snowbird,Utah)
Title: Singularities, mirror symmetry, and the gauged linear sigma model : Summer School on
Crossing the Walls in Enumerative Geometry, May 21-June 1, 2018, Snowbird, Utah / Tyler
J.Jarvis,NathanPriddis,editor.
Description: Providence, Rhode Island : American Mathematical Society, [2021] | Series: Con-
temporary mathematics, 0271-4132 ; volume 763 | “This volume contains lecture notes from
several [Summer School] short courses as well as some additional papers on current topics”–
Preface. |Includesbibliographicalreferences.
Identifiers: LCCN2020042993|ISBN9781470457006(paperback)|ISBN9781470464196(ebook)
Subjects: LCSH:Gromov-Witteninvariants–Congresses. |Singularities(Mathematics)–Congresses.
|Mirrorsymmetry–Congresses. |Intersectiontheory(Mathematics)–Congresses. |Calabi-Yau
manifolds–Congresses. |Modulitheory–Congresses. |AMS:Algebraicgeometry–Surfacesand
higher-dimensionalvarieties–Mirrorsymmetry. |Algebraicgeometry–Projectiveandenumer-
ative geometry – Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invari-
ants, Donaldson-Thomas invariants. | Algebraic geometry– Surfaces and higher-dimensional
varieties–Calabi-Yaumanifolds. |Algebraicgeometry–Localtheory–Singularities. |Alge-
braic geometry – Families, fibrations – Stacks and moduli problems. | Algebraic geometry –
Curves–Families,moduli(algebraic). |Differentialgeometry–Symplecticgeometry,contact
geometry| Differential geometry– Symplectic geometry, contact geometry | Several complex
variables and analytic spaces – Complex manifolds – Calabi-Yau theory. | Several complex
variablesandanalyticspaces–Singularities–Localsingularities.
Classification: LCCQA665.S562021|DDC516.3/6–dc23
LCrecordavailableathttps://lccn.loc.gov/2020042993
ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online)
DOI:https://doi.org/10.1090/conm/763
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Contents
Preface
Yongbin Ruan and Nathan Priddis vii
Quasimaps and some examples of stacks for everybody
Rachel Webb 1
Introduction to the gauged linear sigma model
Emily Clader 19
Localization and mirror symmetry
Dustin Ross 43
A brief introduction to cosection localization and P-fields
Wei-Ping Li 67
Virtual classes for hypersurfaces via two-periodic complexes
Mark Shoemaker 89
Localized Chern characters for 2-periodic complexes and virtual cycles
Jeongseok Oh 117
Singularity theory and mirror symmetry
Todor Milanov 131
Counting points with Berglund–Hu¨bsch–Krawitz mirror symmetry
Ursula Whitcher 155
Variations on the theme of quantum Lefschetz
Honglu Fan and Yuan-Pin Lee 171
Type II extremal transitions in Gromov–Witten theory
Rongxiao Mi 183
A Lecture on Holomorphic Anomaly Equations and Extended Holomorphic
Anomaly Equations
Chiu-Chu Melissa Liu 191
v

Preface
As part of activities supported by a National Science Foundation FRG grant
(DMS-1564502), a summer school was held at Snowbird, Utah, in 2018, offering an
array of short topics courses for graduate students in the general area of Gromov–
Witten theory and enumerative geometry. We felt that it would help graduate
studentstoaccessabroadareaofadvancedtopicswhichtheymaynothavethrough
the normal graduate course offerings of their math departments. This volume
containslecturenotesfromseveraloftheseshortcoursesaswellassomeadditional
papers on current topics.
SeveralyearsearlieranothersummerschoolwasheldinColorado,providingan
extensive treatment of the more basic topics in Gromov–Witten theory. Building
on the success of Colorado summer school, the focus of current summer school was
tocapturethemorerecentdevelopmentsofthesubject, suchasquasimapsandthe
gauged linear sigma model.
The fundamental object of Gromov–Witten theory is the notion of stable map
f : C → X from a nodal curve C to a target X which is a K¨ahler or symplectic
manifold. Butitisverydifficulttodoanycomputationforageneraltarget. During
last thirty years, nearly all of the computations that have been done are for when
X is a complete intersection in a toric variety. Recently, there has been a strong
pushtoextendthecalculationstocompleteintersectionsofnonabeliantargetssuch
as Grassmannians or flag varieties. In all these cases, the target X is embedded in
anambient spaceoftheformofaGITquotientV// G,whereV isarepresentation
θ
of a reductive group G and θ is a polarization.
It was known classically that when we vary the polarization θ, the quotient
V// G undergoes a certain birational transformation called a wall crossing. In this
θ
setting, Gromov–Witten theory can be described as what is known as a gauged
theory. In 1992, Witten proposed a physical model exactly for this situation. He
named his model the gauged linear sigma model or GLSM. During the next twenty
years, the GLSM became popular in physics, and was studied by many physicists.
In retrospect, it is surprising that is has taken so long to get into mathematics.
Let me explain the basic set-up for a GLSM. Take the famous quintic 3-fold
defined by
(cid:2) (cid:4)
(cid:3)
X = [x ,x ,x ,x ,x ]∈P4| x5 =0 .
5 0 1 2 3 4 i
i
Instead of P4, the GLSM target is the total space of the GIT-quotient
(C5−(0,0,0,0,0,0))×C// C∗ =O(−5)
θ
vii
viii PREFACE
over P4, together with a superpotential
(cid:3)
W =p x5.
i
i
In this setting X is interpreted as the critical locus
5
Crit ={x|dW =0}.
W x
Ingeneral,itisalwaysagoodideatoexpresstheobjectofinterestasthecritical
locus of a holomorphic function. An additional key ingredient is an R-charge—a
C∗-action such that W has degree one with respect to this action. The GLSM
can be thought as the quantization of the preceding picture by a 2d quantum field
theory.
The collection of lecture notes in this book explains many ingredients of the
GLSM as well as some related theories. Following is a short description for each
article that the reader can take as a guide.
(1) Rachel Webb. Quasimaps and some examples of stacks for ev-
eryone.
The mathematical GLSM is built upon quasimap theory. This ar-
ticle is an introduction to quasimaps, beginning with the motivation of
allowing basepoints in maps. It also includes a discussion on stack quo-
tients of a reductive group acting on an affine variety and GIT quotients
(as orbifolds). The main goal is to introduce the moduli space of sta-
ble quasimaps together with its virtual cycle. The article ends with a
discussion of nonabelian targets.
(2) Emily Clader. Introduction to the gauged linear sigma model.
Thisarticlebeginswithadiscussionoforbifolds,orbifoldlinebundles,
and orbifold cohomology. It introduces p-fields and variation of GIT and
shows how Gromov–Witten theory and certain other curve-counting the-
oriescanberealizedasaGLSM,endingwithadiscussionofwall-crossing
formulas.
(3) Dustin Ross. Localization and mirror symmetry.
The GLSM contains both Gromov-Witten theory and FJRW theory
as special cases. The relation between them is described by the variation
of polarization of the ambient GIT quotient. This article gives a proof of
mirror symmetry statements for the quintic, using localization. It intro-
duces FJRW theory and Gromov–Witten theory, shows how to construct
theJ-functionsandI-functionsforthesetheoriesandthenprovesthemir-
ror theorems for these two theories.
(4) Wei-Ping Li. A brief introduction to cosection localization and
p-fields.
There are two motivating examples for the GLSM: Chang-Li’s the-
ory of p-fields and Clader’s theory of the hybrid model. In particular,
Chang-Li introduced a fundamental technique called cosection localiza-
tion, originating from Kiem and Li, to construct a virtual cycle in p-field
theory. This technique is also very important for the GLSM. This article
gives a good introduction to this topic as well as the topic of p-fields, us-
ing basic examples as motivation and for understanding the construction.
PREFACE ix
Themain goal ofthisarticleis toshow how thevirtual fundamental class
can be constructed using these techniques.
(5) Mark Shoemaker. Virtual classes via matrix factorizations.
The cosection localization technique only constructs the virtual class
in the case of narrow or compact-type insertions. This article describes
the construction of the virtual class by 2-periodic complexes using the
exampleofthezerosectionofavectorbundle. Themaingoalofthisarticle
is to present the ideas of the construction of the virtual class for the case
of broad insertions using an algebraic method (an analytic approach was
carriedoutbyadifferentgroup). Itshouldbereadasapreparatoryarticle
to the more advanced article by Shoemaker and his coauthors detailing
the algebraic construction of the virtual class.
(6) Jeongseok Oh. Localized Chern characters for 2-periodic com-
plexes and virtual cycles.
Thisarticlereviewstheconstructionofavirtualfundamentalclassvia
a perfect obstruction theory. It describes the virtual fundamental class
vialocalized Chern characters and compares the classical virtual class for
the quintic to the virtual class constructed this way.
(7) Todor Milanov. Singularity theory and mirror symmetry.
This article gives an introduction to Frobenius manifolds and Given-
tal’s R-matrix. These topics are less directly connected to the GLSM,
but are essential in mirror symmetry. It details the construction of the
Frobenius manifold associated tothe Fermat hypersurfaces, including the
I-function, the mirror map, and analytic continuation on the Frobenius
manifold.
Theremainingarticlesaremoreresearchorientedandhavealooserconnection
to the GLSM. They should give readers a broader perspective of the subject.
(8) UrsulaWhitcher. CountingpointswithBerglund-Hu¨bsch-Krawitz
mirror symmetry.
Thisarticlegivesadifferentperspectivetomirrorsymmetry,namelyit
considersCalabi-YauvarietiesoverfinitefieldsandBHKmirrorsymmetry
in this setting. The main result is that if two polynomials are both dual
(underBHKmirrorsymmetry)totwopolynomialswiththesameweights,
thenthezetafunctionsshareacommonfactor. Furthermore, inthiscase,
the number of points on the hypersurfaces over a finite field of order q on
each variety is equivalent modulo q.
(9) HonglongFanandY.P.Lee. Variationsonthethemeofquantum
Lefschetz.
This paper discusses several recursions that allow one to compute
Gromov–Witten invariants via twisted invariants of some master space,
reframing the computations in terms of Givental’s formalism in order to
extend from the case of Calabi–Yau hypersurfaces to more general tar-
gets. It explores the recursions one obtains by varying the master space.
Finally, it considers the case where the Gromov–Witten theory of the
hypersurfaces is trivial (such as for a K3 surface), and thereby obtains
interesting relations on the Gromov–Witten theory of the ambient space.
(10) Rongxiao Mi. Extremal transitions and quantum D-modules.