Introduction to the gauged linear sigma model

Emily Clader

Abstract. We describe the definition of the gauged linear sigma model (GLSM),

focusing specifically on Fan–Jarvis–Ruan–Witten theory and its generaliza-

tion, the hybrid model. These theories are related, via the celebrated Landau–Ginzburg/Calabi–Yau (LG/CY) correspondence, to the Gromov–Witten the-

ory of complete intersections in weighted projective space. We discuss how the

LG/CY correspondence motivates the meaning of the GLSM, and conversely,how a GLSM perspective can be used to prove the correspondence in many

cases.

The goal of these lectures is to motivate the definition of the gauged linearsigma model, or GLSM, a generalization of Gromov–Witten theory as well as other(perhaps less familiar) objects such as Fan–Jarvis–Ruan–Witten (FJRW) theory. Inparticular, we show that, from a GLSM viewpoint, the definition of FJRW theoryis an entirely natural analogue of the definition of Gromov–Witten theory. Thecelebrated Landau–Ginzburg/Calabi–Yau correspondence relates these two theoriesto one another, and placing both in the more general context of the GLSM hintsat a method by which this correspondence could be proven. In the last lecture,we discuss the proof of the Landau–Ginzburg/Calabi–Yau correspondence in genuszero, and we mention ongoing work that points toward a proof in higher genus.

1. Preliminaries on orbifolds and Gromov–Witten theory

Before beginning our discussion of the GLSM, we must recall some basics aboutorbifolds and their cohomology, as well as the definition of Gromov–Witten theoryand quantum cohomology.

1.1. What is an orbifold? Roughly speaking, an orbifold is a space locallyhomeomorphic to the quotient of an open subset of Euclidean space by the actionof a finite group. More precisely, analogously to the definition of a manifold, onecan define:

Definition 1.1. Let X be a topological space. An orbifold chart on X consistsof:

(1) an open set U ⊆ X;

(2) an open set U ⊆ Rn for some n;

(3) a smooth action of a finite group G on U ;

2000 Mathematics Subject Classification. Primary 14H81.

1

2 EMILY CLADER

(4) a continuous, G-invariant map φ : U → U such that the induced map

φ : U/G→ U

is a homeomorphism.

From here, one must prescribe what it means for two overlapping orbifold chartsto be compatible. This is surprisingly subtle, and we encourage the careful readerto attempt to write down a reasonable definition for herself before reading further.

Definition 1.2. We say that an orbifold chart

V /Hψ−→ V ⊆ X

is a subchart of an orbifold chart

U/Gφ−→ U ⊆ X

if V ⊆ U and there is a group homomorphism τ : H → G and a smooth embedding

ι : V ↪→ U such that:

(1) ι(h · x) = τ(h) · ι(x) for all x ∈ V and h ∈ H, where · denotes the group

actions of H on V and G on Y ;

(2) ψ(x) = φ(ι(x)) for all x ∈ V ;

(3) for each x ∈ V , the stabilizer Hx is isomorphic to the stabilizer Gι(x).

Given two orbifold charts

U1/Gφ1−→ U1 ⊆ X and U2/G

φ2−→ U2 ⊆ X,

we say that these charts are compatible if every point of U1 ∩U2 is contained in anopen set U3 for which

U3/Gφ3−→ U3 ⊆ X

is a subchart of both of the two original charts.

Having defined compatibility of charts, we can finally define:

Definition 1.3. A (smooth) orbifold X is a topological space X together witha family of compatible charts covering X.

The key point is that, although U is homeomorphic to U/G, the orbifold chart

remembers more than just the topological quotient U/G: it remembers the actionof G, and specifically, where that action has isotropy. In particular, the definitionof compatibility of orbifold charts implies that each point x ∈ X has an isotropygroup that is well-defined up to isomorphism:

Definition 1.4. Let X be an orbifold and let x ∈ X. The isotropy group (or

stabilizer) of X is the stabilizer Gx of any point x ∈ U such that φ(x) = x, where

φ : U/G → U is any orbifold chart. (Choosing a different x in the same orbifold

chart yields an isomorphic stabilizer because φ is G-invariant, while choosing adifferent orbifold chart yields an isomorphic stabilizer by condition (3) of Definition1.2.)

Let us turn to some of the key examples of orbifolds.

INTRODUCTION TO THE GLSM 3

Example 1.5. Let the cyclic group Zn act on C via multiplication by nth rootsof unity. Then there is an orbifold, denoted

X = [C/Zn],

in which the underlying topological space is X = C and there is a single globalchart

φ : C→ C

φ(z) = zn.

In particular, φ descends to a continuous map φ : C/Zn → C whose inverse z 7→[z1/n] is indeed well-defined and continuous.

The isotropy group of any nonzero element x ∈ C is trivial, in this example,whereas the isotropy group of 0 ∈ C is Zn. As a result, one typically represents theorbifold [C/Zn] graphically as the space C with a copy of the group Zn attached tothe origin.

Example 1.6. Generalizing the above example, let M be any smooth manifoldand let G be a finite group acting smoothly on M . Then there is a “global quotient”orbifold

X = [M/G]

with underlying topological space X = M/G. To construct X as an orbifold, onemust use the fact that any x ∈ M for which the G-action has isotropy group Gxis contained in a (manifold) chart Ux ∼= Ux ⊆ Rn for which Ux is invariant underthe action of Gx. The orbifold charts then consist of the subsets Ux/Gx ⊆ M/G

with the quotient maps φ : Ux → Ux/Gx and the action of Gx on Ux induced by

the homeomorphism Ux ∼= Ux.

A special case of the above, which appears initially trivial but is surprisinglyrich, is the following:

Example 1.7. If M is a single point and G is any finite group (acting in theonly possible way on a point), we denote the resulting global quotient orbifold byBG.

The following example, on the other hand, is provably not (always) a globalquotient:

Example 1.8. Let C∗ act on Cn+1 by

λ · (z0, . . . , zn) = (λw0z0, . . . , λwnzn),

where (λ0, . . . , λn) are coprime natural numbers. Then there is an orbifold

X = P(w0, . . . , wn),

referred to as a weighted projective space, in which X is the topological quotient

X = (Cn+1 \ {0})/C∗.

To form the orbifold charts, first note that, as in the case of ordinary projectivespace, each x ∈ X is contained in one of the subsets

Ui = {[z0, . . . , zn] ∈ (Cn+1 \ {0})/C∗ | zi 6= 0} ⊆ X.

4 EMILY CLADER

Then Ui ∼= Cn, on which the coordinates are denoted (z0, . . . , zi−1, zi+1, . . . , zn)

and the map φi : Ui → Ui is

φi(z0, . . . , zi−1, zi+1, . . . , zn) = [z0 : · · · : zi−1 : 1 : zi+1 : · · · : zn].

The group Zwi of with roots of unity acts on Ui by

ζ · (z0, . . . , zi−1, zi+1, . . . , zn) = (ζw0z0, . . . , ζwi−1zi−1, ζ

wi+1zi+1, . . . , ζwnzn),

and it is straightforward to check that this action makes the above into an orbifoldchart.

Exercise 1.9. Consider the action of Z2 on the torus T = S1×S1 ⊆ C∗×C∗,where the nontrivial element of Z2 acts by

(eit1 , eit2) 7→ (e−it1 , e−it2).

Find an explicit atlas of orbifold charts on X = T/Z2. What, topologically, is X?

Exercise 1.10. Consider the action of the symmetric group S3 on P1×P1×P1

given by permuting the three coordinates; the resulting quotient is the symmetricproduct X = S3(P1). At which points does the action of S3 have nontrivial isotropy,and what is the isotropy group? Find an explicit atlas of orbifold charts on X, andconvince yourself that, topologically, X ∼= P3.

1.2. Orbifold vector bundles and a first pass at orbifold cohomology.In general, all of the geometric constructions one might associate to a manifold canbe extended appropriately to orbifolds. The philosophy, when defining the orbifoldanalogue of a manifold construction, is to apply the manifold construction on the

charts U and insist that it be equivariant with respect to the actions of the localgroups G.

This idea is easiest to make precise in the case when the orbifold is a globalquotient. For example, in that case, the orbifold analogue of a vector bundle canbe defined as follows:

Definition 1.11. Let X = [M/G] be a global quotient orbifold. Then anorbifold vector bundle on X consists of a vector bundle π : E → M together withan action of G on E that takes the fiber over each x ∈ M linearly to the fiberover gx. A section of an orbifold vector bundle on X is a section s of π that isequivariant with respect to the G-actions on E and M .

By carrying out the above construction in orbifold charts, this definition canbe extended to arbitrary orbifolds; we refer the reader to [1] for a careful definition.

Orbifolds also have their own version of the tangent bundle. In the globalquotient case, this is:

Definition 1.12. Let X = [M/G] be a global quotient orbifold. Then thetangent bundle to X , denoted TX , is the orbifold vector bundle defined by theordinary tangent bundle π : TM → M , on which G acts by the derivative of itsaction on M .

The typical operations on vector bundles extend to the orbifold setting, so,having defined the tangent bundle TX , one can define the cotangent bundle, andits wedge powers ∧kT ∗X , and sections of ∧kT ∗X—that is, differential k-forms.This leads to a definition of orbifold de Rham cohomology H∗dR(X ).

INTRODUCTION TO THE GLSM 5

Unfortunately, though, this definition of orbifold de Rham cohomology capturesnone of the information of the orbifold structure. More precisely, one has thefollowing theorem:

Theorem 1.13 (Satake). For any orbifold X with underlying topological spaceX, there is an isomorphism

H∗dR(X ) ∼= H∗(X;R).

So, for example, the orbifold de Rham cohomology of BG is simply the cohomologyof a point; it sees nothing of the group G.

The way out of this sad state of affairs was proposed by Chen and Ruan [4]:one should first define orbifold quantum cohomology, and then restrict to its degree-zero part to obtain a definition of orbifold cohomology that truly takes the orbifoldstructure into account.

Before discussing Chen–Ruan’s definition of orbifold cohomology, we digress torecall the basics of quantum cohomology in the non-orbifold setting.

1.3. Review of Gromov–Witten theory and quantum cohomology.Let X be a smooth projective variety, and for simplicity, fix an embedding i : X ↪→Pr. For any integers n, β ≥ 0, the moduli space of genus-zero stable maps to X isthe parameter space M0,n(X,β) of tuples (C; q1, . . . , qn; f), where:

• C is an algebraic curve of (arithmetic) genus zero with at worst nodalsingularities;

• q1, . . . , qn ∈ C are a collection of distinct smooth points of C;• f : C → X is a morphism for which i∗f∗[C] = β ∈ H2(Pr;Z) ∼= Z;• the data (C; q1, . . . , qn; f) is stable in the sense that it has finitely many

automorphisms, or equivalently, any irreducible component of C on whichf is constant has at least three “special points,” which are either markedpoints qi or nodes.

This moduli space is equipped with evaluation maps

evi :M0,n(X,β)→ X

evi(C; q1, . . . , qn; f) = f(qi).

From here, the idea of Gromov–Witten invariants is to choose n cohomology classesα1, . . . , αn ∈ H∗(X) and study

(1.1)

∫M0,n(X,β)

ev∗1(α1) · · · ev∗n(αn).

In particular, if α1, . . . , αn were Poincare dual to subvarieties Y1, . . . , Yn of X, then(1.1) would, naıvely, capture the number of points in ev−1

1 (Y1) ∩ · · · ev−1n (Yn)—in

other words, the number of degree-β, genus-zero curves passing through these nsubvarieties.

However, M0,n(X,β) can be a very singular space, and it can have differentcomponents of different dimensions, so it is in general unclear how to make sense ofthe integral (1.1) in a meaningful way. It is a deep and difficult foundational factthat, despite its singularity,M0,n(X,β) admits a pure-dimensional homology class

[M0,n(X,β)]vir that enjoys some of the properties that a fundamental class on asmooth space would have. Integrating against this virtual fundamental class givesthe precise definition of Gromov–Witten invariants:

6 EMILY CLADER

Definition 1.14. Let n, β ∈ Z≥0, and let α1, . . . , αn ∈ H∗(X). Then theassociated genus-zero Gromov–Witten invariant is

〈α1 · · ·αn〉X0,n,β :=

∫[M0,n(X,β)]vir

ev∗1(α1) · · · ev∗n(αn).

We are now prepared to define quantum cohomology in the variety setting:

Definition 1.15. Let q denote a formal parameter. The quantum cohomologyof X is the vector space H∗(X)[[q]] equipped with the product ∗ defined by(

α ∗ β, γ)

:=∑β≥0

qβ〈α β γ〉X0,3,β ,

where ( , ) is the Poincare pairing

(ω, ν) :=

∫X

ω ∪ ν.

The fundamental feature of this definition is that its degree-zero part recoversthe usual cohomology:

Exercise 1.16. Verify that setting q = 0 in the quantum product recoversH∗(X) with its usual cup product.

1.4. Orbifold cohomology. We would like to generalize the definition ofGromov–Witten invariants and quantum cohomology to orbifolds, but in order todo so carefully, one must define what is meant by a morphism of orbifolds. Thisdefinition is remarkably difficult to make precise; it can be defined via equivariantmorphisms in local orbifold charts, but the atlas of orbifold charts on the domainorbifold may need to be refined before doing so.

Rather than covering the careful definition of an orbifold morphism, we willcontent ourselves with one key example and one key fact:

Example 1.17. Let X = [M/G] and Y = [N/H] be global quotient orbifolds.Then one example of an orbifold morphism is a pair (f0, f1) consisting of a contin-uous map

f0 : M → N

and a homomorphismf1 : G→ H

such that f0 is equivariant with respect to f1.

Fact 1.18. Let X and Y be any orbifolds, and let f : X → Y be an orbifoldmorphism. Then f induces a map f : X → Y between the underlying topologicalspaces of X and Y, respectively, as well as a homomorphism of isotropy groups

λx : Gx → Gf(x)

for each x ∈ X.

Exercise 1.19. Verify that, in the special case of Example 1.17, one indeedobtains maps f and λx as in Fact 1.18.

The key upshot of Fact 1.18 is the following. Equipped with the notion ofan orbifold morphism, one can define a moduli space M0,n(X , β) of stable mapsf : C → X from genus-zero orbifold curves C to X . In the orbifold setting, though,there are two pieces of local data around each marked point qi:

INTRODUCTION TO THE GLSM 7

• the image f(qi) ∈ X , and• the homomorphism λqi .

Moreover, the isotropy groups of orbifold curves are in fact cyclic, and they have acanonical generator ζi induced by the orientation. Thus, the data of λqi is encodedin a single element λqi(ζi) of the isotropy group of X at f(qi).

As a result of these considerations, the natural evaluation maps onM0,n(X , β)land not in X itself but in a richer orbifold:

Definition 1.20. The inertia orbifold of X is

IX := {(x, g) | x ∈ X, g ∈ Gx}.

As its name suggests, the inertia orbifold can be given the structure of anorbifold; the description in Definition 1.20 is merely the underlying topologicalspace. We illustrate the orbifold structure only in a single example, which is themost important example for what follows. The reader is referred to [1] for a morecareful general definition.

Example 1.21. Let X = [C/Zr]. Then X = C, and any nonzero x ∈ C hastrivial isotropy whereas the isotropy group of 0 ∈ C is Zr. It follows that, on thelevel of the underlying space,

(1.2) I[C/Zr] = C t {0} t · · · t {0}.To give this the structure of an orbifold, let Zr act on the above by sending

(x, g ∈ Gx) 7→ (hx, hgh−1 ∈ Ghx)

for each h ∈ Zr. Because Zr is abelian, this action amounts to sending (x, g) to(hx, g), so it preserves the r components in (1.2). As an orbifold, then,

I[C/Zr] = [C/Zr] t BZr t · · · t BZr.

Having defined the inertia orbifold, the evaluation maps are

evi :M0,n(X , β)→ IX

evi(C; q1, . . . , qn; f) = (f(qi), λqi(ζi)).

From here, the definition of Gromov–Witten invariants carries over verbatim, butthe insertions α1, . . . , αn to these invariants are chosen from H∗(IX ) as opposedto H∗(X ). In particular, quantum cohomology in the orbifold setting is a productstructure on H∗(IX )[[q]].

In light of Exercise 1.16, one should expect the “ordinary” cohomology of orb-ifolds to be obtained from their quantum cohomology by setting q = 0. Ratherthan proving this as a theorem, one takes this result as the definition of orbifoldcohomology:

Definition 1.22. Let X be an orbifold. Then the orbifold (or Chen–Ruan)cohomology of X is the vector space

H∗CR(X ) := H∗(IX ),

where the right-hand side denotes the de Rham (or singular) cohomology of IX ,which is the same as the cohomology of its underlying topological space. The prod-uct structure on H∗CR(X ) is the q = 0 limit of the quantum product on H∗(IX )[[q]].

Exercise 1.23. What is H∗CR([C/Zr]), as a vector space?

8 EMILY CLADER

Exercise 1.24. Consider a weighted projective space X = P(w0, w1) withcoprime weights w0 and w1. Calculate the inertia orbifold and the orbifold coho-mology (as a vector space) of X . Although we have not explicitly defined weightedprojective spaces with non-coprime weights, what do you expect are the inertiaorbifold and the orbifold cohomology of P(3, 3)? Of P(3, 6)?

2. Fan–Jarvis–Ruan–Witten theory

Equipped with the background developed in the previous section, we are pre-pared to define one of our primary objects of study in these lectures: Fan–Jarvis–Ruan–Witten (FJRW) theory.

2.1. The structure of FJRW theory. Structurally, FJRW theory is analo-gous to the Gromov–Witten theory of a smooth projective variety X. In particular,it consists of:

(1) a vector space H (the “state space”), analogous to H∗(X) in Gromov–Witten theory;

(2) a moduli space M equipped with a virtual fundamental cycle, analogousto Mg,n(X,β) in Gromov–Witten theory;

(3) a notion of “correlators”, analogous to Gromov–Witten invariants, whichare integrals against the virtual cycle of M associated to any tuple ofelements in H.

The input data for FJRW theory, though, is not a variety X but a polynomialW ∈ C[x1, . . . , xN ]. This polynomial is required to be of a particular type:

Definition 2.1. A polynomial W ∈ C[x1, . . . , xN ] is quasihomogeneous if thereexist positive integers w1, . . . , wN (the weights) and d (the degree) such that

W (λw1x1, . . . , λwNxN ) = λdW (x1, . . . , xN )

for all λ ∈ C.

Note, in particular, that quasihomogeneity implies that the vanishing of W definesa hypersurface XW in the weighted projective space P(w1, . . . , wN ).

In addition to quasihomogeneity, in what follows, we require our polynomialsW to satisfy two further technical conditions:

(1) Nondegeneracy: The hypersurface XW ⊆ P(w1, . . . , wN ) is nonsingularas an orbifold;

(2) Invertibility: The number of monomials of W is equal to the numberof variables, and the exponent matrix (whose rows correspond to mono-mials in W and whose entries record the exponent of each variable in amonomial) is invertible.

A particularly simple example of a nondegenerate, invertible, quasihomoge-neous polynomial—to which we will return repeatedly in what follows—is:

Example 2.2. Let W (x1, . . . , x5) = x51 + · · ·+x5

5. This is a quasihomogeneouspolynomial with w1 = · · · = w5 = 1 and d = 5.

In the following subsections, we will define the ingredients of the FJRW theoryof W precisely. Before doing so, though, it is worth grounding the discussionby looking ahead to the ultimate goal: in Section 3, we present the Landau–Ginzburg/Calabi–Yau correspondence, which asserts an equivalence between the

INTRODUCTION TO THE GLSM 9

FJRW theory of W and the Gromov–Witten theory of XW in certain cases. Weencourage the reader to keep this correspondence in mind as motivation for thedefinitions that follow.

2.2. The state space of FJRW theory. Let W ∈ C[x1, . . . , xN ] be a (non-degenerate, invertible) quasihomogeneous polynomial with weights w1, . . . , wN anddegree d. Then we define:

Definition 2.3. The FJRW state space associated to W is the Q-vector space

HW := H∗CR([CN/J ],W+∞;Q),

where

J :=⟨(e2πi

w1d , . . . , e2πi

wNd

)⟩⊆ (C∗)N

acts diagonally on CN andW+∞ := W−1(ρ)

for any sufficiently large real number ρ. (The resulting vector space is independentof ρ for ρ� 0, because topologically, W−1(ρ) is eventually independent of ρ.)

We note that the action of J on CN is cooked up so that W (g~x) = W (~x) forall g ∈ J , so W gives a well-defined map out of the quotient CN/J . Furthermore,recalling that H∗CR(X ) := H∗(IX ) and IX = {(x, g ∈ Gx)}, the FJRW state spacecan more explicitly be expressed as

HW =⊕g∈J

H∗(CNg /J,W+∞;Q),

where CNg ⊆ CN is the fixed locus of g. The cohomology groups appearing inthe above are ordinary (for example, de Rham) cohomology groups, for which thecohomology of a quotient Y/G can be calculated as the G-invariant part of thecohomology of Y . Thus:

HW =⊕g∈J

H∗(CNg ,W+∞;Q)J ,

in which the superscript denotes the J-invariant part.

Example 2.4. Let W (x1, . . . , x5)x51 + · · ·+ x5

5. Then

J =⟨(e2πi 15 , . . . , e2πi 15

)⟩∼= Z5,

acting diagonally on C5 by multiplication by fifth roots of unity. Given that theaction of 1 ∈ J fixes all of C5 but the action of any nontrivial g ∈ J fixes only theorigin, we have

I[C5/J ] = [C5/Z5] t BZ5 t BZ5 t BZ5 t BZ5.

It follows that

HW = H∗(C5,W+∞;Q)Z5 ⊕⊕

g 6=1∈Z5

H∗({0}, ∅;Q) = H∗(C5,W+∞;Q)Z5 ⊕Q4.

The state space decomposes into two types of elements:

Definition 2.5. An element g ∈ J (or the corresponding component of HW )is called narrow if it fixes only 0 ∈ CN , meaning that the corresponding componentof HW is a copy of Q. Elements (or components) that are not narrow are calledbroad.

10 EMILY CLADER

For example, in Example 2.4, the element g = 1 is broad and all other g ∈ Jare narrow.

Exercise 2.6. Find an example of a quasihomogeneous polynomial for whichthere exists a nontrivial broad element g ∈ J .

2.3. The moduli space of FJRW theory. Let W be as in the previous

subsection. Then the moduli space of FJRW theory is the parameter space MW

g,n

of tuples (C; q1, . . . , qn;L;φ), where

• (C; q1, . . . , qn) is a genus-g, n-pointed orbifold stable curve;• L is an orbifold line bundle on C;• φ is an isomorphism

φ : L⊗d∼−→ ωlog,

where ωlog := ωC([q1] + · · ·+ [qn]),

subject to a certain stability condition on the actions of the isotropy groups on thefibers of L. The term “orbifold stable curve” here has a precise meaning that wewill not specify; in particular, though, it means that C has nontrivial isotropy onlyat special points.

Example 2.7. When W (x1, . . . , x5) = x51 + · · · + x5

5, the moduli space MW

g,n

is otherwise known as the moduli space of 5-spin structures and is often denoted

M1/5

g,n .

2.4. The correlators of FJRW theory. To define the correlators of FJRWtheory, we must refine the definition of the moduli space by decomposing it intocomponents depending on the “multiplicities” of the bundle L.

Definition 2.8. Let C be an orbifold curve, let L be an orbifold line bundleon C, and let q ∈ C be a point with isotropy group Gq = Zr. Then, locally nearq, the curve C looks like the global quotient [C/Zr], so the data of L consists of

a bundle L on C together with an action of Zr according to which the projection

map L → C is equivariant. In particular, Zr acts on the fiber of L over q. Themultiplicity of L at q is defined as the number

multq(L) := mq ∈ {0, 1, . . . , r − 1}

such that the action of Zr on L is

ζ · (x, v) =(e2πi 1r x, e2πi

mqr v)

in local coordinates around q.

In the case of MW

g,n, the stability condition ensures that all of the isotropygroups have order dividing d. Thus, one can view the multiplicities of L at anyspecial point as an element of {0, 1, . . . , d− 1}. For a tuple

m1, . . . ,mn ∈ {0, 1, . . . , d− 1},

we set

MW

g,(m1,...,mn) := {(C; q1, . . . , qn;L;φ) | multqi(L) = mi for all i} ⊆ MW

g,n.

INTRODUCTION TO THE GLSM 11

The key observation needed to define FJRW correlators, at least in the narrowcase, is that for any g ∈ J ,

g =(e2πim

w1d , . . . , e2πim

wNd

)for some m ∈ {0, 1, . . . , d − 1}. In particular, the components of the FJRW statespace can be indexed by the same set {0, 1, . . . , d− 1} that indexes the componentsof the FJRW moduli space.

More specifically, we define narrow correlators as follows:

Definition 2.9. Decompose the FJRW state space as

HW =⊕

g broad

H∗(CNg ,W+∞;Q)J ⊕⊕

g narrow

Q{eg},

where eg denotes the fundamental class on the component of HW indexed by g.Choose n narrow elements

α1, . . . , αn ∈ HW ,and write αi = ci · egi for ci ∈ Q and

gi =(e2πimi

w1d , . . . , e2πimi

wNd

),

where mi ∈ {0, 1, . . . , d− 1}. Then we define the associated FJRW correlator as

〈α1 · · ·αN 〉Wg,n = c1 · · · cn∫

[MWg,(m1,...,mn)]

vir

1.

Two difficult fundamental facts are needed in order to make this definitioncomplete. First, there indeed exists a virtual cycle [MW

g,(m1,...,mn)]vir against which

the above integral can be defined; we will say some more words about how thiscycle is constructed in later sections of these notes. And second, the definition ofcorrelators can be extended to broad insertions. This second fact is harder, andwe will not address it here; for the cases of interest below, narrow correlators aresufficient.

We conclude this section with several exercises to practice with the notion ofmultiplicity:

Exercise 2.10. Let (C; q1, . . . , qn) be a smooth orbifold curve at which eachmarked point qi has isotropy group Zd, and let L be an orbifold line bundle on C.A fundamental fact about the multiplicities of L is that the bundle

|L| := L⊗O

(−

n∑i=1

mi

d[qi]

)is pulled back from the coarse underlying curve C. Use this fact to explain why themultiplicities must satisfy

(2.1)

n∑i=1

mi ≡ 0 mod d.

Exercise 2.11. Generalizing to nodal curves of compact type, convince your-self that the multiplicity of L at each branch of each node is determined by (2.1)together with the “kissing condition” that m′+m′′ ≡ 0 mod d for the multiplicitiesm′,m′′ at opposite branches of the same node. What happens if the curve is notof compact type—in other words, if it has a non-separating node?

12 EMILY CLADER

Exercise 2.12. Work out the “fundamental fact” from Exercise 2.10 explicitlyin local coordinates, by considering the diagram

L = [(C× C)/Zd]

((QQQQQ

QQQQQQ

QQ[x,v]7→[x,x−mv] // [(C× C)/Zd] = |L|

vvllllll

llllll

l

C = [C/Zd] ,

where L is defined by the action ζ(x, v) = (ζx, ζmv) and |L| is defined by the actionζ(x, v) = (ζx, v) of Zd on C× C. (See [19, Section 2.1.4] for related discussion.)

3. The Landau–Ginzburg/Calabi–Yau correspondence

Two questions would be reasonable for the reader to ask at this point: wheredo all of these FJRW theory definitions come from, and what does all of this haveto do with the Gromov–Witten theory of the hypersurface defined by W?

The answers to both of these questions come from putting FJRW theory in thecontext of the Landau–Ginzburg/Calabi–Yau (LG/CY) correspondence, which wedescribe in this section. In order to tell that story in a more readable form, werestrict our attention to the case where

W = W (x1, . . . , x5) = x51 + · · ·+ x5

5,

whose vanishing locus is the quintic threefold Q ⊆ Pr. We describe, in this case,an isomorphism between the state spaces H∗(Q) and HW of Gromov–Witten andFJRW theory, respectively, and a “matching” between the generating functions ofgenus-zero Gromov–Witten and FJRW correlators under a particular identificationof the state spaces.

3.1. The state space correspondence. The first step in the LG/CY corre-spondence is to explain why the state spaces H∗(Q) and HW are isomorphic. Todo so, we introduce a larger space out of which both of these can be built.

Consider the space C5 × C with coordinates (x1, . . . , x5, p), and let C∗ act onthis space by

λ(x1, . . . , x5, p) = (λx1, . . . , λx5, λ−5p).

Then the polynomial

W (x1, . . . , x5, p) = p(x51 + · · ·+ x5

5)

is C∗-invariant, so it gives a well-defined map out of the quotient (C5 × C)/C∗.This quotient is not separated, but it admits two maximal separated subquo-

tients; in the language of geometric invariant theory, these are the GIT quotients(C5 × C) �θ C∗ associated to the positive and negative characters θ ∈ χ(C∗) ∼= Z.Namely, they are:

X+ :=(C5 \ {0})× C

C∗= OP4(−5)

and

X− :=C5 × (C \ {0})

C∗= [C5/Z5].

INTRODUCTION TO THE GLSM 13

If one calculates the orbifold cohomology of X− relative to a fiber W+∞

ofW : X− → C, the result, by definition, is the FJRW state space HW . On the other

hand, the (orbifold) cohomology of X+ relative to a fiber W+∞

of W : X+ → C is

H∗(OP4(−5),W+∞

) ∼= H∗(P4,P4 \Q) ∼= H∗(Q),

where the first isomorphism follows from the deformation retraction of OP4(−5)

onto P4 (which carries W+∞

bijectively onto P4 \Q) and the second isomorphismis the Thom isomorphism.

Via this argument, the state spaces of Gromov–Witten and FJRW theory canbe viewed as arising in completely analogous ways from the larger quotient (C5 ×C)/C∗. In fact, this framework can be leveraged—with the help of a number ofexact sequences—to prove that the two state spaces are isomorphic:

H∗(Q) ∼= HW .

This is the first piece of the LG/CY correspondence.

Exercise 3.1. Show, furthermore, that there is an isomorphism between theambient part of the cohomology of H∗(Q)—that is, the cohomology classes pulledback from P4—and the narrow part of HW .

3.2. The moduli spaces of Gromov–Witten and FJRW theory. Thesecond, more substantial, piece of the correspondence is a comparison of correlators,and in particular, of the moduli spaces and their virtual cycles.

The basic structure of this comparison is modeled on the argument for thestate space correspondence explained above; namely, just as the state space cor-respondence arose by placing both H∗(Q) and HW in the context of a larger butbadly-behaved quotient (C5 × C)/C∗, the moduli comparison begins by finding a

larger but badly-behaved moduli space out of which Mg,n(Q, β) and MW

g,n canboth be built.

That larger moduli space, denoted Xg,n,β , parameterizes tuples

(C; q1, . . . , qn;L, x1, . . . , x5, p),

in which (C; q1, . . . , qn) is again a genus-g, n-pointed orbifold stable curve, L anorbifold line bundle of degree β on C, and

(x1, . . . , x5, p) ∈ Γ(L⊕5 ⊕ (L⊗−5 ⊗ ωlog)

),

subject to certain stability and compatibility conditions that we will not makeexplicit here.

Just as in the case of the quotient (C5 × C)/C∗, this moduli space is non-separated; furthermore, it is non-compact (as the sections can be scaled by anycomplex number) and not Deligne–Mumford (as scaling the line bundle leads toinfinitely many automorphisms). Nevertheless, it contains two separated Deligne–Mumford substacks, X+

g,n,β and X−g,n,β , by imposing that either ~x or p is nowhere-vanishing.

When ~x is nowhere-vanishing, it is equivalent to the data of a map f : C → P4,and the stability condition on Xg,n,β implies that this map is stable. Furthermore,from this perspective, we have L = f∗O(1), so

X+g,n,β = {(C; q1, . . . , qn; f ; p) | f : C → P4 stable, p ∈ Γ(f∗O(−5)⊗ ωlog)}.

14 EMILY CLADER

On the other hand, when p is nowhere-vanishing, it trivializes L⊗−5 ⊗ ωlog, so

X−g,n,β = {(C; q1, . . . , qn;L;x1, . . . , x5;ϕ) | xi ∈ Γ(L), ϕ : L⊗−5 ∼= ωlog}.These two moduli spaces, while separated and Deligne–Mumford, are still not

compact, since each still includes an unconstrained section of a bundle. Analogouslyto the way in which we passed, in the state space correspondence, to cohomology

relative to the fiber W+∞

, we restrict on the moduli level to the loci Z±g,n,β ⊆ X±g,n,β

on which the section (x1, . . . , x5, p) lands in the critical locus

Crit(W ) ⊆ C5 × C.More specifically:

Exercise 3.2. Viewing W as a function on X+ = OP4(−5), verify that thecritical locus of W is the quintic threefold Q inside the zero section P4 ⊆ X+.Similarly, viewing W as a function on X− = [C5/Z5], verify that the critical locusof W is [{0}/Z5].

In particular, then, we have

Z+g,n,β = {(C; q1, . . . , qn; f ; p) | f : C → Q stable, p = 0} =Mg,n(Q, β),

which is the moduli space of Gromov–Witten theory, whereas

Z−g,n,β = {(C; q1, . . . , qn;L;x1, . . . , x5;ϕ) | x1 = · · · = x5 = 0, ϕ : L⊗−5 ∼= ωlog} =MW

g,n,

which is the moduli space of FJRW theory.We have thus succeeded in constructing the moduli spaces of the two theories in

entirely analogous ways. In fact, this framework can be upgraded to give a uniformconstruction of the virtual fundamental cycles on the two moduli spaces, via thecosection construction of Kiem–Li [20]. The basic idea, which was first applied tothis setting by Chang–Li–Li [2, 3], is thatX±g,n,β has a natural deformation theory—and hence, a natural virtual fundamental cycle—coming from the cohomology ofL⊕5⊕(L⊗−5⊗ωlog), and the cosection construction is designed to produce a virtualfundamental cycle supported on a critical locus inside a larger space.

One caution is in order here: unlike in the state space correspondence, noequivalence between virtual cycles or correlators falls out immediately from the factthat the two moduli spaces arise in analogous ways. Rather, the above argumentshould be viewed, at this point, as merely a heuristic framework for motivating arelationship between the two theories.

Nevertheless, a correspondence of correlators does hold in genus zero, as Chiodoand Ruan proved:

Theorem 3.3 (Chiodo–Ruan). The genus-zero FJRW theory of W can be en-coded in a generating function JFJRW(t) taking values in HW [[z−1, z], and thegenus-zero Gromov–Witten theory of Q can be encoded in a generating functionJGW(q) taking values in H∗(Q)[[z−1, z]. After choosing a specific isomorphismHW ∼= H∗(Q), these two generating functions are related by changes of variables inq and t, an identification q = t−5, and analytic continuation between the t = 0 andq = 0 expansions.

We return in the last section to discuss some ideas in the proof of the genus-zero LG/CY correspondence. Before doing so, however, we address another naturalquestion: if FJRW corresponds to the Gromov–Witten theory of the hypersurface

INTRODUCTION TO THE GLSM 15

Q, what theory corresponds in this way to the Gromov–Witten theory of a completeintersection?

4. The hybrid model and the GLSM

Let

W1, . . . ,Wr ∈ C[x1, . . . , xN ]

be a collection of quasihomogeneous polynomials of the same weights w1, . . . , wNand the same degree d, defining a nonsingular complete intersection

Z = {W1 = · · · = Wr = 0} ⊆ P(w1, . . . , wN ).

The answer to the question posed at the end of the previous section is provided bya theory referred to as the “hybrid model,” whose definition occupies this section.Just as in the FJRW case, the theory consists of a state space, a moduli space(equipped with a virtual fundamental cycle), and a notion of correlators.

4.1. The state space of the hybrid model. To understand what the appro-priate FJRW-type state space associated to Z might be, we return to the variationof GIT perspective from the previous section. Namely, introduce r auxiliary vari-ables p1, . . . , pr, and let C∗ act on CN × Cr by

λ(x1, . . . , xN , p1, . . . , pr) = (λw1x1, . . . , λwN pN , λ

−dp1, . . . , λ−dpr).

Then the polynomial

W (x1, . . . , xN , p1, . . . , pr) = p1W1(~x) + · · ·+ prWr(~x)

gives a well-defined map out of the quotient (CN × Cr)/C∗.Again, while this original quotient is not separated, it admits two maximal

separated subquotients,

X+ = {~x 6= 0} =

r⊕j=1

OP(w1,...,wN )(−d)

and

X− = {~p 6= 0} =

N⊕i=1

OP(d,...,d)(−wi).

Taking cohomology relative to a fiber W+∞

of W yields

H∗CR

r⊕j=1

OP(w1,...,wN )(−d),W+∞

∼= H∗CR(P(w1, . . . , wN ),P(w1, . . . , wN )\Z) ∼= H∗CR(Z).

Here, the second isomorphism is the Thom isomorphism. The first isomorphism

can be deduced from the fact that W+∞

intersects the fibers of OP(w1,...,wN )(−d)⊕r

in an affine space (with trivial cohomology) over points in P(w1, . . . , wN )\Z, whileit does not intersect the fibers of OP(w1,...,wN )(−d)⊕r over points in Z at all.

With this in mind, we define the hybrid-model state space as

HW := H∗CR

(N⊕i=1

OP(d,...,d)(−wi),W+∞).

Note that, by the definition of orbifold cohomology, this space can be decomposedas a direct sum over g ∈ Zd. Analogously to the FJRW case, we refer to g as narrow

16 EMILY CLADER

if its fixed locus is P(d, . . . , d) ⊆ X− or broad otherwise. The narrow elements of Zdeach contribute a copy of H∗(P(d, . . . , d)) = H∗(PN−1) to the hybrid-model statespace.

4.2. The moduli space of the hybrid model. As for the moduli space, werepeat the analogous variation-of-GIT argument once again. Namely, let Xg,n,β bethe (non-compact, non-separated, Artin) stack parameterizing tuples

(C; q1, . . . , qn;L;x1, . . . , xN ; p1, . . . , pr),

in which (C; q1, . . . , qn) is a genus-g, n-pointed orbifold stable curve and

(x1, . . . , xN , p1, . . . , pr) ∈ Γ

N⊕i=1

L⊗wi ⊕r⊕j=1

(L⊗−d ⊗ ωlog)

.

Within Xg,n,β , there are two separated Deligne–Mumford substacks, which we againdenote by X+

g,n,β and X−g,n,β , given by imposing that either ~x or ~p is nowhere-vanishing.

When ~x is nowhere-vanishing, the situation is exactly as in the previous section.Namely,

X+g,n,β = {(C; q1, . . . , qn; f ; p1, . . . , pr) | f : C → P(~w), pj ∈ Γ(f∗O(−d)⊗ ωlog)}.

When ~p is nowhere-vanishing, on the other hand, whereas it led to a trivializationin the setting of FJRW theory, it now corresponds to a map f : C → Pr−1. Thus,we have

X−g,n,β = {(C; q1, . . . , qn;L;x1, . . . , xN ; f) | xi ∈ Γ(L⊗wi), f : C → Pr−1, f∗O(1) ∼= L⊗−d⊗ωlog}.

Remark 4.1. It is perhaps worth pausing to note at this point that X+g,n,β

and X−g,n,β are nearly equal to the moduli spaces of stable maps to X+ and X−,respectively; the only difference is the appearance of the bundle ωlog in variousfactors. For a much more detailed study of how the theories with and without ωlog

relate to one another, we refer the reader to [21] or [11].

Continuing on in the template laid out in the FJRW case, we observe thatX±g,n,β are not compact, and we restrict to the loci Z±g,n,β ⊆ X±g,n,β on which the

section (x1, . . . , xN , p1, . . . , pr) lands in the critical locus

Crit(W ) ⊆ CN × Cr.

Specifically:

Exercise 4.2. Viewing W as a function on X+ =⊕r

j=1OP(~w)(−d), verify that

the critical locus of W is the complete intersection Z in the zero section P(~w) ⊆ X+.

Similarly, viewing W as a function on X− =⊕N

i=1OP(d,...,d)(−wi), verify that the

critical locus of W is the zero section.

As a result, then, we have

Z+g,n,β =Mg,n(Z, β),

whereas we define the hybrid model state space as

Z−g,n,β = {(C; q1, . . . , qn;L; f) | f : C → Pr−1, f∗O(1) ∼= L⊗−d ⊗ ωlog}.

INTRODUCTION TO THE GLSM 17

Perhaps, at this point, the name “hybrid” makes some sense; the above modulispace combines the Gromov–Witten theory of Pr−1 with the FJRW theory of adegree-d polynomial.

4.3. The correlators of the hybrid model. Just as in the FJRW case,because Z±g,n,β arise as critical loci inside a larger moduli space with a naturaldeformation theory, they have induced virtual fundamental cycles via the cosectionconstruction [2, 3]. Somewhat more explicitly:

Exercise 4.3. Let

(C;L;x1, . . . , xN , p1, . . . , pr) ∈ Xhybg,0,β .

Show, using Serre duality, that there is a map

σ :

N⊕i=1

H1(L⊗wi)⊕r⊕j=1

H1(L⊗−d ⊗ ω)→ C

σ(u1, . . . , uN , v1, . . . , vr) =

N∑i=1

∂W

∂xi(~x, ~p) · ui +

r∑j=1

∂W

∂pj(~x, ~p) · vj ,

and that this map fails to be surjective if and only if

(C;L;x1, . . . , xN , p1, . . . , pr) ∈ Zhybg,0,β .

(These fiberwise maps are used to define the cosection on Xhybg,0,β from which the

virtual cycle on Zhybg,0,β is constructed, and they are the reason why the ωlog’s are

needed in the sections pj .)

Furthermore, both of these moduli spaces have evaluation maps

evi : Z±g,n,β → X±

for each i ∈ {1, . . . , n}. To see this, we need the following fundamental fact (see,for example, [25] for further explanation):

Fact 4.4. Let C be the universal curve over Mg,n, and let π : C → Mg,n

be the projection map. If ∆i ⊆ C denotes the divisor corresponding to the ithmarked point, then ωlog|∆i is trivial. Furthermore, the analogous statement is truefor Xg,n,β , X±g,n,β , or Z±g,n,β .

With this fact in mind, consider the moduli space Xg,n,β , with its universalcurve π : C → Xg,n,β , its universal line bundle L on C, and its universal section

σ ∈ Γ

N⊕i=1

L⊗wi ⊕r⊕j=1

(L⊗−d ⊗ ωlog)

.

Then, for any i ∈ {1, . . . , n}, we have

σ|∆i∈ Γ

N⊕i=1

L⊗wi ⊕r⊕j=1

L⊗−d .

18 EMILY CLADER

Exercise 4.5. Use Fact 4.4 and the above discussion to explain how to con-struct evaluation maps

evi : X±g,n,β → X±,

and hence, by restriction, evi : Z±g,n,β → X±.

The essential idea, at this point, is to define correlators in the hybrid model bypulling back elements of H∗CR(X−) under the evaluation maps and integrating themagainst the virtual fundamental cycle on Z−g,n,β . There is an important subtlety,

though: we have defined the hybrid model state space to be not H∗CR(X−) but

H∗CR(X−,W+∞

). It would seem, then, that the insertions to our correlators aredrawn from the wrong vector space.

The solution to this issue is to decompose the hybrid state space into broadand narrow components:

H∗CR(X−,W+∞) =

⊕g broad

H∗(

(X−)g,W+∞)⊕ ⊕

g narrow

H∗(PN−1).

In particular, the narrow part of the hybrid state space can be viewed as a subspaceof

(4.1)⊕g∈Zd

H∗(PN−1) = H∗CR(P(d, . . . , d)) = H∗CR(X−).

We can then define correlators with narrow insertions as follows:

Definition 4.6. Let α1, . . . , αn ∈ HW be narrow, and, via (4.1), view each asan element of H∗CR(X−). Then the associated hybrid model correlator is

〈α1 · · ·αn〉hybg,n,β =

∫[Z−g,n,β ]vir

ev∗1(α1) · · · ev∗n(αn).

Exercise 4.7. Convince yourself that, in the case where N = 1, this recoversthe definition of narrow FJRW correlators.

Remark 4.8. The same construction can be applied to Z+g,n,β = Mg,n(Z, β).

Here, the state space is H∗CR(Z) whereas the insertions are drawn from H∗CR(X+) ∼=H∗(P(~w)); thus, in the same way that hybrid correlators are defined only for narrowinsertions, these correlators are defined only for ambient insertions—those pulledback under the inclusion Z → P(~w). If we define the virtual cycle of Z+

g,n,β via thecosection construction, however, it is not at all obvious that the resulting correlatorsrecover the usual ambient Gromov–Witten invariants of Z. Chang and Li proved, inthe case where Z is the quintic threefold, that these two versions of the correlatorsindeed do agree, up to a sign [2].

Having developed the definition of the hybrid model correlators, we are nowprepared to state the generalization of the LG/CY correspondence to the settingof complete intersections, at least under restrictive assumptions:

Theorem 4.9. Suppose that Z is a Calabi–Yau threefold and w1 = · · · = wN =1. The genus-zero hybrid theory can be encoded in a generating function Jhyb(t)taking values in HW [[z−1, z], and the genus-zero Gromov–Witten theory of Z canbe encoded in a generating function JGW(q) taking values in H∗(Q)[[z−1, z]. Afterchoosing a specific isomorphism HW ∼= H∗(Q), these two generating functions arerelated by changes of variables in q and t, an identification between q and t, andanalytic continuation between the t = 0 and q = 0 expansions.

INTRODUCTION TO THE GLSM 19

In fact, a version of the genus-zero LG/CY correspondence has been proven inmuch greater generality, for (almost) any Calabi–Yau Z in weighted projective spacefor which wi|d for all i; this was carried out by Ross and the author in [11, 12], basedon the work of Lee–Priddis–Shoemaker [21] in the hypersurface case. However, acareful statement of the correspondence in that generality requires more technicalmachinery than we have developed here.

4.4. The general gauged linear sigma model. The story we have told forFJRW theory and the hybrid model can be vastly generalized; it is an instance ofthe gauged linear sigma model (GLSM), which was developed mathematically byFan, Jarvis, and Ruan [15].

In full generality, the GLSM is a theory that depends on three pieces of inputdata:

(1) a GIT quotient X = [V �θ G], where V is a complex vector space andG ⊆ GL(V );

(2) a polynomial function W : X → C known as the “superpotential”;(3) an action of C∗ on V known as the “R-charge.”

The rough idea, from here, is to form a moduli space parameterizing “Landau–Ginzburg maps” from curves into the critical locus of W ; these look nearly the sameas ordinary maps to the critical locus of W , but there are additional occurrences ofωlog dictated by the R-charge.

In order for the resulting moduli space to be compact, it is sometimes necessaryto weaken the notion of maps to “quasimaps”, an idea first introduced by Ciocan-Fontanine and Kim [7, 8]. Quasimaps also play a key role in the proof of thegenus-zero LG/CY correspondence, so we discuss them further in the final sectionof these notes.

5. Wall-crossing and the proof of the LG/CY correspondence

We turn, in this final section, to the proof of the LG/CY correspondence ingenus zero. A number of proof strategies have been developed at this point, in-cluding Chiodo and Ruan’s Lagrangian cone approach [5] (based on ideas of [13])and Lee–Priddis–Shoemaker’s approach via the crepant transformation conjecture[21, 11] (which leverages [14]). In what follows, we describe a proof via quasimapsand wall-crossing, which was first applied to the LG/CY correspondence by Rossand Ruan [23]; one advantage of this approach is that it points toward a strategyfor higher genus.

5.1. Quasimaps. As in the previous section, let W1, . . . ,Wr ∈ C[x1, . . . , xN ]be quasihomogeneous polynomials of the same weights w1, . . . , wN and the samedegree d, defining a nonsingular complete intersection Z ⊆ P(w1, . . . , wN ). Recallthat the moduli spaces of Gromov–Witten theory and the hybrid model were bothconstructed as substacks of the larger (Artin) stack Xg,n,β parameterizing tuples

(C; q1, . . . , qn;L;x1, . . . , xN ; p1, . . . , pr)

with

(x1, . . . , xn, p1, . . . , pr) ∈ Γ

N⊕i=1

L⊗wi ⊕r⊕j=1

(L⊗−d ⊗ ωlog)

,

by first imposing that either ~x or ~p was nowhere-vanishing.

20 EMILY CLADER

Suppose, now, that we weaken this requirement, allowing ~x (on the Gromov–Witten side) or ~p (on the hybrid side) to have zeroes of bounded order. Specifically,on the Gromov–Witten side, we consider the following definition (due to Ciocan-Fontanine and Kim [7, 8]):

Definition 5.1. Fix a positive rational number ε, and define X+,εg,n,β ⊆ Xg,n,β

to be the locus where:

(1) ~x(q) = 0 for at most finitely many points q ∈ C, all of which are nonspecialand satisfy

ordq(~x) ≤ 1/ε,

where ordq(~x) denotes the order of vanishing of ~x;(2) the bundle L⊗ε ⊗ ωlog is ample.

Elements of X+,εg,n,β are referred to as ε-stable quasimaps to X+.

The second condition in Definition 5.1 is equivalent to requiring that deg(L|C′) >1/ε on any genus-zero components C ′ ⊆ C with only one special point (such com-ponents are referred to as “rational tails”) and deg(L|C′) > 0 on any genus-zerocomponents C ′ ⊆ C with only two special points (referred to as “rational bridges”).The intuition behind this condition is the following. Having imposed a bound onthe order of vanishing of ~x, properness of the moduli space forces one to specifywhat happens in the limit as a nowhere-vanishing ~x approaches a zero of orderβ > 1/ε. Just as in the stable maps setting, the limit in this case is a curve witha new rational component on which L has degree β. The second condition in Def-inition 5.1 is concocted to allow precisely such rational components; allowing anymore would lead to non-uniqueness of limits, preventing the moduli space frombeing separated.

Having constructed the moduli space X+,εg,n,β , we mimic the previous section

and restrict further to the locus

Z+,εg,n,β ⊆ X

+,εg,n,β

where (~x, ~p) lands in Crit(W ) = Z ⊆ X—that is, where

Wj(~x) ≡ 0 ∈ Γ(L⊗d) for all j ∈ {1, . . . , r}

and

~p ≡ 0.

This recovers Ciocan-Fontanine and Kim’s moduli space of ε-stable quasimaps toZ.

When ε > 2, condition (1) of Definition 5.1 implies that ~x is nowhere-vanishingand hence gives rise to an honest map f : C → P(w1, . . . , wN ), and condition(2) implies that L has positive degree on all rational components with fewer thanthree special points, so f is stable. Thus, for such ε, the moduli space of ε-stablequasimaps to Z, which we denote by Z+,∞

g,n,β , is nothing but the usual moduli spaceof stable maps.

Taking ε → 0, on the other hand (that is, requiring conditions (1) and (2) ofDefinition 5.1 for all ε > 0) corresponds to allowing arbitrary isolated zeroes of~x and disallowing all rational tails. The resulting moduli space, which we denoteby Z+,0

g,n,β , is the moduli space of stable quotients studied by Marian, Oprea, and

Pandharipande [22].

INTRODUCTION TO THE GLSM 21

All of this can be carried out analogously on the hybrid side. Namely, insideXg,n,β , we replace the requirement that ~p is nowhere zero by a bound on the orderof its zeroes:

Definition 5.2. Fix a positive rational number ε, and define X−,εg,n,β ⊆ Xg,n,β

to be the locus where:

(1) ~p(q) = 0 for at most finitely many points q ∈ C, all of which are nonspecialand satisfy

ordq(~p) ≤ 1/ε,

where ordq(~p) denotes the order of vanishing of ~p;(2) the bundle (L⊗−d ⊗ ωlog)⊗ε ⊗ ωlog is ample.

Then, within X−,εg,n,β , we restrict further to the locus Z−,εg,n,β on which ~p lands

in Crit(W ) = P(d, . . . , d) ⊆ X−, or in other words, where ~x ≡ 0. As above, taking

ε� 0 in the definition of Z−,εg,n,β recovers the hybrid moduli space Z−g,n,β .

5.2. Proof of genus-zero LG/CY. One of the most important reasons forour interest in quasimaps is their intimate connection to mirror symmetry, whichleads ultimately to the LG/CY correspondence.

The idea of genus-zero mirror symmetry—at least, one of its manifestations—isto calculate the genus-zero Gromov–Witten invariants of the complete intersectionZ by relating their generating function to an explicit power series. In particular,Givental showed in [16] that the generating function JGW(q) of genus-zero Gromov–Witten invariants of Z (mentioned in Theorem 4.9) is related by change of variablesto the “I-function”

IGW(q) = z∑β≥0

qβ∏dβb=1(dH + bz)r∏βb=1(H + bz)N+1

,

a power series in q with coefficients in H∗(Z)[[z−1, z], where H ∈ H∗(Z) denotesthe hyperplane class. This is a powerful result that allows the genus-zero Gromov–Witten invariants of Z to be explicitly computed.

The original appearance of the I-function in the literature was in terms ofperiod integrals and Picard–Fuchs equations, but Ciocan-Fontanine and Kim gavean alternative explanation for its role. Namely, they showed that one can define agenerating function JGW,ε(q) of genus-zero ε-stable quasimap invariants for any ε,and JGW,ε(q) coincides with JGW(q) when ε� 0 whereas JGW,0(q) is precisely theI-function. Furthermore, it is apparent from the definition of quasimaps that theinvariants change only when 1/ε crosses an integer value, so JGW,ε(q) changes onlywhen ε crosses one of a discrete set of “walls.” From this perspective, genus-zeromirror symmetry becomes the result of a “wall-crossing” formula exhibiting howJGW,ε(q) varies when ε crosses a wall; this re-proof of mirror symmetry was carriedout by Ciocan-Fontanine and Kim in [7].

Precisely the same story can be told on the hybrid side, using wall-crossing torelate the hybrid J-function Jhyb(t) = Jhyb,∞(t) to the function Jhyb,0(t), which isthe “hybrid I-function” and can be calculated explicitly as a hypergeometric series;this was done by Ross–Ruan for FJRW theory and by Ross and the author for thehybrid model [23, 12].

22 EMILY CLADER

From here, one proves the LG/CY correspondence by relating the two I-functions to one another. More specifically, if one expands IGW(q) in the form

IGW(q) = I0(q) · 1 + I1(q) ·H + I2(q) ·H2 + · · ·+ IN−1−r(q)HN−1−r,

then the coefficients Ii(q) form a basis of solutions to a differential equation inq known as the Picard–Fuchs equation. In the same way, the hybrid I-functionassembles a basis of solutions to a differential equation in t, and these two differentialequations coincide after a particular identification between q and t. (For example, inthe case of the quintic threefold, the identification is q = t−5.) Thus, by analyticallycontinuing the solutions in the q-coordinate patch to solutions in the t-coordinatepatch, and then applying a linear isomorphism of the state space to change fromone basis of solutions to another, one shows that IGW(q) and Ihyb(t) are relatedby analytic continuation and a state space isomorphism. This, combined withthe wall-crossing changes of variables on both sides, proves the genus-zero LG/CYcorrespondence.

5.3. Towards higher-genus LG/CY. The proof of the genus-zero LG/CYcorrespondence outlined in the previous subsection suggests a path toward a proofin all genus: first, prove all-genus wall-crossing theorems exhibiting the dependenceof the generating functions of Gromov–Witten or hybrid invariants on ε, and second,relate the ε→ 0 Gromov–Witten generating function to the ε→ 0 hybrid generatingfunction.

The first step of this program has been successfully carried out, by Ciocan-Fontanine and Kim on the Gromov–Witten side [6] and by Janda, Ruan, and theauthor on the hybrid side [9]:

Theorem 5.3. For any complete intersection Z ⊆ P(w1, . . . , wN ) of hypersur-faces of the same degree, we have

[ZGW/hyb,εg,n,β ]vir =

∑k

β0+β1+...+βk=β

b~β∗

(k∏i=1

ev∗n+i(µGW/hyb,εβi

(−ψn+i)) ∩ [ZGW/hybg,n+k,β0

]vir

).

Here, b~β : ZGW/hybg,n+k,β0

→ ZGW/hybg,n,β is a certain comparison map that converts marked

points to zeroes of ~x or ~p, and µGW/hyb,εβi

is a certain coefficient of the Gromov–Witten or hybrid I-function.

We should also note that the proof of [9] yields an alternative proof of thewall-crossing on the Gromov–Witten side [10], and on the hybrid side, a differentproof of Theorem 5.3 was independently given by Y. Zhou [24] in the case wherer = 1.

Theorem 5.3 represents significant progress toward the higher-genus LG/CYcorrespondence, but the second step of the program—relating the ε → 0 theorieson the two sides—remains almost entirely mysterious. It has been established ingenus zero (as described in the previous subsection) and in genus one [17, 18]; thelatter is the subject of another of the mini-courses at this workshop.

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Department of Mathematics, San Francisco State University, 1600 Holloway Av-enue, San Francisco, CA 94132, USA

E-mail address: eclader@sfsu.edu