Huijun FANTyler JARVISYongbin RUAN

Abstract This is a survey article for the mathematical theory of Witten’s Gauged Linear Sigma Model,as developed recently by the authors.Instead of developing the theory in the most general setting,in this paper the authors focus on the description of the moduli.

Keywords Landau-Ginzburg,Calabi-Yau,Gromov-Witten,GLSM,Moduli,Phase,Variation of geometric invariant theory,Symplectic reduction

1 Introduction

In 1991,in an e ff ort to generalize his famous conjecture regarding the KdV-hierarchy for the intersection theory of the moduli space of Riemann surfaces(see[34–35]),Witten proposed a remarkable first-order,nonlinear,elliptic PDE associated to an arbitrary quasihomogeneous singularity.It has the simple form:

where W is a quasihomogeneous polynomial with an isolated singularity at the origin,and uiis interpreted as a section of an appropriate orbifold line bundle on an orbifold Riemann surface

During the last decade we have carried out a comprehensive treatment of the Witten equation and have used it to construct a theory similar to Gromov-Witten theory(see[16–18]).This so-called FJRW-theory can be viewed as the Landau-Ginzburg phase of a Calabi-Yau hypersurface

in weighted projective space.The relation between the Gromov-Witten theory of XWand the FJRW-theory of W is the subject of the Landau-Ginzburg/Calabi-Yau correspondence,a famous duality from physics.More recently,the LG/CY correspondence has been reformulated as a precise mathematical conjecture(see[30]),and a great deal of progress has been made on this conjecture(see[7–9,26,28]).

A natural question is whether the LG/CY correspondence can be generalized to complete intersections in projective space,or more generally to toric varieties.The physicists’answer is“yes”.In fact,Witten considered this question in the early 90s(see[34])in his e ff ort to give a physical derivation of the LG/CY correspondence.In the process,he invented an important model in physics called the Gauged Linear Sigma Model(GLSM for short).From the point of view of partial differential equations,the gauged linear sigma model generalizes the Witten Equation(1.1)to the Gauged Witten Equation:

where A is a connection of certain principal bundle,andµis the moment map of the GIT-quotient,viewed as a symplectic quotient.In general,both the Gromov-Witten theory of a Calabi-Yau complete intersection X and the LG dual of X can be expressed as gauged linear sigma models.Furthermore,the LG/CY correspondence can be interpreted as a variation of the moment mapµ(or a deformation of GIT)in the GLSM.

During last several years,we constructed a rigorous mathematical theory for the gauged linear sigma model(see[19–20]),and this new model seems to have many applications(see,for example[14,28]).Our new theory is a generalization of FJRW-theory from the case of a finite gauge group to the case where the gauge group is any reductive Lie group.Surprisingly,our older theory(finite group)is one of the more difficult cases in the general theory.

We deal with the Gauged Witten Equation both analytically(see[20])and algebraically(see[19]).But in this paper we focus on the algebraic version of the theory.In the case of a continuous group,we can treat the Gauged Witten Equation algebraically using some stability conditions.It turns out to be very convenient to incorporate the stability conditions from the quasimap theory of Ciocan-Fontanine,Kim,Maulik,and Cheong(see[6,10–11,24]).And indeed,it is natural to view our GLSM-theory as a union of FJRW-theory with quasimap theory.However,it is possible to impose other stability conditions(see[5,12]).

Understanding the details of the mathematical construction of the GLSM can be daunting.This paper is an attempt to help the reader navigate past the more technical constructions and begin to understand the underlying moduli spaces.In the next section we give a brief overview of the main ingredients to the theory,and in the subsequent section we brie fly review the definition of the moduli problem and the most important stability conditions.The remainder of the paper is focused on giving examples of the moduli spaces for various choices of input data.

2 The Basic Setting

The input data of our new theory consists of the following.We discuss these in more detail below.

(1)A finite dimensional vector space V over C.

(2)A reductive algebraic group G⊆GL(V),sometimes called the gauge group.

(3)A G-character θ with the property.We say that it defines a strongly regular phase

(4)A choice of C∗action on V,called the R-charge and denoted by.This action is required to commute with the G-action,and we requireto have finite order d.

(5)A G-invariant quasihomogeneous polynomial W:V→C,called the superpotential,having degree d with respect to theaction.We require that the GIT quotient Crit(W)//θG of the critical locus Crit(W)be compact.

(6)A stability parameter ε.This can be any positive rational number,but in practice,the two most useful cases are the limiting cases of ε→ ∞ or ε→ 0+.Fortunately,these cases are also easier to describe than the positive rational cases.For simplicity,in this paper we will only discuss the limiting cases of 0+and∞.

(7)A Γ-character ϑ,where Γ is the subgroup of GL(V)generated by G andby Γ.We require that ϑ define a lift of θ,meaning that ϑ|G= θ.Except in the case of ε=0+,we also require that this lift be a good lift,meaning that it satisfyA choice of good lift a ff ects the stability conditions for the moduli space.But in the case of ε=0+the lift need not be good,and every lift will produce the same stability conditions and the same moduli space.

With the above input data we construct a theory with the following main ingredients:

(1)A state space,which is the relative Chen-Ruan cohomology of the quotientwith an additional shift by 2q.For each conjugacy class Ψ⊂G,let

(2)The stack of LG-quasimaps.

of(ε,ϑ)-stable Landau-Ginzburg quasimaps to

(3)A virtual cycle:

with virtual dimension

(4)Numerical invariants.Using the virtual cycle,we can define correlators

One can then define a generating function in the standard fashion.These invariants satisfy the usual gluing axioms whenever all insertions are of compact type.

In the rest of this section,we will discuss some of the input data and the state space in more detail.

2.1 GIT and symplectic quotients

The first two pieces of data consist of a reductive algebraic group G(the gauge group)acting on a finite-dimensional vector space V∼=Cn.We do not require G to be connected,but we require that G/G0be finite,where G0is the identity component of G.If the gauge group action on V factors through SL(V),then we say that it satisfies the Calabi-Yau condition.But in general we do not require that G satisfy this condition.

We wish to consider the quotient stack[Z/G]for a closed subvariety Z⊆V,but since the group G may not be compact,the quotient is not generally separated(Hausdor ff).Geometric Invariant Theory(GIT for short)and symplectic reduction each give a way to construct separated quotients.

2.1.1 GIT quotients

The key to constructing a separated quotient using GIT is to choose a linearization of the action of G on Z,i.e.,a lifting of the action of G to a line bundle L over Z.We always assume that the linearization on Z is induced by a linearization on V.Since V=Cn,any line bundle L on V is trivial L=V × C,and the linearization is determined by a character θ:G → C∗.

Definition 2.1For any character θ:G → C∗we write Lθfor the line bundle V ×C with the induced linearization.We also often write Lθto denote the corresponding line bundle on Z.

Geometric Invariant Theory identifies an open subset Zss(θ)of Lθ-semistable points in Z as the set of those points v∈Z for which there exists a positive integer k and a G-invariant sectionWe denote the set of points in Z that are semistable with respect to G and θ byThe GIT quotient stack[Z//θG]is defined to be the stack

Let Z/affG be the affine quotient given by Z/affG=Spec(C[Z∗]G),where C[Z∗]is the ring of regular functions on Z.The GIT quotient stack[Z//θG]is an algebraic stack with an underlying(coarse moduli)space

The linearization Lθinduces a line bundle(a.k.a.a polarization)on,which we denote by Lθ.

Definition 2.2We say that a point v ∈ V is stable with respect to the linearization θ(or θ-stable)if

(1)v is θ-semistable.

(2)The stabilizer StabG(v)={g∈G|gv=v}is finite.

We denote the set of θ-stable points of Z by.We say that a point is unstable if it is not semistable.

The stable locus is important because the quotient stackis a Deligne-Mumford stack,whereasis not necessarily Deligne-Mumford.

Remark 2.1Mumford-Fogarty-Kirwan[27]use the name properly stable to describe what we call stable.

Remark 2.2For any integers

so it makes sense to extend the GIT constructions to fractional linearizations,corresponding to fractional characters in

For a fixed Z,changing the linearization gives a different quotient.The space of(fractional)linearizations is divided into chambers,and any two linearizations lying in the same chamber have isomorphic GIT quotients.We call the isomorphism classes of these quotients phases.If the linearizations lie in distinct chambers,the quotients are birational to each other,and are related by flips(see[15,32]).

Definition 2.3We say that(or the corresponding linearization Lθ)is strongly

For purposes of this paper,all linearizations need to be strongly regular.

Remark 2.3For any strongly regular phase θ,the complex dimension ofn−dim(G).

2.1.2 Symplectic quotients

One may also think of the GIT quotients as symplectic reductions.Takewith the standard K¨ahler formSince G is reductive,it is the complexification of a maximal compact Lie subgroup H,acting on Z via a faithful unitary representation.Denote the Lie algebra of H by h.

We have a Hamiltonian action of H on Z with moment mapµZ:for the action of H on Z,given byis a value of the moment map,then the locusµ−1(Hτ)is an H-invariant set,and the symplectic orbifold quotient of Z at τ is defined as

where Hτis the stabilizer in H of τ.The value τ of the moment map plays the role for the symplectic quotient that the linearization θ plays for the GIT quotient.These are related by the following result.

Theorem 2.1(see[19,Corollary 2.1.8])Letbe a character of G.Taking derivations of the character θ defines a weight.Whenever the coadjoint orbit of τθin h∗is trivial(e.g.,in the case that τ is in the Lie algebra of the center of G,or if G is Abelian),then we have

As with the space of GIT linearizations,the spaceis divided into chambers;and values of τ that lie in the same chamber define isomorphic quotients.The walls between the chambers correspond to the critical points of the moment mapµ.In many cases these are easier to identify in the symplectic formulation than in the GIT formulation.

2.2 Superpotential and critical locus

The next piece of data required for the GLSM is the superpotiential,which is G-invariant polynomial W:V→C.We are especially interested in the critical locus of the superpotential.

Definition 2.4Let θ:G → C∗define a strongly regular phaseThe superpotential W descends to a holomorphic functiondenote the semistable points of the critical locus.The group G acts onand the stack quotient is

2.3 R-charge and the group Γ

The gauged linear sigma model(GLSM for short)requires an additional C∗-action on V called the R-charge.The R-charge is a C∗-action on V of the form

Remark 2.4Our choice of-action differs from what the physics literature calls R-charge by a factor of 2.More precisely,the physicists’R-charge is the C∗-action given by the weights(2c1/d,···,2cn/d);but for our purposes,is the more natural choice.

Definition 2.5We define the exponential grading element

which has order d.

It is sometimes convenient to write

We require the actions of G andto be compatible,by which we mean

(1)They commute:gr=rg for any g ∈ G and any

Definition 2.6The group Γ is the subgroup of GL(V)generated by G and.

The representation γ=gr is unique up to an element of hJi.Moreover,there is a welldefined homomorphism

Moreover,there is another homomorphism

This is also well defined,and gives another exact sequence:

Definition 2.7Let N=n−dim(G).We define the central charge of the theory for the choices V,G,,W to be

2.4 Lifts of the linearization to Γ

Although we are primarily interested in the GIT quotients of V by G,our constructions also depend heavily on the GIT quotients of V by Γ.For this,we need a lift of the G-linearization to a Γ-linearization.That is,we require a Γ-character ϑ that lifts θ,meaning that ϑ|G= θ.It is not hard to prove that lifts always exist,including the trivial lift ϑ(gr)= θ(g).

For a given lift ϑ,we always havebut equality does not necessarily hold.If it does hold,we say that ϑ is a good lift of θ.For the stability parameter ε = ∞ we require the lift to be a good lift.For the choice ε=0+,the lift need not be good,and every lift will produce the same theory.Unfortunately,not every θ∈ bG has a good lift for every choice of(G-compatible)-action,but most of the interesting examples of GLSMs have a good lift.

2.5 Choice of

All of our constructions ultimately depend not onbut rather only on the embeddingson the sumweights,and on a choice of a liftOf course the choice of q and the embedding of Γ in GL(V)put many constraints onbut they still allow some flexibility.For example,when the gauge group G is a torus with a Calabi-Yau weight system(that is,if its weight matrixfor each i),then we have a lot of flexibility.The following lemma is not hard to prove(see[19,Lemma 3.3.1]).

2.6 Hybrid models

A very important subclass of the toric examples—when—consists of the socalled hybrid models.Several examples of the hybrid model have been worked out in detail by E.Clader in[13].

Definition 2.8For a torus,a phase θ of(W,G)is called a hybrid model if

3 The GLSM Moduli Space

3.1 Landau-Ginzburg quasimaps

Definition 3.1For any k-pointed orbicurve C,y1,···,yk,denote by ωlog,Cthe line bundle

Definition 3.2A prestable,k-pointed,genus-g,LG-quasimap tois a tupleconsisting of

(A)a prestable,k-pointed orbicurveof genus g,

(B)a principal(orbifold)Γ-bundle

A prestable,k-pointed,genus-g,LG-quasimap tois a prestable,k-pointed,genus-g,LG-quasimap tosuch that the image of the induced map[σ]:P → V lies in Crit(W).

Definition 3.3The points b occurring in condition(2)above are called base points of the quasimap.That is,is a base point if at least one point(and hence every point)of thefiberover b is mapped by σ into an Lθ-unstable G-orbit of V.

Definition 3.4For any prestable LG-quasimap,a Γ-equivaria-nt line bundledetermines a line bundle,and pulling back along σ gives a line bundle

In particular,any characterdetermines a Γ-equivariant line bundle Lαon V and hence a line bundle

3.2 Stability conditions for the stack of LG-quasimaps

Definition 3.5For anydefine the degree of α on Q to be

Definition 3.6Given a polarizationof θ and a prestable LG-quasimapwe say that Q is 0+stable if

(1)every rational component has at least two special points(a mark yior a node),and

(2)on every irreducible component C′with trivial,the line bundlehas positive degree.

It turns out that condition(2)holds for one lift if and only if it holds for all lifts,because for any two lifts ϑ and ϑ′of θ,the bundlesalways differ by a power of(see[19,Proposition 4.2.14]).Moreover,over Q the “trivial” lift,defined by setting ϑ(gr)= θ(g),is always a valid choice of lift.

Definition 3.7Given a polarization θ∈and a good lift ϑ of θ (see Subsection 2.4)and a prestable LG-quasimapwe say that Q is(∞,ϑ)-stable if

(1)there are no basepoints of σ on C,and

(2)for every irreducible component,the line bundlerestricted tohas nonnegative degree,with the degree only being allowed to vanish on components whereis ample.

Definition 3.8For a given choice of compatible G-and-actions on a closed affine variety,a strongly regular character θ∈and a nondegenerate W,we denote the corresponding stack of k-pointed,genus-g,0+stable LG-quasimaps intoof degree β by

respectively.

If ϑ is a good lift of θ,then the corresponding stack of k-pointed,genus-g,(∞,ϑ)-stable LG-quasimaps intoof degree β is denoted by

respectively.

4 A Hypersurface in Weighted Projective Space

The remainder of this paper is dedicated to giving examples of the stack of stable LG-quasimaps tofor various choices of input data.We begin with an example of a hypersurface in weighted projective space.

4.1 Basic setup

Suppose that G=C∗and F ∈ C[x1,···,xK]is a quasihomogeneous polynomial of G-weights(b1,···,bK)and total G-degree b.Suppose further that F has an isolated singularity at the origin.Let

where the variables x1,···,xKare the first K coordinates and p is the last coordinate.We assign G-weight−b to the variable p,so that W is G invariant.

The critical set of W is given by the equations:

If p 6=0,then the fact that the only singularity of F is at the origin means that(x1,···,xK)=(0,···,0).If p=0,then the only constraint is F(x1,···,xK)=0.So the critical locus is

Suppose thatbi>0 for i=1,···,K and b>0.Ifthen we have a Calabi-Yau weight system,but we do not assume that here.

4.2 Phases for the hypersurface

Recall that different choices of linearization θ or moment map values τ give different quotients,but any two linearizations lying in the same chamber have isomorphic quotients,called phases.

In the hypersurface case,the affine moment map

is a quadratic function whose only critical value is τ=0,and there are two phases τ>0 or τ<0.

Case of τ>0We have

For each choice of p,the set of(x1,···,xK) ∈ CK,such that(x1,···,xK,p) ∈ µ−1(τ),is a nontrivial ellipsoid E,isomorphic to S2K−1;and we obtain a map from the symplectic quotient

The resulting symplectic quotientcan be expressed as the total space of the line bundle O(−b)over WP(b1,···,bK).Ifthis is the canonical bundle

Alternatively,we can consider the GIT quotienthas weight−e,with e>0.One can easily see that the Lθ-semistable points are,and the first projection pr1:induces the map

Case of τ<0We have

For each choice of x1,···,xK∈ CKthe set of p ∈ C such that(x1,···,xK,p)∈ µ−1(τ)is the circle S1⊂C,and we obtain a mapIf we choose the generator of U(1)to

be λ−1,then p can be considered to have positive weight b.Moreover,every p has isotropy equal to the bth roots of unity(isomorphic to Zb).The quotient

In the GIT formulation of this quotient,this corresponds to θ :G → C∗of weight−e,and e<0,the Lθ-semistable points are equal to.The second projectioninduces the map

The critical locus is the single point{(0,···,0)}in the quotient].It is clearly compact,so the polynomial W is nondegenerate.We calla Landau-Ginzburg phase or a pure Landau-Ginzburg phase(see[36]).This example underlies Witten’s physical argument of the Landau-Ginzburg/Calabi-Yau correspondence for Calabi-Yau hypersurfaces of weighted projective spaces.

4.3 GLSM moduli space for the hypersurface

4.3.1 Geometric phase

For the phase τ>0,we choose(thus giving a hybrid model),so W has-weight d=1.The element J is trivial,and the group

There are two ingredients to the moduli space:An Artin stack of geometric data and a more subtle stability condition.

The geometric data correspond to sections of the vector bundle.This bundle can be written as a direct sum of line bundles

So LG-quasimaps to the geometric phasecorrespond to the following data:

For all stability conditions,the LG-quasimaps can fail in at most a finite number of points to map to the θ-semistable locus.This corresponds to the locus where s1,···,sKdo not simultaneously vanish.Moreover,has p=0.But if p=0 at all but a finite number of points,then p is always 0.

Thus,without further specifying stability conditions,LG-quasimaps to the geometric phasecorrespond to the following data

Now we consider the stability conditions.

Case ofε = ∞ We must find a good lift of θ.Let ℓ be a generator ofoverwith G acting on ℓ with weight −e(and e>0 in the geometric phase).The trivial lift ϑ0of θ corresponds toacting trivially on ℓ.A monomial of the formis Γ-invariant and does not vanish on points with xi6=0,so every point of CN×C with xi6=0 is in.Letting i range from 1 to N shows that.Thus ϑ0is a good lift of the character θ.

(1)the section σ has no basepoints(the sinever vanish simultaneously);

(2)the line bundle A has positive degree on every componentwherehas nonpositive degree.

Case of ε =0+The 0+stable LG-quasimaps must have the section σ take its values infor all but a finite number of basepoints y

Letting ϑ be the trivial lift of θ,we haveand so the 0+stable LG quasimaps toare the data of(4.2)or(4.1),respectively,satisfying the stability conditions:

(1)There are at most a finite number of basepoints(where the sivanish simultaneously),and these only occur away from nodes and marked points.

(2)Every rational component has at least two special points(node or marked point).

(3)On rational components with exactly two special points,the line bundle A has positive degree.

Thus 0+stable LG-quasimaps intoare stable quotients intoand 0+stable LG-quasimaps intoare stable quotients into XFwith p-fields.

Remark 4.1There is a parallel theory of quasimaps into XF.Both theories have the same moduli spaces,but the virtual cycle constructions are different.For ε= ∞,Chang-Li[3]proved the equivalence using a sophisticated degeneration argument.A similar argument probably works for other ε-theories.

4.3.2 LG phase

For the phase τ <0,we chooseto have weights cxi=biand cp=0,which again gives a hybrid model.But now W has-weight d=b,and,where ξ=exp(2πi/d).We have

Thus,LG-quasimaps to XLGagain consist of exactly the same data as(4.1):

The base points of these quasimaps occur precisely at the zeros of p,and the base locus forms an e ff ective divisor D inwith,so the section p gives an isomorphismand can be viewed as a weighted b-spin condition(see[29]).So we can reformulate the moduli problems as

where each D is an e ff ective divisor that is disjoint from the nodes and marked points of

For the LG-quasimaps to lie in the critical locus requires every si=0,so LG-quasimaps toconsist of

Case ofε = ∞ Again,the trivial lift is the only good lift of θ.To see this,let ℓ be a generator ofwith G acting on ℓ with weight −e(and e<0 in the LG phase).The trivial liftcorresponds toacting trivially on ℓ,and a monomial of the formis Γ-invariant and does not vanish on points with p 6=0,soThusis a good lift of the character θ.Any other lift ϑ must have nontrivial-action on ℓ,and hence any Γ-invariant function must have at least one factor of xiin every monomial,which implies that any point of V withis not ϑ-semistable.Thusis the only good lift.

Since ε= ∞,no base points are permitted,so D=0 andFor convenience,let us assume that e=−cb for some c>0.The stability condition is now that

can only have degree 0 on components whereis ample,thusmust be a stable orbicurve.

So in this case(∞,ϑ0)-stable LG-quasimaps tocorrespond to stable b-spin curves

studied in[1,23].

Case of ε=0+ In this case basepoints are permitted,so D is not necessarily 0.The bundle

must have positive degree on any component whereis not ample.

Thus 0+stable LG quasimaps tocorrespond to the data of(4.6)satisfying the conditions:

(1)Every rational component has at least two special points,and

(2)On every irreducible component C′with trivial,there is at least one basepoint.

5 Complete Intersection in Weighted Projective Space

5.1 Basic setup

Suppose that G=C∗and we have several quasihomogeneous polynomials F1,F2,···,FM∈C[x1,···,xK]of G-degree(d1,···,dM),where each variable xihas G-weight bi>0.We assume that the Fjintersect transversely in WP(b1,···,bK)and define a complete intersection.Let

where we assign G-weight−dito pi.In the special case thatThen the complete intersection defined by F1= ···=FM=0 is a Calabi-Yau orbifold in WP(b1,···,bK).But we do not assume the Calabi-Yau condition here.

The critical set of W is defined by the following equations:

Since the Fjintersect transversely,an argument similar to that for the hypersurface shows that the critical locus consists of those(x,p)where either p=0 and x satisfies Fi(x)=0 for all i,or p is unconstrained and x=0:

5.2 Phases for a complete intersection

The moment map is

Again,there are two phases,τ>0 and τ<0.

Case of τ >0When τ>0,we again call this the geometric phase.Any choice of p=(p1,···,pM)determines a nontrivial ellipsoid E ⊂ CKof points x=(x1,···,xK)such

that(x,p)lies inµ−1(τ).Quotienting by U(1),the first projection pr1:E×CM→ E induces a mapThe full quotient is

In the GIT formulation,this again corresponds to θ:G → C∗having weight−e,with e>0.The semistable points of this phase are those with x 6=0,and the semistable points of the critical locus correspond to points in

Case of τ <0 When τ>0,we again call this the Landau-Ginzburg phase.The quotient

In the GIT formulation,this corresponds to θ:G → C∗having weight−e,with e<0.The semistable points of this phase are those with p 6=0,and the semistable points of the critical locus correspond to the image of the zero section of

5.3 GLSM moduli space for a complete intersection

5.3.1 Geometric phase

We choose the R-charge to act on CK×CMwith weights(0,···,0,1,···,1),which gives a hybrid model.And W hasd=1.The element J is trivial,and the group

The geometric data correspond to sections of the vector bundlewhich can be

written as a direct sum of line bundles

where A corresponds to the G-action and B corresponds to theaction.And we have an isomorphism

So LG-quasimaps to the geometric phasecorrespond to the data:

Case of ε = ∞ Again ϑ0is easily seen to be a good lift,andThus,just as in the hypersurface case,we have that(∞,ϑ0)-stable LG-quasimaps tocorrespond to stable maps to the complete intersection(∞,ϑ0)-stable LG-quasimaps toare stable maps to X with p-fields.

Case of ε=0+The arguments given in the hypersurface case are easily adapted to show that 0+stable LG-quasimaps intoare stable quotients into X ⊂ WP(b1,···,bK),and 0+stable LG-quasimaps intoare stable quotients into XFwith p-fields.

5.3.2 LG phase

Assume that d1= ···=dr=d and choose the R-charge weights cxi=biand cpj=0.Now W has-weight d,and

Case of ε = ∞ Again,the trivial lift is the only good lift of θ.The line bundleagain

And so the stability condition is that

can only have degree 0 on components where ωlog,Cis ample.And since ε= ∞,no base points are permitted,so the picannot all simultaneously vanish.

So in this case(∞,ϑ0)-stable LG-quasimaps tocorrespond to stable maps to WP(d,d,And for each(p1,···,pn)∈ WP(d,···,d),we have a pure LG-model of superpotentialOne can view this as a family of pure LG-theories.

Remark 5.1An LG-phase of a complete intersection of differing degrees(i.e.,di6=djfor some 1≤i,j≤r)does not admit a hybrid model structure and will generally have no good lift.

Case of ε=0+Now basepoints are permitted,and every rational component must have at least two special points.Again,the stability condition is that

can only have degree 0 on components where ωlog,Cis ample.

Remark 5.2As mentioned in Remark 5.1,without the condition d1=d2= ···=dr=d there is usually no good lift of θ.But in the ε=0+case,we do not need a good lift,sofixing any d>0 we can take cpj=d − dj,which again gives W the-weight of d and a corresponding proper DM stack of 0+stable LG-quasimaps.

6 Graph Moduli Space

The graph moduli space is very important in Gromov-Witten theory.We can construct it in the GLSM setting as follows.

Suppose that we have a phase θ,a superpotential Wwith a certain R-chargedefining Γ and a lift ϑ of θ.We construct a new GLSM as follows.

Let V′=V ×C2,and let C∗act on C2with weights(1,1).Letwith the product action,so G acts trivially on the last two coordinates and C∗acts trivially on thefirst n coordinates.

Let θ′:G′→ C∗be given by sending any(g,h)∈ G × C∗to θ(g)h−kfor some k>0.The GIT quotient is the productLet W′be defined on V′by the same polynomial as W,so that the critical locus of W′in V′is C2times the critical locus of W,and the GIT quotient of the critical locus is the product of P1and the corresponding quotient in the original GLSM.

Keeping the same R-charge(that is,lettingacts trivially on the last two coordinates of V′),we have Γ′= Γ×C∗,and we construct a liftby sendingIt is easy to see that ϑ′is a good lift of θ′if ϑ is a good lift of θ.

In the ε = ∞ case,no basepoints can occur,and projecting to the two new coordinates(z0,z1)induces a stable mapTherefore,the new GLSM in this case can be reformulated as the usual GLSM forwith the additional data of a stable map f:

7 Generalized Graph Space

We can generalize slightly the graph moduli space to obtain a new moduli space with a remarkable property.Let us take the quintic GLSM as an example.Now,we consider a new GLSM on,given byacting onwith weights

for an integer d>0.Let the coordinates on Vcorresponding the columns in the weight matrix above.The moment maps are

There are three chambers.We are interested primarily in the chamber 0< µ1

Taking the superpotential we have=,and the map ζ takes

There is no good lift of θ,so we restrict to the case of ε =0+.We must choose some lift for the stability condition,so we take the trivial lift.Any other lift will give the same stability conditions.

The resulting moduli problem consists ofand the R-charge of weight(0,0,0,0,0,1,0,0),

satisfying the stability condition thatis ample on all components where ωlog,Chas degree 0.

The critical locus is

And so,in the chamber we are interested in,the GIT quotient of the critical locus has two componentswhere C1corresponds to x0= ···=x4=0,so z06=0 can be scaled to 1 by the second C∗action and so the quotient C1is isomorphic to P(5,1),with coordinates p,z1.The component C2corresponds to the locus

The GLSMs for the three fixed loci correspond,respectively to(1)a weighted FJRW theory,(2)0+stable quasimaps to the quintic threefold,and(3)the theory of Hassett stable curves with light points given by the vanishing of z0and z1.This remarkable property gives us the hope that we can extract a relation between Gromov-Witten theory and FJRW-theory geometrically by using localization techniques on this moduli space.A program is being carried out right now for the ε=0+theory(see[14]).

The same theory with a different ε = ∞ stability condition was discovered and the localization argument was carried out independently by Chang-Li-Li-Liu(see[5]).

8 Non-Abelian Examples

The subject of gauged linear sigma models for non-Abelian groups is a very active area of research in physics and is far from complete.Here,we discuss complete intersections in a Grassmannian or flag variety.

All of this should work similarly in the setting of complete intersections of quiver varieties,although the details have not been worked out.It would be very interesting to explore mirror symmetry among Calabi-Yau complete intersections in quiver varieties.

8.1 Complete intersections in a Grassmanian

The space Gr(k,n)can be constructed as a GIT quotientis the space of k×n matrices and GL(k,C)acts as matrix multiplication on the left.

The Grassmannian Gr(k,n)can also be embedded into PKforby the Pl¨ucker embedding

The group G=GL(k,C)acts on the Pl¨ucker coordinatesby the determinant,that is,for any U∈G and A∈Mk,n,we have

denote the corresponding complete intersection.

be the superpotential.We assign an action of G=GL(k,C)on

The phase structure is similar to that of a complete intersection in projective space.The moment map is given byAlternatively,to construct a linearization for GIT,the only characters of GL(k,C)are powers of the determinant,so θ(U)=det(U)−efor some e,and τ will be positive precisely when e is positive.

8.1.1 Geometric phase

Assume that e>0.In order to be G-invariant,a monomial must havewhich implies that any points with everymust be unstable,but for each m>0 and each k-tuple(i1,···,ik)the monomialis G invariant,so every point with at least one nonzeromust be θ-semistable.Thusis isomorphic to the bundleover Gr(k,n).

As in the toric case,the critical locus in this phase is given by p1= ···=ps=0=F1=···=Fs,so we recover the complete intersection F1= ···=Fsin Gr(k,n),and we call this phase the geometric phase.

Just as for the toric complete intersection,we choose the-action to have weight 0 on the space Mn,kand weight 1 on all of the pj,so that W hasweight 1 and

The trivial lift ϑ0is a good lift because each monomial of the formℓ is Γ invariant for the action induced by ϑ0.

The prestable moduli problem of LG-quasimaps to the critical locusconsists of maps from prestable orbicurves to the complete intersection

8.1.2 LG-phase

We call the case where e<0 the LG-phase.In order to be G-invariant,a monomialwhich implies that any points with every pj=0 must be unstable,but for each m>0 and each j the monomialis G-invariant,so every point with at least one nonzero pjis θ-semistable.ThereforeAgain,since the Fjand the image of the Pl¨ucker embedding are transverse,the equationsimply that the critical locus isinside

This phase does not immediately fit into our theory because we have an infinite stabilizer SL(k,C)for any points of the form(0,p1,···,ps).This means that the quotient[V//θG]is an Artin stack(not Deligne-Mumford).

Hori-Tong[21]analyzed the gauged linear sigma model of the Calabi-Yau complete intersection Z1,···,1⊂Gr(2,7)which is defined by seven linear equations in the Pl¨ucker coordinates.They gave a physical derivation that its LG-phase is equivalent to the Gromov-Witten theory of the so-called Pfaffian variety

It is interesting to note that the Pfaffianis not a complete intersection.For additional work on this example,see[2,22,25,31].

8.2 Complete intersections in a flag variety

Another class of interesting examples is that of complete intersections in partial flag varieties.The partial flag variety Flag(d1,···,dk)parametrizes the space of partial flags

such that dimVi=di.The combinatorial structure of the equivariant cohomology of Flag(d1,···,dk)is a very interesting subject in its own right.

For our purposes,Flag(d1,···,dk)can be constructed as a GIT or symplectic quotient of the vector space

To consider the gauged linear sigma model for the complete intersection F1= ···=Fs=0 of such sections,we again consider the vector space

We may describe the polarization as

This gives a phase structure similar to a complete intersection in a product of projective spaces.

For example,when ei>0 for all i∈ {1,···,k − 1},we can choose a compatibleaction with weight 1 on pjand weight 0 on each Ai,and again the trivial lift ϑ0is a good lift of θ in this phase.

This example should be easy to generalize to complete intersections in quiver varieties.It would be very interesting to calculate the details of our theory for these examples.

9 General Comments

When G is non-Abelian or the R-charge is not integral,G andinteract in a nontrivial way and the description of moduli space is more complicated.For more details,we encourage readers to consult[19].

An important technique that we have not touched on here is cosection localization,which is the main tool for constructing a virtual cycle for the GLSM.Starting from the noncompact stack of LG-quasimaps tothe cosection localization technique enables us to construct a virtual cycle supported on the compact substack of LG-quasimaps to the critical locus of W.

Finally,we remark that the choice of stability condition in our paper is by no means unique.There are other choices of stability conditions that result in different moduli spaces.Please see[5,12]for examples.