Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds

An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

In this paper, we work over C.

Gromov-Witten invariants of a smooth projective variety.
Let X be a smooth projective variety. Gromov-Witten invariants of X are virtual counts of parametrized algebraic curves of X. More precisely, let M g,n (X, β) be the Kontsevich's moduli space of n-pointed, genus g, degree β stable maps to X, where β ∈ H 2 (X, β) is an effective curve class. It is a proper Deligne-Mumford stack with a perfect obstruction theory of virtual dimension where stands for the pairing between the (rational) homology and cohomology. There is a virtual fundamental class [30,4,2] [M g,n (X, β)] vir ∈ A d vir (M g,n (X, β); Q) which can also be viewed as an element in H 2d vir (M g,n (X, β); Q). The virtual fundamental class defines a Q-linear map For i = 1, . . . , n, let ev i : M g,n (X, β) → X be the evaluation map at the i-th marked point. Genus g, degree β descendant Gromov-Witten invariants of X are defined by (2) τ a 1 (γ 1 ), . . . , τ a n (γ n ) X g,β := where γ i ∈ H * (X; Q), a i ∈ Z ≥0 , and ψ i ∈ H 2 (M g,n (X, β); Q) are ψ-classes (to be defined in Section 3.4). If γ i ∈ H d i (X; Q) then the (2) vanishes unless If X is a smooth algebraic variety which is not projective, then M g,n (X, β) is usually not proper, so [M g,n (X, β)] vir is not defined. If X is not projective but M g,n (X, β) is proper for some particular g, n, β, then [M g,n (X, β)] vir exists and the invariants in (2) are defined for such g, n, β. Genus g, degree β T-equivariant descendant Gromov-Witten invariants of X are defined by

Equivariant Gromov-Witten invariants and virtual localization.
(3) τ a 1 (γ T 1 ), . . . , τ a n (γ T n ) X T g,β := [M g,n (X,β)] vir where γ T i ∈ H * T (X; Q), and ψ T i ∈ H 2 T (M g,n (X, β); Q) is a T-equivariant lift of ψ i ∈ H 2 (M g,n (X, β); Q) (see Section 3.5). If γ T i ∈ H d i T (X; Q) then τ a 1 (γ T 1 ), . . . , τ a n (γ T n ) X T g,β ∈ H In particular, (3) The T-equivariant Gromov-Witten invariants (3) are related to the Gromov-Witten invariants (2) as follows. Let γ i be the image of γ T i under the ring ho- Then The torus fixed part of the restriction of the T-equivariant perfect obstruction theory to the T fixed substack M g,n (X, β) T ⊂ M g,n (X, β) defines a perfect obstruction theory on M g,n (X, β) T and a virtual class [M g,n (X, β) T ] vir T ∈ A T * (M g,n (X, β) T ; Q); the torus moving part defines the virtual normal bundle N vir of the inclusion M g,n (X, β) T ⊂ M g,n (X, β). By localization of virtual fundamental class [17,3], the RHS of (3) is equal to Suppose that X is non-compact, and for some g, n, β, M g,n (X, β) is not proper but M g,n (X, β) T is. Then the RHS of (3) is not defined, but (4) is. In this case, we define T-equivariant Gromov-Witten invariants τ a 1 (γ T 1 ), . . . , τ a n (γ T n ) X T g,β by (4), which is an element in Q(u 1 , . . . , u m ) instead of Q[u 1 , . . . , u m ].

Algebraic GKM manifolds and their Gromov-Witten invariants.
In this paper, an algebraic GKM manifold, named after Goresky-Kottwitz-MacPherson, is a non-singular algebraic variety equipped with an algebraic action of T = (C * ) m , such that there are finitely many torus fixed points and finitely many one-dimensional orbits. Examples of algebraic GKM manifolds include toric manifolds, Grassmanians, flag manifolds, etc.
If X is an algebraic GKM manifold then each connected component of M g,n (X, β) is, up to some quasi-finite map, a product of moduli stacks of pointed stable curves, and the RHS of (4) can be expressed in terms Hodge integrals on moduli stacks of pointed stable curves. This algorithm was first described by Kontsevich for genus zero Gromov-Witten invariants of P r in 1994 [28], before the construction of virtual fundamental class and the proof of virtual localization. The moduli spaces M 0,n (P r , d) of genus zero stable maps to P r are proper smooth DM stacks, so there exists a fundamental class [M 0,n (P r , d)] ∈ H * (M 0,n (P r , d); Q), and one may apply the classical Atiyah-Bott localization formula [1] in this case. H. Spielberg derived a formula of genus zero Gromov-Witten invariants of toric manifolds in his thesis [43]. Localization computations of all genus equivariant Gromov-Witten invariants of toric manifolds can be found in [32]. The main purpose of this paper is to provide details of the virtual localization calculations of all genus equivariant Gromov-Witten invariants for general algebraic GKM manifolds.
1.4. Outline. In Section 2, we define algebraic GKM manifolds and their GKM graphs, following [18,19]. In Section 3, we give a brief review of Gromov-Witten theory. In Section 4, we compute all genus equivariant descendant Gromov-Witten invariants of an arbitrary algebraic GKM manifold by virtual localization. of generalizing the computations for toric manifolds in [32] to GKM manifolds. The second author would like to thank the Columbia University for hospitality during his visits. This work is partially supported by NSF DMS-1159416 and NSF DMS-1206667.

ALGEBRAIC GKM MANFOLDS
In this section, we review the geometry of algebraic GKM manifolds, following [18], and introduce the GKM graph associated to an algebraic GKM manifold, following [19]. The GKM graph in this paper can be non-compact since we consider algebraic GKM manifolds which are not necessarily compact. In Section 4, we will see that the GKM graph contains all the information needed for computing Gromov-Witten invariants and equivariant Gromov-Witten invariants of the GKM manifold.
2.1. Basic notation. Let X be a non-singular algebraic variety of dimension r. We say X is an algebraic GKM manifold if it is equipped with an algebraic action of a complex algebraic torus T = (C * ) m with only finitely many torus fixed points and finitely many one-dimensional orbits.
Let N = Hom(C * , T) ∼ = Z m be the lattice of 1-parameter subgroups of T, and let M = Hom(T, C * ) be the lattice of irreducible characters of T. Then M = Hom(N, Z) is the dual lattice of N. Let N R = N ⊗ Z R and M R = M ⊗ Z R, so that they are dual real vector spaces of dimension k. Let K = U(1) m be the maximal compact subgroup of T. Then N R can be canonically identified with the Lie algebra of K. Let N Q = N ⊗ Z Q and let M Q = M ⊗ Z Q. Then M Q can be canonically identified with H 2 T (point; Q). We make the following assumption on X.

Assumption 5.
(1) The set X T of T fixed points in X is non-empty. (2) The closure of a one-dimensional orbit is either a complex projective line P 1 or a complex affine line C.
Note that (1) and (2) hold when X is proper. Indeed, if X is proper then the closure of any one-dimensional orbit is P 1 .

Example 6.
If X is a non-singular toric variety defined by a finite fan, then X is an alegebraic GKM manifold. Given t ∈ T, let φ t : C m → C m be defined by φ t (z) = t · z. Let T act on Gr(k, m) by t · V = φ t (V), where V is a k-dimensional linear subspace of C m . Given J ⊂ {1, . . . , m}, let J c := {1, . . . , m} \ J, and define The torus-fixed points in Gr(k, m) are So there are ( m k ) torus-fixed points in G(k, m). Let C J and C J ′ be distinct T-fixed points in Gr(k, m). Then There is a torus-fixed line connecting C J and C J ′ if and only of |J ∩ J ′ | = k − 1. In this case, |J ∪ J ′ | = k + 1. The T-fixed lines in G(k, m) are Let X be an algebraic GKM manifold of dimenison r, so that T = (C * ) m acts algebraically on X.
(1) (Vertices) We assign a vertex σ to each torus fixed point p σ in X.
(2) (Edges) We assign an edge ǫ to each one-dimensional O ǫ in X. Let ℓ e be the closure of O ǫ . (3) (Flags) The set of flags in the graph Υ is given by The Assumption 5 can be rephrased in terms of the graph Υ.
Given a flag (ǫ, σ) ∈ F(Υ), let w(ǫ, σ) ∈ M = Hom(T, C * ) be the weight of T-action on T p σ ℓ ǫ , the tangent line to ℓ ǫ at the fixed point p σ , namely, This gives rise to a map w : F(Υ) → M satisfying the following properties.
(1) (GKM hypothesis) Given any σ ∈ V(Υ), and any two distinct edges ǫ, ǫ ′ ∈ E σ , w(ǫ, σ) and w(ǫ ′ , σ) are linearly independent in M R . (2) Any edge ǫ ∈ E σ connecting the vertices σ, σ ′ ∈ V(Υ) satisfies the property that: . In particular, ǫ ′ r = ǫ r = ǫ and a r = 2. Let ǫ be as in (2). The normal bundle of ℓ ǫ ∼ = P 1 in X is given by We define the 1-skeleton of X to be the union of 1-dimensional orbit closures: The formal completionX of X along the 1-skeleton X 1 , together with the T-action, can be reconstructed from the graph Υ and w : F(Υ) → M. We call (Υ, w) the GKM graph of X with the T-action. If ρ : T ′ → T is a homomorphism between complex algebraic tori, then T ′ acts on X by

GROMOV-WITTEN THEORY
In this section, we give a brief review of Gromov-Witten theory and equivariant Gromov-Witten theory.

Moduli of stable curves and Hodge integrals.
An n-pointed, genus g prestable curve is a connected algebraic curve C of arithmetic genus g together with n ordered marked points x 1 , . . . , x n ∈ C, where C has at most nodal singularities, and x 1 , . . . , x n are distinct smooth points. An n-pointed, genus g prestable curve (C, x 1 , . . . , x n ) is stable if its automorphism group is finite, or equivalently, Let M g,n be the moduli space of n-pointed, genus g stable curves, where n, g are nonnegative integers. We assume that 2g − 2 + n > 0, so that M g,n is nonempty. Then M g,n is a proper smooth Deligne-Mumford stack of dimension 3g − 3 + n [8,26,24,25]. The tangent space of M g,n at a moduli point [(C, x 1 , . . . , x n )] ∈ M g,n is given by Since M g,n is a proper Deligne-Mumford stack, we may define We now introduce some classes in A * (M g,n ). There is a forgetful morphism π : M g,n+1 → M g,n given by forgetting the (n + 1)-th marked point (and contracting the unstable irreducible component if there is one): where (C st , x 1 , . . . , x n ) is the stabilization of the prestable curve (C, x 1 , . . . , x n ). π : M g,n+1 → M g,n can be identified with the universal curve over M g,n .
The Hodge bundle E = π * ω π is a rank g vector bundle over M g,n whose fiber over the moduli point • (ψ classes) The i-th marked point x i gives rise a section s i : M g,n → M g,n+1 of the universal curve. Let L i = s * i ω π be the line bundle over M g,n whose fiber over the moduli point Hodge integrals are top intersection numbers of λ classes and ψ classes: By definition, (10) is zero unless a 1 + · · · + a n + k 1 + 2k 2 + · · · + gk g = 3g − 3 + n.

3.2.
Moduli of stable maps. Let X be a nonsingular projective or quasi-projective variety, and let β ∈ H 2 (X; Z). An n-pointed, genus g, degree β prestable map to X is a morphism f : The notion of stable maps was introduced by Kontsevich [28].
The moduli space M g,n (X, β) of n-pointed, genus g, degree β stable maps to X is a Deligne-Mumford stack which is proper when X is projective [5].

Obstruction theory and virtual fundamental classes.
The tangent space T 1 and the obstruction space fit in the tangent-obstruction exact sequence:  Let X be a nonsingular projective variety. We say X is convex if H 1 (C, f * TX) = 0 for all genus 0 stable maps f . Projective spaces P n , or more generally, generalized flag varieties G/P, are examples of convex varieties. When X is convex and g = 0, the obstruction sheaf T 2 = 0, and the moduli space M 0,n (X, β) is a smooth Deligne-Mumford stack.
In general, M g,n (X, β) is a singular Deligne-Mumford stack equipped with a perfect obstruction theory: there is a two term complex of locally free sheaves E → F on M g,n (X, β) such that is an exact sequence of sheaves. (See [4] for the complete definition of a perfect obstruction theory.) The virtual dimension d vir of M g,n (X, β) is the rank of the virtual tangent bundle T vir = F ∨ − E ∨ .
Suppose that M g,n (X, β) is proper. (Recall that if X is projective then M g,n (X, β) is proper for any g, n, β.) Then there is a virtual fundamental class The virtual fundamental class has been constructed by Li-Tian [30], Behrend-Fantechi [4] in algebraic Gromov-Witten theory. The virtual fundamental class allows us to define Let ev i : M g,n (X, β) → X be the evaluation at the i-th marked point: ev i sends These are known as the primary Gromov-Witten invariants of X. More generally, we may also view [M g,n (X, β)] vir as a class in H 2d (M g,n (X, β)). Then (15) is defined for ordinary cohomology classes γ 1 , . . . , γ n ∈ H * (X), including odd cohomology classes which do not come from A * (M g,n (X, β)).
Let π : M g,n+1 (X, β) → M g,n (X, β) be the universal curve. For i = 1, . . . , n, let s i : M g,n (X, β) → M g,n+1 (X, β), be the section which corresponds to the i-th marked point. Let ω π → M g,n+1 (X, β) be the relative dualizing sheaf of π, and let L i = s * i ω π be the line bundle over M g,n (X, β) whose fiber at the moduli point [ f : (C, x 1 , . . . , x n ) → X] ∈ M g,n (X, β) is the cotangent line T * x i C at the i-th marked point x i . The ψ-classes are defined to be We use the same notation ψ i to denote the corresponding classes in the ordinary cohomology group H 2 (M g,n (X, β)).
Suppose that γ i ∈ H d i (X). Then (16) is zero unless

Equivariant Gromov-Witten invariants.
Let X be a non-singular projective or quasi-projective algebraic variety, equipped with an algebraic action of T = (C * ) m . Then T acts on M g,n (X, β) by where (t · f )(z) = t · f (z), z ∈ C. The evaluation maps ev i : M g,n (X, β) → X are T-equivariant and induce ev * i : A * T (X; Q) → A * T (M g,n (X, β); Q). Suppose that M g,n (X, β) is proper, so that there are virtual fundamental classes Given γ i ∈ A d i (X; Q) = H 2d i (X; Q) and a i ∈ Z ≥0 , define τ a i (γ 1 ) · · · τ a n (γ n ) X g,β as in Section 3.4: By definition, (18) is zero unless ∑ n i=1 d i = d vir . In this case, In this paper, we fix a choice of ψ T i as follows. A stable map f : (C, x 1 , . . . , x n ) → X induces C-linear maps T x i C → T f (x i ) X for i = 1, . . . , n. This gives rise to The T-action on X induces a T-action on TX, so that TX is a Tequivariant vector bundle over X, and ev * i TX is a T-equivariant vector bundle over M g,n (X, β). Let T act on L i such that where Q[u 1 , . . . , u m ](k) is the space of degree k homogeneous polynomials in u 1 , . . . , u m with rational coefficients. In particular, where Let M g,n (X, β) T ⊂ M g,n (X, β) be the substack of T-fixed points, and let i : M g,n (X, β) T → M g,n (X, β) be the inclusion. Let N vir be the virtual normal bundle of substack M g,n (X, β) T in M g,n (X, β); in general, N vir has different ranks on different connected components of M g,n (X, β) T . By virtual localization, If M g,n (X, β) T is proper but M g,n (X, β) is not, we define When M g,n (X, β) is not proper, the right hand side of (22) is a rational function (instead of a polynomial) in u 1 , . . . , u m . It can be nonzero when ∑ d i < d vir , and does not have a nonequivariant limit (obtained by setting u i = 0) in general.

VIRTUAL LOCALIZATION
In this section, we compute all genus equivariant descendant Gromov-Witten invariants of any algebraic GKM manifold by virtual localization. This generalizes the toric case in [32,Section 5].
Let X be an algebraic GKM manifold of dimension r, with an algebraic action of T = (C * ) m .
We first give a formal definition.

Definition 23.
A decorated graph Γ = (Γ, f , d, g, s) for n-pointed, genus g, degree β stable maps to X consists of the following data.
We now describe the geometry and combinatorics of a stable map f : (C, x 1 , . . . , x n ) → X which represents a T-fixed point in M g,n (X, β).
For any t ∈ T, there exists an automorphism φ t : (C, There are two possibilities: We define a decorated graph Γ associated to f : (C, x 1 , . . . , x n ) → X as follows.
(1) (Vertices) We assign a vertex v to each connected component (2) (Edges) For any ǫ ∈ E(Υ), let O ǫ ∼ = C * be the 1-dimensional orbit whose closure is ℓ ǫ . Then where the right hand side is a disjoint union of connected components. We assign an edge e to each connected component O e ∼ = C * of f −1 (X 1 \ X T ).
The above (1), (2), (3) define a decorated graph Γ = (Γ, f , d, g, s) satisfying the constraints (i) and (ii) in Definition 23. Therefore Γ ∈ G g,n (X, β). This gives a map from M g,n (X, β) T to the discrete set G g,n (X, β). Let F Γ ⊂ M g,n (X, β) T denote the preimage of Γ. Then M g,n (X, β) T = Γ∈G g,n (X,β) F Γ where the right hand side is a disjoint union of connected components. We next describe the fixed locus F Γ associated to each decorated graph Γ ∈ G g,n (X, β). For later convenience, we introduce some definitions.

Definition 24. Given a vertex v ∈ V(Γ), we define
the set of edges emanating from v, and define There are three types of unstable vertices: The set of stable flags is defined to be Given a decorated graph Γ = (Γ, f , d, g, s), the curves C e and the maps f | C e : The automorphism group A Γ for any point [ f : (C, x 1 , . . . , x n ) → X] ∈ F Γ fits in the following short exact sequence of groups: where Z d e is the automorphism group of the degree d e morphism f | C e : C e ∼ = P 1 → ℓ ǫ e ∼ = P 1 , and Aut( Γ) is the automorphism group of the decorated graph Γ = (Γ, f , d, g, s).
There is a morphism i Γ : M Γ → M g,n (X, β) whose image is the fixed locus F Γ associated to Γ ∈ G g,n (X, β).

Virtual tangent and normal bundles.
Given a decorated graph Γ ∈ G g,n (X, β) and a stable map f : (C, x 1 , . . . , x n ) → X which represents a point in the fixed locus F Γ associated to Γ, let T acts on B 1 , B 2 , B 4 , B 5 . Let B m i and B f i be the moving and fixed parts of B i . We have the following exact sequences: The irreducible components of C are The nodes of C are We have .
Recall that To unify the stable and unstable vertices, we use the following convention for the empty sets M 0,1 and M 0,2 . Let w 1 , w 2 be formal variables.
The following lemma shows that the conventions (i) and (ii) are consistent with the stable case M 0,n , n ≥ 3.
(b) The cases n = 1 and n = 2 follow from the definitions (28) and (30), respectively. For n ≥ 3, we have We twist the above short exact sequence of sheaves by f * TX. The resulting short exact sequence gives rise a long exact sequence of cohomology groups where for i = 0, 1. We have Lemma 34. Let σ ∈ V(Υ), so that p σ is a T-fixed point in X. Define Then (35) w(σ) = ∏ ǫ∈E σ w(ǫ, σ).
From the above discussion, we conclude that where w(σ), h(σ, g), and h(ǫ, d) are defined by (35) We conclude that . . , u r ] be induced by the inclusion i σ : p σ → X. Then To unify the stable vertices in V S (Γ) and the unstable vertices in V 1,1 (Γ) , we use the following convention: for a ∈ Z ≥0 , In particular, (30) is obtained by setting a = 0. With the convention (41), we may rewrite (40) as The following lemma shows that the convention (41) is consistent with the stable case M 0,n , n ≥ 3.
4.4. Sum over graphs. Summing over the contribution from each graph Γ given in Section 4.3.4 above, we obtain the following formula.
Then we have the following formula for the generating function (47).