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Quantum K-Theory I: Foundations

This work is devoted to the study of the foundations of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth projective variety X, analogous to the relation between quantum cohomology and ordinary cohomology. This new quantum product also gives a new class of Frobenius manifolds.


Introduction
This work is devoted to the study of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth variety X, analogous to the relation between quantum cohomology and ordinary cohomology.
In order to understand quantum K-theory, it is helpful to give a brief review of quantum cohomology theory. Quantum cohomology studies the intersection theory (of tautological classes) on M g,n (X, β), the moduli spaces of stable maps from curves C to a smooth projective variety X. The intersection numbers, or Gromov-Witten invariants, look like [Mg,n(X,β)] vir where γ i ∈ H * (X), α ∈ H * (M g,n ) and ψ i are the cotangent classes. These notations will be defined in the next section. For now these numbers are pairings between natural cohomology classes and the (virtual) fundamental class of M g,n (X, β). These invariants at genus(C) = 0 determines a deformation of the product structure of the ordinary cohomology ring. The associativity of the quantum cohomology ring structure is equivalent to the WDVV equation, which is basically a degeneration argument by considering a flat family of solutions parameterized by P 1 and equating two different degeneration points of the solutions.
We propose to apply the same philosophy to K-theory. The functorial interpretation of the integration over the (virtual) fundamental classes [M g,n (X, β)] vir is the push-forward of cohomologies from M g,n (X, β) to the point Spec C (via the orientation defined by virtual fundamental class). In fact the push-forward operation can be performed in the relative setting for any proper morphisms. With this in mind, we define quantum K-invariants as the K-theoretic pushforward to Spec C of some natural vector bundles on M g,n (X, β) (via the orientation defined by the virtual structure sheaf). More precisely, where γ i ∈ K(X), α ∈ K(M g,n ) and L i are the cotangent line bundles. The K-theoretic push-forward χ will be defined in Section 2.1 and Section 4.1 for algebraic and topological K-theory respectively. The quantum K-product is defined by quantum K-invariants and a deformed metric (equation (17)). The associativity of quantum Kproduct will be established by a sheaf-theoretic version of WDVV-type argument. A considerable difference in K-theory and intersection theory arises here. Our main motivations for studying this theory come from two sources. The first is to use this theory to study the geometry of the moduli space of maps. For example, in joint works with R. Pandharipande and with I. Ciocan-Fontanine, a computation of quantum Kinvariants of X = P 1 are used to show that the Gromov-Witten loci ∩ i ev −1 i (pt i ) ⊂ M 0,n (P 1 , d) are not rational when n is large. This thus answers a question raised in [24] Section 2.2.
The other main motivation comes from the relation between Gromov-Witten theory and integrable systems. There are many celebrated examples of this sort in quantum cohomology theory. Two of them are most relevant. The first one is Witten-Kontsevich's theory of two dimensional gravity [18] [28]. This theory builds a link between (full) Gromov-Witten theory of a point to KdV hierarchy. The second example is Givental-Kim's theory: It states that genus zero Gromov-Witten theory on flag manifolds is governed by quantum Toda lattices [13] [16]. Our goal is therefore to extend this relation to quantum Ktheory. The analogous relation between (full) quantum K-theory of a point and discrete KdV hierarchy is yet to be spelled out. See [21] [22] for some computations in this direction. The relation between genus zero quantum K-theory on flag manifolds and finite difference Toda lattices turns out to be a very interesting one. The construction of finite difference Toda lattices involved quantum groups. The interested reader is referred to [14] for the details.
Besides these, we believe that the quantum K-invariants for symplectic manifolds are symplectic invariants. This assertion, if verified, could have some implications in symplectic geometry. Also, this theory could be understood as one manifestation of J. Morava's quantum generalized cohomology theory [27].
Acknowledgements. I am grateful to A. Givental and R. Pandharipande for much advice in the course of this work. The referees' reports have greatly improved the exposition of this work. In particular, the proof of Proposition 11 is drastically simplified. Thanks are also due to T. Graber, G. Kennedy, J. Morava, Y. Ruan and V. Srinivas for useful discussions. Part of the work is written during my stay in NCTS, whose hospitality is appreciated.

Virtual structure sheaf
The main purpose of this section is to define an element O vir in the Grothendieck group of coherent sheaves on the moduli space of stable maps to a smooth projective variety X.
2.1. Preliminaries on algebraic K-theory. The basic reference for algebraic K-theory discussed here is [1].
Let K • (V ) denote the Grothendieck group of locally free sheaves on V and K • (V ) denote the Grothendieck group of coherent sheaves on V . There is a cap-product For any proper morphism f : Convention. If f is an embedding, we will sometimes denote f * ([F ]) simply by [F ].
It is easy to see that regular embeddings and flat morphisms are perfect. Let can be defined as follows. Since v is a regular embedding, v * (O B ′ ) has a finite resolution of vector bundles on B. That is, O B ′ determines an element K • B ′ (B), the Grothendieck group of vector bundles on B whose homologies are supported on B ′ . This then determines an element, denoted by to be the Grothendieck group of the coherent sheaves on V with homologies supported on V ′ (i.e. exact off V ′ ). There is a canonical isomorphism ϕ : . One can introduce a generalized cap product by composing the isomorphism ϕ and the tensor product. The refined Gysin map is: . Remark 1. One can also apply the deformation to the normal cone argument to define the Gysin pull-back. That is, one can replace the fibre square (1) with are the normal cone and normal bundle of the embeddings u, v. The specialization to the normal cone argument works for K-theory as well, i.e. one has a map such that the Gysin map can be defined by combining σ V with the Gysin map v ! N of the above fibre square.
to emphasize the symmetry. If v : B ′ → B can be factored through a regular embedding i : B ′ → Y followed by a smooth morphism p : Y → B, i.e. v is a local complete intersection morphism. One can form the following fibre diagram Then q is smooth and we define v ! ([F ]) := i ! (q * [F ]). This definition is actually independent of the factorization.
Convention. To simplify the notation, we will often denote the class [ Definition. A stable map (C; x, f ) from a pre-stable curve (C; x 1 , . . . , x n ) of genus g curves with n marked points to X, which represents a class β ∈ H 2 (X), is a morphism f : C → X satisfying: 1. The homological push-forward of C satisfies f * ([C]) = β.

Stability:
The following inequality holds on the normalization E ′ of each irreducible component E: It is obvious from the definition that the notion of stable maps is a natural generalization of the notion of stable curves. The essential point being that they only allow only finite automorphisms. This makes the following theorem possible: The stack M g,n (X, β) of stable maps to X, a smooth projective scheme of finite type over C, is a proper Deligne-Mumford stack.
Between the various spaces in Gromov-Witten theory there are many interesting morphisms. The first such class of morphisms is called the forgetful morphisms ft k : M g,n+1 (X, β) → M g,n (X, β) by forgetting k-th marked points and stabilizing if necessary.
The second class is called the stabilization morphisms where M g,n is the moduli stack of stable curves. It is defined by forgetting the map and retaining the pre-stable curve, then stabilizing the curve. Of course one can just forget the map without stabilization. This leads to the prestabilization morphism [6] pres : M g,n (X, β) → M g,n , with M g,n being the moduli stack of prestable curves. The third class is called the evaluation morphisms ev i : M g,n (X, β) → X by evaluating the stable map at the i-th marked point. That is, Another object which we will use quite often is the universal cotangent line bundle L i on M g,n (X, β). It is defined as L i = x * i (ω C/M ), where ω C/M is the relative dualizing sheaf of the universal curve C → M g,n (X, β) and x i are the marked points.
There are combinatorial generalizations of M g,n (X, β) by incorporating the modular graphs τ with degree β [6]. The idea is to use the graphs to keep track of the combinatorics of degeneration of curves. This bookkeeping simplifies the notations in the proof of functorial properties of the virtual structure sheaf. We briefly recall the definitions here.
A graph τ is a quadruple (F τ , V τ , j τ , ∂ τ ) where F τ (flags) and V τ (vertices) are finite sets, j : F τ → F τ is an involution and ∂ : is the set of tails (or marked points) and is the set of edges. A modular graph is a pair (τ, g), where τ is a graph and g : V τ → Z ≥0 is a map. We call g(v) the genus of the vertex v. A modular graph is stable if and #F (v) is the valence of v. One defines the moduli stack of prestable curves associated to a modular graph (τ, g) as where M g,n is the usual moduli stack of prestable curves. The moduli stack of stable curves M τ,g associated to a stable modular graph (τ, g) is defined similarly Geometrically, the moduli stack of (pre)stable curves associated to a (connected) modular graph (τ, g) which has at least one edge consists of (connected) singular curves with singularity prescribed by τ . In other words, the stack of universal curves over the stack of the (pre)stable curves has singular generic fibres. For details, see [6].
Let (τ, g) be a modular graph and β : V τ → H 2 (X) be the degree. 1 (τ, g, β) is a stable modular graph with degree if The moduli stack of stable maps M (X, τ, g, β) 2 associated to (τ, g, β) is defined by the following three conditions: (1) If τ is a graph with only one vertex v, and set of flags F v are all tails, then M (X, τ, g, β) := M g(v),#Fv (X, β).
where τ 1 × τ 2 is the disjoint union of two graphs and τ stands for (τ, g, β), by abusing the notation.
(3) If σ is obtained from τ by cutting an edge, then M (X, τ ) is defined by the following fibre square where ∆ is the diagonal morphism, and i, j are the corresponding marked points where the edge is cut. It is easy to see that the morphisms and operations on M g,n (X, β) generalize to M(X, τ ). See [6] for the details.
2.3. The construction of virtual structure sheaf. We shall now introduce the notion of virtual structure sheaf, whose construction is parallel to that of virtual fundamental class as defined in [25] [5].
Let V → B be a morphism from a DM-stack to a smooth Artin stack with constant dimension. A relative perfect obstruction theory is a homomorphism φ in the derived category (of quasi-coherent sheaves bounded from above) of a two term complex of vector bundles If V admits a global embedding i : V → Y over B into a smooth Deligne-Mumford stack, then the two term cutoff of L • V /B at −1 is given by the first two terms in second exact sequence of relative Kähler differential: Let us now recall the definition of relative intrinsic normal cone.
Note that even if V does not allow a global embedding, one can still define the intrinsic normal cone by (étale) local embedding. It is proved in [5] that the local construction glues together to form a global cone stack.
Remark 3. One could also consider this as an element in the derived category of V defined to be where ⊗ der means the derived tensor. Either way, virtual structure sheaf is in general not a bona fide sheaf.
Apply the above setting to Gromov-Witten theory. Let M τ be the corresponding moduli stack associated with modular graph τ . Recall that there is a prestabilization morphism pres : M (X, τ ) → M τ , which will be our structure morphism V → B. Consider the universal stable map One then get a complex R • π * f * T X , which can be realized as a two term which is a relative perfect obstruction theory. One therefore defines the virtual structure sheaf O vir M (X,τ ) . Note that in the present case M (X, τ ) admits a global embedding (Appendix A of [15]). One might identify L • M (X,τ )/Mτ with its two-term cutoff. 2.4. Basic properties of virtual structure sheaf. Some basic properties of virtual structure sheaf is established here as a K-theoretic version of [5]. Because B will always be M τ in all our applications, the readers should feel free to make this assumption, although all the results will hold for B any smooth equidimensional Artin stack.
The first step to check the consistency of this definition is the following two propositions: Proof. The proof of this proposition is exactly the same as the proof of [4] Proposition 5.3.
, then V is a local complete intersection and the virtual structure sheaf is equal to ordinary structure sheaf. Note that the dimension is called "expected dimension" because it is the dimension in the ("generic") case of no obstruction.
Proof. The question can be reduced to the absolute case by the argument given in [15] Appendix B. Thus we will set B = pt. Since the question is local (inétale topology), one may assume that In fact, it is enough to consider the local ring with maximal ideal m. Denote I = (g 1 , . . . , g s ), then the two-term cut-off of cotangent complex is This induces a complex of R p modules One may assume that g i contains no linear terms in x i , or otherwise the presentation of R p could be changed to eliminate terms linear in x i . That is, I ⊂ m 2 and the above ϕ is a zero map. This implies that the homology of the above complex is Because E • → L • is a perfect obstruction theory, where the last equality is obvious. This shows, combining with the assumption, that dim Spec R p ≤ r −s. However, dim Spec R p is obviously greater or equal to r − s. Thus dim Spec R p = r − s and {g i } form a regular sequence. Therefore V is a local complete intersection.
On the other hand, dim Spec R p = r −s implies that h 1 (E •∨ ⊗C) = s and E • → L • is an isomorphism. Hence C = E 1 , and virtual structure sheaf is the ordinary structure sheaf on V .
The following two propositions form the technical heart of the virtual structure sheaves.
Consider a fibre square . Let E be a perfect relative obstruction theory for U over B and E ′ be a perfect relative obstruction theory for U ′ over B, where U, U ′ are DM stacks and B is a smooth (Artin) stack of constant dimension. 3 Consider the following fibre square of DM B-stacks Suppose that the morphism ν is a local complete intersection morphism of B-stacks that have finite unramified diagonal over B. Recall ( [5] Sect. 7) that E and E ′ are said to be compatible over ν if there exists a homomorphism of distinguished triangles The compatibility of obstruction theory means that we have the following short exact sequence of vector bundle stacks (6) α In general a vector bundle stack can be locally represented as a quotient of a vector bundle by another vector bundle.) In particular when ν is a local complete intersection morphism, one can find a smooth stack Y and V ′ → V factors through Y : when ν is smooth or V ′ and V are smooth.
The rest of this subsection is devoted to the proof of the above two propositions. It can be skipped without losing the logic flow.
Consider the following fibre diagram and C 2 is similarly defined by switching indices 1 and 2. C i are natu- Proof. The proof of Proposition 4 in [20] also proves this lemma: It is constructed there two principal Cartier divisors D and . Now apply the above to the diagram (5). (Similar arguments applies to diagram (4).) Assuming that ν is a regular embedding and β is an embedding, one has where the ρ * is the pull-back on cones induced from ρ. Let 0 : D ′ → N × V D be the zero section. By the deformation to normal cone argument For the readers who are familiar with the arguments in [5], the above lemmas easily implies the propositions. In the following we reproduce their arguments in K-theory.
Let us start with the proof of Functoriality. For simplicity, let B = pt. We will follow closely the proof of Proposition 5.10 in [5]. By Proposition 1 we may choose the global resolution [E ′ 0 → E ′ 1 ] of E ′ such that the right square of the following diagram commutes and φ i are surjective. F i := ker(φ i ) are vector bundles. This induces short exact sequence of vector bundle stacks . The compatibility of obstruction theory means that we have the following short exact sequence of vector bundle stacks (6) When ν is smooth or a regular embedding, Proposition 3.14 of [5] implies that we have a fibre square If ν is smooth, ν ! = µ * and we have If ν is a regular embedding of smooth stacks, then by Lemma 2. In general, when only smoothness of V ′ and V are assumed, one factors v as Apply the smooth case to the right square and the regular embedding case to the left square.
One can generalize the functoriality property to the case β an arbitrary morphism by applying the arguments in [5] Lemma 5.9. That is, to replace the morphism j : This ends the proof of Proposition4.
The proof of Proposition 3 is similar. Let E • = E −1 → E 0 . In the case v is flat, by the standard properties of (intrinsic) normal cones in K-group and proceed as above.
3. Axioms of virtual structure sheaves in quantum K-theory The axioms listed here are K-theoretic version of the Behrend-Manin axioms [6].

3.1.
Mapping to a point. Suppose that β = 0. In this case M (X, τ ) = M τ × X and where the notations are defined in (3). Then which is a complex of locally free sheaves with zero differential. Since M τ × X is smooth,

See Section 2.2.
Proof. It is obvious that the perfect obstruction theory of the product is the product of perfect obstruction theories on each component.

3.3.
Cutting edges. Let σ be a modular graph obtained from τ by cutting an edge as in the fibre diagram (2). In this case M σ = M τ =: M. Consider the fibre square where i, j are the markings created by the cut. The cutting edges axiom says that Proof. It is proved in [4] that the perfect obstruction theories of M (X, σ) and M (X, τ ) are compatible with respect to ∆. It remains to apply Proposition 4.
3.4. Forgetting tails. Let σ be obtained from τ by forgetting a tail (marked point) and let be the forgetful map. Then τ ) . Proof. The proof in [4] p. 610 shows that   [6] . The last one, isogenies axiom, is not literally true in K-theory. The reason is that even though one can identify fundamental classes of two cycles by a birational morphism, it is not true that the structure sheaves of these two cycles are identified. We therefore have to modify the isogenies axiom in quantum K-theory. Four types of isogenies are stated in K-theory in three axioms. The contractions axiom has to be modified, or "quantized", in K-theory. (The meaning of "quantization" will be clear in the next section.) 3.5. Fundamental classes. Let σ be obtained from τ by forgetting a tail. The degree of σ is induced from τ in the obvious way. Consider the commutative diagram Proof. The proof is the same as the proof in [4] pp. 611-3, with the (bivariant) cycle classes changed to (bivariant) K-classes. We summarize the spaces involved in the following diagram: where the last equality follows from Proposition 4.
. The proof is obvious.

3.7.
Contractions. Let φ : τ → σ be a map of stable modular graphs by contracting one edge or one loop, such that genus(σ) = genus(τ ). Denote e the edge or loop in question (in τ ). Letσ be a fixed modular graph obtained from σ by adding k tails and (σ, β) is a stable modular graph with degree.
(τ i 1 , β i j 1 ) are the stable modular graphs with degrees of the following form. τ i 1 are obtained from τ by adding k tails in ways compatible with σ → σ. Thus there is a natural map from τ i 1 toσ by contracting e, and the following diagram is commutative: For example, if M τ = M 0,4 ×M 0,5 and M σ = M 0,7 . Then M (X,σ, β) = M 0,7+k (X, β) and M (X, τ 2 , β 2 ) are of the form M 0,4+ka (X, β a )× X M 0,2+k b (X, β b )× X M 0,5+kc (X, β c ) such that k a + k b + k c = k and β a + β b + β c = β 2 . Here the notation M g 1 ,n 1 +1 (X, β 1 ) × X M g 2 ,n 2 +1 (X, β 2 ) is the fibre product Proof. Consider the following commutative diagram We will show an analogue of (10) in this setting. Namely, (11) Note that the vertical maps only contracts rational components with two marked points (or less). That is the reason why only chains of rational curves are relevant.
It is easy to see that M τ × M σ Mσ → Mσ is a divisor D τ of normal crossing in Mσ, and the "smoothing map" , the difference is supported on the normal crossing subset. The following lemma shows how one could obtain the structure sheaf O Dτ by the inclusion-exclusion principle.
Lemma 3. Let D = ∪ k i=1 D i be a divisor with normal crossing, such that D i are smooth, disjoint away from the origin. Furthermore locally at origin D is defined by x 1 . . . x k = 0. Then is an exact sequence.
Proof. Equation (12) is equivalent to the study the exactness of the following sequence locally at the origin This equation holds because, e.g. k = 2, the following sequence is exact. In the general case this follows from the inclusion-exclusion principle. For the exactness of (12) at points away from the origin, where l divisors intersect (l < k), it follows from the induction as the divisor locally defined by the equation y i 1 . . . y i l .
It remains to note that the normal crossing substack is stratified by the image of µ m : M τ i m → D τ with codimension m in Mσ. This concludes the proof of (11). Now consider the following diagram Let V ֒→ C N be a topological closed embedding. Define . (Remember that there is a Bott periodicity.) Note that this definition is independent of the closed embedding.
There is a cap product between homology and cohomology theories. In algebraic K-theory, it is just the tensor product. In topological Ktheory, it is defined as follows. Let V ⊂ C N be the closed embedding as above and U be a neighborhood of V in C N such that every element in K * (V ) extends to an element in K * (U).
Then the cap product is defined as where the second arrow is the cup product and the last arrow is the Bott periodicity.
One can define the push-forward is a closed embedding. Choose a closed embedding φ : V → C N as above. The push-forward is the composition where the ← is the Thom isomorphism, i.e. multiplying the canonical There is an obvious homomorphism K • (V ) → K 0 (V ) as every algebraic vector bundle is a topological one. One can also construct a homomorphism (14) T op : as follows. Choose a closed embedding of V to a smooth scheme i : V ֒→ Y . (In our applications, all schemes are quasi-projective.) For a bounded complex of coherent sheaves α • on V , i * (α • ) is quasiisomorphic to a bounded complex E • of locally free sheaves on Y , which is exact out of V . Regard E • as a a topological element and embed Y in some C n . By Thom isomorphism E • is identified as a bounded complex of topological vector bundles on C n , exact off V , which is then an element in K 0 (V ). A relation between the push-forward maps in two K-homology theories is given by the following theorem of Baum-Fulton-MacPherson: ) For a proper morphism g : X → Y , the following diagram: There is again a bivariant topological K-theory which unifies the above setting. The homomorphism (14) is also valid in the bivariant setting [9].
Remark 5. Although the K-theory of a moduli stack is quite different from the K-theory of its coarse moduli space, one can define the quantum K-invariants via coarse moduli space in the following way. Let p : M → M be the canonical map from the moduli stack to its coarse moduli space. First construct the virtual structure sheaf O vir M on the moduli stack, then push-forward p * (O vir ) to its coarse moduli space, which is a topological space. One gets ) by projection formula as ev : M → X factors through M. E is any Kelement on M, like L or H (Hodge bundle). This means that one gets the same quantum K-invariants on coarse moduli space if one takes the suitable virtual structure sheaf.
Therefore, one can define quantum K-invariants for topological Ktheory on topological spaces rather than orbispaces, by using the topological K-homology for topological space associated to coarse moduli space and T op(p * (O vir )) as the virtual structure sheaf. This eliminates the difficulty of constructing topological K-homology on orbispaces, which will be addressed in another paper.

4.2.
Definition of Quantum K-invariants. In this subsection we will propose a construction of K-theoretic invariants 5 of the Gromov-Witten type, which we call quantum K-invariants.
Let K • (V ) be the (algebraic or topological) K-cohomology of V and K • (V ) be the (algebraic or topological) K-homology of V . When V is smooth, the notation K(V ) is used to denote both K • (V ) and K • (V ) as they are isomorphic.
The quantum K-invariants is defined to be γ 1 , γ 2 , . . . , γ n ; F g,n,β := χ M g,n (X, β), where γ i ∈ K(X), F ∈ K(M g,n ) and χ is the push-forward to a point (Spec C). Note that the triple (g, n, β) is chosen so that M g,n (X, β) is defined. Note that due to Baum-Fulton-MacPherson Theorem 2 topological invariants will be equal to algebraic invariants (whenever applicable). We will not make distinctions. One could also include the gravitational descendents and define: From this point we assume the K-theory is the topological K-theory. The modification to algebraic K-theory is in most places straightforward. A key point will be discussed in Remark 10.
Let X be a smooth projective variety, and E ⊂ H 2 (X, Z) denote the semigroup of effective curve classes. Let C[E] be the semigroup ring determined by E. Since 0 ∈ E, C[E] has a unit element. For β ∈ E, the corresponding element of C[E] will be denoted by Q β .
Let m be the maximal ideal in C[E] generated by the nonzero elements of E. The Novikov ring N(X) is defined to be the completion of C[E] in the m-adic topology. Alternatively, N(X) may be defined by formal series in Q β : Let e 0 = O, e 1 , e 2 , . . . be a basis of K(X) Q and let t i be its dual coordinates. Define t := i t i e i . The K-theoretic Poincaré pairing (·, ·) : K(X) Q ⊗ K(X) Q → Q is defined to be (e i , e j ) = χ(e i ⊗ e j ). This is a perfect pairing whenever X is smooth. This defines a metric g ij := (e i , e j ) on K(X).
The quantum K-potential of genus 0 is a generating series of genus zero quantum K-invariants: . . , t 0,n,β .
Here (0, n, β) is a triple such that M 0,n (X, β) exists. It is obvious that We will see later that the metric in quantum K-theory ought to be "quantized" due to the modified contraction axiom ( § 3.7). Define the quantum K-metric to be (17) ((e i , e j )) = G ij := ∂ t i ∂ t j G(t).
G ij is the quantization of g ij because G ij | q=0 = g ij by definition. Define G ij to be the inverse matrix of G ij . One could also define the quantum K-classes for X to be a family of linear maps (cf. [19]) ). The equivalence of these two definitions follows from the fact that the Poincaré duality in K(M g,n , Q) is a perfect pairing.
Remark 6. 1. It is obvious how to include the modular graphs into our data and define quantum K-invariants associated to a modular graph.
2. Finally we mention that there is an equivariant version of these invariants. If there is an algebraic group (or compact Lie group) G acting on X, then the moduli space M g,n (X, β) (and its virtual structure sheaf) has a G action too. We can talk about the G-equivariant quantum K-invariants, completely parallel to equivariant quantum cohomology theory.

4.3.
On Kontsevich-Manin axioms in K-theory. The quantum K-invariants (without descendents) as defined above obviously can not satisfy all axioms of cohomological Gromov-Witten invariants [19]. The effectivity and motivic axioms follow from the construction. The grading and divisor axioms are missing, due to the lack of "dimension counting". We will discuss briefly the remaining five axioms below.
S n -covariance: The quantum K-class is covariant under S n action on the marked points.
Fundamental class: Let e 0 be the (K-class of) structure sheaf of X and π n : M g,n → M g,n−1 . Then (19) I X g,n,β (u 1 ⊗u 2 ⊗. . .⊗u n−1 ⊗e 0 ) = π * n (I X g,n−1,β (u 1 ⊗u 2 ⊗. . .⊗u n−1 )). Mapping to a point: Suppose that β = 0. Consider the following diagram M g,n × X where p i are the projection morphisms. Then (20) I X g,n,0 (u 1 ⊗. . .⊗u n ) = (p 1 ) * p * 2 (u 1 ⊗. . .⊗u n ⊗λ −1 (R 1 π * (O C ⊗T V )) ∨ ) Splitting: Fix g 1 , g 2 and n 1 , n 2 such that g = g 1 + g 2 , n = n 1 + n 2 . Let Φ : M g 1 ,n 1 +1 × M g 2 ,n 2 +1 → M g,n be the contraction map which glues the last marked point of M g 1 ,n 1 +1 to the first marked point of M g 2 ,n 2 +1 (contracting a non-looping edge). Then where the notation ft k : M g,n+k → M g,n stands for the forgetful map which forget the additional k marked points. Genus reduction: Let Φ : M g−1,n+2 → M g,n be the contraction map which glues the last two marked points (contracting a loop). Then The proofs of these five axioms follows from the corresponding axioms of virtual structure sheaves. For example, Contractions, Cutting Edges and Products imply the splitting and genus reduction axioms. The only novelty is the appearance of G ij in Splitting and genus reduction axioms. This has its origin in Contractions Axiom (Section 3.7). Let us write G ij (t, q) = g ij + F ij (t, q). The inverse matrix G ij (t, q) is therefore due to the matrix geometric series Note that the contributions from the contracted chains (denoted c m ) in Proposition 11 are exactly F a 1 b 1 g b 1 a 2 . . . F a m−1 b m−1 . The fact that the combinatorics involved can be simplified by introducing G ij is first observed in [12].
Proof. First notice that the virtual structure sheaves in the above equations can be simultaneously erased from both sides by the Fundamental class and Forgetting tails axioms. First the case g = 0. Let D i be the divisors on M 0,n+1 (X, β) such that the generic curves have two components: one contains the i-th and n + 1-th marked points, and the other contains the rest. It is well-known that (24) O(L i ) = π * l i ⊗ O(D i ) (see e.g. [28]), where for the notational convenience we have used l i and L i for i-th universal cotangent line bundles on M 0,n (X, β) and M 0,n+1 (X, β) respectively. It is easy to see that ( [21] Lemma 2) It remains to compute H 0 := R 0 π * O( i d i D i ), which is a vector bundle because R 1 = 0.
To compute H 0 , we use some local arguments. The fibre C x of π is a rational curve with n marked points x 1 , . . . , x n . An element of H 0 (C x , O( i d i D i )) is a rational function with poles of order no more than d i at x i . Therefore H 0 (C x , O( i d i D i )) is filtered by the degrees of poles at x i : It is by definition that the graded piece F k+1 /F k are isomorphic to T ⊗k x i . Summarizing, where "K-theory" above means the equation is valid only in K-theory as we have used the graded pieces.
It is now a matter of elementary computation (see Proposition 1 in [21]) to deduce the genus zero string equation from (26).
In the case g ≥ 1 the Hodge bundles H will naturally occur. Notice that H is actually equal to π * (H), and this implies that the factor 1/(1 − qH) commutes with π * due to the projection formula. By Grothendieck-Riemann-Roch formula, (23) is a rational function of q and q i 's. Therefore (27) π The above equality can be proved using the same arguments as in the genus zero case. By (25) and projection formula is a vector bundle. Now by Serre duality ( where ω is the dualizing sheaf. Fibrewisely, H 0 (C, ω( i d i x i )) is the holomorphic differential with poles of order at most d i at x i . Thus we have a filtration F 1 ⊂ F 2 ⊂ . . . ⊂ F d i of degrees of poles at each marked point x i as above and the graded bundles F k+1 /F k is isomorphic to l ⊗−k i and F 0 = H. Therefore Notice again that the last equality holds only in K-theory (using graded objects). This means which is equivalent to (23).
Proof. The proof is similar to the above proof of string equation. In the proof we again use l i for L i on M g,n (X, β). We will use Forgetting tail axiom, (24)(25) and projection formula Because of the relation L n+1 = ω(x 1 + . . . + x n ), It is easy to see that this is equivalent to (28).
This resembles the cohomological dilaton equation in the sense that the rank of the RHS is 2g − 2 + n.
Remark 8. The techniques used in proving String and Dilaton equations can also be used to find the push-forward of negative powers of L.

Quantum K-ring and Frobenius manifold
In this section we generalize Givental's treatment [12] to non-convex algebraic manifolds. Many statements already appeared in [12] (albeit with the convexity condition there) and are included here for completeness.
5.1. The ring structure in quantum K-theory. The definition of the ring structure of quantum K-theory is analogous to that of quantum cohomology theory.
Let {e i } be a basis of K(X) and t i its coordinates. t := i t i e i as in Section 4.2.
The quantum K-product * is defined to be: (29) ((e i * e j , e k )) : where G(t) is the genus zero potential defined in (16) and ((·, ·)) is the quantized metric defined in equation (17). The main result of this section is Proof. The deformation properties follow immediately from the definition. Setting q = 0 would make * the ordinary tensor product. The (super-)commutativity also follows from the definition.
The associativity follows from the Splitting Axiom (21). It is easy to see that the K-theoretic WDVV equation is equivalent to the associativity of the quantum K-ring. The WDVV equation follows from Splitting Axiom: Let n = 4, n 1 = 2, n 2 = 2 in (21). Different ways of splitting amount to the same invariants.
In the case t = 0, we have defined a deformation by quantum threepoint function, which is similar to the pair-of-pants structure. However, in this case the metric ought to be quantized as well (and is replaced by two-point function at t = 0).
Remark 9. For convex X, a proof can be found in [12].
Remark 10. The above description is good for topological K-theory. For algebraic K-theory, a finite basis does not necessarily exist and the K-theoretic Poincaré pairing has a huge kernel. However, one could proceed by using a presentation which is basis-free and does not rely on Poincaré pairing. Namely, one could define the quantum K-product (at t = 0) as is the alternating sum of virtual structure sheaves appeared in (10).

5.2.
Quantum K-theory and Frobenius (super-)manifolds. In this section, we show that the construction of quantum K-ring produces a (new) class of Frobenius manifolds, in the sense of [26]. This in particular gives a positive answer to the question raised by Bayer and Manin ([3] 1.1.1 Question). Note that in this case the identity element e 0 is not flat. The data of K-theoretic Frobenius structure includes: (1) A metric ((e i , e j )) = G ij := ∂ i ∂ j G(t) on tangent spaces to K(X) C .
on the tangent bundle.