A note on quantum K-theory of root constructions

We consider K-theoretic Gromov-Witten theory of root constructions. We calculate some genus $0$ K-theoretic Gromov-Witten invariants of a root gerbe. We also obtain a K-theoretic relative/orbifold correspondence in genus $0$.


INTRODUCTION
1.1.Étale gerbes.Let X be a smooth projective variety over the complex numbers.An étale gerbe G over X, may be thought of as a fiber bundle over X whose fibers are the classifying stack BG of a certain finite group G. Geometric properties of G are of purely stack-theoretic nature.
In [17], physical theories on an étale gerbe G are considered, leading to the formulation of decomposition conjecture (also known as gerbe duality).Interpreted in mathematics, the decomposition conjecture for G asserts that the geometry of G is equivalent to the geometry of a disconnected space Ĝ equipped with a C * -gerbe.The decomposition conjecture has been proven in several mathematical aspects in [22].
1.2.Gromov-Witten theory.Gromov-Witten theory of a target Z is defined using moduli stacks K g,n (Z, d) of stable maps to Z. Gromov-Witten invariants of Z are integrals of natural cohomology classes on K g,n (Z, d) against the virtual fundamental class of K g,n (Z, d).
1.3.Quantum K-theory.Quantum K-theory, introduced in [13], [20], is the K-theoretic counterpart of Gromov-Witten theory.K-theoretic Gromov-Witten invariants of a target Z are Euler characteristics of natural K-theory classes on K g,n (Z, d) tensored with the virtual structure sheaf O vir Kg,n(Z,d) .An extension of quantum K-theory to target Deligne-Mumford stacks is given in [24].Quantum Hirzebruch-Riemann-Roch theorems [16], [24], [14], [15] imply that quantum Ktheory can be determined by (cohomological) Gromov-Witten theory.Since (cohomological) Gromov-Witten theory of étale gerbes has been shown to satisfy the decomposition conjecture in many cases, it is natural to ask if quantum K-theory of an étale gerbe G can be studied with a viewpoint towards the decomposition conjecture.This note contains an attempt to address this for root gerbes over X in genus 0.
1.4.Root gerbes.Given a line bundle L → X and an integer r > 0, one can associate the stack r L X of r-th roots of L, which is a smooth Deligne-Mumford stack whose points over an Xscheme The coarse moduli space of r L X is X.Furthermore, the natural map ρ ∶ r L X → X has the structure of a µ r -gerbe.
The strategy employed to study quantum K-theory of r L X in this note is the same as that of [5].Namely, we examine the structure of moduli stacks of genus 0 stable maps to r L X and apply pushforward results for virtual structure sheaves.The main result of this note is Proposition 2.2.1.5.Root stacks.Given a smooth irreducible divisor D ⊂ X and an integer r > 0, one can associate the stack X D,r of r-th roots of X along D. In [1], genus 0 relative Gromov-Witten invariants of (X, D) and Gromov-Witten invariants of X D,r are shown to be the same when r is sufficiently large.Their proof uses pushforwards of virtual fundamental classes and an intermediate moduli space.In Section 2.6, we explain how to adapt their argument to obtain a similar result for genus 0 K-theoretic Gromov-Witten invariants, see (2.13).
1.6.Outline.The rest of this note is organized as follows.Section 2.1 recalls notations used the definition of K-theoretic Gromov-Witten invariants of Deligne-Mumford stacks.In Section 2.2 we discuss properties of the structure morphism for moduli stacks of genus 0 stable maps to a root gerbe.In Section 2.3 we discuss pushforwards of virtual structure sheaves.Section 2.4 contains the proof of our main result and Section 2.5 discusses an extension of the main result to a more general class of gerbes.In Section 2.6, we discuss a K-theoretic version of relative/orbifold correspondence.In Section 3 we discuss some related questions.1.7.Acknowledgment.This note is inspired by the results on virtual pushforwards in K-theory in [10] and [11].It is a pleasure to thank the authors Y.-C.Chou, L. Herr, and Y.-P.Lee.The author also thanks E. Sharpe for discussions.The author is supported in part by Simons Foundation Collaboration Grant.

RESULTS
2.1.Quantum K-theory of target stacks.We begin with recalling the definition of K-theoretic Gromov-Witten invariants of Deligne-Mumford stacks, as given in [24].Let Z be a smooth proper Deligne-Mumford stack with projective coarse moduli space Z.The moduli stack of n-pointed genus g degree d stable maps to Z is denoted by K g,n (Z, d).The detailed definition can be found in [4].It is known that K g,n (Z, d) is a proper Deligne-Mumford stack equipped with a perfect obstruction theory, see [4], [3].Applying the recipe of [20] to this perfect obstruction theory yields a virtual structure sheaf O vir Kg,n(Z,d) .There are evaluation maps ev i ∶ K g,n (Z, d) → ĪZ, where ĪZ is the rigidified inertia stack of Z. See [3] for more details on the construction of evaluation maps.
K-theoretic Gromov-Witten invariants of Z are Euler characteristics of the following form: We examine (2.2) in the special case Z = r L X and g = 0.
As explained in [5, Section 3.1], the rigidified inertia stack of r L X is a disjoint union of components Ī( r L X) g indexed by g ∈ µ r .As in [5, Definition 3.3], for g 1 , ..., g n ∈ µ r , put In order for K 0,n ( r L X, d) ⃗ g to be non-empty, the elements g 1 , ..., g n are required to satisfy certain condition, see [5,Section 3.1].
We consider the restriction of (2.2) to K 0,n ( r L X, d) ⃗ g : The structure of the map p has been analyzed in [5].We reproduce [5,Diagram (26)] as follows: (2.4) Here M 0,n is the stack of n-pointed genus 0 prestable curves (see e.g.[7] for a discussion), and M tw 0,n is the stack of n-pointed genus 0 prestable twisted curves (see [21]).M 0,n,d and M tw 0,n,d are variants of M 0,n and M tw 0,n parametrizing prestable (twisted) curves weighted by d ∈ H 2 (X, Z), see [5, Section 3.2] for an introduction and [7] and [26] for further details.
In Example 2.1.When X is a point, the line bundle L is necessarily trivial.In this case r L X = Bµ r .The moduli stacks K 0,n (Bµ r ) ⃗ g and K 0,n (pt) = M 0,n are smooth of expected dimensions.The morphism (2.3) in this case has been studied in [8].It is shown in [8] that there is a factorization K 0,n (Bµ r ) ⃗ g → N → M 0,n , where K 0,n (Bµ r ) ⃗ g → N is the stack of r-th roots of certain line bundle, and N → M 0,n is a root construction.

2.3.
Pushforward.We now examine obstruction theories.Since the map s ′′ is étale, the standard obstruction theory on K 0,n (X, d) relative to M 0,n can be viewed as a obstruction theory d) relative to the morphism q.The stack P can be equipped with an obstruction theory relative to the morphism q ′ by pulling back ,d) .The stack P ⃗ g n can be equipped with an obstruction theory relative to Y ⃗ g 0,n,d by pulling back the obstruction theory on P .Since both maps s ′ and Y ⃗ g 0,n,d → M tw 0,n,d are étale [5,Lemma 3.15], the standard obstruction theory on K 0,n ( r L X, d) ⃗ g relative to M tw 0,n can be viewed as an obstruction theory g .We then have the following results on virtual structure sheaves.
2.5.Banded abelian gerbes.Suppose G is a finite abelian group.Suppose G → X is a gerbe banded by G. Then the isomorphism class of G → X is classified by the cohomology group H 2 (X, G), where G is viewed as a constant sheaf on X.We say that G → X is essentially trivial if the image of its class is trivial for maps H 2 (X, G) → H 2 (X, C * ) induced by group homomorphisms G → C * .Examples of essentially trivial gerbes include toric gerbes [25].
Let G → X be an essentially trivial gerbe over X.Then by [5,Lemma A.2], G is of the form where L 1 , ..., L k are line bundles over X and r 1 , ..., r k are natural numbers.Consider the morphism (2.2) in this case: By the alaysis of [5, Appendix A], (2.8) also fits into diagram like (2.4), with a factorization (2.9) Here ⃗ g is defined in [5,Definition A.5].The map P ⃗ g n → K 0,n (X, d) is by construction virtually birational, hence we can apply [11, Theorem 1.12] to it.By [5,Theorem A.6], the map K 0,n (G, d) ⃗ g → P ⃗ g n is also a gerbe, so we can apply [11, Proposition 1.9] to it.Therefore, we may repeat the arguments in Section 2.4 to extend Proposition 2.2 to essentially trivial banded abelian gerbes G → X.
2.6.Root stacks.Let D ⊂ X be a smooth irreducible divisor.For an integer r > 0, one can construct the stack X D,r of r-th roots of X along D, see [9] and [3,Appendix B].The natural map (2.10) X D,r → X is an isomorphism over X ∖ D and is a µ r -gerbe over D. Denote by D r ⊂ X D,r the inverse image of D under (2.10).
It is shown in [1] that genus 0 relative Gromov-Witten invariants of the pair (X, D) are the same as Gromov-Witten invariants of X D,r for r sufficiently large.Here we explain how their method can be adapted to K-theoretic Gromov-Witten theory.
3.1.1.Root gerbes.For h > 0, the genus-h version of the morphism (2.3), has been studied in [6].The map (3.1) is understood well enough so that a result on the pushforward of virtual fundamental classes is proven in [6].However, pushforward of virtual structure sheaves under (3.1) appears to be difficult.The key issue is that, in order to apply [11, Proposition 1.9, Theorem 1.12], we need (3.1) be factored into virtual birational maps and gerbes.A factorization of (3.1) was obtained for more general banded gerbes in [6,Diagram (41)].In our setting this gives For a root gerbe r L X → X, one can check that P ⃗ g h,n → K h,n (X, d) is also virtually birational.However, by the discussion of [6, Section 6.2], the map K h,n ( r L X, d) ⃗ g → P ⃗ g h,n is a composition of two maps, one has degree 1 r and the other had degree r 2h > 1.The degree r 2h -map cannot possibly be a gerbe.Hence [11, Proposition 1.9] is not applicable to K h,n ( r L X, d) ⃗ g → P ⃗ g h,n .This prevents us from obtaining genus-h version of Proposition 2.2.
3.1.2.Root stacks.The relative/orbifold correspondence in cohomological Gromov-Witten theory has been extended to higher genus in [27].A K-theoretic relative/orbifold correspondence in higher genus is an interesting question.It is unlikely that virtual pushforwards used in genus 0 will be enough in higher genus.Some foundational work in K-theoretic Gromov-Witten theory is required in order to follow the arguments in [27].

On virtual pushforward.
There are many situations in cohomological Gromov-Witten theory in which "virtually birational" maps occur, see [18] for a detailed list.In addition, we note that the morphism u in [12,Lemma 4.16] is virtually birational.Hence we can apply [11,Theorem 1.12] to obtain a calculation of the K-theoretic J-function of weighted projective spaces.Since such a result is a special case of the work [28] on quantum K-theory of toric stacks, we do not persue it in details.
3.3.On decomposition conjecture.Consider an étale gerbe G → X.As defined in (2.1), Ktheoretic Gromov-Witten invariants of G have insertions coming from the K-theory K * ( ĪG) of the rigidified inertia stack ĪG.Since G ⊂ ĪG is a connected component, the K-theory K * (G) of G is a direct summand of K * ( ĪG).The proof of Proposition 2.2 only allows classes in K * ( Ī r L X) pulled back from X. Studying K-theoretic Gromov-Witten invariants of r L X with other kinds of insertions requires new ideas.
For root gerbes G → X arising in toric geometry, e.g.weighted projective spaces and more general toric gerbes, it may be possible to study the decomposition conjecture by analyzing the K-theoretic I-functions calculated in [28] in a manner similar to [25].An additive decomposition of the K-theory K( ĪG) is a basic question.
(2.4), the stack Y ⃗ g 0,n,d is constructed in [5, Definition 3.12] by applying the root construction to a certain divisor of M 0,n,d .It follows that the composition Y ⃗ g 0,n,d → M tw 0,n,d → M 0,n,d is proper and birational.The stacks P and P ⃗ g n are defined by cartesian squares.The map s is defined by [5, Lemma 3.18].