Quantum K-theory of grassmannians and non-abelian localization

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of K-theoretic mirrors.


Introduction.
Just as quantum cohomology theory deals with intersection numbers between interesting cycles in moduli spaces of stable maps of holomorphic curves in a given target (say, a Kähler manifold), quantum K-theory studies sheaf cohomology (e.g. in the form of holomorphic Euler characteristics) of interesting vector bundles over these moduli spaces. While the beginnings of the subject can be traced back to the 20-year-old note [9] by the first-named author, the foundational work by Y.-P. Lee [24], and their joint paper on complete flag manifolds, q-Toda lattices and quantum groups [20], the interest to quantum Ktheory expanded in more recent years due to the discovery of its more diverse relations to representation theory, integrable systems, and qhypergeometric functions.
Apparently the interest was initiated by the 2012 preprint [27] by D. Maulik and A. Okounkov, who connected equivariant quantum cohomology of quiver varieties with R-matrices. In 2014, this led V. Rimányi, V. Tarasov and A. Varchenko [30] to a conjectural description of the quantum K-ring of the cotangent bundle of a partial flag variety. In even more recent literature motivated by representation theory (see e.g. [28,29,23]), certain q-hypergeometric series, interesting from the point of view of the theory of integrable systems, appeared as generating functions for K-theoretic Gromov-Witten (GW) invariants of symplectic quiver varieties. In this literature, K-theoretic computations are based, however, on the quasimap (rather than stable map) compactifications [4] of spaces of rational curves in the GIT quotients of linear spaces. Based on the experience with mirror symmetry and quantum K-theory of toric varieties [13] one anticipates the q-hypergeometric generating functions arising from quasimap spaces to nevertheless represent the "genuine" (i.e. based on stable map compactifications) K-theoretic GW invariants, yet such invariants of a different kind, or more complicated ones than naively expected. In any case, this brings up the question of comparison (first attempted by H. Liu [25]) of the two approaches.
In this paper, we examine in substantial detail the genus-0 quantum K-theory of grassmannians Gr n,N pCq. The grassmannians can be described as the GIT quotients HompC n , C N q{{GL n pCq, and are perhaps the simplest among homogeneous spaces or, more generally, quiver varieties outside the toric class. Most of our methods carry over to other quiver varieties (and all -to any partial flag manifolds), but we prefer to illustrate the available techniques by way of simplest representative examples, trading generality for simplicity of notation.
In Sections 1 and 2 we show how the technique of fixed point localization in moduli spaces of stable maps can be used in order to compute the so-called "small J-function" of the grassmannian -the generating function for simplest genus-0 K-theoretic GW-invariants of it.
In Section 3 we combine the same technique with the idea known as "non-abelian localization" [2] in order to prove the invariance of the genus-0 quantum K-theory of the grassmannian under a suitable infinite dimensional group of pseudo-finite-difference operators. A key point here (inspired by the appendix in the paper [22] by K. Hori and C. Vafa) is to begin with the toric quotient HompC n , C N q{{T n " pCP N´1 q n by the maximal torus T n Ă GL n pCq, and use Weyl-group invariant finite-difference operators on n Novikov's variables of the toric manifold. Just as in the case of toric manifolds [15], this infinite dimensional group of symmetries is large enough in order to reconstruct "all" genus-0 invariants of the grassmannian from the small J-function.
In Section 4, we address the aforementioned comparison problem by interpreting (in several somewhat different ways) the q-hypergeometric series arising from quasimap theory of the cotangent bundle spaces T˚Gr n,N as certain "genuine" K-theoretic GW-invariants, and in particular show that, contrary to a naive belief articulated in the literature, these series fail to represent "small" J-functions (of anything).
In Section 5, we apply the invariance result from Section 4 to illustrate the "non-abelian quantum Lefschetz" principle which characterizes genus-0 quantum K-theory of a complete intersection in (or a vector bundle space over) the grassmannian.
In Section 6, we show how our techniques can be used to extend (to the case of grassmannians) the toric results obtained by Y. Ruan and M. Zhang [31] about the level structures in quantum K-theory. As a byproduct, we clarify (hopefully) the phenomenon of level correspondence between "dual" grassmannians Gr n,N " Gr N´n,N discovered recently by H. Dong and Y. Wen [7].
In Section 7, we exhibit a Jackson-type integral formula for the small J-function in the quantum K-theory of the grassmannian, inspired by the "non-abelian localization" framework from Section 4. Our logic is the same as in the aforementioned appendix [22] by K. Hori and K. Vafa, where cohomological mirrors of Gr n,N were proposed. However, our mirror formula looks different (and possibly new, see [26]) even in the cohomological GW-theory.
Namely, our cohomological mirror of the grassmannian has the form of the complex oscillating integral Here X Q is the complex torus in the linear space with coordinates tx ij u, i " 1, . . . , n, j " 1, . . . , N and ty ii 1 u, i, i 1 " 1, . . . , n, i ‰ i 1 , given by n equations ź and Γ is a Lefschetz thimble in X Q , invariant under the Weyl group S n acting on the coordinates tx ij u, ty ii 1 u by simultaneous permutations of the indices i and i 1 .
As a mirror symmetry test, let us examine the critical set of the phase function ("superpotential") using Lagrange multipliers p 1 , . . . , p n : The critical points are determined from where the third set of equations comes from the constraints. The algebra of functions on the critical set (which is a finite lattice Z n N Ă T n ) invariant under permutations of pp 1 , . . . , p n q is indeed isomorphic to the "small" quantum cohomology algebra of the grassmannian (as described by formula (3.39) in [33]). by the National Science Foundation under Grant DMS-1906326. We are thankful to P. Koroteev and A. Smirnov for their effort in educating us about their work on quantum K-theory of symplectic quiver varieties, and to H. Liu and Y. Wen for sharing and discussing their preprints.
1. The small J-function of Gr n,N .
Let X :" Gr n,N be the grassmannian of n-dimensional subspaces V Ă C N . Its K-ring K 0 pXq is generated by the exterior powers Ź k V of the tautological bundle, k " 1, . . . , n. Using the splitting principle, we will often write them as elementary symmetric functions ř 1ďi 1 ă¨¨¨ăi k ďn P i 1¨¨¨P i k of K-theoretic Chern roots of V " P 1`¨¨¨`Pn . Proposition. The K-theoretic Poincaré pairing on K 0 pXq is given by residue formula where Φ is any symmetric Laurent polynomial of P 1 , . . . , P n .
The formula is obtained as the non-equivariant limit Λ Ñ 1 from its T N -equivariant counterpart, where T N is the torus of diagonal matrices diagpΛ 1 , . . . , Λ N q acting on C N .
T pXq pairing is given by Here Φ is a Laurent polynomial in P and Λ, symmetric in P , χ T is the T -equivariant holomorphic Euler characteristic, taking values in the representation ring ZrΛ˘s of the torus, and the residue sum is taken over all poles P 1 " Λ i 1 , . . . , P n " Λ in (with this ordering of the equations, and i α ‰ i β ). The formula is proved by the direct application of Lefschetz' holomorphic fixed point formula.
The following q-hypergeometric series has emerges from a study of spaces of rational curves in the grassmannian based on their quasimap compactifications: Remark 1. The product on the right can be rearranged as ź and therefore may contain nilpotent factors P j´Pi in the denominator. It is not hard to see, however, that transposing P i and P j does not change the sum of terms with a fixed d 1`¨¨¨`dn , implying that after clearing the denominators, the numerator becomes divisible by P j´Pi (namely, it changes sign under the transposition, and hence vanishes when P i " P j ). Remark 2. Another consequence of the above rearrangement is that, with the exception of the term Q 0 , the series consists of reduced rational functions of q. Namely, the factor at Q d 1`¨¨¨`dn has no pole at q " 0, and the q-degree of the denominator exceeds that of the numerator by N´n`1 ě 2.
Recall that the genus-0 quantum K-theory of a target space X, the "big J-function" is defined as Here Q is the Novikov's variable, tφ α u and tφ a u are Poincaré-dual bases in K 0 pXq, t " ř k t k q k is a Laurent polynomial in q with vector coefficients t k P K 0 pXq b QrrQss, and the correlator represents the Ktheoretic GW-invariant computes a suitable holomorphic Euler characteristic on the moduli spaces of stable maps X g,m`1,d :" M g,m`1 pX, dq with the input (or "insertion") at the marked point (with the index i " 0, . . . , m in the above formula) of the form ř k pevi t k qL k i , where L i stands for the universal cotangent line bundle at the ith marked point. From among several flavors of such K-theoretic GW-invariants (ordinary as in [21], or permutation-equivariant as in [16]), we will currently use the permutation-invariant ones (as the superscript S m indicates), i.e. computing the super-dimension of the part of the sheaf cohomology on the moduli space X 0,m`1,d which is invariant under permutations of the m marked points with the indices i " 1, . . . , m carrying the symmetric inputs tpL i q.
The "small J-function" is obtained from J by setting the input t " 0. In particular, this eliminates the role of the permutation group, and so J p0q represents the "ordinary" K-theoretic GW-invariants. Thus, according to Theorem 1, Theorem 1 is obtained as the non-equivariant limit Λ Ñ 1 from the following result about the T N -equivariant version J T "big J-function" of the grassmannian.
Theorem 1 1 (cf. [32,34] Note that by the very definition (the same as for J with the correlators taking values in the representation ring ZrΛ˘s), the function J T is a Q-series with coefficients which are rational functions of q with vector values in K :" K 0 T pXq b QrrQss. Abusing the language we call such series rational functions of q, denote the space they form by K :" Kpq˘q, and call it the loop space. The part p1´qq`tpqq ("dilaton shift"`"input") belongs to the subspace K`consisting of Laurent polynomials (they can have poles only at q " 0, 8), while the sum of the correlators belongs (as it is not hard to see) to the complementary subspace K´" tf P K | fp0q ‰ 8, fp8q " 0.u. It follows from Remark 2 above (which applies to J T as well) that p1´qqJ T " 1´q mod K´. Thus, the non-obvious statement of Theorem 1 1 is that p1´qqJ T represents a value of J T at all.
The technique of fixed point localization we intend to use goes back to paper [3] by J. Brown, and was adapted to the K-theoretic situation in [10]. The technique, applicable whenever the target carries a torus action with isolated fixed points and isolated 1-dimensional orbits, completely characterizes all values of the big J-function J T as the set of those rational functions f P K which pass two tests, (i) and (ii). They are formulated in terms of specializations f α to the fixed point of the torus action in X. In the case of the grassmannian, take for example the fixed point V 1,...,n " Spanpe 1 , . . . , e n q where (we may assume by choosing the ordering) pP 1 , . . . , P n q " pΛ 1 , . . . , Λ n q: Note that the 2nd factor contains the product (coming from i " j): m"1 p1´q m q with poles at roots of unity, while all other poles are elsewhere (at q " pΛ i {Λ j q´1 {m ), and can be expanded near the roots of unity into power series, e.g.
Criterion (i) stipulates that f α , when expanded as a meromorphic function in a neighborhood of the roots of unity, must represent a value (over a suitable ground ring) of the big J-function of the point target space. We will return to this criterion in a subsequent section and explain how it can be verified.
Criterion (ii) controls residues of f α pqqdq{q at the poles originating from T -equivariant covers of 1-dimensional orbits. Namely, the tangent space to the grassmannian at the fixed point V p1,...,nq carries the torus action with the distinct eigenvalues Λ j {Λ i , i " 1, . . . , n, j " n`1, . . . , N. Consequently, for each choice of i and j there is a 1-dimensional orbit, which compactifies into CP 1 connecting this fixed point with another one. For instance, taking i " 1 and j " n`1, we find such an orbit connecting V p1,...,nq with V p2,...,n`1q . Let φ : CP 1 Ñ CP 1 be the map z Þ Ñ z m 0 ramified at z " 0, 8 (representing the two fixed points (call them α and β). Criterion (ii) has the form of the recursion relation: where Eu are equivariant K-theoretic Euler classes: of the tangent space to X at α, and to the moduli space of degree-m 0 stable maps with 2 marked points at the point represented by the m 0 -fold cover φ respectively.
On the m 0 -fold cover φ : CP 1 Ñ CP 1 , the contributions to the Euler class of the T -modules H 0 pCP 1 ; φ˚L´1bSpanpe j qq and H 0 pCP 1 ; Combing all the contributing factors, we find Checking that this expression matches exactly the recursion coefficient for J T (the middle line in the formula for the residue) is the matter of a straightforward (though somewhat cumbersome) rearrangement of the factors.
Remark 3. Note that the structure of the recursion relations (ii) and the values of the recursion coefficients completely characterize the big J-function of a particular theory, since criterion (i) does not involve any additional choices.
The paper [11] completely describes genus-0 permutation-equivariant GW-invariants of the point target space. Namely, given a ground λalgebra R, the range of the big J-function J pt , (which is a semi-infinite cone that we will denote L pt ) in the loop space Rpq˘q of R-valued rational functions of q is explicitly parameterized as It is assumed here that R is complete in the adic topology defined by a certain ideal R`Ă R, which is respected by the Adams operations Ψ k , k ą 0, in the sense that Ψ k pR`q Ă R k , and that τ P R`, while R`rq˘s means the completion of the space of Λ`-valued Laurent polynomials in q.
The ground ring R we will use is QrΛ1 , . . . , ΛN srrQ 1 , . . . , Q n ss, with the Adam operations acting by Ψ k pL˘1 j q " Λ˘k j , Ψ k pQ i q " Q |k| i , and the ideal R`" pQ 1 , . . . , Q n q. In particular, taking τ " Q 1`¨¨¨`Qn , we find that L pt Q p1´qqJ pt , where J pt is the following product of q-exponential functions: As it is shown in [12,17], L pt is invariant under a large group of linear transformations in the loop space Rpq˘q, which can be described as (pseudo) finite-difference operators. Namely, interpreting Rpq˘q as a space of functions in Q, pick a finite-difference operator Dpq QB Q , Qq which is a non-commutative expression of the operators of multiplication by Q i and operators of translation q Q i B Q i (the latter acts on Q i as multiplication by q) with coefficients which could be Laurent polynomials of q and Λ j . It is almost obvious that the linear vector field f Þ Ñ Df in Rpq˘q is tangent to L pt (and hence e ǫD preserves L pt ). Moreover, according to a result from [17], L pt is preserved by the operator For instance, p1´qqJ pt P L pt is obtained this way from the "small J-function" 1´q P L pt , when D " Q 1`¨¨¨`Qn (the multiplication operator, or order 0). Following [12], we are going to use Q-independent operators of the form D l,Λ :"´qΛp1´q l¨QB Q q, where Λ is a Laurent monomial from QrΛ˘s, and l¨QB Q :" This expression is in fact the asymptotical expansion near the unit circle on the q-plane of the ratio of two q-Gamma functions: The ratio and its asymptotical expansion act on monomials Q d " dn n the same way: In order to complete our fixed point localization proof of Theorem 1 1 , we apply to J pt the following operators (where l " 1 i contains 1 in the 1th position and 0 everywhere else): The terms of the last sum are interpreted as meromorphic functions in the neighborhood of the unit circle, i.e. with poles (which come from the factors with i " j in the left product) at the roots of unity only, while all other factors are expanded into power series. When multiplied by 1´q, the latter series lies in L pt over the ground λ-algebra R " QrΛ˘srrQ 1 , . . . , Q n ss. The substitution Q 1 "¨¨¨" Q n " Q (which induces a homomorphism of λ-algebras R Ñ R 0 :" QrΛ˘srrQss) yields therefore a series which lies in the range J pt over R 0 . It actually coincides with the localization p1´qqJ T p1,...,nq of J T at the indicated fixed point. Therefore p1´qqJ T p1,...,nq , when interpreted as a series of meromorphic functions near the unit circle, satisfies criterion (i). Due to the Weyl group symmetry between all fixed points, and between all 1-dimensional orbits connecting them, this finishes the proof of Theorem 1 1 .

Non-abelian localization and explicit reconstruction
The approach to computing GW-invariants of GIT quotients via nonabelian localization (and eventually quasimap compactifications) was proposed by A. Bertram, I. Ciocan-Fontanine and B. Kim [2] following their proof [1] of the Hori-Vafa conjecture. The conjecture (formulated in the appendix to [22]) gave a novel proposal for the mirrors of GIT quotients C M {{G. The idea, illustrated by K. Hori and C. Vafa in the example of the grassmannians, was to replace the factorization by a semi-simple G with the (cohomologically equivalent to it) succession of the factorizations by its maximal torus T and then by its Weyl group W " NpT q{T . The first step yields a toric manifold, whose mirror and genus-0 GW-invariants are well-understood. In the case of the grassmannian Gr n,N " HompC n , C N q{{GL n pCq, it is the product X :" pCP N´1 q n of projective spaces. Its small where P i are the Hopf bundles over the factors. The second step can be described this way: The Γ-operators here, correspond to the roots of g, i.e. in our case of g " gl n pCq to the line bundles According to [13], p1´qqJ T Πg{t represents a value of the big J-function of the super-space ΠE, where E is a vector bundle over X, equal to ' i‰j P j {P i in the case at hands, which is associated with the adjoint action of the maximal torus on g{t, and Π indicates the parity change of the fibers. By definition, the quantum K-theory of such a super-space is obtained by systematically replacing the virtual structure sheaves O g,m,d of the moduli spaces X g,m,d with O g,m,d b Eu Cˆp ft˚ev˚Eq, where the subscript in the K-theoretic Euler class indicates it is equivariant with respect to the scalar action of Λ 0 P Cˆon the fibers of E, and ft : X g,m`1,d Ñ X g,m,d and ev : X g,m`1,d Ñ X are respectively the forgetting of and evaluation at the last marked point.
On the other hand, the explicit reconstruction results of [15] tell us how to parameterize the entire big J-function of a toric manifold (or super-manifold) from one value of it. Namely, the range of the big Jfunction, L Πg{t in our example, is invariant under the action of a huge group, P, of pseudo-finite-difference operators in Novikov's variables Q 1 , . . . , Q n . It is generated by the exponentials e D of any (R`-adically small) finite-difference operators DpP q Q 1 B Q , Q, qq, and by operators of the form 1 e ř ką0 Ψ k pDpP q kQB Q , Q, qq{kp1´q k q . The orbit of p1´qqJ T X under this group is the whole of L X (and moreover, picking suitable operators as described in [15] one obtains an explicit parameterization of L X ).

Theorem 3. Elements of the orbit of p1´qqJ T
Πg{t under the subgroup P W of the operators invariant with respect to the Weyl group, in the specialization Q 1 "¨¨¨" Q n " Q, Λ 0 " 1 turn into values of the big J-function of the grassmannian.
We conjecture that a similar result holds universally for non-singular GIT quotients, i.e. that L C{{G is obtained from L W Πg{t (where Πg{t is the super-space over the base C{{T defined as explained above) by specializing the Novikov ring to its W -invariant part, and passing to the limit Λ 0 " 1.
Proof of the theorem is based on T N -fixed point localization. It should be obvious after Section 2 that the criterion (i) of the fixed point method is invariant under P even before the specialization to Q i " Q, Λ 0 " 1. To verify (ii), take localizations pJ T Πg{t q p1,...,nq and pJ T Πg{t q pn`1,2,...,nq of J T Πg{t (thinking of the fixed points Spanpe 1 , . . . , e n q and Spanpe 2 , . . . , e n`1 q in the grassmannian). The ambiguity in the ordering of the values P i " Λ i 1 becomes irrelevant in the limit Q 1 " 1 Note that above operators Γ 1i´1j ,Λ0Pi{Pj are the compositions of the operators of multiplication by e ř ką0 Ψ k pΛ0Pi{Pj q{kp1´q´kq , whose cumulative effect, according to the Adams-Riemann-Roch (see [17]), is to transform L X to L Πg{t , and of the operators of this form with D " qΛ 0 P i q QiBQ i {P j q Qj BQ j which preserve L Πg{t . Q n " Q due to the W -symmetry. Before the limit, we take here P 1 " Λ 1 , . . . , P n " Λ n for the first fixed point, and P 1 " Λ n`1 , P 2 " Λ 2 , . . . , P n " Λ n for the second. For x " pΛ 1 {Λ n`1 q´1 {m 0 , we have ..,n`1q p1,...,nq pm 0 q pJ T Πg{t q pn`1,2,...,nq | q"x .
This is simply the recursion relation (ii) for the target space X " pCP N´1 q n corresponding to the 1-dimensional T N -orbit connecting two fixed points, Spanpe 1 q and Spanpe n`1 q in projection to the first factor CP N´1 , and constant (and equal to Spanpe i q, i " 2, . . . , n) in the other projections. The recursion coefficient here turns into the correct one for the grassmannian in the limit Λ 0 " 1 and Q 1 " Q. Note that operators P k i q kQ i B Q i specialize to Λ k i q kQ i B Q i at the fixed point Spanpe 1 , . . . , e n q, and for i ą 1 commute with Q m 0 1 , while for i " 1 we have This implies that for any finite difference operator D regular at q " pΛ 1 {Λ n`1 q´1 {m 0 , the localizations pDJ T Πg{t q p1,...,nq and pDJ T Πg{t q pn`1,2,...,nq of DJ T Πg{t also satisfy the above recursion relation with the same recursion coefficient. In fact this direct verification is not even necessary, since it simply elucidates in terms of fixed point localization the general fact that L Πg{t is P-invariant.
We conclude that when D is W -invariant, p1´qqDJ T Πg{t specializes at Q 1 "¨¨¨" Q n " Q and Λ 0 " 1 into a point in the loop space K (corresponding to the grassmannian) which satisfies the correct recursion relation, and hence belongs to L Gr n,N .
Remark 4. Of course, the above argument applies more generally than the grassmannian example, and works whenever a torus (T N in this case) acts on C{{T n and C{{G with isolated fixed points and isolated one-dimensional orbits. In particular, it applies to twisted quantum K-theories studied in [18] and generalizing the above transition from X " pCP N´1 q n to Πg{t. Namely, let E " EpP 1 , . . . , P n q P K 0 T N pGr n,N q be a virtual vector bundle (for this, E needs to be symmetric in P i ). It can be used to "twist" the virtual structure sheaves of the moduli spaces of stable maps -for both targets, Πg{t and Gr n,N : where µ k are some prefixed elements of the ground ring R (and, abstractly speaking, should better be taken from R`as a precaution lest the modifying expression diverges). Then the big J-functions in the twisted quantum K-theories of Πg{t and Gr n,N are related the same way as described in the theorem: For any W -invariant value of the twisted J-function of Πg{t (in place of J T Πg{t ), the elements of its orbit under P W in the limit Q 1 "¨¨¨" Q n " Q specialize into values of the big J-function in the twisted quantum K-theory of the grassmannian. 4. Balanced I-functions and T˚Gr n,N .
In some recent literature motivated by representation theory (see e.g. [28,29,23]), quantum K-theory of symplectic quiver varieties plays a role, and among them, the cotangent bundles of the grassmannians (rather than the grassmannians per se) take the place of the target spaces. K-theoretic computations in the quasimap compactifications of spaces of rational curves in such targets lead A. Okounkov and his followers to q-hypergeometric functions quite interesting from the point of view of the theory of integrable systems. To illustrate one specific property (apparently important in their theory for technical reasons) consider the series Here Y P Cˆ(denoted in [28] and elsewhere by ) represents the circle acting by scalar multiplication on the fibers of a vector bundle over the compact base (which is meant to be T˚Gr n,N in our example). Note that the series is formed of fractions p1´q m Y Xq{p1´q m Xq, which are bounded both as q Ñ 0 and q Ñ 8. In the fixed-point computations on quasimap spaces of symplectic targets, this property of generating functions being balanced (in terminology of [25]) is a by-product of tensoring the virtual structure sheaf with the square root of the determinant bundle of the moduli space (i.e. in effect computing indexes of real Dirac operators rather than holomorphic Euler characteristics). The questions we will address here are about the place of the series I T and its close counterparts in the "genuine" (i.e. based on stable map compactifications) quantum K-theory of the grassmannian: Does I T represent a value of the big J-function of any version of quantum K-theory, and if so, then what version and on which space? Is it the small J-function in that theory? We will give several different affirmative answers to the first question, and negative to the second. Initially the interest in GW-theory of ΠE for a bundle E over a compact base is motivated by the fact that in the non-equivariant limit Y Ñ 1, GW-invariants of ΠE, when the limit exist, turn into GWinvariants of the zero locus of a generic section of E (which in the case of E " T X consists of χpXq isolated points).
Proof can follow the same route as that of Theorem 1 1 . The localization of I T at the fixed point p1, . . . , nq in the grassmannian, together with such localizations at other fixed points, pass the test (i) of the fixed point theory, and the residues at the poles q " pΛ j {Λ i q 1{m 0 satisfy the recursion relation of the familiar form (ii) with suitable recursion coefficients. This should be obvious after our analysis of the series J T in Section 2 and Section 1 respectively. Moreover, according to Remark 3, it only suffices to match the values of these recursion coefficients with those in GW-theory of ΠT Gr n,N . Using the notation x " pΛ n`1 {Λ 1 q 1{m 0 and our result from Section 1, we find the recursion coefficient corresponding to the pole q " x in the forḿ EupT p1,...,nq Xq EupT φ X 0,2,m 0 q as before, times the modifying factor where the target X " Gr n,N . Unsurprisingly, the modifying factor is almost reciprocal to the expression for EupT p1,...,nq Xq{ EupT φ X 0,2,m 0 q. They differ by the presence of Y in each factor, and by the extra factor 1´x m 0 Y Λ 1 {Λ n`1 (actually equal to 1´Y , and excluded from the expression in Section 1 where Y " 1). In our computation of H 0 pCP 1 ; φ˚pT Xqq, the latter (zero) factor represents the line spanned by the vector field z m 0 B z m 0 (infinitesimally rescaling the target CP 1 ), and falls out of T φ X 0,2,m 0 because of the infinitesimal automorphism zB z of the source CP 1 . Thus, the factor 1´Y remains present in Eu Cˆp ft˚φ˚pT Xqq. Note that Y was introduced as the character of Cˆ-action on T˚X. The action on T X is given therefore by Y´1, but the definition of the K-theoretic Euler class as the exterior algebra of the dual bundle restores the factors Y everywhere. Thus, the modifying factor coincides with Eu Cˆp ft˚φ˚pT Xqq{ Eu Cˆp T p1,...,nq Xq, and the recursion coefficient altogether has the required forḿ Recall that the Hirzebruch χ´Y -genus of a compact complex manifold M is defined by where the rightmost interpretation assumes that Y P Cˆacts fiberwise on the tangent bundle by Y´1. More generally, one can define the (classical) Hirzebruch K-theory by replacing the structure sheaves O M with O M b Eu Cˆp T Mq. The quantum Hirzebruch K-theory of a target X is defined by similarly modifying the virtual structure sheaves of the moduli spaces X g,m,d using their virtual tangent bundles: According to a result from [19], the theory thus obtained can be expressed via the ordinary quantum K-theory, implying in particular Corollary 1 (see Remark 4 below). However, it also follows from our fixed point approach. Namely, the big J-function of (permutationinvariant) quantum Hirzebruch K-theory has the form 1´q where the correlators are defined using the virtual structure sheaves of the Hirzebruch K-theory. This is not an ad hoc definition, but is dictated by the general formalism; the dilaton shift and the first input embody respectively: the Euler class (of the universal line bundle q´1) corresponding to the genus, and the reciprocal of the equivariant Euler class of L´1 0 . Consequently, the recursion coefficient of the fixed point theory acquires a new factor 1´Y : the residue of 1´qY L 0 1´qL 0 dq q at the pole q " L´1 0 (equal in our computations to pΛ 1 {Λ n`1 q´1 {m 0 ). But the above explanation why this factor belongs to Eu Cˆp ft˚φ˚pT Xqq means it does not belong to Eu Cˆp T φ X 0,2,m 0 q. The latter occurs in Lefschetz' fixed point formula for the modified virtual structure sheaf. The net result is that the recursion relation (ii) remains the same as in the theory of ΠT X. Note that a scalar factor, such as 1{1´qY in p1´qqI T {p1´qY q has no effect on the recursion relation (a fact indicating that the range L Ă K of the big J-function is an "overruled cone"). The role of this factor is to guarantee that modulo Q, the series equals the dilaton shift, and hence the rest of the series is Q-adically small as required.
Remark 5. By the way, 1{p1´qY q " ř mě0 Y m q m is considered a "Laurent polynomial" in q i.e. an element of K`in Hirzebruch Ktheory, as it doesn't have poles relevant in localization theory. The correlator part of the big J-function in the quantum Hirzebruch Ktheory clearly satisfies J | q"8 " Y J | q"0 , and this condition defines the new space K´. The general result of [19], which applies to the allgenera permutation-equivariant quantum K-theory, says that the total descendant potential D Y X for the Hirzebruch version of the theory is obtained from the "ordinary" one, D 0 X , by three transformations: the Eulerian twisting corresponding to the bundle E " T X´1 (in genus 0, this has practically the same effect as the twisting by T X, producing the big J-function of ΠT X), and the above changes in the dilaton shift and polarization K " K`' K´. The transformations correspond to the three summands in the virtual tangent bundles: where L is the universal cotangent line over X g,m`1,d at the m`1-st marked point, and j : Z Ñ X g,m`1,d is the inclusion of the nodal locus. Here ft˚ev˚T X represents variations of stable maps from pointed curves with a fixed complex structure, ft˚pL´1q represents variations of the complex structure of the curves, while the last term is supported on the virtual divisor ftpZq Ă X g,m,d where the combinatorics of the curves changes, and accounts for the difference between the virtual tangent bundle and the sheaf of vector fields tangent to this divisor.
Another form of Theorem 4 can be derived from Serre's duality. The cotangent line bundle L of a pointed nodal curve and its canonical bundle K are related by L " KpDq, where D :" ř m i"1 σ i is the divisor of the marked points (i.e. away from the nodes, a section of L is a differential allowed to have 1st order poles at the markings). Given a bundle E over X, on X g,m,d we have Applying the quantum Adams-Riemann-Roch (Theorem 2 in [18]), we find that tensoring of O g,m,d with Eu´1 Cˆp pft˚p1´Lq ev˚E _ q _ q in the correlators of permutation-equivariant quantum K-theory is equivalent to the change p1´qq Þ Ñ p1´qq Eu´1 Cˆp Eq in the dilaton shift. The same change of this inputs: t Þ Ñ Eu´1 Cˆp Eqt, is effected by tensoring with the Euler classes of´evi E. In other words, the dilaton-shifted total descendant potential of the theory twisted by Eu´1ppft˚ev˚E _ q _ q is obtained from the one twisted by Eu Cˆp ft˚ev˚Eq by the transformation D ΠE pqq Þ Ñ D ΠE pEu´1 Cˆp Eqqq. The potentials are considered as quantum states in suitable Fock spaces, and the transformation is induced by the map f Þ Ñ Eu´1 Cˆp Eqf between two copies of the loop space K (equipped with two different symplectic structures: based on the Poincare pairing χpX; Eu´1 Cˆp Eqabq on the source space, and χpX; Eu Cˆa bq on the target. Consequently, the big J-functions in the genus-0 theory are related by the inverse transformation: J ΠE Þ Ñ Eu Cˆp EqJ ΠE . Applying all this to E " T Gr n,N , we arrive at the following conclusion. Corollary 2 (cf. [25]). The series Eu Cˆp T Gr n,N qp1´qqI T represents a value of the big J-function in the torus-equivariant, permutationinvariant quantum K-theory of the grassmannian Gr n,N twisted by One more way of modifying virtual structure sheaves, which was recently introduced and explored by Y. Ruan and M. Zhang [31], consists in tensoring O g,m,d with a power of the determinant line bundle pdetpft˚ev˚Eqq´l, thereby bringing the level structure (of level l) into the quantum K-theory. Note that in terms of K-theoretic Chern roots L 1 , . . . , L M of a vector bundle E, So, we take E " T˚Gr n,N , E " ft˚ev˚pT˚Gr n,N q, and describe the modification of O g,m,d used in Corollary 2 as tensoring with both Eu´1 Cˆp Eq and pdet Eq´1. After the first operation we land in the theory of the noncompact bundle space T˚Gr n,N , and after the second in the level 1 version of this theory. The Poincaré pairing changes accordingly into χpX; Eu´1 Cˆp T˚Gr n,N qpdet T˚Gr n,N q´1abq. By the Riemann-Roch formula, dim E " p1´gq dim X`ş X c 1 pT˚Xq, which for g " 0 yields p´1q dim E " p´1q dim Gr n,N p´1q N d . The first sign is absorbed by the ratio of the Euler classes (of T and T˚) in the Poincaré pairings, and the second by the change Q Þ Ñ p´1q N Q, leading to the following conclusion.
Corollary 3. The series detpT˚Gr n,N q´1 Eu Cˆp T˚Gr n,N q p1´qq I T pp´1q N Qq represents a value of the big J-function in the level 1, torus-equivariant, permutation-invariant quantum K-theory of the cotangent bundle space T˚Gr n,N . Explicitly, the product of the determinant and the Eulerian prefactor differs by the sign p´1q dim Gr n,N from Eu Cˆp T Gr n,N q " Because of this pre-factor, the series even modulo Q is not equal the dilaton shift 1´q (as well as in Corollary 2), which already disqualifies it for the role of the "small" J-function.
In fact the q-hypergeometric series which arises in the K-theoretic computations on the spaces of quasimaps to the grassmannian is slightly different from the one in Corollaries 2 and 3. It has the form which differs from I T in that in the products of the factors 1´q m Y P i {Λ j , the range of m is not from 1 to d i (as for the factors without Y ) but from 0 to d i´1 (and similarly for the factors 1´q m Y P i {P j ). We claim, however, that p1´qq r I and p1´qqp´1q dim Gr n,N r I T pp´1q N Qq) represent some values of the big J-functions of the same theories as described in Corollary 2 and Corollary 3 respectively.
Namely, consider the version of r dn n , and apply to it the operator ś N j"1 This results in restoring the "missing" factors with m " d i or m " d i´dj , and in the limit Q 1 "¨¨¨" Q n " Q yields the series of Corollary 3 (modulo to the sign p´1q dim Gr n,N and the change Q Þ Ñ p´1q N Q. On the other hand, the operator can be written as and hence belongs to the group P W , which justifies our claim due to Theorem 3 and Remark 4. The series p1´qq r I T appears to have better chances to pose for the "small" J-function, because the term with Q 0 is 1´q, and other terms are reduced rational functions of q. And indeed, H. Liu [25], looking for a stable-map K-theory interpretation of the q-hypergeometric series arising in the quasimap K-theory of quiver varieties, shows that in the case n " 1 of projective spaces, the series p1´qq r I T is the small Jfunction in the theory described by Corollary 2. However, he falls short of sticking to this interpretation, because he finds as example (namely the manifold of flags in C 3 ) where the similarly twisted small J-function is unbalanced.
In fact none of p1´qq r I T with n ą 1 (and none of other I-series featuring in this section) represent "small" J-functions, and not because some rational functions are not reduced, but for much more dramatic reasons. Namely, in our fixed point characterization of the big J-function, the poles participating in the recursion relations come from the characters of the torus action on the grassmannian per se: q " pΛ i {Λ j q´1 {m . The terms of a balanced I-series containing the factors 1{p1´q m Y P i {P j q lead to the poles at q " pY Λ i {Λ j q´1 {m , which cannot come from fixed point localization. Such fractions should therefore be interpreted as elements of K`, i.e. as geometric series ř kě0 q mk pY P i {P j q k converging in the Y -adic topology. Thus, representing p1´qq r I T as p1´qq`tpqq mod K´results in a very complicated value of tpqq, meaning that the series represents the value of the big J-function with the inputs tpL i q which are rather far from 0. What makes the effect even more dramatic is that it is the input in the permutation-invariant quantum K-theory, no counterpart of which has been discussed so far in the context of quasimap spaces.
Apart from this, the interpretation of the series given in Corollary 3 is quite parallel to its definition [28,29,23] in the quasimap theory as a generating function capturing some K-theoretic GW-invariants of the cotangent bundle of the grassmannian based on the virtual structure sheaves "symmetrized" by the determinant factors.
Finally, we would like to stress that, although we have formulated Theorem 4 and its corollaries as statements about the particular Ifunction, modified slightly in one way or another, in fact these modifications affect the recursion coefficients in a simple and controllable way, implying that the whole big J-functions of the respective theories coincide up to these minor modifications. In particular, Corollary 3 is connected to Theorem 4 by a general phenomenon called the "nonlinear (or quantum) Serre duality" [5]. It relates GW-invariants of the super-space ΠE and bundle space E _ , and was first observed in [8] (for cohomological GW-invariants) via fixed point localization. For the full treatment (including higher genus) of the K-theoretic reincarnation of the quantum Serre duality we refer to [35].

Non-abelian quantum Lefschetz.
A somewhat different proof of Theorem 4 could be derived from Theorem 3 together with Remark 4, applied to GW-invariants of the grassmannian Euler-twisted by the tangent bundle Here we illustrate this approach using as an example Eulerian twistings applied to the dual tautological bundle E " P´1 1`¨¨¨`P´1 n . The Euler-twisted theory (of both Πg{t and Gr n,N ) is defined by where Y P Cˆacts by multiplication on the fibers of E. According to the quantum Adams-Riemann-Roch theorem [18], the twisted theory is obtained from the untwisted one by the multiplication: L ΠE " l´1L Gr n,N , where This is a convenient moment to address one general technical issue. Values of big J-functions are supposed to lie in R`-neighborhood of the dilaton shift 1´q. The terms containing Novikov's variables are R`-small, and remain such after multiplication by anything like l. Moreover, for Laurent polynomials tpqq with R`-small coefficients, l t contains only finitely many non-reduced terms, and so modulo K´it remains a Laurent polynomial (with R`-small coefficients). However, the product lp1´qq " p1´qq`Y E _ mod K´seems to present a problem. One way to resolve it is to postulate that R`Q Y . Here is a better way to deal with this issue, which is especially useful if one also needs to use Y´1 in the same context. Consider the operator which is l times the pseudo-finite-difference operator from the group P W corresponding to the finite-difference operator´qY ř n i"1 P i q Q i B Q i . Therefore l´1L Πg{t " D´1L Πg{t , which by Theorem 3 (or rather its generalization explained in Remark 4) turns into L ΠE in the limit Q 1 "¨¨¨" Q n " Q, Λ 0 " 1. The advantage of using D instead of l is that D does not change terms constant in Q 1 , . . . , Q n : Dp1´qq " 1´q. Proof. Apply D´1 and use Corollary. The small J-function of ΠE equals p1´qqI where I "

Level structures and dual grassmannians
Of course, the approach illustrated by Theorem 5 applies to any bundle E over the grassmannian, since E can always be written as a symmetric combination of monomials ś i P l i i . We are going to use this together with the observation (see Section 4) that det´1p´Eq " EupEq{ EupE _ q in order to describe the effect of the level structure on the genus-0 quantum K-theory of the grassmannian. For the sake of illustration, we take E to be the tautological bundle V " P 1`¨¨¨`Pn . With E :" ft˚ev˚V , we have level-l twisted structure sheaves The Adams-Riemann-Roch theorem from [18] yields the multiplication operator l " e´l ř k‰0 Ψ k pV q{kp1´q k q , and the respective pseudo-finite-difference operator Applying it to Q d i i , we find: The above calculation is somewhat formal. The initial determinantal twisting of O g,m,d and the finial modifying factors are well-defined, but in order to justify intermediate steps, one needs add to R`two variables Y, Y 1 , and replace E and E _ with Y E and Y 1 E _ . This will lead to the product of fractions p1´Y P i q m q{p1´Y 1 P´1 i q´mq, where one can pass to the limit Y " Y 1 " 1, thus obtaining the following result.
Here L

pV,lq
Gr n,N the range of the big J-function in the level-l permutationinvariant genus-0 quantum K-theory of the grassmannian Gr n,N , where V is its tautological bundle, and the level-l twisted structure sheaves are defined as in [31]: O g,m,d b det´lpft˚ev˚V q. Note, that the spurious signs p´1q l dim E " p´1q l ř d i`l dim V initially introduced in the determinantal twisting disappears from our ultimate formulation. The first part of it can be absorbed by the change Q Þ Ñ p´1q l ř d i Q of the Novikov variable, which is offset by the signs of p´P i q ld i in our computation. The second part, p´1q l dim V , is the discrepancy in Poincaré pairings (it affects the notion of dual bases tφ α u, tφ α u in the definition of J-functions) which correspond to the two twistings of O Gr n,N : by EuplV q{ EuplV _ q " ś p´P i q´l and det´lpV q " ś P´l i . The theorem is a non-abelian counterpart of the result of Ruan-Zhang for toric manifolds, obtained in [31] on the basis of adelic characterization. Both can also be derived by fixed point localization.
Corollary. The series p1´qqI T pV,lq , where I T pV,lq " ÿ 0ďd 1 ,...,dn represents a point in L plq Gr n,N , and for´n ă l ď N´n`1 is the small J-function of the level-l theory.
Proof. The formula itself is obtained, of course, from the nonabelian representation of the small J-function p1´qqJ T " p1´qqI T pV,0q by the recipe described in the theorem. For l ě 0, terms of p1´qqI T pV,lq have no pole at q " 0, and for l ď N´n`1 can be shown to be reduced rational functions of q (except the Q 0 -term 1´q). Indeed, the difference between the q-degrees of the denominator and numerator of the coefficient of I T pV,lq indexed by pd 1 , . . . , d n q is When l ď N´n`1, the binomial sum is non-negative, since for each i the number of j with d j ă d i does not exceed n´1. The linear term is ą 1 unless all d i ‰ 1. Note that in this case the whole expression doesn't depend on l, and is still ě N´n`1 ą 1. Thus, even after multiplication by 1´q the rational function remains reduced.
For l ă 0, the terms of the series are therefore also reduced, but can have a pole at q " 0. However, even when this happens, the pole disappears after summing the terms with the same degree d 1`¨¨¨`dn -at least when l ą´n. This follows from a non-trivial combinatorial result of H. Dong and Y. Wen [7], according to which I T pV,lq " r I T pv _ ,´lq for´n ă l ă N´n, where r I T p r V _ ,´lq is the similar series corresponding to the dual grassmannian Gr N,N´n :" HompC N´n , C N _ q{{GL N´n pCq, and the bundle r V _ dual to the tautological one: Here r P i are K-theoretic Chern roots of the tautological N´n-dimensional bundle r V . Note the characters Λ´1 j of the torus T N action on C N _ . Also note the binomial coefficient`d i`1 2˘( instead of`d i 2˘) : this is the effect of using r V _ rather than r V in the construction of the determinantal twistings. In the previous section we already had the experience of using E " V _ , from which it is easy to infer the origin of the modifying factors: The dual grassmannians are canonically identified by pV Ă C N q Þ Ñ pV K Ă C N _ q, and the result of [7] identifies the two expressions as Qseries with coefficients in K 0 T pGr n,N q " K 0 T pGr N,N´n q-valued rational functions of q when´n ă l ă N´n. By the previous estimates of the q-degrees (where this time the linear term l ř d i isn't present), p1´qq r I T p r V _ ,´lq passes the requirements to be a small J-function when 0 ď´l ă N´pN´nq`1, i.e. 0 ě l ě´n. Therefore (though this is not apparent) so does p1´qqI T V,l at least for 0 ą l ą´n.
Example. For Gr 1,N " CP N´1 , we have Obviously the series is the small J-function only when´1 ă l ď N, i.e. the boundaries given by the corollary are sharp.
Proposition. L pV,lq Gr N,N´n .
Proof. From r V _ " C N {V we find detpft˚ev˚V q b detpft˚ev˚r V _ q " detpft˚ev˚C N q which over moduli spaces of rational curves equals det C N " ś N j"1 Λ j " det V b det r V _ , the factor absorbed by the discrepancy in Poincaré pairings between the two theories.

Mirrors
Consider the improper Jackson integral (or q-integral), defined as in the example The infinite product converges for |q| ą 1, and near |q| " 1 has the asymptotical expansion Since the integrand vanishes as X " q´d with d ă 0, the q-integral can be computed as ÿ .
The last expression is closely related to the q´1-gamma function (see [6] for a modern treatment of it, including the above q-integral representation). In particular, the application of the translation operator q ΛB Λ : Λ Ñ qΛ results in the multiplication of the whole expression by 1´Λ´1. This also follows from the property of the integrand p1´q XB X {Λqf pXq " Xf pXq " q ΛB Λ f pXq, since the q-integral is obviously preserved by the translation q XB X of the integrand. The latter property of q-integrals will be more useful to us than the previous explicit calculation of their values in terms of q-gamma functions. Our goal will be to represent the small J-function J T X of the grassmannian X " G n,N by suitable Jackson-like integrals in a fashion similar to representing cohomological J-functions by complex oscillating integrals in the mirror theory of, say, toric or flag manifolds.
To maintain visual resemblance with complex oscillating integrals, we will denote X´1d q X as d ln q X, and use the asymptotical expressions e ř ką0 X k {kp1´q k q and e´ř ką0 Y k q k {kp1´q k q for ś 8 m"1 p1´X{q m q and 1{ ś 8 m"0 p1´Y {q m q respectively. The latter product (multiplied by Y ln Λ{ ln q ) can be q-integrated [6] from 0 to´8: ş´8 0 gpY q Y´1d q Y :" ř dPZ gp´q´dq. Let us recall from Section 3 that J T X is obtained from J T Πg{t by passing to the limit Λ 0 " 1, Q 1 "¨¨¨" Q n " Q, and takes values in K 0 T pXq consisting of symmetric functions of P 1 , . . . , P n . Before the limit, J T Πg{t is the J-function of a toric superspace. Their K-theoretic mirrors were proposed in [14]. We begin with setting up these toric mirrors and studying their properties.
By the "cycle" Γ we understand a q-lattice of rank nN´n 2´2 n suitable for multi-dimensional q-integration; we'll meet some examples later. 2 According to [14], this is the K-theoretic mirror to what we denoted in Section 3 by Πg{t: the toric super-bundle over pCP N´1 q n " C N n {{T n with the fiber g{t associated with the adjoint action of the maximal torus T n in GL N pCq on Lie GL N pCq{ Lie T n . Namely, in the torusnon-equivariant limit J Πg{t , the "small J-function" J T Πg{t introduced in Section 3 satisfies (as it is not hard to check) the system of finite difference equations ź So, the claim is that our mirror q-integral satisfies the same system (for scalar-valued rather than K 0 pXq-valued functions): ź To check this, we note that translations operators q X ij B X ij and q Y ii 1 B Y ii 1 project to the Q-space into respectively q Q i B Q i and q Q i 1 B Q i 1´Q i B Q i . Applying q X ij B X ij to the factor △ ij :" e ř ką0 X k ij {kp1´q k q in the integrand of I containing X ij , we obtain Therefore, applying ś j p1´q X ij B X ij q, we find the integrand multiplied by ś j X ij . Similarly, applying q´Y ii 1 B Y ii 1 to we obtain p1´Y ii 1 qq´1∇ ii 1 , and hence applying 1´q q´Y ii 1 B Y ii 1 we find the integrand multiplied by p1´qp1´Y ii 1 qq´1q " Y ii 1 . Since ś i 1 ‰i Y i 1 i ś j X ij " Q i ś i 1 ‰i Y ii 1 for i " 1, . . . , n, the promised finite difference equations follow.
The torus-equivariant counterpart I T of I is obtained by inserting into the integrand the factor ź By repeating the above computations, we find that I T satisfies finite difference equations ź p1´qΛ 0 q Q i B Q i´Qi 1 B Q i 1 q I T , i " 1, . . . , n.