The Hirzebruch–Riemann–Roch Theorem in true genus-0 quantum K-theory
We completely characterize genus-0 K-theoretic Gromov–Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov–Witten invariants of this manifold. This is done by applying (a virtual version of) the Kawasaki–Hirzebruch–Riemann–Roch formula for expressing holomorphic Euler characteristics of orbibundles on moduli spaces of genus-0 stable maps, analyzing the sophisticated combinatorial structure of inertia stacks of such moduli spaces, and employing various quantum Riemann–Roch formulas from fake (i.e. orbifold–ignorant) quantum K-theory of manifold and orbifolds (formulas, either previously known from works of Coates–Givental, Tseng, and Coates–Corti–Iritani–Tseng, or newly developed for this purpose by Tonita). The ultimate formulation combines properties of overruled Lagrangian cones in symplectic loop spaces (the language, that has become traditional in description of generating functions of genus-0 Gromov-Witten theory) with a novel framework of adelic characterization of such cones. As an application, we prove that tangent spaces of the overruled Lagrangian cones of quantum K-theory carry a natural structure of modules over the algebra of finite-difference operators in Novikov’s variables. As another application, we compute one of such tangent spaces for each of the complete intersections given by equations of degrees in a complex projective space of dimension .
K-theoretic Gromov–Witten invariants of a compact complex algebraic manifold are defined as holomorphic Euler characteristics of various interesting vector bundles over moduli spaces of stable maps of compact complex curves to . They were first introduced in  (albeit, in a limited generality of genus-0 curves mapped to homogeneous Kähler spaces), where it was shown that such invariants define on a geometric structure resembling Frobenius structures of quantum cohomology theory.
At about the same time, it was shown  that simplest genus-0 K-theoretic GW-invariants of the manifold of complete flags in are governed by the finite-difference analogue of the quantum Toda lattice. More precisely, a certain generating function for K-theoretic GW-invariants, dubbed in the literature the J-function (and depending on variables, namely, Novikov’s variables introduced to separate contributions of complex curves in by their degrees) turns out to be a common eigenfunction (known in representation theory as Whittaker’s function) of commuting finite-difference operators, originating from the center of the quantized universal enveloping algebra . In quantum cohomology theory, the corresponding fact was established by B. Kim , who showed that the cohomological J-function of the flag manifold of a complex simple Lie algebra is a Whittaker function of the Langlands-dual Lie algebra . The K-theoretic generalization involving quantized universal enveloping algebras was stated in  as a conjecture, and still remains an open problem.
Foundations for K-theoretic counterpart of GW-theory were laid down by Y.-P. Lee  in the reasonable generality of arbitrary complex algebraic target spaces (and holomorphic curves of arbitrary genus). While the general structure and universal identifies (such as the string equation, or topological recursion relations) of quantum cohomology theory carry over to case of quantum K-theory, the latter is still lacking certain computational tools of the former one, and for the following reason.
The so-called divisor equations in quantum cohomology theory tell that the number of holomorphic curves of certain degree with an additional constraint, that a certain marked point is to lie on a certain divisor , is equivalent to (more precisely, differs by the factor from) the number of such curves without the marked point and without the constraint. Consequently, the dependence of J-functions on Novikov’s variables is redundant to their behavior as functions on introduced through constraints at marked points. In particular, differential equations satisfied by the J-function in Novikov’s variables (e.g. the Toda equations in the case of flag manifolds) are directly related to the quantum cup-product on .
In K-theory, however, any analogue of the divisor equation seems missing, and respectively the K-theoretic analogue of the quantum cup-product, and differential equations of the Frobenius-like structure on are completely detached from the way the J-functions depend on Novikov’s variables. Because of this lack of structure with respect to Novikov’s variables, it appears even more perplexing that in examples (such as projective spaces, or flag manifolds) the J-functions of quantum K-theory turn out to satisfy interesting finite-difference equations.
The idea of computing K-theoretic GW-invariants in cohomological terms is naturally motivated by the classical Hirzebruch–Riemann–Roch formula 
The problem (which is at least a decade old) of putting this idea to work encounters the following general difficulty. The HRR formula needs to be applied to the base which, being a moduli space of stable maps, behaves as a virtual orbifold (rather than virtual manifold). The HRR formula for orbibundles on orbifolds was established by Kawasaki  and expresses the holomorphic Euler characteristic (which is an integer) as an integral over the inertia orbifold (rather than itself). The latter is a disjoint union of strata corresponding to points with various types of local symmetry (and being one of the strata corresponding to the trivial symmetry).
When is a moduli space of stable maps, the strata of the inertia stack parametrize stable maps with prescribed automorphisms. It is reasonable to expect that individual contributions of such strata can be expressed as integrals over moduli spaces of stable maps of quotient curves, and thus in terms of traditional GW-invariants. However, the mere combinatorics of possible symmetries of stable maps appears so complicated (not mentioning the complexity of the integrands required by Kawasaki’s theorem), that obtaining a “quantum HRR formula” expressing K-theoretic GW-invariants via cohomological ones didn’t seem feasible.
In the present paper, we give a complete solution in genus-0 to the problem of expressing K-theoretic GW-invariants of a compact complex algebraic manifold in terms of its cohomological GW-invariants. The solution turned out to be technology-consuming, and we would like to list here those developments of the last decade that made it possible.
One of them is the Quantum HRR formula [6, 3] in fake quantum K-theory. One can take the RHS of the classical Hirzebruch–Riemann–Roch formula for the definition of on an orbifold . Applying this idea systematically to moduli spaces of stable maps, one obtains fake K-theoretic GW-invariants, whose properties are similar to those of true ones, but the values (which are rational, rather than integer) are different. The formula expresses fake K-theoretic GW-invariants in terms of cohomological ones.
Another advance is the Chen–Ruan theory [1, 2] of GW-invariants of orbifold target spaces, and the computation by Jarvis–Kimura of such invariants in the case when the target is the quotient of a point (or more generally a manifold) by the trivial action of a finite group.
Next is the theorem of Tseng  expressing twisted GW-invariants of orbifold target spaces in terms of untwisted ones.
Finally, our formulation of the Quantum HRR Theorem in true quantum K-theory is based on a somewhat novel form of describing generating functions of GW-theory, which we call adelic characterization. For a general and precise formulation of the theorem, the reader will have to wait until Section 6, but here we would like to illustrate the result with an example that was instrumental in shaping our understanding.
Here is unipotent, and stands for the Hopf bundle on ,
satisfying the relation in .
It is a power series in Novikov’s variable with vector coefficients
which are rational functions of , and take values in .
It was shown
On the other hand, one can use quantum Riemann–Roch and Lefschetz’ theorems of  and  to compute, starting from the cohomological J-function of , a value of the J-function of the fake quantum K-theory. The result (see Section 10) turns out to be the same: . This sounds paradoxical, since — one can check this directly for in low degrees! — contributions of non-trivial Kawasaki strata neither vanish no cancel out.
In fact this is not a contradiction, for as it turns out, coefficients of the series do encode fake K-theoretic GW-invariants, when is expanded into a Laurent series near the pole . Furthermore, when is expanded into a Laurent series near the pole , where is a primitive -th root of unity, the coefficients represent certain fake K-theoretic GW-invariants of the orbifold target space . Moreover, according to our main result, these properties altogether completely characterize those -series (whose coefficients are vector-valued rational functions of ) which represent true genus-0 K-theoretic GW-invariants of a given target manifold.
This fact is indeed the result of application of Kawasaki’s HRR formula to moduli spaces of stable maps. Namely, the complicated combinatorics of strata of the inertia stacks can be interpreted as a certain identity which, recursively in degrees, governs the decomposition of the J-function into the sum of elementary fractions of with poles at all roots of unity. The theorem is stated in Section 6 (after the general notations, properties of quantum K-theory, Kawasaki’s HRR formula, and results of fake quantum K-theory are described in Sections 1–5), and proved in Sections 7 and 8.
In Section 10, we develop a technology that allows one to extract concrete results from this abstract characterization of quantum K-theory. In particular, we prove (independently of ) that the function is indeed the J-function of , as well as similar results for codimension- complete intersections of degrees satisfying .
Let denote the operator of translation through of the variable . It turns out that for every ,
also represent genus-0 K-theoretic GW-invariants of . This example illustrates a general theorem of Section 9, according to which J-functions of quantum K-theory are organized into modules over the algebra of finite-difference operators in Novikov’s variables. This turns out to be a consequence of our adelic characterization of quantum K-theory in terms of quantum cohomology theory, and of the -module structure (and hence of the divisor equation) present in quantum cohomology theory.
1. K-theoretic Gromov–Witten invariants
Let be a target space, which we assume to be a nonsingular complex projective variety. Let denote Kontsevich’s moduli space of degree- stable maps to of complex genus- curves with marked points. Denote by the line (orbi)bundles over formed by the cotangent lines to the curves at the respective marked points. When , and , we use the correlator notation
for the holomorphic Euler characteristic over of the following sheaf:
Here are the evaluation maps, and is the virtual structure sheaf of the moduli spaces of stable maps. The sheaf was introduced by Yuan-Pin Lee . It is an element of the Gröthendieck group of coherent sheaves on the stack , and plays a role in K-theoretic version of GW-theory of pretty much similar to the role of the virtual fundamental cycle in cohomological GW-theory of . According to , the collection of virtual structure sheaves on the spaces satisfies K-theoretic counterparts of Kontsevich–Manin’s axioms  for Gromov–Witten invariants.
Note that, in contrast with cohomological GW-theory, where the invariants are rational numbers, K-theoretic GW-invariants are integers.
The following generating function for K-theoretic GW-invariants is called the genus-0 descendant potential of :
Here denotes the monomial in the Novikov ring, the formal series completion of the semigroup ring of the Mori cone of , where the monomial represents the degree of rational curves in , and stands for any Laurent polynomial of one variable, , with vector coefficients in . Thus, is a formal function of with Taylor coefficients in the Novikov ring.
2. The symplectic loop space formalism
Let be the Novikov ring. Introduce the loop space
By definition, elements of are -series whose coefficients are vector-valued rational functions on the complex circle with the coordinate . It is a -module, but we often suppress Novikov’s variables in our notation and refer to as a linear “space.” Moreover, abusing notation, we write , where . We call elements of “rational functions of with coefficients in ,” meaning that they are rational functions in the -adic sense, i.e. modulo any power of the maximal ideal in the Novikov ring.
We endow with symplectic form , which is a -valued non-degenerate anti-symmetric bilinear form:
Here stands for the K-theoretic intersection pairing on :
It is immediate to check that the following subspaces in are Lagrangian and form a Lagrangian polarization, :
i.e. is the space of Laurent polynomials in , and consists of rational functions vanishing at and regular at .
The following generating function for K-theoretic GW-invariants is defined as a map and is nick-named the big J-function of :
The first summand, , is called the dilaton shift, the second, , the input, and the sum of the two lies in . The remaining part consists of GW-invariants, with and being any Poincare-dual bases of . It is a formal vector-valued function of with Taylor coefficients in .
Indeed, the moduli space
is a “virtual orbifold” of finite dimension. In
particular, in the K-ring of it, the line bundle satisfies
a polynomial equation, , with .
Proposition. The big J-function coincides with the differential of the genus-0 descendant potential, considered as the section of the cotangent bundle which is identified with the symplectic loop space by the Lagrangian polarization and the dilaton shift :
Proof. To verify the claim, we compute the symplectic inner product of the -part of , with a variation, , of the input, and show that it is equal to the value of the differential on . Note that, since has no poles other than or , we have
Therefore the symplectic inner product in question is equal to
3. Overruled Lagrangian cones
A Lagrangian variety, , in the symplectic loop space is called an overruled Lagrangian cone if is a cone with the vertex at the origin, and if for every regular point of , the tangent space, , is tangent to along the whole subspace . In particular: (i) tangent spaces are invariant with respect to multiplication by , (ii) the subspaces lie in (so that is ruled by a finite-parametric family of such subspaces), and (iii) the tangent spaces at all regular points in a ruling subspace are the same and equal to .
Theorem (). The range of the big J-function of quantum K-theory of is a formal germ at of an overruled Lagrangian cone.
Proof. As explained in , this is a consequence of the relation between descendants and ancestors.
The ancestor correlators of quantum K-theory
are defined as formal power series of holomorphic Euler characteristics
where , the “ancestor” bundles, are pull-backs of the universal cotangent line bundles on the Deligne-Mumford space by the contraction map . The latter map involves forgetting the map of holomorphic curves to the target space as well as the last marked points.
The genus-0 ancestor potential is defined by
and depends on and . The graph of its differential is identified in terms of the ancestor version of the big J-function:
Here , and
In the ancestor version of the symplectic loop space formalism, the loop space and its polarization are the same as in the theory of descendants, but the symplectic form is based on the pairing tensor rather than the constant Poincare pairing .
Let and be Lagrangian submanifolds defined by the descendant and ancestor J-functions and . Then
where is an isomorphism of the symplectic loop spaces, defined by the following matrix :
It is important that the genus-0 Deligne-Mumford spaces are manifolds (of dimension ). Consequently, the line bundles are unipotent. Moreover, at the points with the ancestor potential has all partial derivatives of order equal to . In geometric terms, the cone is tangent to along . This means that the cone is swept by ruling subspaces parametrized by , an that each Lagrangian subspace is tangent to along the corresponding ruling subspace. The theorem follows.
The proof of the relationship is based on comparison of the bundles and , and is quite similar to the proof of the corresponding cohomological theorem given in Appendix 2 of . It uses the K-theoretic version of the WDVV-identity introduced in , as well as the string and dilaton equations.
The genus-0 dilaton equation can be derived from the geometric fact about the K-theoretic push-forward along the map forgetting the first marked point. It leads to the relation
The latter translates into the degree-2 homogeneity of with respect to the dilaton-shifted origin, and respectively to the conical property of .
The string equation is derived from (thanks to rationality of the fibers of the forgetting map) and relationships between and for (see for instance ). It can be stated as the tangency to the cone of the linear vector field in defined by the operator of multiplication by . The operator of multiplication by
is anti-symmetric with respect to and thus defines a linear Hamiltonian vector field. Since is a cone, this vector field is also tangent to , which lies therefore on the zero level of its quadratic Hamilton function. This gives another, Hamilton-Jacobi form of the string equation.
4. Hirzebruch–Riemann–Roch formula for orbifolds
Given a compact complex manifold equipped with a holomorphic vector bundle , the Hirzebruch–Riemann–Roch formula  provides a cohomological expression for the super-dimension (i.e. Euler characteristic) of the sheaf cohomology:
The generalization of this formula to the case when is an orbifold and an orbibundle is due to T. Kawasaki . It expresses as an integral over the inertia orbifold of :
By definition, the structure of an -dimensional complex orbifold on is given by an atlas of local charts , the quotients of neighborhoods of the origin in by (linear) actions of finite local symmetry groups (one group for each point ).
By definition, charts on the inertia orbifold have the form , where is the fixed point locus of , and is the centralizer of in . For elements from the same conjugacy class, the charts are canonically identified by the action of . Thus, locally near , connected components of the inertia orbifold are labeled by conjugacy classes, , in . Integration over the fundamental class involves the division by the order of the stabilizer of a typical point in (and hence by the order of at least).
Near a point , the tangent and normal orbibundles and are identified with the tangent bundle to and normal bundle to in respectively.
The Kawasaki’s formula makes use of the obvious lift to of the orbibundle on . By , we denoted the K-theoretic Euler class of , i.e. the exterior algebra of the dual bundle, considered as a -graded bundle (the “Koszul complex”).
The fiber of an orbibundle on at a point carries the direct
decomposition into the sum of eigenspaces of .
By we denote the trace bundle
The denominator in Kawasaki’s formula is invertible because does not have eigenvalue on the normal bundle to its fixed point locus.
Finally, and denote the Todd class and Chern character.
When is a global quotient, , of a manifold by a finite group, and is a -equivariant bundle over , Kawasaki’s result reduces to Lefschetz’ holomorphic fixed point formula for super-traces in the sum
The orbifold is contained in its inertia orbifold as the component corresponding to the identity elements of local symmetry groups. The corresponding term of Kawasaki’s formula is
We call it the fake holomorphic Euler characteristic of . It is generally speaking a rational number, while the “true” holomorphic Euler characteristic is an integer.
Note that the right hand side of Kawasaki’s formula is the fake holomorphic Euler characteristic of an orbibundle, , on the inertia orbifold.
Our goal in this paper is to use Kawasaki’s formula for expressing genus-0 K-theoretic GW-invariants in terms of cohomological ones. We refer to  (see also the thesis ) for the virtual version of Kawasaki’s theorem, which justifies application of the formula to moduli spaces of stable maps.
The moduli spaces of stable maps are stacks, i.e. locally are quotients of spaces by finite groups. The local symmetry groups are automorphism groups of stable maps. A point in the inertia stack is specified by a pair: a stable map to the target space and an automorphism of the map. In a sense, a component of the inertia stack parametrizes stable maps with prescribed symmetry.
The components themselves are moduli spaces naturally equipped with virtual fundamental cycles and virtual structure sheaves. In fact, they are glued from moduli spaces of stable maps of smaller degrees — quotients of symmetric stable maps by the symmetries. Thus the individual integrals of Kawasaki’s formula can be set up as certain invariants of fake quantum K-theory, i.e. fake holomorphic Euler characteristics of certain orbibundles on spaces glued from usual moduli spaces of stable maps.
Our plan is to identify these invariants in terms of conventional ones and express them — and thereby the “true” genus-0 K-theoretic Gromov-Witten theory — in terms of cohomological GW-invariants.
5. The fake quantum K-theory
Fake K-theoretic GW-invariants are defined by
i.e. as cohomological GW-invariants involving the Todd class of the virtual tangent bundle to the moduli spaces of stable maps.
The Chern characters are unipotent, and as a result, generating function for the fake invariants are defined on the space of formal power series of . In particular, the big J-function
takes an input
The symplectic form is defined by
Expand into a series of powers of :
According to , we obtain a Darboux basis:
Taking to be spanned over by , we obtain a Lagrangian polarization of . As before, the big J-function coincides, up to the dilaton shift , with the graph of the differential of the genus-0 descendant potential: .
The range of the function forms (a formal germ at of) an overruled Lagrangian cone, . The proof is based on the relationship  between gravitational descendants and ancestors of fake quantum K-theory, which looks identical to the one in “true” K-theory (although the values of fake and true GW-invariants disagree).
In fact the whole setup for fake GW-invariants can be made purely topological, extended to include , and moreover, generalized to all complex-orientable extraordinary cohomology theories (i.e. complex cobordisms). In this generality, the quantum Hirzebruch–Riemann–Roch theorem of [3, 6] expresses the fake GW-invariants (of all genera) in terms of the cohomological gravitational descendants. The special case we need is stated below, after a summary of the symplectic loop space formalism of quantum cohomology theory.
Take , and . Let denote the space of power -series whose coefficients are Laurent series in one indeterminate, . Abusing notation we write: , (remembering that elements of are Laurent series only modulo any power of ). Define in the symplectic form
and Lagrangian polarization
Using Poincare-dual bases of , and the notation , we define the big J-function of cohomological GW-theory
It takes inputs from , takes values
where for and , we have:
The range of the function is a Lagrangian cone, , overruled in the sense that its tangent spaces, , are tangent to along (see Appendix 2 in ).
Identify with using the Chern character isomorphism and . Then is obtained from by the pointwise multiplication on by :
Remarks. (1) Given a function , the Euler–Maclaurin asymptotics of is obtained by the formal procedure:
where , is the anti-derivative , and are Bernoulli numbers. Taking to be the Todd series, , and summing over the Chern roots of the tangent bundle , we get:
where the coefficients hide another occurrence of Bernoulli numbers:
(2) Note that neither nor is symplectic: the former because , the latter because of the factor . However the composition is symplectic.
(3) The transformation between cohomological and K-theoretic J-functions (or descendant potentials) encrypted by the theorem, involves three aspects. One is the transformation , while the other two are the changes of the polarization and dilaton shift. Namely, maps to but does not map to , and there is a discrepancy between the dilaton shifts: .
(4) Since is an overruled cone, it is invariant under the multiplication by the ratio . This shows one way of correcting for the discrepancy in dilaton shifts.
(5) The proof of the theorem does not exploit any properties of overruled cones. One uses the family of “extraordinary” Todd classes to interpolate between cohomology and K-theory, and establishes an infinitesimal version of the theorem. For this, the twisting classes of the moduli spaces are expressed in terms of the descendant classes by applying the Gröthendieck–Riemann–Roch formula to the fibrations .
We refer for all details to the dissertation . However, in Section 8, we indicate geometric origins of the three changes described by the theorem: the change in the position of the cone, in the dilaton shift, and in the polarization.
6. Adelic characterization of quantum K-theory
Recall that point is a series in the Novikov variables, , with vector coefficients which are rational functions of . For each , we expand (coefficients of) in a Laurent series in and thus obtain the localization near . Note that for , the localization lies in the loop space of fake quantum K-theory. The main result of the present paper is the following theorem, which provides a complete characterization of the true quantum K-theory in terms of the fake one.
Theorem. Let be the overruled Lagrangian cone of quantum K-theory of a target space . If , then the following conditions are satisfied:
(i) has no pole at unless is a root of .
(ii) When , the localization lies in .
In particular, the localization at of the value of the J-function with the input lies in . In the tangent space to at the point , make the change , , and denote by the resulting subspace in . Let denote the Euler–Maclaurin asymptotics as of the infinite product:
(iii) If be a primitive -th root of 1, then .
Conversely, if satisfies conditions (i),(ii),(iii), then .
Remarks. (1) The cone is a formal germ at . The statements (direct and converse) about “points” are to be interpreted in the spirit of formal geometry: as statements about families based at .
(2) K-theoretic Chern roots are characterized by where are cohomological Chern roots of .
(3) After the substitution the infinite product becomes
The Euler–Maclaurin expansion has the form
where also depends on as a parameter:
Note that since are nilpotent, is polynomial in with coefficients which expand into power series of . The scalar factor of is since for each of Chern roots,
(4) The (admittedly clumsy) definition of subspace can be clarified as follows. The tangent space to at the point is the range of the linear map , where is a matrix Laurent series in with coefficients in the Novikov ring (see Section 3). Let be obtained from by the change , . Then .
(5) The condition (iii) seems ineffective, since it refers to a tangent space to the cone at a yet unknown point . However, we will see later that the three conditions together allow one, at least in principle, to compute the values for any input , assuming that the cone is known, in a procedure recursive on degrees of stable maps. In particular, this applies to . The cone , in its turn, is expressed through , thanks to the quantum HRR theorem of the previous section, by a procedure which in principle has a similar recursive nature. Altogether, our theorem expresses all genus-0 K-theoretic gravitational descendants in terms of the cohomological ones. Thus this result indeed qualifies for the name: the Hirzebruch–Riemann–Roch theorem of true genus-0 quantum K-theory.
We describe here a more geometric (and more abstract) formulation of the theorem using the adelic version of the symplectic loop space formalism.
For each , let be the space of power -series with vector Laurent series in as coefficients. Define the symplectic form
and put . The adele space is defined as the subset in the Cartesian product:
consisting of collections such that, modulo any power of Novikov’s variables, for all but finitely many values of . The adele space is equipped with the product symplectic form:
Next, there is a map , which to a rational function of assigns the collection of its localizations at . Due to the residue theorem, the map is symplectic:
Given a collection of overruled Lagrangian cones such that modulo any power of Novikov’s variables, for all but finitely many values of , the product becomes an adelic overruled Lagrangian cone in the adele symplectic space.
In fact, “overruled” implies invariance of tangent spaces under multiplication by . Since is invertible at , all with must be linear subspaces.
According to the theorem, the image of the cone under the map followed by a suitable adelic (pointwise) completion, is an adelic overruled Lagrangian cone: