Given a projective smooth complex variety X, one way to associate to it numerical

invariants is by taking holomorphic Euler characteristics of interesting vector bundles on

the moduli spaces of genus 0, degree d stable maps with n marked points to X. We

call these numbers genuine quantum K-theoretic invariants of X. Their generating series is

called the genus 0 K-theoretic descendant potential of X and can be viewed as a function

on a suitable infinite dimensional vector space K+. Its graph is a uniruled Lagrangian cone

in the cotangent bundle of K+.

We give a complete characterisation of points on the cone, proving a Hirzebruch Riemann

Roch type theorem for the genuine K-theory of X. In particular, our result can be used

to recursively express all genus 0 K-theoretic invariants of X in terms of cohomological

ones (usually known as Gromov-Witten invariants). The main technical tool we use is the

Kawasaki Riemann Roch theorem of [Ka], which reduces the computation of holomorphic

Euler characteristic of a bundle on an orbifold to the computation of a cohomological integral

on the inertia orbifold.

In the process, we need to study more general cohomological Gromov-Witten invariants

of an orbifold X, which we call twisted invariants. These are obtained by capping the virtual

fundamental classes of the moduli spaces with certain multiplicative characteristic

classes. We twist the Gromov-Witten potential by three types of twisting classes and we

allow several twistings of each type. We use a Mumford's Grothendieck-Riemann-Roch

computation on the universal curve to give closed formulae which show the eect of each

type of twist on their generating series (the twisted potential). This generalizes earlier results

of [CG] and [TS].

Quantum K-theory studies the K-theoretic Gromov-Witten (GW) invariants, which, for a given target space $X$, are defined as virtual holomorphic Euler characteristics of vector bundles over the moduli space $\overline{\mathcal{M}}_{g,m}(X,d)$ of stable maps. Roughly speaking, the invariants count the number of curves on $X$ satisfying specific topological and geometric constraints. Since the emergence of the theory at the beginning of this century as an analogue of quantum cohomology, these invariants have not only provided new insight into enumerative geometry, but have inspired applications in representation theory, integrable systems, and even theoretic physics as well.

In this dissertation, we study the genus-$0$ quantum K-theory of partial flag varieties using the idea of abelian/non-abelian correspondence. Regarding any flag variety as a GIT quotient of vector space by a non-abelian group, we relate its quantum K-theory to that of its associated abelian GIT quotient, a toric variety whose quantum K-theory is well-understood. As a consequence, we provide eventually for the flag variety an explicit parameterization of the image cone of a generating function commonly known as the ``big $\mathcal{J}$-function'', which encodes all of its genus-$0$ K-theoretic GW invariants. This generalizes existing results for toric varieties.

The main technical tool employed in the process is a recursive characterization of the image cone developed in a series of papers by Givental. The characterization exists for any target space equipped with a torus action with isolated fixed points as well as isolated connecting orbits, and is obtained from a brilliant observation of the fixed-point localization computations of the invariants. Since the toric fixed points on the associated abelian quotient $Y$ of a flag variety are, however, seldom isolated, a substantial part of this dissertation is dedicated to extending the characterization to non-isolated cases. We demonstrate a vanishing result of the contribution from the non-isolated fixed points to the recursive formulae.

As applications of the techniques developed in this dissertation, we consider various twisted quantum K-theories. In particular, we first derive a quantum Lefschetz theorem which characterizes genus-$0$ K-theoretic GW invariants of a complete intersection in (or a vector bundle space over) the flag variety. Various generating functions of quasi-map invariants that arise in literature are thus realized as values of twisted big $\mathcal{J}$-functions. Moreover, we study the level structures, which are twistings of determinantal type on the quantum K-theory, and explain the level correspondence phenomenon between dual flag varieties. Finally, we establish the quantum Serre duality for general target spaces, which relates two different types of twistings involving the level structures.

At the end of this dissertation, we construct the K-theoretic mirror of the flag variety as $q$-integrals on a family of tori, following the pattern in quantum cohomology. When the integration cycles are carefully chosen, the $q$-integrals are proven to recover, up to some scalars, components of the small $J$-function which is the specific value of the big $\mathcal{J}$-function with vanishing input.

We prove a twisting theorem for nodal classes in permutation-equivariant quantum K-theory, and combine it with existing theorems of Givental to obtain a twisting a theorem for general characteristic classes of the virtual tangent bundle of the moduli space of stable maps. Using this result, we develop complex cobordism-valued Gromov-Witten invariants defined via K-theory, and relate those invariants to K-theoretic ones via the quantization of suitable symplectic transformations. This procedure is a K-theoretic analogue of the quantum cobordism theory developed by Givental and Coates. Using the universality of cobordism theory, we give an example of these results in the context of ``Hirzebruch K-theory", which is the cohomology theory determined by the Hirzebruch y-genus.

We then introduce Euler-theoretic Gromov-Witten invariants, which are based on the ordinary topological Euler characteristics of loci of curves. These invariants have useful enumerative properties. They are integer valued, there is a reliable way to remove boundary contributions, and when the moduli spaces are smooth orbifolds, have a concrete geometric intepretation. In this case, the invariants also encompass the ordinary (i.e. integer) topological Euler characteristic of the moduli space of stable maps and its compactification.

We give a Wick-type formula computing these invariants in terms of twisted cohomological Gromov-Witten invariants, and we also show that they can be regarded as the limit as y approaches 1 of the Hirzebruch-theoretic invariants introduced earlier in this work.

Specializing to the case of a point target space, our formulas yield similar results to the computation of the Euler characteristics of the moduli spaces of n-pointed curves, given by Bini-Harer.

We prove a conjecture of Artur Elezi in a generalized form suggested by Givental. Namely, our main result relates genus-0 Gromov--Witten invariants of a bundle space with such invariants of the base, provided that the fiber is a toric manifold. When the base is the point, a new proof of mirror theorems by A. Givental and H. Iritani for toric manifolds is obtained.