The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory
By Alexander Givental, Valentin Tonita. 2011
69 citations recorded by the source corpus.
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 in separate papers 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 $l_1,...,l_k$ in a complex projective space of dimension $\geq l_1^2+...+l_k^2-1$.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5ccbb11026678d7f997975352d0722849ab0d77f
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Elliptic stable envelopes (2016, background): “theoretic computation on the moduli spaces M0,n(X) and M0,n(X×P1) of stable pointed rational maps to respective targets. Givental and his collaborators studied difference equations in quantum K-theory [34,35], and their theory is very general — Xcan be an arbitrary nonsingular algebraic variety, or e”
- Lectures on K-theoretic computations in enumerative geometry (2015, methodology): “Many yeas ago, Givental initiated a very broad program of developing quantum K-theory and, in particular, the paper [37] gives a very general proof of the existence of quantum difference equations in quantum K-theory.”