The Hirzebruch-Riemann-Roch theorem in true genus-0 quantum \(K\)-theory. (English) Zbl 1335.19002

Eguchi, Tohru (ed.) et al., Symplectic, Poisson, and noncommutative geometry. Selected papers based on the presentations at the conferences: conference on symplectic and Poisson geometry in interaction with analysis, algebra and topology, Berkeley, CA, USA, May 4–7, 2010, conference on symplectic geometry, noncommutative geometry and physics, Berkeley, CA, USA, May 10–14, 2010 and Kyoto, Japan, November 1–5, 2010. Cambridge: Cambridge University Press (ISBN 978-1-107-05641-1/hbk). Mathematical Sciences Research Institute Publications 62, 43-91 (2014).
Summary: 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 manifolds 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’ variables. As another application, we compute one of such tangent spaces for each of the complete intersections given by equations of degrees \(l_1,\dots,l_k\) in a complex projective space of dimension \(\geq l^2_1+ \cdots+l^2_k-1\).
For the entire collection see [Zbl 1320.53003].


19L10 Riemann-Roch theorems, Chern characters
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
81T45 Topological field theories in quantum mechanics
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C40 Riemann-Roch theorems
14J81 Relationships between surfaces, higher-dimensional varieties, and physics
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