In the present paper we extend the Riemann–Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and push-forwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann–Roch theorem for moment graphs. As an application, we obtain the Riemann–Roch type theorem for the equivariant K‑theory of some Kac–Moody flag varieties.

A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.

We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M_ {g, 1}. Finally in Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brezin and Hikami.

Let the vector bundle E be a deformation of the tangent bundle over the Grassmannian G(k, n). We compute the ring structure of sheaf cohomology valued in exterior powers of E, also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [6] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples.

We conjecture that appropriate K-theoretic Gromov-Witten invariants of complex flag manifolds G/B are governed by finite-difference versions of Toda systems constructed in terms of the Langlands-dual quantized universal enveloping algebras U_q(g'). The conjecture is proved in the case of classical flag manifolds of the series A. The proof is based on a refinement of the famous Atiyah-Hirzebruch argument for rigidity of arithmetical genus applied to hyperquot-scheme compactifications of spaces of rational curves in the flag manifolds.