Quantum Riemann–Roch, Lefschetz and Serre


Given a holomorphic vector bundle $E$ over a compact Kähler manifold $X$, one defines twisted Gromov-Witten invariants of $X$ to be intersection numbers in moduli spaces of stable maps $f:\Sigma \to X$ with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle $H^0(\Sigma,f^* E) \ominus H^1(\Sigma,f^* E)$. Using the formalism of quantized quadratic Hamiltonians [25], we express the descendant potential for the twisted theory in terms of that for $X$. This result (Theorem $1$) is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants.

When $E$ is concave and the $\mathbb{C}^\times$-equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of $X$ give Gromov-Witten invariants of the total space of $E$. “Nonlinear Serre duality” [21], [23] expresses Gromov-Witten invariants of $E$ in terms of those of the super-manifold $\Pi E$: it relates Gromov-Witten invariants of $X$ twisted by the inverse Euler class and $E$ to Gromov-Witten invariants of $X$ twisted by the Euler class and $E^*$. We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2).

When the bundle $E$ is convex and a submanifold $Y\subset X$ is defined by a global section of $E$, the genus-zero Gromov-Witten invariants of $\Pi E$ coincide with those of $Y$. We establish a “quantum Lefschetz hyperplane section principle” (Theorem $2$) expressing genus-zero Gromov-Witten invariants of a complete intersection $Y$ in terms of those of $X$. This extends earlier results [4], [9], [18], [29], [33] and yields most of the known mirror formulas for toric complete intersections.