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.


Thomas Coates

Department of Mathematics, Harvard University, Cambridge, MA 02138, United States

Alexander Givental

Department of Mathematics, University of California Berkeley, Berkeley CA 94720, United States