  Given a holomorphic vector bundle $E:EX X$ over a compact K\"ahler manifold,
one introduces twisted GW-invariants of $X$ replacing virtual fundamental
cycles of moduli spaces of stable maps $f: \Sigma \to X$ by their cap-product
with a chosen multiplicative characteristic class of $H^0(\Sigma, f^* E) -
H^1(\Sigma, f^*E)$. Using the formalism of quantized quadratic hamiltonians, we
express the descendent potential for the twisted theory in terms of that for
$X$. The result (Theorem 1) is a consequence of Mumford's Riemann -- Roch --
Grothendieck formula applied to the universal stable map.
  When $E$ is concave, and the inverse $\CC^{\times}$-equivariant Euler class
is chosen, the twisted theory yields GW-invariants of $EX$. The ``non-linear
Serre duality principle'' expresses GW-invariants of $EX$ via those of the
supermanifold $\Pi E^*X$, where the Euler class and $E^*$ replace the inverse
Euler class and $E$. We derive from Theorem 1 the 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, the genus 0 GW-invariants of $\Pi E X$ coincide with those of
$Y$. We prove a ``quantum Lefschetz hyperplane section principle'' (Theorem 2)
expressing genus 0 GW-invariants of a complete intersection $Y$ via those of
$X$. This extends earlier results of Y.-P. Lee and A. Gathmann and yields most
of the known mirror formulas for toric complete intersections.
