Using the formalism of quantized quadratic Hamiltonians [{\it A. B. Givental}, Mosc. Math. J. 1, No. 4, 551--568 (2001;

Zbl 1008.53072)], the authors are able to prove quantum versions of three classical theorems in algebraic geometry; namely, the Riemann-Roch theorem, Serre duality, and the Lefschetz hyperplane section theorem. The key ingredient consists in introducing a notion of twisted Gromov-Witten invariants of a compact projective complex manifold $X$; the quantum version of the aforementioned theorems can then be seen as relations between the twisted and the nontwisted Gromov-Witten theory of $X$.
More precisely, let $X_{g,n,d}$ be the moduli space of genus $g$, $n$-pointed stable maps to $X$ of degree $d$, where $d$ is an element in $H_2(X;\mathbb{Z})$, and let $E$ be a holomorphic vector bundle on $X$. Since a point in $X_{g,n,d}$ is represented by a pair $(\Sigma,f)$, where $\Sigma$ is a complex curve and $f:\Sigma\to X$ a holomorphic map, one can use $f$ to pull back $E$ on $\Sigma$ and then consider the $K$-theory Euler character of $f^*E$, i.e., the virtual vector space $H^0(\Sigma,f^*E)\ominus H^1(\Sigma,f^*E)$, as the fiber over $[(\Sigma,f)]$ of a virtual vector bundle $E_{g,n,d}$ over $X_{g,n,d}$. This intuitive construction is made completely rigorous by considering $K$-theory push-pull $K^0(X)\to K^0(X_{g,n,d})$ along the diagram $$\CD X_{g,n+1,d}@> {\text{ev}_{n+1}}>> X\\ @V{\pi}VV\\ X_{g,n,d}\endCD$$ A rational invertible multiplicative characteristic class of a complex vector bundle is an expression of the form $$ \bold{c}(\cdot)=\exp\left(\sum_{k=0}^\infty s_k \text{ch}_k(\cdot)\right), $$ where $\text{ch}_k$ are the components of the Chern character, and the $s_k$ are arbitrary parameters. These data determine a cohomology class $\bold{c}(E_{g,n,d})$ (actually, a formal family of cohomology classes parametrized by the $s_k$) in $H^*(X_{g,n,d};\mathbb{Q})$, and one can define the total $(\bold{c},E)$-twisted descendant potential $\mathcal{D}_{\bold{c},E}^g$ as $$ \mathcal{D}_{\bold{c},E}(t_0,t_1,\dots)=\exp\left(\sum_{g\geq 0}\hbar^{g-1}\mathcal{F}^g_{\bold{c},E}(t_0,t_1,\dots)\right), $$ where $$ \mathcal{F}_{\bold{c},E}^g(t_0,t_1,\dots)=\sum_{n,d}\frac{Q^d}{n!}\int_{[X_{g,n,d}]}\bold{c}(E_{g,n,d}) (\sum_{k_1=0}^\infty(\text{ev}_1^*t_k)\psi_1^{k_1}) \cdots (\sum_{k_1=0}^\infty(\text{ev}_n^*t_k)\psi_n^{k_n}). $$ Here $Q^d$ is the representative of $d$ in the semigroup ring of degrees of holomorphic curves in $X$, $t_0,t_1,\dots$ are rational cohomology classes on $X$, and $\psi_i$ is the first Chern class of the universal cotangent bundle over $X_{g,n,d}$ corresponding to the $i$-th marked point of $X$. For $E$ the zero element in $K^0(X)$, the twisted potential $\mathcal{D}_{\bold{c},E}^g$ reduces to $\mathcal{D}_X$, the total descendant potential of $X$.
At this point the formalism of quantized quadratic hamiltonians enters the picture. One considers the symplectic space $\mathcal{H}=H^*(X;\mathbb{Q})((z^{-1}))$ of Laurent polynomials in $z^{-1}$ with coefficients in the cohomology of $X$, endowed with the symplectic form $$ \Omega(\bold{f},\bold{g})=\frac{1}{2\pi i}\oint \left(\int_X\bold{f}(-z)\bold{g}(z)\right)\,dz. $$ The subspace $\mathcal{H}_+=H^*(X;\mathbb{Q})[z]$ is a Lagrangian subspace, and $(\mathcal{H},\Omega)$ is identified with the canonical symplectic structure on $T^*\mathcal{H}_+$. Finally, given an infinitesimal symplectic transformation $T$ of $\mathcal{H}$, one can consider the differential operator $\hat{T}$ of order $\leq 2$ on functions on $\mathcal{H}_+$, which is associated by quantization to the quadratic Hamiltonian $\Omega(T\bold{f},\bold{f})/2$ on $\mathcal{H}$. By the inclusion $\mathcal{H}_+\hookrightarrow H^*(X;\mathbb{Q})[[z]]$, the operator $\hat{T}$ acts on asymptotic elements of the Fock space, i.e., on functions of the formal variable $\bold{q}(z)=q_0+q_1z+q_2z^2+\cdots$ in $H^*(X;\mathbb{Q})[[z]]$. By the dilaton shift, i.e., setting $\bold{q}(z)=\bold{t}(z)-z$, with $\bold{t}(z)=t_0+t_1z+t_2z^2+\cdots$, the operator $\hat{T}$ acts on any function of $t_0,t_1,\dots$, notably on the descendant potentials.
Having introduced this formalism, the authors are able to express the relation between twisted and untwisted Gromov-Witten invariants in an extremely elegant way: up to a scalar factor, $$ \mathcal{D}_{\bold{c},E}=\hat{\Delta}\mathcal{D}_X, $$ where $\Delta:\mathcal{H}\to \mathcal{H}$ is the linear symplectic transformation defined by the asymptotic expansion of $$ \sqrt{\bold{c}(E)}\prod_{m=1}^\infty \bold{c}(E\otimes L^{-m}) $$ under the identification of the variable $z$ with the first Chern class of the universal line bundle $L$. This is the quantum Riemann-Roch theorem; it explicitly determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants. The result is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family $\pi:X_{g,n+1,d}\to X_{g,n,d}$. If $E=\mathbb{C}$ is the trivial line bundle, then $E_{g,n,d}=\mathbb{C}\ominus \bold{E}_g^*$, where $\bold{E}_g$ is the Hodge bundle, and one recovers from quantum Riemann-Roch results of {\it D. Mumford} [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271--328 (1983;

Zbl 0554.14008)] and {\it C. Faber, R. Pandharipande} [Invent. Math. 139, No.1, 173--199 (2000;

Zbl 0960.14031)] on Hodge integrals.
If $\bold{c}^*$ is the multiplicative characteristic class $$ \bold{c}^*(\cdot)=\exp\left(\sum_{k=0}^\infty(-1)^{k+1} s_k\mathrm{ch}_k(\cdot)\right), $$ then $\bold{c}^*(E^*)=1/\bold{c}(E)$, and one the following quantum version of Serre duality: $$ \mathcal{D}_{\bold{c}^*,E^*}(\bold{t}^*)=(\mathrm{sdet}\,\bold{c}(E))^{-\frac{1}{24}}\mathcal{D}_{\bold{c},E}(\bold{t}), $$ where $\bold{t}^*(z)=\bold{c}(E)\bold{t}(z)+(1-\bold{c}(E))z$.
Finally, if $E$ is a convex vector bundle and a submanifold $Y\subset X$ is defined by a global section of $E$, then the genus zero Gromov-Witten invariants of $Y$ can be expresssed in terms of the invariants of $X$ twisted by the Euler class of $E$. These are in turn related to the untwisted Gromov-Witten invariants of $X$ by the quantum Riemann-Roch theorem, so the authors end up with a quantum Lefschetz hyperplane section principle, expressing genus zero Gromov-Witten invariants of a complete intersection $Y$ in terms of those of $X$. This extends earlier results [{\it V. V. Batyrev, I. Ciocan-Fontanine, B. Kim} and {\it D. van Straten}, Acta Math. 184, No. 1, 1--39 (2000;

Zbl 1022.14014); {\it A. Bertram}, Invent. Math. 142, No. 3, 487--512 (2000;

Zbl 1031.14027); {\it A. Gathmann}, Math. Ann. 325, No. 2, 393--412 (2003;

Zbl 1043.14016); {\it B. Kim}, Acta Math. 183, No. 1, 71--99 (1999;

Zbl 1023.14028); {\it Y.-P. Lee}, Invent. Math. 145, No. 1, 121--149 (2001;

Zbl 1082.14056)], and yields most of the known mirror formulas for toric complete intersections. The idea of deriving mirror formulas by applying the Grothendieck-Riemann-Roch theorem to universal stable maps is not new: according to the authors it can be traced back at least to Kontsevich’s investigations in the early 1990s, and to Faber’s and Pandharipande’s work on Hodge integrals.