## Quantum Riemann-Roch, Lefschetz and Serre.(English)Zbl 1189.14063

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Using the formalism of quantized quadratic Hamiltonians [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 $\begin{tikzcd} X_{g,n+1,d}\rar["\mathrm{ev}_{n+1}"]\dar["\pi" '] &X\\ X_{g,n,d}\end{tikzcd}$ A rational invertible multiplicative characteristic class of a complex vector bundle is an expression of the form $\mathbf{c}(\cdot)=\exp\left(\sum_{k=0}^\infty s_k \text{ch}_k(\cdot)\right),$ where $$\text{ch}_k$$ are the components of the Chern character, and the $$s_k$$ are arbitrary parameters. These data determine a cohomology class $$\mathbf{c}(E_{g,n,d})$$ (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 $$(\mathbf{c},E)$$-twisted descendant potential $$\mathcal{D}_{\mathbf{c},E}^g$$ as $\mathcal{D}_{\mathbf{c},E}(t_0,t_1,\dots)=\exp\left(\sum_{g\geq 0}\hbar^{g-1}\mathcal{F}^g_{\mathbf{c},E}(t_0,t_1,\dots)\right),$ where $\mathcal{F}_{\mathbf{c},E}^g(t_0,t_1,\dots)=\sum_{n,d}\frac{Q^d}{n!}\int_{[X_{g,n,d}]}\mathbf{c}(E_{g,n,d}) (\sum_{k_1=0}^\infty(\text{ev}_1^*t_k)\psi_1^{k_1}) \cdots (\sum_{k_1=0}^\infty(\text{ev}_n^*t_k)\psi_n^{k_n}).$ 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 $$\mathcal{D}_{\mathbf{c},E}^g$$ 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 $\Omega(\mathbf{f},\mathbf{g})=\frac{1}{2\pi i}\oint \left(\int_X\mathbf{f}(-z)\mathbf{g}(z)\right)\,dz.$ 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 $$\Omega(T\mathbf{f},\mathbf{f})/2$$ 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 $$\mathbf{q}(z)=q_0+q_1z+q_2z^2+\cdots$$ in $$H^*(X;\mathbb{Q})[[z]]$$. By the dilaton shift, i.e., setting $$\mathbf{q}(z)=\mathbf{t}(z)-z$$, with $$\mathbf{t}(z)=t_0+t_1z+t_2z^2+\cdots$$, 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, $\mathcal{D}_{\mathbf{c},E}=\hat{\Delta}\mathcal{D}_X,$ where $$\Delta:\mathcal{H}\to \mathcal{H}$$ is the linear symplectic transformation defined by the asymptotic expansion of $\sqrt{\mathbf{c}(E)}\prod_{m=1}^\infty \mathbf{c}(E\otimes L^{-m})$ 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 $$E_{g,n,d}=\mathbb{C}\ominus \mathbf{E}_g^*$$, where $$\mathbf{E}_g$$ is the Hodge bundle, and one recovers from quantum Riemann-Roch results of D. Mumford [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271–328 (1983; Zbl 0554.14008)] and C. Faber, R. Pandharipande [Invent. Math. 139, No.1, 173–199 (2000; Zbl 0960.14031)] on Hodge integrals.
If $$\mathbf{c}^*$$ is the multiplicative characteristic class $\mathbf{c}^*(\cdot)=\exp\left(\sum_{k=0}^\infty(-1)^{k+1} s_k\mathrm{ch}_k(\cdot)\right),$ then $$\mathbf{c}^*(E^*)=1/\mathbf{c}(E)$$, and one the following quantum version of Serre duality: $\mathcal{D}_{\mathbf{c}^*,E^*}(\mathbf{t}^*)=(\mathrm{sdet}\,\mathbf{c}(E))^{-\frac{1}{24}}\mathcal{D}_{\mathbf{c},E}(\mathbf{t}),$ where $$\mathbf{t}^*(z)=\mathbf{c}(E)\mathbf{t}(z)+(1-\mathbf{c}(E))z$$.
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 [V. V. Batyrev, I. Ciocan-Fontanine, B. Kim and D. van Straten, Acta Math. 184, No. 1, 1–39 (2000; Zbl 1022.14014); A. Bertram, Invent. Math. 142, No. 3, 487–512 (2000; Zbl 1031.14027); A. Gathmann, Math. Ann. 325, No. 2, 393–412 (2003; Zbl 1043.14016); B. Kim, Acta Math. 183, No. 1, 71–99 (1999; Zbl 1023.14028); 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.

### MSC:

 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14C40 Riemann-Roch theorems 14J33 Mirror symmetry (algebro-geometric aspects)
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