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Applications of polyfold theory. I: The polyfolds of Gromov-Witten theory. (English) Zbl 1434.53095

Memoirs of the American Mathematical Society 1179. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2203-5/pbk; 978-1-4704-4060-2/ebook). v, 218 p. (2017).
This paper uses polyfold theory to construct Gromov-Witten invariants in symplectic manifolds. It is the first expected application of the authors’ theory of polyfolds, which they developed to establish the analytic foundations of symplectic field theory (SFT).
The first author et al. began publishing their new theory of polyfolds in the papers [J. Eur. Math. Soc. (JEMS) 9, No. 4, 841–876 (2007; Zbl 1149.53053); Geom. Funct. Anal. 19, No. 1, 206–293 (2009; Zbl 1217.58005); Geom. Topol. 13, No. 4, 2279–2387 (2009; Zbl 1185.58002); Math. Ann. 346, No. 1, 139–198 (2010; Zbl 1210.58004)]. Simply put, their theory of polyfolds extends concepts from smooth differential geometry, including bundles, Fredholm sections of bundles, Cauchy-Riemann operators, and integration of differential forms to spaces that are much more general than traditional smooth manifolds. For instance, their theory assigns the structure of a smooth M-polyfold to an open 2-disk in the plane with a 1-dimensional arc attached [the first author et al., Discrete Contin. Dyn. Syst. 28, No. 2, 665–788 (2010; Zbl 1211.53099)].
In the paper under review, the authors apply their polyfolds theory to the spaces of \(J\)-holomorphic stable curves used to construct Gromov-Witten invariants. They construct an equivalence class of polyfold structures on the ambient space \(Z\) of stable curves from noded Riemann surfaces to a symplectic manifold \((Q,\omega)\), and they construct a bundle \(p:W \rightarrow Z\) which they equip with an sc-smooth strong polyfold bundle structure. They show that the Cauchy-Riemann operator \(\bar{\partial}_J\) defines an sc-smooth proper Fredholm section of \(p:W \rightarrow Z\), and they produce generic perturbations of \(\bar{\partial}_J\) such that the solution sets of the perturbed Fredholm problem are compact, weighted, smooth branched sub-orbifolds with a natural orientation. This allows them to apply the theory from [the first author et al., Math. Ann. 346, No. 1, 139–198 (2010; Zbl 1210.58004)], and they obtain the Gromov-Witten invariants as integrals of sc-differential forms over the perturbed solution sets of the Cauchy-Riemann operator.
The results in this paper can be stated more precisely as follows. Let \((S,j,M,D)\) be a noded Riemann surface with marked points \(M\), where \((S,j)\) is an oriented closed smooth surface with a smooth almost complex structure \(j\) and \(D\) is a finite collection of unordered nodal pairs \(\{x,y\}\) in \(S\) so that \(x \neq y\) and any two pairs that intersect are identical. The noded Riemann surface \((S,j,M,D)\) is called connected if the topological space obtained by identifying the points in the nodal pairs is connected, and it is called stable if its automorphism group is finite. A stable map of class \((m,\delta)\), where \(m \geq 3\) and \(\delta \geq 0\), is a tuple \(\alpha = (S,j,M,D,u)\), where \(u:S \rightarrow Q\) is a continuous map to a symplectic manifold \((Q,\omega)\), meeting a list of conditions given in Definition 1.4 of the paper. In particular, near each nodal point in \(D\), \(u\) has weak partial derivatives up to order \(m\) which when weighted by \(e^{\delta s}\) are \(L^2\)-functions, and \(u\) is of class \(m\) around all other points. Moreover, \((S,j,M,D)\) is connected and \(u(x) = u(y)\) for every nodal pair \(\{x,y\} \in D\).
There is a natural notion of equivalence between stable maps, and equivalence classes of stable maps are called stable curves. Fixing a \(\delta_0 \in (0,2\pi)\) the collection of all equivalence classes of stable curves of class \((3,\delta_0)\) is denoted by \(Z = Z^{3,\delta_0}(Q,\omega)\). A basis for the topology on \(Z\) involves gluing noded Riemann surfaces using small disk structures around the nodal points and a gluing profile. One of the main theorems in the paper, proved in Chapter 3, can now be stated as follows.
Theorem 1.7. Given a strictly increasing sequence \((\delta_m)\), starting at the previously chosen \(\delta_0\) and staying below \(2\pi\), and the gluing profile \(\varphi(r) = e^{\frac{1}{r}} - e\), the space \(Z = Z^{3,\delta_0}(Q,\omega)\) of stable curves in \(Q\) has in a natural way the structure of a polyfold for which the \(m\)-th level consists of equivalence classes of stable maps \((S,j,M,D,u)\) in which \(u\) is of class \((m + 3, \delta_m)\).
The bundle \(W\) over \(Z\) consists of tuples \(\hat{\alpha} = (\alpha,\xi) = (S,j,M,D,u,\xi)\), where \(Q\) is equipped with an almost complex structure \(J\), and \(\xi\) is a continuous section along \(u\) such that the map \[ \xi(z):T_zS \rightarrow T_{u(z)}Q \] is complex anti-linear for \(z \in S\). The authors prove the following theorems concerning the bundle \(p:W \rightarrow Z\) in Chapters 3 and 4.
Theorem 1.10. Let \(Z = Z^{3,\delta_0}(Q, \omega)\) be the previously introduced space of stable curves with its polyfold structure associated with the increasing sequence \((\delta_m) \subset (0, 2\pi)\) and the exponential gluing profile \(\varphi\). Then the bundle \(p: W \rightarrow Z\) has in a natural way the structure of a strong polyfold bundle in which the \((m,k)\)-bi-level (for \(0 \leq k \leq m+1\)) consists of elements of base regularity \((m+3, \delta_m)\) and of fiber regularity \((k+2, \delta_k)\).
Theorem 1.11. The Cauchy-Riemann section \(\bar{\partial}_J\) of the strong polyfold bundle \(p:W \rightarrow Z\) is an sc-smooth component-proper Fredholm section, which is naturally oriented. On the component \(Z_{A,g,m}\) of the polyfold \(Z\) the Fredholm index of \(\bar{\partial}_J\) is equal to \[ \text{Ind}(\bar{\partial}_J) = 2c_1(A) + (2n - 6)(1 - g) + 2m, \] with \(2n = \text{dim }Q\), where \(g\) is the arithmetic genus of the noded Riemann surfaces, and \(m\) the number of marked points and \(A \in H_2(Q)\).
In the previous theorem, \(Z_{A,g,m}\) denotes the subset of \(Z=Z^{3,\delta_0}(Q,\omega)\) consisting of equivalence classes \([S,j,M,D,u]\) where the underlying noded Riemann surface is of genus \(g\) with \(m\) marked points and \(u\) represents the homology class \(A \in H_2(Q,\mathbb{Z})\). Similarly, \(Z_{g,m} \subset Z\) is the analogous subset without any restriction on the homology class represented by \(u\). There are evaluation maps \[ \mathrm{ev}_i:Z_{g,m} \rightarrow Q \] given by evaluating \(u\) at the marked points for \(i=1,\ldots ,m\), and a map \[ \gamma:Z_{g,m} \rightarrow \overline{\mathcal{M}}_{g,m} \] to the Deligne-Mumford compactification of the space of ordered marked stable Riemann surfaces with its holomorphic orbifold structure, given by “forgetting” the \(u\) in \([S,j,M,D,u] \in Z_{g,m}\). The authors prove that in the cases of interest for GW-invariants these maps are sc-smooth.
Theorem 1.8. If \(2g+m \geq 3\), then the maps \(\mathrm{ev}_i:Z_{g,m} \rightarrow Q\) for \(1 \leq i \leq m\) and \(\gamma:Z_{g,m} \rightarrow \overline{\mathcal{M}}_{g,m}\) are sc-smooth.
Theorems 1.7, 1.8, 1.10, and 1.11 show that the authors can apply the polyfold theory from their previous papers to construct the Gromov-Witten invariants. In particular, they have an abstract perturbation theory involving sc\(^+\)-multisections \(\lambda\) that they can apply to the Fredholm section \(\bar{\partial}_J\) to achieve transversality. Moreover, they can use the maps \(\mathrm{ev}_i\) and \(\gamma\) to pull back differential forms to sc-differential forms and integrate them over the solution sets \[ S(\bar{\partial}_J,\lambda) = \{z \in Z_{A,g,m}| \lambda(\bar{\partial}_J(z)) > 0\} \] using a weight function \(\vartheta:Z_{A,g,m} \rightarrow \mathbb{Q}^+\). This establishes the main theorem of the paper, which proves the existence of the Gromov-Witten invariants using polyfold theory, as a direct consequence of Theorem 4.23 of [the first author et al., Geom. Topol. 13, No. 4, 2279–2387 (2009; Zbl 1185.58002)].
Theorem 1.12. Let \((Q, \omega)\) be a closed symplectic manifold of dimension \(2n\). For a given homology class \(A \in H_2(Q)\) and for given integers \(g,m \geq 0\) satisfying \(2g + m \geq 3\), there exists a multi-linear map \[ \Psi^Q_{A,g,m}: H^\ast(Q;\mathbb{R})^{\otimes m} \rightarrow H^\ast(\overline{\mathcal{M}}_{g,m};\mathbb{R}) \] which, on \(H^\ast(Q;\mathbb{R})^{\otimes m}\), is super-symmetric with respect to the grading by even and odd forms. This map is uniquely characterized by the following formula. For a given compatible almost complex structure \(J\) on \(Q\) and a given small generic perturbation by an sc\(^+\)-multisection \(\lambda\), we have the representation \[ <\Psi^Q_{A,g,m}([\alpha_1] \otimes \cdots \otimes [\alpha_m]),[\tau]> \ = \int_{(S(\bar{\partial}_J,\lambda),\vartheta)} \gamma^\ast(PD(\tau))\wedge \mathrm{ev}_1^\ast(\alpha_1) \wedge \cdots \wedge \mathrm{ev}_m^\ast(\alpha_m) \] in which \(\alpha_1,\ldots, \alpha_m \in H^\ast(Q), \tau \in H^\ast(\overline{\mathcal{M}}_{g,m})\), \(PD\) denotes the Poincaré dual, and \(\gamma:Z_{g,m} \rightarrow \overline{\mathcal{M}}_{g,m}\) is the map in Theorem 1.8.

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
58B99 Infinite-dimensional manifolds
58C99 Calculus on manifolds; nonlinear operators
57R17 Symplectic and contact topology in high or arbitrary dimension
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