Polyfold Gromov–Witten theory
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Notes to self
This documents my mathematical accomplishments over the last decade. Explain it as such! Everything in my mathematical life revolved around a proof of the Gromov–Witten axioms.
History of the Gromov–Witten axioms
In 1985 Gromov published the paper “Pseudo holomorphic curves in symplectic manifolds”, laying the foundations for the modern study of pseudo holomorphic curves (also know as \(J\)-holomorphic curves) in symplectic topology (Gromov 1985). In this paper, Gromov proved a compactness result for the moduli space of \(J\)-holomorphic curves in a fixed homology class. This paper contained antecedents to the modern notion of the Gromov–Witten invariants in the proofs of the nonsqueezing theorem and the uniqueness of symplectic structures on \(\mathbb{C}P^2\).
Around 1988, inspired by Floer’s study of gauge theory on three manifolds, Witten introduced the topological sigma model (Floer 1988; Witten 1988). The invariants of this model are the “\(k\)-point correlation functions”, another precursor to the modern notion of the Gromov–Witten invariants. Witten also observed some of the relationships between these invariants and possible degenerations of Riemann surfaces (Witten 1991). Further precursors to the notion of the Gromov–Witten invariants can also be seen in McDuff’s classification of symplectic ruled surfaces (McDuff 1991).
In 1993 Ruan gave a modern definition of the genus zero Gromov–Witten invariants for semipositive symplectic manifolds (Ruan 1996, 1994). At the end of 1993, Ruan and Tian established the associativity of the quantum product for semipositive symplectic manifolds, giving a mathematical basis to the composition law of Witten’s topological sigma model (Ruan and Tian 1995).
In 1994 Kontsevich and Manin stated the Gromov–Witten axioms, given as a list of formal relations between the Gromov–Witten invariants (Kontsevich and Manin 1994). At the time it was not possible for Kontsevich and Manin to give a proof of the relations they listed; the definition of the Gromov–Witten invariant (complete with homology classes from a Deligne–Mumford space) would require in addition new ideas involving “stable maps” (Kontsevich 1995). Hence they used to term “axiom” with the presumed meaning “to take for assumption without proof”/ “to use as a premise for further reasoning”. And indeed, from these starting assumptions they were able to establish foundational results in enumerative geometry, answers to esoteric questions such as:
Kontsevich’s recursion formula. Let \(d\geq 1\). How many degree \(d\) rational curves in \(\mathbb{C}P^2\) pass through \(3d - 1\) points in general position?
Moreover, in this paper they outlined some of the formal consequences of the axioms by demonstrating how to combine the invariants into a Gromov–Witten potential, and interpret the axioms as differential equations which the potential satisfies.
To varying extents, this work has predated the construction of a well-defined Gromov–Witten invariant in symplectic geometry for \(J\)-holomorphic curves of arbitrary genus, and for all closed symplectic manifolds. Efforts to construct a well-defined Gromov–Witten invariant constitute an ever growing list of publications, including but not limited to the following: (Li and Tian 1998; Fukaya and Ono 1999; Fukaya et al. 2012; Siebert 1996; Cieliebak and Mohnke 2007; McDuff and Wehrheim 2012, 2018, 2017; Ionel and Parker 2013; Pardon 2016). A discussion of some of the difficulties inherent in these approaches can be found in (Fabert et al. 2016). Similarly, there have been several efforts to prove the Gromov–Witten axioms (Fukaya and Ono 1999; McDuff and Salamon 2012; Castellano 2016).
Over the past two decades, Hofer, Wysocki, and Zehnder have developed a new approach to resolving transversality issues that arise in the study of \(J\)-holomorphic curves in symplectic geometry called polyfold theory (Hofer, Wysocki, and Zehnder 2007, 2009a, 2009b, 2017a, 2010b, n.d., 2010a, 2017b). This approach has been successful in constructing a well-defined Gromov–Witten invariant (Hofer, Wysocki, and Zehnder 2017a).
What is a Gromov–Witten invariant?
Let \((M,\omega)\) be a closed symplectic manifold of dimension \(\dim M = 2n\), and let \(J\) be a \(\omega\)-compatible almost complex structure. For a fixed homology class \(A\in H_2(M,{\mathbb Z})\), and for fixed integers \(g\geq 0\), \(k\geq 0\), we consider the set of \(J\)-holomorphic curves:
\[ {\mathcal M}_{A,g,k}(J) := \left\{ \begin{array}{c} u: (\Sigma_g,j) \to M \\ \{z_1,\ldots,z_k\}\in \Sigma_g \end{array} \biggm| \begin{array}{c} \tfrac{1}{2} (du+J\circ du\circ j)=0 \\ u_*[\Sigma_g] = A \end{array} \right\} \biggm/ \begin{array}{l} u \sim u\circ \phi, \\ \phi\in \text{Aut} \end{array} \]
consisting of smooth maps \(u:(\Sigma_g,j)\to M\) which satisfy the Cauchy–Riemann equation modulo reparametrization; here \((\Sigma_g,j)\) is a genus \(g\) Riemann surface and \(\text{Aut}\) is the automorphism group of the Riemann surface \((\Sigma_g,j)\) which preserves the ordering of the marked points.
Gromov proved this set has a natural compactification in (Gromov 1985), which was later refined into the stable map compactification of Kontsevich in (Kontsevich 1995), and thus we may also consider its compactification, the Gromov–Witten moduli space:
\[ \bar{\mathcal{M}}_{A,g,k} (J) := {\mathcal M}_{A,g,k} (J) \sqcup \{\text{stable nodal $J$-holomorphic curves}\}. \]
We seek to use this space to construct invariants of the symplectic manifold \(M\). To this end, we define the evaluation map which evaluates a stable curve on each marked point:
\[ ev: \bar{\mathcal{M}}_{A,g,k} (J) \to M\times \cdots \times M. \]
On the top stratum of non-noded stable \(J\)-holomorphic curves it is given by
\[ ev\left([(u,z_1,\ldots,z_k)] \right): = (u(z_1),\ldots, u(z_k)). \]
With the fixed integers \(g \geq 0\), \(k \geq 3\) we also consider the associated Deligne–Mumford space, the natural compactification of the space of configurations of a complex structure and \(k\)-marked points on a genus \(g\) Riemann surface modulo biholomorphic equivalence:
\[ \bar{\mathcal{M}}_{g,k} := \text{cl} \left(\{ j, \{z_1,\ldots,z_k\}\in \Sigma_g \mid j \text{ complex structure on } \Sigma_g, z_i\neq z_j \text{ if } i \neq j\} / \text{Aut} \right). \]
When \(g = 0\) this space is a finite-dimensional manifold, and when \(g\neq 0\) this space is a finite-dimensional orbifold, in either case of dimension \(\text{dim}\) \(\bar{\mathcal{M}}_{g,k} = 6g - 6 + 2k\) We may define a projection map from the GW-moduli space to the DM-space which forgets the curve which maps to \(M\) and which stabilizes the resulting unstable domain components:
\[ \pi: \bar{\mathcal{M}}_{A,g,k} (J) \to \bar{\mathcal{M}}_{g,k}. \]
On the top stratum of non-noded stable \(J\)-holomorphic curves it forgets the map \(u\) and is given by
\[ \pi\left([(u,j,z_1,\ldots,z_k)]\right) := [(j,z_1,\ldots,z_k)]. \]
The traditional interpretation of a Gromov–Witten invariant is the (supposedly) finite count of \(J\)-holomorphic curves which at the \(i\)th-marked point pass through a submanifold \({\mathcal X}_i \subset M\) and whose marked point configuration is restricted by the projection map to a suborbifold \({\mathcal B}\subset \bar{\mathcal{M}}_{g,k}\).
Such an intersection number should depend only on the homology classes of the submanifolds, and should be independent of the almost complex structure. This count can be packaged algebraically as a homomorphism:
\[ \mathop{\mathrm{GW}}_{A,g,k} : H_*(M;{\mathbb Q})^{\otimes k} \times H_*(\bar{\mathcal{M}}_{g,k};{\mathbb Q}) \to {\mathbb Q}. \]
A foundational problem in symplectic geometry is to actually show that such a GW-invariant is well-defined. Ideally, we would like to define a GW-invariant rigorously via an intersection number:
\[ \mathop{\mathrm{GW}}_{A,g,k} ([{\mathcal X}_1],\ldots,[{\mathcal X}_k];[{\mathcal B}]) = (ev\times \pi) (\bar{\mathcal{M}}_{A,g,k} (J)) \cdot ({\mathcal X}_1\times \cdots \times {\mathcal X}_k \times {\mathcal B}), \]
or as an integral:
\[ \mathop{\mathrm{GW}}_{A,g,k} ([{\mathcal X}_1],\ldots,[{\mathcal X}_k];[{\mathcal B}]) = \int_{\bar{\mathcal{M}}_{A,g,k}(J)} ev^* (\mathop{\mathrm{PD}}[{\mathcal X}_1]\wedge \cdots \wedge \mathop{\mathrm{PD}}[{\mathcal X}_k]) \wedge \pi^* \mathop{\mathrm{PD}}[{\mathcal B}], \]
or as a pairing with a (virtual) fundamental class:
\[ \mathop{\mathrm{GW}}_{A,g,k} ([{\mathcal X}_1],\ldots,[{\mathcal X}_k];[{\mathcal B}]) = \left\langle (ev\times\pi)_* [\bar{\mathcal{M}}_{A,g,k}(J)], \mathop{\mathrm{PD}}[{\mathcal X}_1\times\cdots\times{\mathcal X}_k\times{\mathcal B}] \right\rangle. \]
Such definitions require additional structure on the GW-moduli space; an intersection number requires tangent spaces and notions of transversal intersection, an integral requires smooth partitions of unity and notions of differential forms, and a (virtual) fundamental class requires a distinguished homology class on the topological space.
However, a priori, the GW-moduli space only has the structure of a compact topological space, and this alone is insufficient to define any of the above. More structure is needed.
The classical pseudocycle Gromov–Witten invariant for genus-zero and semipositive symplectic manifolds
The polyfold Gromov–Witten invariant for arbitrary genus and general symplectic manifolds
Polyfold theory, developed by Hofer, Wysocki, and Zehnder, is a modern new approach to resolving transversality issues that arise in attempts to solve moduli space problems in symplectic geometry. The polyfold theoretic approach to solving a moduli space problem is to recast the problem into familiar terms from differential geometry. To do this, we may construct a “Gromov–Witten polyfold” \({\mathcal Z}_{A,g,k}\)—a massive, infinite-dimensional ambient space, designed to contain the entire unperturbed Gromov–Witten moduli space \(\bar{\mathcal{M}}_{A,g,k}(J)\) as a compact subset. We may furthermore construct a “strong polyfold bundle” \({\mathcal W}_{A,g,k}\) over \({\mathcal Z}_{A,g,k}\); the Cauchy–Riemann operator then defines a “scale smooth Fredholm section” of this bundle, \(\bar{\partial}_J:{\mathcal Z}_{A,g,k} \to {\mathcal W}_{A,g,k}\), such that \(\smash{\bar{\partial}_J}\vphantom{\partial}^{-1}(0) = \bar{\mathcal{M}}_{A,g,k}(J)\). We can construct “abstract perturbations” \(p\) of this section such that \(\bar{\partial}_J+p\) is transverse to the zero section and such that \((\bar{\partial}_J+p)^{-1}(0)\) is a compact set. In this way, we may take a scale smooth Fredholm section and “regularize” the unperturbed Gromov–Witten moduli space yielding a perturbed Gromov–Witten moduli space \({\mathcal S}_{A,g,k}(p):= (\bar{\partial}_J+p)^{-1}(0)\) which has the structure of a compact oriented “weighted branched orbifold”.
\[ \begin{array}{c} \bar{\mathcal{M}}_{A,g,k}(J) = \bar{\partial}_J^{-1}(0) \\ \text{\small{compact topological space}} \\ \end{array} \xrightarrow{\text{"polyfold regularization"}} \begin{array}{c} {\mathcal S}_{A,g,k}(p):=(\bar{\partial}_J+p)^{-1}(0) \\ \text{\small{compact "weighted branched orbifold"}} \end{array} \]
This approach has been successful in giving a well-defined Gromov–Witten invariant for curves of arbitrary genus, and for all closed symplectic manifolds. Suppose that \(2g+k\geq 3\), and consider the following diagram of smooth maps between the perturbed Gromov–Witten moduli space \({\mathcal S}_{A,g,k}(p)\), the \(k\)-fold product manifold \(M^k\), and the Deligne–Mumford orbifold \(\smash{\bar{\mathcal{M}}}\vphantom{\mathcal{M}}^{\text{log}}_{g,k}\):
\[ \begin{align*} &{\mathcal S}_{A,g,k}(p) \xrightarrow{ev_1\times\cdots\times ev_k} M^k\\ &\pi \bigg{\downarrow} \\ &\smash{\bar{\mathcal{M}}}\vphantom{\mathcal{M}}^{\text{log}}_{g,k} \end{align*} \]
Here \(ev_i\) is evaluation at the \(i\)th-marked point, and \(\pi\) is the projection map to the Deligne–Mumford space which forgets the stable map solution and stabilizes the resulting nodal Riemann surface by contracting unstable components.
Consider homology classes \(\alpha_1,\ldots, \alpha_k \in H_* (M;{\mathbb Q})\) and \(\beta\in H_* (\smash{\bar{\mathcal{M}}}\vphantom{\mathcal{M}}^{\text{log}}_{g,k};{\mathbb Q})\). We can represent the Poincar'e duals of the \(\alpha_i\) and \(\beta\) by closed differential forms in the de Rahm cohomology groups, \(\mathop{\mathrm{PD}}(\alpha_i)\in H^*_{\mathop{\mathrm{dR}}} (M)\) and \(\mathop{\mathrm{PD}}(\beta)\in H^*_{\mathop{\mathrm{dR}}}(\smash{\bar{\mathcal{M}}}\vphantom{\mathcal{M}}^{\text{log}}_{g,k})\). By pulling back via the evaluation and projection maps, we obtain a closed \(\text{sc}\)-smooth differential form
\[ ev_1^* \mathop{\mathrm{PD}}(\alpha_1) \wedge \cdots \wedge ev_k^* \mathop{\mathrm{PD}}(\alpha_k) \wedge\pi^* \mathop{\mathrm{PD}}(\beta) \in H^*_{\mathop{\mathrm{dR}}} ({\mathcal Z}_{A,g,k}). \]
Theorem. (Hofer, Wysocki, and Zehnder 2017a, Thm. 1.12) The polyfold Gromov–Witten invariant is the homomorphism
\[ \mathop{\mathrm{GW}}_{A,g,k} : H_* (M;{\mathbb Q})^{\otimes k} \otimes H_* (\smash{\bar{\mathcal{M}}}\vphantom{\mathcal{M}}^{\text{log}}_{g,k}; {\mathbb Q}) \to {\mathbb Q} \]
defined via the “branched integration” of (Hofer, Wysocki, and Zehnder 2010a):
\[ \mathop{\mathrm{GW}}_{A,g,k} (\alpha_1,\ldots,\alpha_k;\beta) : = \int_{{\mathcal S}_{A,g,k}(p)} ev_1^* \mathop{\mathrm{PD}}(\alpha_1) \wedge \cdots \wedge ev_k^* \mathop{\mathrm{PD}}(\alpha_k) \wedge\pi^* \mathop{\mathrm{PD}}(\beta). \]
This invariant does not depend on the choice of perturbation.