Research Statement Netanel Rubin-Blaier

1. Introduction

”Algebra is the offer made by the devil to the mathematician...” Michael Atiyah In 1872, Felix Klein suggested the ”Erlangen program”: studying geometry by looking at the group of symmetries. Manifolds are no different : every oriented manifold X has an associated group of diffeomorphism Diff+(X); every symplectic manifold (X, ω) has a group of symplectic automorphism Symp(X, ω), and we can (almost) reconstruct the geometry completely from the group-theoretic picture. However, in turns out that in some sense these groups behave in opposite ways: • In the smooth case, there isn’t a single example (!) of a smooth closed 4-manifold X for + which π0Diff (X) is known, while in higher dimensions we have the miracle of surgery and + rational homotopy in our disposal to tame Diff (X) (at least when π1(X) is trivial) • In the symplectic case, one of Gromov’s original application of the technique of J-holomorphic 2 1 1 curve was to prove that Symp(CP , ωstd) = PU(3) and Symp(P × P , ω ⊕ ω) = (SO(3) × SO(3)) n Z2. But in higher dimensions virtually nothing is known. The main tool in symplectic topology is the study of J-holomorphic curves. This is a result of Gro- mov and Floer foundational who told us that much of classical enumerative geometry can be recast as the study of solutions to (perturbed) Cauchy-Riemann equations (and therefore generalized to the symplectic setting). Ultimately, the counts of this solutions are organized into two families of algebraic gadgets: • Open string. Fukaya showed us how to encode all the inter-dependencies in Lagrangian Floer theory in a single A∞-category; • Closed string. Witten interpreted and generalized Gromov’s invariants as n-point correla- tion functions in a topologically twisted version of string theory. Guided by the uniqueness of 2D quantum gravity, he conjectured an integrable hierarchy of equations governing them. This was subsequently proved by Kontsevich. My research can be characterized as ”applied” Homological Mirror Symmetry. Namely he (conjectured) existence of open–closed and closed–open isomorphisms [Kon95]

OC • CO • HH•−n(F uk(M, ω)) −→ QH (M, ω) −→ HH (F uk(M, ω)), combined with the dg-PROP action of chains on Kimura–Stasheﬀ–Voronov moduli spaces on the left-hand side (seen as the closed state space of an open-closed TCFT) [Cos07], as well as the fun- damental Koszul duality in genus zero between the Gravity and Deligne-Mumford operads all point to the fact that quantum cohomology and Gromov-Witten invariants are just the tip of a much richer chain-level iceberg forming a ”derived version” of the classical invariants of mirror symmetry. I use this and the formal similarities between the resulting invariant and some classical ones from the ﬁelds of low-dimensional topology, and rational homotopy, to answer some concrete question about the symplectic mapping class group.

Although the origin of my work is in symplectic topology, the results should have a broader appeal: Kontsevich’s celebrated HMS conjecture allows us to transfer symmetries from the A-side to actions on bounded derived categories, and categorical MCG actions are interesting from a representation theoretical and categoriﬁcation point of view. Moreover, if we take X to be the moduli of stable bundles over a curve of genus g, this mapping class group action has intriguing connections 1 to Monopole and Instanton Floer theories (Atiyah-Floer conjectures), and maybe even S-duality and geometric langlands (via Kapustin-Witten and Nadler-Zaslow).

2. Symplectic Mapping Class Groups and the Quantum Johnson homomorphism

”Groups, as men, will be known by their actions” Guillermo Moreno

2.1. Summary of Past work [RB17]. We are confronted by a group we don’t understand, so we do what mathematicians have always done: search for something it naturally acts upon. Of course (given the introduction), the obvious thing to consider is the natural map π (1) π0Symp(X, ω) → AuteqCY (D F uk(X, ω))/[2] taking each symplectic isotopy class to a Calabi-Yau A∞-auto-equivalence of the split-closed Fukaya category. However, this is a problematic choice – the Fukaya category of a closed manifold is an extremely complicated object and we do not know how to compute it (almost always). So we look at the first (and easiest) strata of chain-level Gromov-Witten theory: the A∞-enhancement of the usual star product ?. This is classical, and goes back to Fukaya’s first papers in 1997. We proceed to explicitly constructing a family version of it (which should morally correspond to Hochschild cohomology of a ”family of Fukaya categories over the circle” via some parametrized version of the closed–open map) and show that it already has some non-trivial applications to symplectic isotopy problems: Following Kaledin, we look at the obstruction class of the resulting A∞-structure, and argue that it can be related to a quantum version of Massey products on the one hand (which, in nice cases, can be related to actual counts of rational curves) and to the classical Andreadakis- Johnson theory of the Torelli group on the other hand. This are the main technical results (see Theorem 1.1 for a summary and sections 2,3,10,11,12 for the Floer theory and homological algbra that goes into the construction). In the second part of the paper, we apply this machine to go hunting for exotic symplectomorphism: these are elements of infinite order in the kernel + K(M, ω) := π0Symp(M, ω) → π0Diff (M, ω) of the forgetful map from the symplectic mapping class group to the ordinary MCG. We demon- strate how we can apply the theory above to prove the existence of such elements ψY for certain a 3 Fano 3-fold obtained by blowing-up P at a genus 4 curve.

Unlike the four-dimensional case, no power of a Dehn twist around Lagrangian 3-spheres can be exotic (because they have infinite order in smooth MCG). In the final part of the paper, the classical connection between our Fano varieties and cubic 3-folds allows us to prove the existence of a new phenomena: ”exotic relations” in the subgroup generated by all Dehn twists. Namely, it turns out we can factor some power of [ψY ] in π0Symp(Y, ω) into 3-dimensional Dehn twists. So the isotopy class of the product in the ordinary MCG is torsion, but of infinite order in the symplectic MCG. The main result in the paper is Theorem 0.5 which states: 3 Theorem 2.1. Let C be a smooth, non-hyperelliptic curve of genus 4, embedded in P . Let X be 3 the Fano 3-fold obtained by blowing up C in P . We equip X with a monotone symplectic form ωX . 0 0 00 00 There exists Lagrangian spheres {L1, ··· ,L5} and {L1, ··· ,L5} in X and an integer N > 0, such that the symplectic isotopy class κ of the following product of generalized Dehn twists N 6N 6N ψ := (τ 0 ◦ · · · ◦ τ 0 ) ◦ (τ 00 ◦ · · · ◦ τ 00 ) : X → X L1 L5 L1 L5 is of infinite order in the kernel of the natural map + π0Symp(X, ωX ) → π0Diff (X). 2 2.2. Current projects. First, the technical apparatus described in the previous section admits a generalization to the second strata of chain-level GW theory: B∞-algebras (i.e., homotopy Gersten- haber). This are essentially described by picking a suitable chain model for quantum cohomology and looking at Getzler-Neuwirth cells, see [Vor00]. As a consequence, we can define a schematic homotopy types of symplectic manifolds using [KPT08] and make connection with the group- theoretic definition of Johnson homomorphism in terms of lower central sequence by looking at the monodromy or parallel transport of a family of B∞-algebras. Second, I am working on using the technical result above to prove the faithfulness action of the entire Torelli group on (some eigen- component under c1? a ) of the Fukaya categories of various Fano n-folds with intimate connection to curves. Such ”2-linear” representations are also interesting from a categorification point of view.

The most important one to consider is probably the moduli of stable bundles M(Σg) of curves of genus g. Ivan Smith [Smi12] has proven there is a faithful action in the case g = 2 and there are classical descriptions of the cohomology, quantum cohomology, rational homotopy, and a very nice (and recursive in genus) construction of a perfect Morse function by Thaddeus, making the problem very amenable to the computation of Quantum Massey products. Remark 2.2. The Fukaya categories of compact group representation varieties are part of Don- aldson theory (N = 2 super Yang-Mills). A related problem (perhaps a pipe-dream?) is to study the mapping class group on the Fukaya categories of the moduli space of Higgs bundles, i.e., A-side of N = 4 super Yang-Mills. In that context, there are some interesting and encourging results by Ben–Zvi, Brochier and Jorda which prove faithfulness of the categorical MCG action on the B-side. Two other related reseach questions I am working on (motivated by results in low-dimensional topology) are the construction of quantum Miller-Morita-Mumford classes for locally Hamiltonian bundles over surfaces, and what is the relation between the entropy of the auto-equivalence as deﬁned in Dimitrov-Haiden-Katzarkov-Kontsevich which measures dynamical complexity and the algebraic complexity of the dynamics as seen by the quantum Johnson homomorphism (is there some kind of an asymptotic bound?)

Finally, in a joint with Bong H. Lian and Jingyu Zhau) we trying to prove some physics conjecturs regarding the B-side monodromy around this loci of bad N = (2, 2)-SCFT theories in the categorical level. Eventually, the purpose is to categorify the GKZ/Tautological system on the B-side following some recent predictions by physicists.

References [Vor00] A. A. Voronov, Homotopy Gerstenhaber algebras, ConférenceMoshéFlato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 307–331. ↑3 [KPT08] L. Katzarkov, T. Pantev, and B. Toën, Schematic homotopy types and non-abelian Hodge theory, Compos. Math. 144 (2008), no. 3, 582–632. ↑3 [Smi12] I. Smith, Floer cohomology and pencils of quadrics, Invent. Math. 189 (2012), no. 1, 149–250. ↑3 [Kon95] M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser,Basel, 1995, pp. 120–139. ↑1 [Cos07] K. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), no. 1, 165–214. ↑1 [RB17] N. Rubin-Blaier, The quantum Johnson homomorphism and symplectomorphism of 3-folds, arXiv preprint (2017). ↑2