Research StatementNetanel 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 thegroup of symmetries. Manifolds are no different : every oriented manifold X has an associatedgroup of diffeomorphism Diff+(X); every symplectic manifold (X,ω) has a group of symplecticautomorphism Symp(X,ω), and we can (almost) reconstruct the geometry completely from thegroup-theoretic picture [Tak79, Ban98]. However, in turns out that in some sense these groupsbehave in opposite ways:

• In the smooth case, there isn’t a single example (!) of a smooth closed 4-manifold X forwhich π0Diff+(X) is known, while in higher dimensions we have the miracle of surgery andrational 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

curve was to prove that Symp(CP2, ωstd) = PU(3) and Symp(P1 × P1, 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 recastas the study of solutions to (perturbed) Cauchy-Riemann equations (and therefore generalized tothe symplectic setting). Ultimately, the counts of this solutions are organized into two families ofalgebraic gadgets:

• Open string. Fukaya showed us how to encode all the inter-dependencies in LagrangianFloer 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 uniquenessof 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 (conjec-tured) existence of open–closed and closed–open isomorphisms [Kon95]

HH•−n(Fuk(M,ω))OC−→ QH•(M,ω)

CO−→ HH•(Fuk(M,ω)),

combined with the dg-PROP action of chains on Kimura–Stasheff–Voronov moduli spaces on theleft-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 pointto the fact that quantum cohomology and Gromov-Witten invariants are just the tip of a muchricher chain-level iceberg forming a ”derived version” of the classical invariants of mirror symmetry.

We apply this to the study of symplectic mapping class group1 by looking at the inducedaction

G ↪→ π0Symp(M,ω)→ Auteq(DπFuk(M,ω))→ Aut(HH•(DπFuk(M,ω)))

OC→ Aut(QH•(M,ω)))

for various algebraic varieties and naturally defined subgroups G ↪→ π0Symp(M,ω). This actionis often exotic (which makes it more interesting from a symplectic point-of-view), i.e., there are

1When M = Σg a closed surface, this is the same as the usual mapping class group.

1

elements of infinite order in the symplectic mapping class group which become torsion when con-sidered in π0Diff+(M) and are very hard to detect using ordinary methods.

In past work [RB17], we have applied these ideas to:

(1) There is an map (called the quantum Johnson homomorphism) attaching ”character-istic class” in Hochschild cohomology to symplectic isotopy classes and argued that it canbe 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-Johnsontheory of the Torelli group on the other hand.

(2) Using this technique, we can prove the existence of a certain exotic symplectomorphism ψYin the symplectic mapping class group of the Fano 3-fold obtained by blowing up P3 at agenus 4 curve (equipped with the monotone symplectic form).

(3) Unlike the four-dimensional case, no power of a Dehn twist around Lagrangian 3-spheres canbe 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 provethe existence of a new phenomena: the existence of ”exotic relations” in the subgroupgenerated by all Dehn twists. Namely, it turns out we can factor some power of [ψY ] inπ0Symp(Y, ω) into 3-dimensional Dehn twists. So the isotopy class of the product in theordinary MCG is torsion, but of infinite order in the symplectic MCG.

A detailed description appears below in section 2. Furthermore, the method generalizes to provethe existence of faithful MCG actions on chain-level GW theory for other Fano 3-folds where wehave a natural embedding

Modg ↪→ π0Symp(M,ω)

and these should lift to faithful MCG actions on Fukaya categories (work-in-progress).

Although the origin of my work is in symplectic topology, the results should have a broaderappeal: Kontsevich’s celebrated HMS conjecture [Kon95] allows us to transfer symmetries fromthe A-side to actions on bounded derived categories, and 2-linear representation or categoricalMCG actions are interesting from a representation theoretical and categorification point of view[KS02, RZ03, LQ17]. Moreover, if we take M to be the moduli of stable bundles over a curve ofgenus g, this mapping class group action has intriguing connections to Monopole and InstantonFloer theories (Atiyah-Floer conjectures), and maybe even S-duality and geometric langlands (viaKapustin-Witten [KW07] and Nadler-Zaslow [NZ09,Nad09]).

One disadvantage of homological mirror symmetry conjecture is that it only tells us how torelate asymptotic expansions around the cusps (large volume, maximal unipotent monodromy) ofthe moduli space of N = (2, 2)-superconformal field theories. However, physical expectation isthat mirror symmetry extends to ”global mirror symmetry”: there exists an identification of thestringy Kahler moduli space of the A-side with the complex moduli space of the B-side. Further,the moduli space of conformal field theories may be compactified by including the ”discriminantlocus” consisting of badly-behaved theories. We (usually) expect a theory to go bad because someD-brane becomes massless. So one consequence of mirror symmetry is that if X and Y are a mir-ror dual to each other then the (quantum corrected) complexified Kahler form moduli space of Xshould match the moduli space of complex structures on Y . In particular, the discriminants (andtheir multiplicities!) should agree. This implies the monodromy around them should coincide aswell.

2

A second, related part of my work (part of an ongoing project, joint with Bong H. Lian andJingyu Zhau) is trying to prove some conjectural formulas for the monodromy around this lociof bad theories. This involves looking at ”global mirror symmetry frameworks”, e.g., via analyticcontinuation of the quantum connection on the A-side, or by embedding into moduli spaces ofthe Bridgeland stability conditions on the B-side. Eventually, the purpose is to categorify theGKZ/Tautological system on the B-side. See section 3 for more.

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 results.

2.1.1. Main result: exotic symplectomorphism of Fano 3-fold. Let C be a smooth, non-hyperellipticcurve of genus 4, embedded in P3. Let X be the Fano 3-fold obtained by blowing up C in P3. Weequip X with a monotone symplectic form ωX .

Theorem 2.1 (0.5 in [RB17]). There exists Lagrangian spheres {L′1, · · · , L′5} and {L′′1, · · · , L′′5}in X and an integer N > 0, such that the symplectic isotopy class κ of the following product ofgeneralized Dehn twists

ψN := (τL′1◦ · · · ◦ τL′

5)6N ◦ (τL′′

1◦ · · · ◦ τL′′

5)6N : X → X

is of infinite order in the kernel of the natural map

π0Symp(X,ωX)→ π0Diff +(X).

Let us try and explain the idea behind the result. Denote {Ai, Bi} where i = 1, . . . , 4 are fixedcycles defining a standard symplectic basis of H1(C,Z) ∼= Z⊕8 equipped with the intersection form.See Figure 1 for an illustration.

B4B3B2B1

A1 A2 A3 A4

Figure 1. The standard basis

Q. Let γ be one of the standard basis elements, How can we show that the Dehn twist φ := τγ ,is not isotopic to the identity?

A. We can form the mapping torus Cφ, and compare the structure of the cohomology algebraH•(Cφ) to that of the trivial mapping torus H•(Cid) ∼= H•(C)[t]/(t2), where t is a formal variableof degree one.

Q. Now let γ be a bounding simple closed curve (BSCC) of genus greater the one and γ′ abounding pair, How can we show that the Dehn twist φ := τγ , is not isotopic to φ′ := τγ′?

A. The cohomology algebra won’t help us this time – they are both identical to H•(C)[t]/(t2).The second Johnson homomorphism [Joh80, Kit96] τ2 distinguishes them based on the differencebetween the triple Massey product on the respective mapping tori. More generally, there is a filtra-tion Ig[k] and a sequence of homomorphism (τk)k≥1 such that Ker(τk) = Ig[k + 1]. These higher

Johnson homomorphisms are known to express the change in (k + 1)-fold Massey products in3

the mapping tori.

Consider the following following situation: let C0 be a genus 4 curve, smooth everywhere exceptat a single separating node p, such that the normalization Cν0 consists of two smooth curves withgenus 2. Denote ∆ :=

{z ∈ C

∣∣ |z| < 1}

the open unit disc, and ∆∗ := ∆\ {0}. We consider a

family of curves C → ∆ in which C0 is the central fiber and Ct is smooth and non-hyperellipticwhen t 6= 0. Fix a reference fiber C over z = 1

2 . Then the monodromy ψC : C → C is the separating

Dehn twist Tγ from before. Denote the restriction of the family to the punctured disc C∗ → ∆∗.By Riemann-Roch, we get an embedding

(1) C∗ ↪→ P3 ×∆∗

Blowing up the family of curves inside the projective bundle yields a family of simply connected,smooth, Fano 3-folds Xt over the punctured disc, which we denote

(2) X ∗ → ∆∗.

Restricting to the circle{|z| = 1

2

}⊂ ∆∗ and fixing a reference fiber X over z = 1

2 , we get a locallyHamiltonian fibration

(3) X → S1

with monotone fibers. The parallel transport map along the circle, which we denote

(4) ψ : X → X

is a symplectomorphism by Moser’s argument. Up to isomorphism, X depends only on the sym-plectic isotopy class of ψ : X → X. It is easy to see why Johnson’s argument does not generalize:the relevent triple Massey product should now land in

(5) H3 ⊗H3 ⊗H3 → H3+3+3−1 = H8,

which vanishes. This is a feature, not a bug - the loss of information is reflected in genuinetopological behavior. Indeed, a high enough power of the monodromy ψNX becomes isotopic tothe identity. The phenomena described in theorem 2.1 can be understood as follows: symplectictopology retains the information that differential topology forgets, leading to ”exotic” behavior.

2.1.2. Technique : quantum Johnson homomorphism. The key idea is that we can rephrase John-son’s original result as the obstruction classes coming from the deformation theory ofA∞-homomorphismbetween the rigid A∞-algebras representing the rational homotopy type of surfaces. Namely,

Proposition 1. If φ is non-trivial in the mapping class group, then the A∞-algebra of Morsecochains of the mapping torus is non-formal, i.e., not quasi-isomorphic to its own cohomologyalgebra viewed as an A∞-algebra.

Johnson’s τk essentially measure how is far is the A∞-algebra of chains on the mapping torusfrom being formal.

Now we associate to (M,ωM ) the quantum cohomology (equivalently, self-Floer cohomology ofthe diagonal ∆ ⊂M ×M)

(6) QH•M := QH•(M,ω) ∼= HF •(∆,∆),

regarded as an A∞-algebra. Then every symplectic automorphism ψ : M → M induces an A∞-quasi-equivalence

(7) Ψ : QH•M → QH•M .4

and we can construct a geometric model for the “A∞-algebra mapping torus” QH•ψ coming

from a S1-parametrized version of Fukaya’s seminal work in [Fuk97, FO97] (and considered laterby [RT95], [BC07], [FOOO09] and many others).

Remark 2.2. This should morally correspond to Hochschild cohomology of a ”family of Fukayacategories over the circle” via some parametrized version of the closed–open map. In particular, toobtain a fuller theory of the Johnson homomorphism realizing the connection with Magnus expan-sions, we should use the homotopy Gerstenhaber B∞-structure on quantum cohomology.

Instead of trying to directly prove non-formality of the specific (and very complicated) A∞-structure of QH•ψ, we consider the obstruction to constructing a quasi-isomorphism with the co-homology algebra, which gives us a universal quantum Massey product. This assignment iscalled a second quantum Johnson homomorphism, and the values of qτ2 are triple quantum(matrix) Massey products. Here is a summary of its properties.

Theorem 2.3 (0.1 in [RB17]). Let (M,ω) be a closed, monotone and simply connected symplecticmanifold.

• We associate to every symplectomorphism φ : M →M an A∞-algebra ”mapping torus”

(8) φ 7→ QHφand to every symplectic isotopy φt, an A∞-quasi-isomorphism

(9) {φt} 7→ (Φ : QHφ0 → QHφ1)

• Given a spherical homology class A ∈ HS2 (M ;Z), and for suitable φ, we assign to it a

”characteristic class”

(10) qτ2 : φ 7→ QHφ 7→ o3,Aφ ∈ HH•(..., ...)

in the Hochschild cohomology of a certain algebra which is:

(1) Natural w.r.t to the quasi-isomorphism (10);

(2) Vanishes for φ = id, and

(3) Can be ”evaluated” on suitable cohomology classes.

Remark 2.4. Like τ2, this is actually a part of a bigger story: there is a filtration {QJk} of thesymplectic mapping class group and a system of higher obstructions qτk, where each term in thefiltration is defined by the vanishing of all previous obstruction classes. Moreover (work-in-progress),the right structure to study is probably the schematic homotopy type of quantum cohomology obtainedby considering the B∞-structure, passing to the bar construction which is a DG-Hopf algebra andapplying the machinery of Katzarkov-Pantev-Toen [KPT08].

For the family X , we give an explicit computation of these classes (and show that there is anon-vanishing Massey product). It is based on the proof of 1, and a careful choice of Morse-Smaleand coherent perturbation data allows us to prove that

Proposition 2.5 (N. Blaier). For every N > 0, QH•X(ΨNX) is non-formal, i.e., not quasi-isomorphic

to its own cohomology, considered as an A∞-algebra.

2.1.3. Degeneration of the cubic. We still haven’t mentioned how to get the exotic factorization.In order to do so, we must complete the central fiber and realize the exotic element we constructedexplicitly as the monodromy of a degenerating family of projective varieties. The problem is thatthe central fiber of original family of algebraic curves does not have a canonical embedding into P3.In the spirit of the Hasset-Keel program (MMP for moduli spaces), we seek to find some alternatecompactification of M4 that trades the loci of curves with genus 2 tails for a loci of curves with

5

more complicated singularities (but which we can embed in P3, and then blow up). The new centralfiber and the generic fiber of our family

C∗ → ∆

appears in figure 2 and 3. The central fiber is now a tree of three rational curves attached at twoA5-singularities. The smoothing is a smooth genus 4 curve as usual. The monodromy factorizationof the separating twist is obtained by applying the chain relation to both genus 2 subsurfaces. Using

L3

L1 L2

Figure 2. The singular fiber

L3

L1

L2

Figure 3. The generic fiber

Allcock–Carlson–Toledo’s analysis relates the Ball quotient compactification of the the moduli ofcubic 3-fold and the related GIT for genus 4 curves ([ACT11],[CMJL12]). We prove that we can

extend X ∗ → ∆∗ to a family over ∆, with a central fiber X ∗0 which is the blow-up at the intersectionof a degenerate cubic (with two three-fold A5-singularities) and a quadric, see figure 4 for a schematicdescription of how the central fiber sits inside the P1×P1 quadric: L1 and L2 are (1, 0)-ruling linesand L3 is a curve of bidegree (1, 3) tangent to each of them with multiplicity 3.

L3L2L1

Figure 4. Maximally degenerate 2A5-curve in the quadric

Combined with 2.5 and some work with the explicit equations defining the family this proves themain theorem.

2.2. Current projects.

2.2.1. Representation varieties and Moduli spaces of flat bundles. Smooth intersections of twoquadrics [AL00, Don80] are deeply related to moduli spaces of hyperelliptic curves: to any suchsmooth Fano n-fold one can naturally associate a genus two curve by taking the ramified doublecover branching over the zeros of the discriminant of the pencil. When n = 3, one can furtherrecover the original Fano as a moduli of stable rank 2 vector bundles of odd degree on the curve.This is a well-know instance of a Fourier-Mukai duality between moduli space of genus two curves

6

and moduli space of our Fano manifolds. So if we take six points λ0, . . . , λ5 ∈ C, and consider theintersection of the two quartics

(11)∑

x2k = 0 ,

∑λkx

2k = 0

in CP5. Varying the λj in C, one obtains a natural action, by parallel transport, of the hyperellipticmapping class group Γg,1 of a once-pointed curve on the curve Σ2 and the base loci Q0∩Q1. There

is a certain finite extension Γg,1 which acts by symplectomorphism on Q0 ∩Q1.

Theorem 2.6 (Corollary 1.3 in [Smi12]). The natural representation

(12) Γg,1 → π0Symp(Q0 ∩Q1)

is faithful, but the composition

(13) Γg,1 → π0Symp(Q0 ∩Q1)→ π0Diff (Q0 ∩Q1)

has infinite kernel.

This setting is very similar to the previous one: the complete intersection of a quadric and acubic is a canonically embedded genus 4 curve. The idea is the same – compute relevent terms inthe Kaledin class of the induced A∞-functor. However, the connection with the classical theory ofthe Johnson homomorphism is a little different here, a key point being the Jacobian interpetationof the Johnson homomorphism – in a word, the basic idea (which I learned from Nick Salter) isthat you should replace jacobians with k-step nilmanifolds. Just as a Jacobian is a K(π, 1) forthe abelianization of π = π1(Σg), you should think of these nilmanifolds as being K(π, 1)’s forthe k-th universal nilpotent quotient of π; as well as the isomorphism between the Jacobian of thehyperelliptic curve and the intermediate Jacobian of the base loci (this was proved in Miles Reid’sthesis). One then uses the explicit correspondence found in [Don80] (and used to establish theisomorphism above) along with fact that the Fano variety of lines is isomorphic to the Jacobian viathe Abel-Jacobi map to compute the quantum Massey products on the mapping torus.

But what about g > 2? If we fix a point p ∈ Σg and denote byM(Σg) the moduli space of ranktwo, fixed odd determinant stable bundles on a complex curve of genus g, equivalently the spaceof conjugacy classes of SU(2)-representations of π1(Σg\p) which have holonomy I at the puncture.

M(Σg) admits a natural symplectic structure. In the topological setting, there is an extension Γgof Γg = π0Diff+(Σg) by Z2g

2 = H1(Σg;Z2) which acts on the moduli space via a homomorphismΓg → Symp(M(Σg)). Denote the induced representation on connected components as

(14) ρ : Γg → π0Symp(M(Σg))

Donaldson’s former student Michael Callahan proved (unfinished Oxford D. Phil thesis, circa 1993)that ρ distinguished the Dehn twist on a separating curve γ ⊂ Σ2 from the identity. Even more istrue, and the following result was proved by Ivan Smith.

Theorem 2.7 (Theorem 1.5 in [Smi12]). For g ≥ 2, ρ does not factor through the symplectic group(i.e. ρ is non-trivial on the Torelli group). When g = 2, ρ is faithful.

I’m currently trying to to use the classical results [ST95, Zag95, Mun99, BM08, Tha00] to provethe following compute quantum Massey products in the mapping torus for general g and prove thefollowing

Theorem 2.8 (N. Blaier, In progress). For any g ≥ 2, the composition

Γg → π0Symp(M(Σg))→ Auteq(DπFuk(M(Σg)))

is faithful, and therefore, ρ is faithful as well.7

Remark 2.9. 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 studythe mapping class group on the Fukaya categories of the moduli space of Higgs bundles, i.e., A-sideof N = 4 super Yang-Mills. In that context, there are some interesting and encourging results[BZBJ15] by Ben–Zvi, Brochier and Jorda which prove faithfulness of the categorical MCG actionon the B-side.

2.2.2. From MCG to RAAG’s. Ordinary mapping class groups are probably the simplest ”compo-nent” of symplectic MCG’s to detect, but they are also highly specialized. The next logical stepwould to be to try and detect embeddings of right-angeled Artin groups AΓ into π0Symp(M,ω) ofalgebraic varieties. As it turns out [Wad15], the Johnson-Andreadakis theory and Magnus expan-sions generalizes well to this setting. It was proved in [Kea17, Corollary 3.11] that there exist highdimensional Milnor fibers W with Artin groups embedded into Auteq(Fuk(W,ω)), so we can startby considering smoothing of projective algebraic varieties with suitable singularities. In particu-lar, note that there is no known analogue2 of [Sei08, Corollary 6.8] or [Kea17, Corollary 3.16] forhigh-dimensional projective algebraic varieties.

2.3. Future directions.

2.3.1. Dynamical properties. Let φ : (M,ω) → (M,ω) be a symplectomorphism. It is a classicalfact that HF ∗(φ) is a module over the quantum cohomology QH∗(M). In fact, more is true. PaulSeidel observed that HF •(φ) is module over QH•(M) in the A∞-sense, and is isomorphic to itspullback by the induced A∞-functor φ∗.

It seems interesting to explore the consequences of such a result: HF •(φ) contains importantdynamical information, but is known to be difficult to compute in general. More generally, onemight ask if there is a connection between the following three quanitites:

(1) The level of the isotopy class in the quantum Johnson filtration, which measures algebraiccomplexity.

(2) The entropy of the auto-equivalence as defined in Dimitrov-Haiden-Katzarkov-Kontsevich[DHKK14] which measures dynamical complexity

Asymptotic bounds where one quantity dominates the other are known in the case on the mappingclass group, see [FLM08], and it is a general ideology that they should exist in general.

2.3.2. Quantum characteristic classes. As a variation of the approach via the A∞-algebra structureof the mapping cone sketched above, we can also define the “mapping cone” as an A∞-bimodule3.The advantage of this approach is that the deformation theory is “smaller”4; indeed, there is a goodclassification theory

Theorem 2.10. A∞-bimodules which are equal to the diagonal bimodule on the cohomology level,have “characteristic classes” in Hochschild cohomology HH1.

While defining the quantum Massey products, it became apparent that there is no reason torestrict ourselves to fibrations over B = S1 exclusively. The logical next step seems to be toassociate “secondary characteristic classes” in quantum cohomology to symplectic fibrations E over

2At least, none that I’m aware of!3The reader may wonder why are we not just looking at the A∞-functor itself ... the problem is that while

this space contains more information then both approachs, the operadic structure that controls it is encoded in themultiphedra - which are not simple polytopes. As a result, the gluing parameters are not independent of each otherwhich makes the construction substantially more complicated.

4The deformation theory of a mapping cone of an A∞-functor f : A → A is similar to the deformation theory ofthe pair (A, f), while the deformation theory of the A∞-bimodule is closer to the defomation theory of the functor f .

8

surfaces B = Σg. The surface can be described as the bouquet of 2g circles {a1, b1, . . . , ag, bg} withan additional two-disc imposing the single relation

(15)

g∏i=1

[ai, bi] = 1.

To every circle, we associate an A∞-bimodule as before, and to the 2-disc we associate a fixedchoice of trivialization of the composition of all A∞-bimodules given by (15). The idea is to usethis invariant to construct some “exotic” classes in π2(Symp(M,ω)), where (M,ω) is the fiber of E .

Remark 2.11. There is already a (seemingly?) unrelated different theory of quantum character-istic classes for Hamiltonian fiber bundles [Sav08] extending previous early foundational work ofSeidel in this area [Sei97]. This classes are known to be sensitive to interesting geometric structureunderlying the symplectomorphism group (like the Hofer’s metric for example). It seems like aninteresting question to ask whether similar connections would hold for our invariants as well.

2.3.3. Detecting classes in H∗(BDiff +(M)). Instead of trying to detect elements in the kernel ofπk(Symp(M,ω)) → πk(Diff +(M)), we can take the opposite approach and use our knowledge ofsymplectic topology to shed some light on manifold bundles. We may consider the result of applyingthe richer algebraic structure of symplectic topology, reflected in quantum Massey products andquantum Steenrod operations (see [McD08]) to characteristic classes of symplectic fibrations. In away similar to [Sei15], we can speculate that some of the classes may survive to H∗(BDiff +(M)),thus giving interesting new characteristic classes for smooth manifold bundles.

3. Navigation rules, D-branes, and categorified GKZ

As we have mentioned, the Homological Mirror Symmetry conjecture only relates asymptotic ex-pansions around the cusps (large volume, large complex structure) of the moduli space ofN = (2, 2)-superconformal field theories. However, physical expectation is that mirror symmetry extends to”global mirror symmetry”: there exists an identification of the stringy Kahler moduli space of theA-side with the complex moduli space of the B-side. Thus, a consequence of mirror symmetry isthat if X and Y are a mirror dual to each other then the (instanton corrected) complexified Kahlerform moduli space of X should match the moduli space of complex structures on Y . Further, themoduli space of conformal field theories may be compactified by including the ”discriminant locus”consisting of badly-behaved theories. Thus, these discriminants (complete with their multiplicities!)should agree between these moduli spaces.

Let us demonstrate this with the example which inspired the whole field of mirror symmetry:the quintic 3-fold. On one side, we have a

X ={x5

1 + . . .+ x55 = 0

}equipped with a natural holomorphic 3-form ω = dx1 ∧ dx2 ∧ dx3/x

44 (this is affine coordinates

with x5 = 1). On the A-side, we consider the (complexified) Kahler moduli space a con-tractible one-dimensional complex space. On the B-side, we consider the (complex structure)deformations. Greene and Plesser describe the mirror as a resolution of singularities of a fam-ily of hypersurfaces in the quotient P4/G where the G here is defined to be all automorphism[x1, . . . , x5] 7→ [α1x1, . . . , α5x5] where α5

i = 1 and det =∏i αi = 1 as well. We can mod out by

the kernel (transformations with α1 = . . . = α5 = α a fifth root of unity) to obtain a group G,non-canonically isomorphic to (Z5)3. This family of hypersurfaces is the Dwork family

Yψ :=

{x5

1 + . . .+ x55 + 5ψ(

5∏i=1

xi) = 0

}9

This family is invariant under G and descends to P4/G. The mirror is a crepant resolution Yψ ofthe family. In fact, the action of the diagonal symmetry identifies the fiber over ψ with the fiberover αψ where α is a root of unity. So after resolution, we get a family of CY 3-folds over thequotient [P1/Z5]. The stringy Kahler moduli space is expected to coincide with the moduli spaceof t = ψ5 which is the suborbifold {

t ∈ C∣∣ t 6= 1

}/Z5 ⊂ [P1/Z5](16)

The family is regular away from t = ∞ (large complex structure point) and t = 1(Gepner point).These limit points are the discriminant locus. Alongside with the stack-theoretic point t = 0(conifold point) they are what is usually referred to as special limit points.

Expanding around the large volume/maximal degenerate point, physicist are able to expressthe monodromy as spherical functors acting on the Fukaya category/bounded derived category ofcoherent sheaves. One major obstracle that makes sense of this is that we do not have a definitionof what the stringy Kahler moduli space is! In order for an object in the A-side/B-side categoryto represent a physical D-brane it must be ”Π-stable”. In particular if you have reason to believethat a certain set of a D-branes are stable at a given point in the moduli space then one can movealong a path in moduli space and see how the spectrum of stable states changes. String theoristsstrongly believe that changes in Π-stability depend only on the homotopy class of the path in themoduli space of conformal field theories. The fact that changes in Π-stability do depend on thehomotopy class of such paths was used by Kontsevich to conjecture a picture of monodromy atleast in the case of the quintic CalabiYau threefold. The space of Bridgeland stability conditionswas introduced in [AD02] as a mathematical counterpart to Π-stability. In particular, it serves asa substitute for the missing definition of the stringy Kahler space. Physicist predict that given apair of CY mirrors X and Y , there should be an embedding

MKah(X) ↪→ Auteq(DbCoh(X)) Stab(DbCoh(X))/C(17)

where MKah(X) is the stringy Kahler space; this was indeed confirmed in some important cases,including elliptic curves, K3’s, and split-abelian surfaces. From this methods we can derive anexplicit conjecture due to Kontsevich [Kon] for the nature of these spherical functors, which wassubsequently generalized in [Hor99, AD02] and [Joc07, BJR09]. For example, in the case of thequintic 3-fold,

• Around the large complex structure limit point, the monodromy acts via by shifting theB-field of the complexified Kahler form. If we consider the A-side Fukaya category withoutthe B-field, that should roughly correspond to a shift of the grading.• Around the Gepner point, the brane corresponding to the structure sheaf becomes massless.

That means the monodromy should be the auto-equivalence

Φ(−) = STOX◦ ((−)⊗OX(H))(18)

where H is the hyperplane divisor and STOXdenotes the Seidel-Thomas spherical twist

with respect to the structure sheaf. Note that it satisfies Φ5 = [2] which descends to theidentity Φ5 = id on the level of K0.

More generally, in the case of closed string B-side, we have a very good description found byGelfand, Kapranov and Zelevinsky. Some conjectures regarding a categorified version of this hasappeared in a recent preprint [PSAMRPKW17]. Our goal is to formulate an explicit conjecturein terms of Picard-Fuchs equations and Gromov-Witten invariants, prove it, and generalize to thecase of the tautological system.

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A different (but related) project is to look at ”cousins families” of the mirror quintic. That is,within the class of irreducible differential operators of Calabi-Yau type5 the symplectically rigiddifferential operators constitute an important subclass with three regular singularities and wereclassified in [BR17]. This class includes all operators whose associated monodromy representationis symplectically rigid. They are essentially the simplest CY’s families from the point of viewof Picard-Fuchs equations. A recent iterative construction, due to Doran and Malmendier [CF-DAM17], constructs all such 1-parameter families of CY in a geometrically explicit way whichrealize those variations of Hodge structures. We are trying to prove the close string implicationsof the monodromy conjecture for all such families using analytic continuation (in the same way asLG-CY correspondence done for the quintic family in [CR11]).

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