Coisotropic Branes and Homological Mirror Symmetry for Tori Denis - - PowerPoint PPT Presentation

Coisotropic Branes and Homological Mirror Symmetry for Tori

Denis Auroux

Harvard University

November 14, 2019

Annual Meeting of the Simons Collaboration on HMS (based on Yingdi Qin’s PhD thesis)1

partially supported by NSF and by the Simons Foundation (Simons Collaboration on Homological Mirror Symmetry)

1except for the sign mistakes, which are entirely mine Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 1 / 14

Jacobi theta functions and counting triangles

Jacobi theta function on the elliptic curve E = C / Z+τZ

All doubly periodic holomorphic functions are constant, but we can ask for quasi-periodic functions: s(z + 1) = s(z), s(z + τ) = e−πiτ−2πizs(z) (section of deg. 1 line bundle L → E) Only one up to scaling! s(z) = ϑ(τ; z) =

n∈Z

exp(πin2τ + 2πinz).

(Jacobi, 1820s)

Counting triangles in T 2 = R2/Z2 (weighted by area)

Lx L1 L0

. . .

L0 L1 Lx

x

s e1 e0

? = · · · + q(x−1)2/2+qx2/2 +q(x+1)2/2 + . . . = qx2/2

n∈Z q

1 2n2+nx = eπiτx2ϑ(τ; τx)

(q = e2πiτ )

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 2 / 14

Homological mirror symmetry (Kontsevich 1994)

Algebraic (or analytic) geometry

Coherent sheaves (eg: OV , vector bundles E → V , skyscrapers Op∈V , ...) Morphisms (+ extensions): H∗hom(E, F) = Ext∗(E, F). Derived category = complexes 0 → · · · → Ei

di

− → Ei+1 → · · · → 0 / ∼ Eg: functions, intersections, cohomology...

Mirror symmetry: DbCoh(V ) ≃ DπF(X, ω) in general: over Novikov field

here: over C

Symplectic geometry: Fukaya category F(X, ω)

(X, ω) loc.≃ (R2n, dxi ∧dyi), Lagrangian submanifolds L (dim. n, ω|L = 0) + rk 1 loc. system ∇.

Floer cohomology measures intersections

(physicists’ version: over C instead of Novikov field)

CF ∗(L, L′) = C|L∩L′|

q p L L′

∂p = exp(2πi

• (B + iω)) hol∇ q

(⊗ local coefficients)

Product CF(L′, L′′) ⊗ CF(L, L′) → CF(L, L′′): p′ · p = exp(2πi

• (B + iω)) hol∇ q

q p p′ L L′′ L′

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 3 / 14

Homological mirror symmetry for T 2

(Polishchuk-Zaslow 1998)

T 2

τ = R2/Z2, B+iω = τ dr ∧dθ

vs. Eτ = C / (Z+τZ)

Lz L1 L0 L0 L1 Lz=τx+y = {x}×S1

θ

(∇ = d + 2πiy dθ)

x

s e1 e0

In F(T 2

τ ), e1 · s = ? e0 s ∼ section of L, e0 ∼ evaluation O → Oz

L0

s

− → L1

e1

− → Lz

(mirror to: O → L → Oz on Eτ) ? =

n∈Z

eπiτ(x+n)2+2πi(x+n)y + ... = eπiτx2+2πixyϑτ(z) ϑτ(z) =

n∈Z

exp(πin2τ + 2πinz) ∈ H0(Eτ, L):

• ϑ(z + 1) = ϑ(z),

ϑ(z + τ) = e−πiτ−2πizϑ(z) (Jacobi, 1820s) Similarly for rest of F(T 2

τ ) ≃ Coh(Eτ ). In higher dim., τ ∈ Matn×n(C) (Fukaya, Kontsevich-Soibelman, Abouzaid-Smith) Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 4 / 14

What is the mirror of multiplication by i?

Over C, the Fukaya category of T 2

i = (R2/Z2, ω0) has an autoequivalence mirror to complex

multiplication by i on Ei = C/(Z + iZ). This should “exchange position and holonomy”: Lz=ix+y = ({x} × S1

θ , ∇ = d + 2πiy dθ) ←

→ Liz=iy−x = ({y} × S1

θ , ∇ = d − 2πix dθ).

No Lagrangian correspondence in (T 2 × T 2, ω = −dr1 dθ1 + dr2 dθ2) induces such a functor.

A coisotropic correspondence ?

Consider the line bundle (ξC → T 2 × T 2, ∇C = d − 2πi(r1 dθ2 + r2 dθ1)). Then the “mult. by i” functor maps to (ξ → L, ∇) to (π2)∗(ξC ⊗ π∗

1ξ) (check: this has Lagrangian support).

We think of C = (ξC → T 2 × T 2, ∇C) as a coisotropic correspondence. The curvature F = dr1 dθ2 + dr2 dθ1 satisfies (ω−1F)2 = −1. (Similarly for F(T 2

τ ) ≃ F(T 2 −1/τ)). Q: How does C fit into F(T 2 × T 2)?

Note: none of this occurs over the Novikov field / for non-archimedean abelian varieties!

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 5 / 14

Coisotropic branes

(Kapustin-Orlov 2001)

Kapustin-Orlov observed: for (T 4, ω0), the image of ch : F(T 4) → H2(T 4), L → [L] has rank 5, while for V = Ei × Ei, ch : Db(V ) ։

p Hp,p(V ) which has rank 6. So F(T 4) ≃ Db(V ).

Fix: take split-closure TwπF(T 4) (Abouzaid-Smith), or add coisotropics (Kapustin-Orlov).

Definition (Coisotropic branes) (without B-field) (Kapustin-Orlov)

A coisotropic brane consists of a coisotropic submanifold C n+k ⊂ (X 2n, ω) with a U(1)-bundle (ξ, ∇) such that ˜ F =

1 2πi F∇ satisfies:

(i) ˜ F = 0 on the isotropic leaves TC n−k

iso

= ker ω|TC, (ii) (ω−1 ˜ F)2 = −1 complex structure on TC/TCiso.

(⇒ ˜ F + iω holom. symplectic, and k is even).

Proposal: End(C, ∇) = H∗(C, OC ) (loc. constant in TCiso, holomorphic in TC/TCiso).

(Note: in X 4, ˜ F ∧ ˜ F = ω ∧ ω and [ ˜ F] ∈ H2(X, Z) ⇒ coisotropic branes only exist at special locus in K¨ ahler moduli space)

In (T 4, ω0), the “missing” generator is mirror to OΓ, where Γ = {(z, iz)} ⊂ V (mult. by i). Candidate: C = T 4, ∇ = d − 2πi(r1 dθ2 − r2 dθ1).

Question: how to enlarge F(T 2n) to include coisotropic branes? (hom(L, C)? compositions?) Theorem (Yingdi Qin): this can be done by a doubling construction.

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 6 / 14

Reformulating the theta function

Classical fact: f (z) = eπz2/2ϑi(z) = eπz2/2

n∈Z e−πn2e2πinz satisfies f (iz) = f (z) !

Slogan: understanding mult. by i in HMS is difficult because this invariance property is not obvious!

Alternative, invariant expression: c f (z) =

• (m,n)∈Z2(−1)mne−π(m2+n2)/2 eπ(m+in)z

(c =

√ 2 ϑ(0))

Symplectic interpretation: Floer product in T 2 × T 2

In T = T 2 × T 2, with ω = 1

2(dr ∧dθ + d ˆ

r ∧d ˆ θ), B = 1

2(d ˆ

r ∧dr + d ˆ θ∧dθ)

L0 L1

Lz=ix+y = {x}×S1

θ × S1 ˆ r × {y}

(∇ = d + 2πiy dθ)

x

s e1 e0

×

y

s e1 e0

⇒ e1 · s ∼ f (z) e0. In general, (−2iτ)1/2ϑτ(0) eπiz2/2τ ϑτ(z) =

m,n(−1)mneπin2τ/2e−πim2/2τ exp(πi(m τ + n)z)

Similar interpretation in T 2 × T 2 with mutually inverse symplectic areas!

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 7 / 14

Lifting branes to the doubled torus

(Yingdi Qin)

Definition

Given a symplectic torus (T = V /Λ, ω), the linear dual torus (with inverse symplectic form) (T ∗ = V ∗/Λ∗, ω−1), and the standard pairing σ0 =

i d ˆ

xi ∧ dxi, define T = T × T ∗, with symplectic form 1

2(ω ⊕ ω−1) and B-field 1 2σ0.

So: B B + iω ω = 1

2σ0 + 1 2(ω ⊕ ω−1). (Eventually T should be allowed to carry a B-field!)

Definition

Given a brane (L, ∇, ε) in T, let L = {(x, x∗) ∈ T ×T ∗ | x ∈ L and ∀γ ∈H1(L)⊂Λ, exp(2πix∗, γ) = (−1)ε(γ)hol∇(x + S1

γ)},

with ∇ ∇ = π∗∇. This is a Lagrangian brane in T.

ε : H1(L, Z) → Z/2 such that ε(γ + γ′) − ε(γ) − ε(γ′) = c1(∇)(γ ∧ γ′) mod 2 (⇔ rel. spin structure)

When L is Lagrangian, L = L × (L⊥ + hol∇) (translated) conormal. For L = T space-filling coisotropic, L = “graph of ∇” (Lagr. for ω

ω = 1

2(ω ⊕ ω−1) using (ω−1F)2 = −1). Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 8 / 14

The mirror of the doubled torus

(Yingdi Qin) For T = (R/Z)2n, ω = a dr ∧ dθ (τ = ia ∈ Mn×n(C)), with SYZ fibers F = {r} × T n

θ ,

the mirror abelian variety is E = Cn/(Zn + τ(Zn)). For T ∗ = (R/Z)2n, ω−1 = a−1 dˆ r ∧ d ˆ θ, with SYZ fibers F ∗ = T n

ˆ r × {ˆ

θ}, the mirror abelian variety is ˆ E = Cn/(Zn + τ−1(Zn)) (≃ E).

For T = (R/Z)4n, B B + iω ω = 1

2(σ0 + ia dr ∧ dθ + ia−1 d ˆ

r ∧ d ˆ θ), F = {r} × T n

θ × T n ˆ r × {ˆ

θ}, the mirror is E = (Cn × Cn)/(Z2n + τ τ(Z2n)), τ τ = 1

2

• τ

1 −1 −τ −1

• .

Using coordinates u = z + τ ˆ z, v = z − τ ˆ z, we have: E ≃ E × E.

There are simpler mirrors to E × E; this one has the property that, even if a sheaf E ∈ Coh(E) corresponds to a coisotropic in T, a closely related sheaf on E × E corresponds to a Lagrangian in T.

SYZ fibers F ⊂ T lift to fibers F ⊂ T which correspond to points in E × 0, i.e. v = 0.

{r} × T n

θ × T n ˆ r × {ˆ

θ}, ∇ = d + 2πi(y dθ + ˆ y dˆ r) ↔ (z = τ

2 r + y − ˆ θ 2 , ˆ

z = τ−1

2

ˆ θ + ˆ y + r

2).

Lifts from T have ˆ θ = y and ˆ y = 0, so (z = 1

2 (τr + y), ˆ

z = 1

2(τ −1y + r)), hence (u = τr + y, v = 0).

L ∼ Op ⇒ L ∼ Op ⊠ O0. Similarly, L ∼ E ⇒ L ∼ E ⊠ E0. (if E ∈ Picd(E), then E0 = origin of Picd)

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 9 / 14

HF(L, L′): first examples

L = {(x, x∗) ∈ T ×T ∗ | x ∈ L and ∀γ ∈H1(L), exp(2πix∗, γ) = (−1)ε(γ)hol∇(x + S1

γ)},

∇ ∇ = π∗∇. (For L Lagrangian, L = L × (L⊥ + hol∇).) If T is mirror to E then T is mirror to E × E, and L ∼ E ⇒ L ∼ E ⊠ E0. HF(L, L′) works well for deg. 1 line bundles

L0 L1 Lz=τx+y = {x}×S1

θ ×S1 ˆ r ×{y}

(∇=d+2πiy dθ)

x

s e1 e0

×

y

s e1 e0

⇒ e1 · s ∼ ϑτ(z) e0.

and for graph of mult. by i in square torus: C = T 4, ∇ = d − 2πi(r1 dθ2 − r2 dθ1) C={ˆ r1 =θ2, ˆ θ1 =r2, ˆ r2 =−θ1, ˆ θ2 =−r1}, ∇

Lz: ri = Im zi, ˆ θi = Re zi, ∇ = d + 2πi (Re zj) dθj

⇒ HF(C, Lz) = 0 iff z2 = iz1.

(but then... HF(C, Lz) ≃ H∗(T 2) instead of H∗(S1))

In general, Hom spaces are too large: Ext∗(E ⊠ E0, F ⊠ F0) = Ext∗(E, F) ⊗ Ext∗(E0, F0). For Lagrangians, dim HF(L, L′) = (dim HF(L, L′))2; restrict to “u-part”?

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 10 / 14

The u-part: HF(L, L)u ⊂ HF(L, L)

(Yingdi Qin)

HF(L, L) ≃ H∗(L; C) = H1(L; C). Among all deformations of L in F(T), only consider those which are lifted from Def (L ⊂ T). On mirror: Ext1(E ⊠ E0, E ⊠ E0) ⊃ Ext1(E, E) ⊗ 1. This has a nice geometric characterization:

A complex structure on T

On T, let B B + iω ω± = 1

2σ0 + i 2(ω ⊕ ±ω−1). Then J = B

B−1ω ω− is a complex structure (J2 = −1), mapping v ∈ T(T) = V to ιvω ∈ V ∗ = T(T ∗). Fact: L ⊂ T is a complex submanifold! (but not complex Lagrangian w.r.t. holom. sympl. structure) The deformations of L which come from lifting correspond to H0,1

J (L) ⊂ H1(L; C).

Definition

HF(L, L)u := H0,∗

J (L) ⊂ H∗(L; C) = HF(L, L). Similarly for the “continuous” part of L ∩ L′ when L, L′ not transverse.

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 11 / 14

The u-part: HF(L, L′)u

(Yingdi Qin)

In T 2

τ , consider L0 : {θ = 0} and Ld : {θ = −d r} (mirrors to O, L⊗d).

The generators sk = ( k

d , 0) ∈ L0 ∩ Ld of HF(L0, Ld) correspond to the ϑ-basis of H0(E, L⊗d):

ϑk/d(z) =

n∈Z exp

• πid(n + k

d )2τ + 2πid(n + k d )z

• L0L0

LdLd LzLz

s0 s1

×

L0 Ld Lz

ˆ s0 ˆ s1

In T, the generator sj ⊗ ˆ sk ∈ CF(L0, Ld) corresponds to

• ℓ∈Z/d

e2πikℓ/d ϑ(j−ℓ)/d(u) ϑℓ/d(v) ∈ H0(E × E, L⊗d ⊠ L⊗d).

Definition

HF(L0, Ld) ⊃ HF(L0, Ld)u = span of sj ⊗ (

k ˆ

sk) (j ∈ Z/d) (↔ ϑj/d(u) ϑ0/d(v))

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 12 / 14

Main result

For L, L′ linear (Lagrangian or coisotropic) branes in (T, ω), consider their lifts L, L′ to Lagrangian branes in (T, B B + iω ω), and define HF(L, L′)u = T ∗-translation-invariant, (0, 1)J part of HF(L, L′). with product x · y = πT(i(x) · i(y)) using inclusion and projection HF(L, L′)u

πT

⇆

i

HF(L, L′).

Theorem (Yingdi Qin)

Let H∗F(T)u be the category whose objects are linear (Lagrangian or coisotropic) branes in T, with hom(L, L′) = HF(L, L′)u, and composition x · y = πT(i(x) · i(y)). The Donaldson- Fukaya category H∗F(T) of linear Lagrangians in T embeds fully faithfully into H∗F(T)u. This gives a version of the Fukaya category which includes coisotropic branes!

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 13 / 14

H∗F(T, ω) ≃ H∗F(T ∗, ω−1)

Given dual tori (T, ω) and (T ∗, ω−1), the doubled tori T = T × T ∗ ≃ T ∗ × T = T∗ carry the same symplectic form 1

2(ω ⊕ ω−1), but opposite B-fields ± 1 2σ0 = ± 1 2

• i d ˆ

xi ∧ dxi. [σ0] ∈ H2(T, Z): the tautological bundle (ξT , ∇T = d + 2πi ˆ xi dxi) has curvature 2πiσ0. Hence F(T∗) ≃ F(T) via B-twist β = − ⊗ (ξT , ∇T ). Under β, branes lifted from T ← → branes lifted from T ∗. The subspaces HF(L, L′)u are different ((u, v)-splitting is the same, but ϑτ −1, 0

d (τ −1v) = ϑτ, 0 d (v))

but the projections πT, πT ∗ induce isomorphisms. Hence H∗F(T)u ≃ H∗F(T∗)u. Restricting to lifts of Lagrangians, H∗F(T, ω) ≃ H∗F(T ∗, ω−1).

(e.g. T 2’s of inverse areas). Note: F(T) ≃ F(T ∗) is induced by a coisotropic corresp. in T × T ∗, which lifts to the diagonal in T × T∗ ≃ T × T!)

This also works for partial dualization. E.g., in T = T 4 × T 4, C={ˆ r1 =θ2, ˆ θ1 =r2, ˆ r2 =−θ1, ˆ θ2 =−r1}, (with ∇ dependent on B-field twist) is lifted from any of the following branes: C = T 4

r1,θ1,r2,θ2, ∇ = d − 2πi(r1 dθ2 − r2 dθ1)

C = T 4

ˆ r1,ˆ θ1,ˆ r2,ˆ θ2, ∇ = d + 2πi(ˆ

r1 d ˆ θ2 − ˆ r2 d ˆ θ1)

(mirror to OΓ, Γ = {z2 = iz1})

L = {ˆ r1 = θ2, ˆ θ1 = r2} ⊂ T 4

ˆ r1,ˆ θ1,r2,θ2

L = {ˆ r2 = −θ1, ˆ θ2 = −r1} ⊂ T 4

r1,θ1,ˆ r2,ˆ θ2

(on mirror, rotate 2nd factor: O∆, ∆ = {z2 = z1}).

Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 14 / 14