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 + τ Z All doubly periodic holomorphic functions are constant, but we can ask for quasi-periodic functions: s ( z + 1) = s ( z ), s ( z + τ ) = e − π i τ − 2 π iz s ( z ) (section of deg. 1 line bundle L → E ) exp( π in 2 τ + 2 π inz ). s ( z ) = ϑ ( τ ; z ) = � Only one up to scaling! (Jacobi, 1820s) n ∈ Z Counting triangles in T 2 = R 2 / Z 2 (weighted by area) L 1 L x x e 0 L 0 L 1 L 0 s L x e 1 . . . ? = · · · + q ( x − 1) 2 / 2 + q x 2 / 2 + q ( x +1) 2 / 2 + . . . 1 2 n 2 + nx = e π i τ x 2 ϑ ( τ ; τ x ) = q x 2 / 2 � n ∈ Z q ( q = e 2 π i τ ) 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: O V , vector bundles E → V , skyscrapers O p ∈ V , ...) Morphisms (+ extensions): H ∗ hom ( E , F ) = Ext ∗ ( E , F ). d i → E i +1 → · · · → 0 / ∼ Derived category = complexes 0 → · · · → E i − Eg: functions, intersections, cohomology... � Mirror symmetry: D b Coh ( V ) ≃ D π F ( X , ω ) in general: over Novikov field here: over C Symplectic geometry: Fukaya category F ( X , ω ) ( X , ω ) loc. ≃ ( R 2 n , � dx i ∧ dy i ), Lagrangian submanifolds L ( dim . n , ω | L = 0) + rk 1 loc. system ∇ . Floer cohomology measures intersections (physicists’ version: over C instead of Novikov field) L ′ CF ∗ ( L , L ′ ) = C | L ∩ L ′ | � ∂ p = exp(2 π i ( B + i ω )) hol ∇ q q p p ′ L ′ ( ⊗ local coefficients) L ′′ L q p p ′ · p = exp(2 π i Product CF ( L ′ , L ′′ ) ⊗ CF ( L , L ′ ) → CF ( L , L ′′ ): � ( B + i ω )) hol ∇ q 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 = R 2 / Z 2 , B + i ω = τ dr ∧ d θ vs. E τ = C / ( Z + τ Z ) L z = τ x + y = { x }× S 1 ( ∇ = d + 2 π iy d θ ) L 1 θ e 0 x L 0 s L 0 L 1 L z e 1 s e 1 In F ( T 2 τ ), L 0 − → L 1 − → L z (mirror to: O → L → O z on E τ ) e 1 · s = ? e 0 s ∼ section of L , e 0 ∼ evaluation O → O z e π i τ ( x + n ) 2 +2 π i ( x + n ) y + ... = e π i τ x 2 +2 π ixy ϑ τ ( z ) ? = � n ∈ Z � ϑ ( z + 1) = ϑ ( z ) , ϑ τ ( z ) = � exp( π in 2 τ + 2 π inz ) ∈ H 0 ( E τ , L ): (Jacobi, 1820s) ϑ ( z + τ ) = e − π i τ − 2 π iz ϑ ( z ) n ∈ Z Similarly for rest of F ( T 2 τ ) ≃ Coh ( E τ ). In higher dim., τ ∈ Mat n × 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 = ( R 2 / Z 2 , ω 0 ) has an autoequivalence mirror to complex multiplication by i on E i = C / ( Z + i Z ). This should “exchange position and holonomy”: L z = ix + y = ( { x } × S 1 → L iz = iy − x = ( { y } × S 1 θ , ∇ = d + 2 π iy d θ ) ← θ , ∇ = d − 2 π ix d θ ) . No Lagrangian correspondence in ( T 2 × T 2 , ω = − dr 1 d θ 1 + dr 2 d θ 2 ) induces such a functor. A coisotropic correspondence ? Consider the line bundle ( ξ C → T 2 × T 2 , ∇ C = d − 2 π i ( r 1 d θ 2 + r 2 d θ 1 )). Then the “mult. by i” functor maps to ( ξ → L , ∇ ) to ( π 2 ) ∗ ( ξ C ⊗ π ∗ 1 ξ ) (check: this has Lagrangian support). We think of C = ( ξ C → T 2 × T 2 , ∇ C ) as a coisotropic correspondence. The curvature F = dr 1 d θ 2 + dr 2 d θ 1 satisfies ( ω − 1 F ) 2 = − 1. − 1 /τ )). Q: How does C fit into F ( T 2 × T 2 )? (Similarly for F ( T 2 τ ) ≃ F ( 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 , ω 0 ), the image of ch : F ( T 4 ) → H 2 ( T 4 ), L �→ [ L ] has rank 5, while for V = E i × E i , ch : D b ( V ) ։ � p H p , p ( V ) which has rank 6. So F ( T 4 ) �≃ D b ( V ). Fix: take split-closure Tw π F ( 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 2 n , ω ) with a U(1)-bundle ( ξ, ∇ ) such that ˜ 1 F = 2 π i F ∇ satisfies: (i) ˜ F = 0 on the isotropic leaves TC n − k = ker ω | TC , iso (ii) ( ω − 1 ˜ F ) 2 = − 1 complex structure on TC / TC iso . ( ⇒ ˜ F + i ω holom. symplectic, and k is even). Proposal: End ( C , ∇ ) = H ∗ ( C , O C ) (loc. constant in TC iso , holomorphic in TC / TC iso ). (Note: in X 4 , ˜ F ∧ ˜ F = ω ∧ ω and [ ˜ F ] ∈ H 2 ( X , Z ) ⇒ coisotropic branes only exist at special locus in K¨ ahler moduli space) In ( T 4 , ω 0 ), the “missing” generator is mirror to O Γ , where Γ = { ( z , iz ) } ⊂ V (mult. by i ). Candidate: C = T 4 , ∇ = d − 2 π i ( r 1 d θ 2 − r 2 d θ 1 ). Question: how to enlarge F ( T 2 n ) 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 n ∈ Z e − π n 2 e 2 π inz satisfies f ( iz ) = f ( z ) ! Classical fact: f ( z ) = e π z 2 / 2 ϑ i ( z ) = e π z 2 / 2 � Slogan: understanding mult. by i in HMS is difficult because this invariance property is not obvious! ( m , n ) ∈ Z 2 ( − 1) mn e − π ( m 2 + n 2 ) / 2 e π ( m + in ) z √ Alternative, invariant expression: c f ( z ) = � ( c = 2 ϑ (0) ) Symplectic interpretation: Floer product in T 2 × T 2 L z = ix + y = { x }× S 1 θ × S 1 r × { y } ( ∇ = d + 2 π iy d θ ) ˆ L 1 In T = T 2 × T 2 , with x e 0 e 0 L 0 s e 1 × r ∧ d ˆ ω = 1 2 ( dr ∧ d θ + d ˆ θ ), ⇒ e 1 · s ∼ f ( z ) e 0 . y e 1 r ∧ dr + d ˆ s B = 1 2 ( d ˆ θ ∧ d θ ) In general, ( − 2 i τ ) 1 / 2 ϑ τ (0) e π iz 2 / 2 τ ϑ τ ( z ) = � m , n ( − 1) mn e π in 2 τ/ 2 e − π im 2 / 2 τ exp( π i ( 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 σ 0 = � x i ∧ dx i , define T = T × T ∗ , with i d ˆ symplectic form 1 2 ( ω ⊕ ω − 1 ) and B-field 1 2 σ 0 . ω = 1 2 σ 0 + 1 2 ( ω ⊕ ω − 1 ). So: B B + i ω (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 ∈ H 1 ( L ) ⊂ Λ , exp(2 π i � x ∗ , γ � ) = ( − 1) ε ( γ ) hol ∇ ( x + S 1 γ ) } , ∇ = π ∗ ∇ . This is a Lagrangian brane in T . with ∇ ε : H 1 ( L , Z ) → Z / 2 such that ε ( γ + γ ′ ) − ε ( γ ) − ε ( γ ′ ) = c 1 ( ∇ )( γ ∧ γ ′ ) mod 2 ( ⇔ rel. spin structure) When L is Lagrangian, L = L × ( L ⊥ + hol ∇ ) (translated) conormal. 2 ( ω ⊕ ω − 1 ) using ( ω − 1 F ) 2 = − 1) . For L = T space-filling coisotropic, L = “graph of ∇ ” (Lagr. for ω ω = 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 ) 2 n , ω = a dr ∧ d θ ( τ = ia ∈ M n × n ( C )), with SYZ fibers F = { r } × T n θ , the mirror abelian variety is E = C n / ( Z n + τ ( Z n )). For T ∗ = ( R / Z ) 2 n , ω − 1 = a − 1 d ˆ θ , with SYZ fibers F ∗ = T n r ∧ d ˆ r × { ˆ θ } , ˆ E = C n / ( Z n + τ − 1 ( Z n )) the mirror abelian variety is ˆ ( ≃ E ). 2 ( σ 0 + ia dr ∧ d θ + ia − 1 d ˆ ω = 1 r ∧ d ˆ r × { ˆ For T = ( R / Z ) 4 n , B θ ), F = { r } × T n θ × T n B + i ω θ } , ˆ the mirror is E = ( C n × C n ) / ( Z 2 n + τ � � τ 1 τ ( Z 2 n )), τ τ = 1 . − τ − 1 2 − 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. ˆ z = τ − 1 r × { ˆ r ) ↔ ( z = τ θ ˆ y + r { r } × T n θ × T n θ } , ∇ = d + 2 π i ( y d θ + ˆ y d ˆ 2 r + y − 2 , ˆ θ + ˆ 2 ). ˆ 2 Lifts from T have ˆ y = 0, so ( z = 1 z = 1 2 ( τ − 1 y + r )), hence ( u = τ r + y , v = 0). θ = y and ˆ 2 ( τ r + y ) , ˆ L ∼ O p ⇒ L ∼ O p ⊠ O 0 . Similarly, L ∼ E ⇒ L ∼ E ⊠ E 0 . (if E ∈ Pic d ( E ), then E 0 = origin of Pic d ) Denis Auroux (Harvard University) Coisotropic branes and HMS November 14, 2019 9 / 14
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