An invitation to homological mirror symmetry Denis Auroux Harvard - PowerPoint PPT Presentation

An invitation to homological mirror symmetry Denis Auroux Harvard University IMSA Inaugural Conference, Miami, September 6, 2019 partially supported by NSF and by the Simons Foundation (Simons Collaboration on Homological Mirror Symmetry) Denis Auroux (Harvard University) Homological mirror symmetry September 6, 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 ) exp( π in 2 τ + 2 π inz ). s ( z ) = ϑ ( τ ; z ) = � Only one up to scaling! n ∈ Z (Jacobi, 1820s) 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 . ? = · · · + T ( x − 1) 2 / 2 + T x 2 / 2 + T ( x +1) 2 / 2 + . . . 1 2 n 2 + nx = e π i τ x 2 ϑ ( τ ; τ x ) = T x 2 / 2 � n ∈ Z T ( T = e 2 π i τ ) Denis Auroux (Harvard University) Homological mirror symmetry September 6, 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 ) 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). Intersections (mod. Hamiltonian isotopy) = Floer cohomology L ′ CF ∗ ( L , L ′ ) = K | L ∩ L ′ | ∂ p = T a q q p p ′ � ω = a L ′ L ′′ ( ⊗ local coefficients) L q p p ′ · p = T a q Product CF ( L ′ , L ′′ ) ⊗ CF ( L , L ′ ) → CF ( L , L ′′ ): L Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 3 / 14

Example: elliptic curve (Polishchuk-Zaslow) L = C 2 / ( z , v ) ∼ ( z + 1 , v ) ∼ ( z + τ, e − π i τ − 2 π iz v ) E = C / Z + τ Z , � dim H 0 ( E , L ) = 1, exp( π in 2 τ + 2 π inz ). s ( z ) = ϑ ( τ ; z ) = n ∈ Z X = T 2 = R 2 / Z 2 L 1 L x e 0 x L 0 s L 0 L 1 e 1 . . . L x s e 1 L 0 − → L 1 − → L x e 1 · s = ? e 0 e 0 ∼ evaluation O → O x ? = · · · + T ( x − 1) 2 / 2 + T x 2 / 2 + T ( x +1) 2 / 2 + . . . 1 2 n 2 + nx = e π i τ x 2 ϑ ( τ ; τ x ) = T x 2 / 2 � ( T = e 2 π i τ ) n ∈ Z T Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 4 / 14

Homological mirror symmetry: towards a general setting Projective Calabi-Yau varieties ( c 1 = 0): 1 T 2 (Polishchuk-Zaslow) , T 2 n (Kontsevich-Soibelman, Fukaya, Abouzaid-Smith) , K3 surfaces (Seidel, Sheridan-Smith) , X d = n +2 ⊂ CP n +1 (Sheridan) , . . . Fano case: CP n , del Pezzo, toric varieties ... (LG models) 2 (Kontsevich, Seidel, Auroux-Katzarkov-Orlov, Abouzaid, FOOO ...) General type case, affine varieties, etc. 3 Riemann surfaces, compact (Seidel, Efimov) or non-compact (Abouzaid-Auroux-Efimov-Katzarkov-Orlov, Lee, ...) hypersurfaces ⊂ ( C ∗ ) n or toric varieties (Gammage-Shende, Abouzaid-Auroux, ...) ... and beyond Goal of talk: give a flavor of this program � HMS for all Riemann surfaces starting with (focusing on HMS itself, ignoring developments from Strominger-Yau-Zaslow, skeleta, family Floer theory, etc.) Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 5 / 14

� � ≃ D b � � Example 1: F c (classical) c L r = { r } × S 1 (+ local system ξ ) X = R × S 1 , ω = dr ∧ d θ , ⇒ HF ∗ ( L r , L r ) ≃ H ∗ ( S 1 , K ), ⇒ HF ∗ ( L r , L r ′ ) = 0. r ∂ p = q − q = 0 q p M pt = { ( L r , ξ ) ∈ F ( X ) } / ∼ has a natural analytic structure Coordinate: z ( L r , ξ ) = T r hol ( ξ ) ∈ K ∗ . ( ∀ L ′ , CF (( L r , ξ ) , L ′ ) has analytic dependence on z ) → O z ∈ D b ( X ∨ = K ∗ ) ( L r , ξ ) ∈ F ( X , ω ) ← Strominger-Yau-Zaslow: X CY, π : X → B Lagrangian torus fibration ⇒ mirror X ∨ = {O p , p ∈ X ∨ } = ( L b = π − 1 ( b ) , ξ ) ∈ F ( X ) � � / ∼ Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 6 / 14

� � ≃ D b � � Abouzaid-Seidel Example 1: F wr “wrapped Fukaya category” X = R × S 1 ⊃ L 0 = R × { 0 } non-compact Lagrangian. Hamiltonian perturbation: H = 1 2 r 2 , φ 1 H ( r , θ ) = ( r , θ + r ). ( → intersections ∈ X int + Reeb flow at boundary). CW ∗ ( L 0 , L 0 ) := CF ∗ ( φ 1 � H ( L 0 ) , L 0 ) = K x i . i ∈ Z φ 1 H ( L 0 ) ˜ x 2 L 0 x − 1 x 0 x 1 ˜ x 1 φ 2 H ( L 0 ) r X int p ′ φ 1 ( L ) L Product: q ˜ q ∈ φ 2 ( L ) ∩ L ↔ q ∈ φ 1 ( L ) ∩ L via r �→ 2 r ) p (˜ φ 2 ( L ) x k · x l = x k + l ⇒ End( L 0 ) ≃ K [ x ± 1 ]. ( x k � x k ) Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 7 / 14

� � ≃ D b ( K ∗ ) Abouzaid-Seidel Example 1: F wr “wrapped Fukaya category” X = R × S 1 ⊃ L 0 = R × { 0 } ⇒ End( L 0 ) ≃ K [ x ± 1 ] ≃ End( O X ∨ ) . φ 1 H ( L 0 ) ˜ x 2 L 0 x − 1 x 0 ˜ x 1 x 1 φ 2 H ( L 0 ) L 0 generates F wr ( X ). Yoneda: L �→ Hom( L 0 , L ) gives an embedding F wr ( X ) ֒ → End( L 0 )-mod . Example: ( L r , ξ ) �→ HF ( L 0 , ( L r , ξ )) ≃ K [ x ± 1 ] / ( x − z ) ( z = T r hol( ξ )) Theorem F wr ( X ) ≃ K [ x ± 1 ] -mod ≃ D b Coh ( X ∨ ) . Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 8 / 14

� � Example 2: F wr (Abouzaid-A.-Efimov-Katzarkov-Orlov) X = S 2 \ {− 1 , 0 , ∞} = C ∗ \ {− 1 } , L 0 = R + ⇒ CW ( L 0 , L 0 ) = � i ∈ Z K x i . φ 1 H ( L 0 ) x 2 ˜ L 0 x − 1 x 0 ˜ x 1 x 1 φ 2 H ( L 0 ) − 1 � x i + j if ij ≥ 0 x j · x i = 0 if ij < 0 ⇒ End( L 0 ) ≃ K [ x , y ] / ( xy = 0). ... X ∨ = Spec K [ x , y ] / ( xy = 0) = { xy = 0 } ⊂ A 2 ? F wr ( X ) ֒ → End( L 0 )-mod ?? Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 9 / 14

� � ≃ D b ( { xy = 0 } ) Example 2: F wr (A-A-E-K-O) X = C ∗ \ {− 1 } : L 0 = (0 , ∞ ), L 1 = ( − 1 , 0), L 2 = ( −∞ , − 1) generate φ 1 H ( L 0 ) L 0 y 1 x y x yu y u y v x L 1 L 2 0 z ∞ − 1 K [ x , y ] / ( xy ) u y · v y = y v y · u y = y L 0 u x K [ x ] u y K [ y ] v x K [ x ] v y K [ y ] v z K [ z ] L 1 L 2 u z K [ z ] + exact triangles K [ y , z ] / ( yz ) K [ x , z ] / ( xz ) u y u x u z L 2 − → L 0 − → L 1 − → L 2 [1] v y v x v z − → L 0 − → L 2 − → L 1 [1] L 1 Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 10 / 14

� � ≃ D b ( { xy = 0 } ) Example 2: F wr (A-A-E-K-O) X = C ∗ \ {− 1 } ⊃ L 0 , L 1 , L 2 X ∨ = { xy = 0 } = A ∪ B ⊂ A 2 K [ x , y ] / ( xy ) K [ x , y ] / ( xy ) =: R O L 0 u x K [ x ] u y K [ y ] K [ y ] x K [ x ] K [ x ] v x K [ x ] y K [ y ] v y K [ y ] v K [ z ] v z K [ z ] O A O B L 1 L 2 u z K [ z ] u K [ z ] K [ y , z ] / ( yz ) K [ y , z ] / ( yz ) K [ x , z ] / ( xz ) K [ x , z ] / ( xz ) Hom( O A , O A ) ≃ K [ y ], Ext 2 k ( O A , O A ) ∋ z k . u y u x u z x 1 u − → L 0 − → L 1 − → L 2 [1] O B − → O − → O A − → O B [1] L 2 v y v x v z y 1 v − → L 0 − → L 2 − → L 1 [1] L 1 O A − → O − → O B − → O A [1] Theorem ( A-A-E-K-O) ⇒ F wr ( X ) ≃ D b Coh ( X ∨ ) Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 11 / 14

� � ≃ D b sing ( C 3 , − xyz ) Example 2: F wr (A-A-E-K-O) X = P 1 \ {− 1 , 0 , ∞} ← → X ∨ = { xy = 0 } : F wr ( X ) ≃ D b Coh ( { xy = 0 } ) lacks symmetry in x , y , z . how to extend to higher genus? – gluing ? Stabilization: X ≃ { x + y + 1 = 0 } ⊂ ( C ∗ ) 2 . ( X = Bl (( C ∗ ) 2 × C , X × 0) , W = p C ) ← → ( X ∨ = C 3 , W ∨ = − xyz ) . Theorem ( A-A-E-K-O) F wr ( X ) ≃ D b sing ( X ∨ , W ∨ ) := D b Coh ( { xyz = 0 } ) / Perf . (Orlov) ( L 0 , L 1 , L 2 ) ← → ([ O { z =0 } ] , [ O { x =0 } ] , [ O { y =0 } ]) This result extends to all Riemann surfaces (AAEKO, Seidel, Efimov, H. Lee). Mirror ( X ∨ , W ∨ ), dim X ∨ = 3. (Hori-Vafa, A-A-K) Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 12 / 14

Geometry of ( X ∨ , W ∨ ) (Hori-Vafa, Clarke, Abouzaid-A-Katzarkov, ...) For an affine plane curve Σ = { f ( x , y ) = 0 } ⊂ ( C ∗ ) 2 , mirror: X ∨ = toric CY 3-fold determined by tropicalization of f , W ∨ ∈ O ( X ∨ ), Z := { W ∨ = 0 } = � toric strata. sing( Z ) = crit( W ∨ ) = � 1-dim. strata = union of P 1 and A 1 . → ( X ∨ , W ∨ ) = � ( C 3 , − xyz ) Mirror decompositions : Σ = � ← Jeff Koons, Balloon Dog (photo Librado Romero - The New York Times) Theorem (Heather Lee) Denis Auroux (Harvard University) Homological mirror symmetry September 6, 2019 13 / 14