Curved Dimers Raf Bocklandt August 13, 2020 1 / 13 The Fukaya - PowerPoint PPT Presentation

Curved Dimers Raf Bocklandt August 13, 2020 1 / 13

The Fukaya category of a closed surface • Objects: closed curves on the surface, • Morphisms: Linear combinations of intersection points, 2 / 13

Products 3 / 13

Problems • Monogons, digons, polygons • Signs • Self-intersections • Convergence • Huge Different approaches to solve all these problems (Seidel, Abouzaid, Fukaya, etc). 4 / 13

An approach using dimers 5 / 13

An approach using dimers: Gentle algebra The gentle A ∞ -algebra of an arc collection is the path algebra of Q A divided by the ideal generated by the face like paths. C Q A Gtl ± A = � αβ | αβ ∈ P − � We give the algebra a Z 2 -grading using the rule below. | α | = 0 | α | = 0 | α | = 1 | α | = 1 6 / 13

An approach using dimers: The higher products On Gtl ± A we put an A ∞ -structure defined by the rule that µ ( αβ 1 , . . . , β k ) = α and µ ( β 1 , . . . , β k γ ) = ( − 1 ) deg γ γ if β 1 , . . . , β l are the consecutive angles of an immersed polygon without internal marked pounts bounded by arcs. β 3 β 2 β 4 β 1 γ α 7 / 13

An approach using dimers: The twisted completion A twisted complex A over C • consists of a pair ( M , δ ) where M = ⊕ i A i [ k i ] is a direct sum of shifed objects and δ is a degree 1 element in Hom ( M , M ) . Additionally we assume that δ is strictly lower triangular satisfies the Maurer-Cartan equation : µ 1 ( δ ) + µ 2 ( δ, δ ) + µ 3 ( δ, δ, δ ) + · · · = 0 . Given a sequence of twisted complexes ( M 0 , δ 0 ) , · · · , ( M k , δ k ) we can introduce a twisted k -ary product by taking a sum over all possible ways to insert δ ′ s between the entries � µ k ( a 1 , · · · , a k ) = ˜ µ • ( δ 0 , · · · , δ 0 , a 1 , · · · , a k , δ k , · · · , δ k ) . � �� � � �� � m 0 , ··· , m k ≥ 0 m 0 m k C • ⊂ Tw C • 8 / 13

Strings and Bands α 1 α 2 α 3 L �� a j , � k − 1 � ( L , δ ) := j = 1 α u a 1 a 4 a 3 a 2 a 1 α 4 α 2 α 3 α 1 �� k � j = 1 a j , δ = � k ( B , δ ) = j = 1 λ j α j 9 / 13

Deformation theory • A deformation of an A ∞ -algebra A , µ is a k [[ � ]] -linear Z 2 -graded curved A ∞ -structure µ � on A [[ � ]] that reduces to A , µ if we quotient out � . • Deformation theory of ( A , µ ) is described by its Hochschild cohomology. • A deformed twisted complex is a pair M = ( ⊕ i A i [ j i ] , δ + ǫ ) where ( ⊕ i A i [ j i ] , δ ) is an ordinary twisted complex and ǫ ∈ � End ( ⊕ i A i [ j i ])[[ � ]] . The curvature of M is µ ( δ + ǫ ) + µ ( δ + ǫ, δ + ǫ ) + . . . • The deformed twisted complexes form a defomation of (a category equivalent to) the twisted complexes of A . See also Lowen-Van den Berg (htps://arxiv.org/abs/1505.03698), FOOO. 10 / 13

Deformation of gentle A ∞ -algebras Theorem The Hochschild cohomology of the gentle A ∞ -algebra Gtl ± A is equal to • HH 0 ( Gtl ± A ) = � m ∈ M C [ ℓ m ] ℓ m ∂ m ⊕ C n + 2 g − 1 , C [ ℓ m | m ∈ M ] • HH 1 ( Gtl ± A ) = ( ℓ i ℓ j | i � = j , i , j ∈ M ) . Proof. Either via direct computation using Bardzel’s bimodule resolution of Gtl ± A or using mirror symmetry and matrix factorizations (Lin-Pomerleano/Wong). Qestion Can we describe the deformations explicitely? (Joint with Van de Kreeke) 11 / 13

� Curved gentle A ∞ -algebras For each deformation class of Gtl ± A we can find a nice curved A ∞ -algebra deformation of Gtl ± A (over k [ � ] instead of k [[ � ]]) . E.g. if f = � i λ i ℓ i with λ i ∈ ( � ) then we set µ 0 ( a ) = λ m 1 ℓ m 1 + λ m 2 ℓ m 2 if m 1 m 2 and if β 1 , . . . , β l are the consecutive angles of an immersed polygon with internal marked points bounded by arcs. β 3 β 2 m 2 m 3 β 4 m 1 β 1 γ α we set µ ( αβ 1 , . . . , β k ) = λ 1 λ 2 λ 3 α and µ ( β 1 , . . . , β k γ ) = ( − 1 ) deg γ λ 1 λ 2 λ 3 γ. 12 / 13

Deforming strings and bands • Afer deforming the gentle algebra, every string and band becomes curved. • Band objects that do not enclose a disk are isomorphic to an uncurved object. • In general we cannot keep � inside k [ � ] if we go to the twisted completion. • Solution: look at a fixed number of band objects depending on the dimer. • Behaviour depends on the genus of the surface and dimer. 13 / 13