BV pushforwards and exact discretizations in topological field - PowerPoint PPT Presentation

BV pushforwards and exact discretizations in topological field theory Pavel Mnev Max Planck Institute for Mathematics, Bonn Antrittsvorlesung, University of Zurich, February 29, 2016

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Manifold − − − − → Invariants of the manifold

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Manifold M − − − − → Algebra associated to M     � � Field theory on M − − − − → Invariants of M

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Manifold M − − − − → Algebra associated to M     � � Field theory on M − − − − → Invariants of M Upper right way: algebraic topology (Poincar´ e, de Rham,...) Lower left way: mathematical physics/topological field theory (Schwarz, Witten, Kontsevich,...)

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Manifold M − − − − → Algebra associated to M     � � Field theory on M − − − − → Invariants of M Upper right way: algebraic topology (Poincar´ e, de Rham,...) Lower left way: mathematical physics/topological field theory (Schwarz, Witten, Kontsevich,...) What happens when we replace M with its combinatorial description? (E.g. a triangulation)

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Pushforward in probability theory: y = F ( x ) x has probability distribution µ implies y has probability distribution F ∗ µ . Examples : Throw two dice. What is the distribution for the sum? 1 Benford’s law. 2 Pushforward in geometry: fiber integral.

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Plan. From discrete forms on the interval to Batalin-Vilkovisky formalism Effective action (BV pushforward) Application to topological field theory

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Algebra of ”discrete forms” on the interval Appetizer/warm-up problem: discretize the algebra of differential forms on the interval I = [0 , 1] . De Rham algebra Ω • ( I ) ∋ f ( t ) + g ( t ) · dt with operations d, ∧ satisfying d 2 = 0 Leibniz rule d ( α ∧ β ) = dα ∧ β ± α ∧ dβ Associativity ( α ∧ β ) ∧ γ = α ∧ ( β ∧ γ ) Also: super-commutativity α ∧ β = ± β ∧ α .

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Algebra of ”discrete forms” on the interval The problem: construct the algebra structure on “discrete forms” (cellular cochains) C • ( I ) = Span( e 0 , e 1 , e 01 ) ∋ a · e 0 + b · e 1 + c · e 01 with same properties. Represent generators by forms i : e 0 �→ 1 − t, e 1 �→ t, e 01 �→ dt And define a projection �� 1 � p : f ( t ) + g ( t ) · dt �→ f (0) · e 0 + f (1) · e 1 + g ( τ ) dτ · e 01 0 Construct d and ∧ on C • : d = p ◦ d ◦ i , i.e. d ( e 0 ) = − e 01 , d ( e 1 ) = e 01 , d ( e 01 ) = 0 α ∧ β = p ( i ( α ) ∧ i ( β )) , i.e. e 0 ∧ e 0 = e 0 , e 1 ∧ e 1 = e 1 , e 0 ∧ e 01 = 1 2 e 01 , e 1 ∧ e 01 = 1 2 e 01 , e 01 ∧ e 01 = 0

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Algebra of ”discrete forms” on the interval d, ∧ satisfy d 2 = 0 , Leibniz, but associativity fails: e 0 ∧ ( e 0 ∧ e 01 ) � = ( e 0 ∧ e 0 ) ∧ e 01 However, one can introduce a trilinear operation m 3 such that α ∧ ( β ∧ γ ) − ( α ∧ β ) ∧ γ = = d m 3 ( α, β, γ ) ± m 3 ( dα, β, γ ) ± m 3 ( α, dβ, γ ) ± m 3 ( α, β, dγ ) – “associativity up to homotopy”. m 3 itself satisfies [ ∧ , m 3 ] = − [ d, m 4 ] for some 4-linear operation m 4 etc. – a sequence of operations ( m 1 = d, m 2 = ∧ , m 3 , m 4 , . . . ) satisfying a sequence of homotopy associativity relations – an A ∞ algebra structure on C • ( I ) .

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Aside: A ∞ algebras Aside: A ∞ algebras Definition (Stasheff) An A ∞ algebra is: a Z -graded vector space V • , 1 a set of multilinear operations m n : V ⊗ n → V , n ≥ 1 , 2 satisfying the set of quadratic relations � m q + s +1 ( • , · · · , • , m r ( • , · · · , • ) , • , · · · , • ) = 0 � �� � � �� � � �� � q + r + s = n q s r

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Aside: A ∞ algebras Aside: A ∞ algebras Definition (Stasheff) An A ∞ algebra is: a Z -graded vector space V • , 1 a set of multilinear operations m n : V ⊗ n → V , n ≥ 1 , 2 satisfying the set of quadratic relations � m q + s +1 ( • , · · · , • , m r ( • , · · · , • ) , • , · · · , • ) = 0 � �� � � �� � � �� � q + r + s = n q s r Remark: Case m � =2 = 0 – associative algebra. Case m � =1 , 2 = 0 – differential graded associative algebra (DGA).

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Aside: A ∞ algebras Aside: A ∞ algebras Definition (Stasheff) An A ∞ algebra is: a Z -graded vector space V • , 1 a set of multilinear operations m n : V ⊗ n → V , n ≥ 1 , 2 satisfying the set of quadratic relations � m q + s +1 ( • , · · · , • , m r ( • , · · · , • ) , • , · · · , • ) = 0 � �� � � �� � � �� � q + r + s = n q s r Remark: Case m � =2 = 0 – associative algebra. Case m � =1 , 2 = 0 – differential graded associative algebra (DGA). Examples: Singular cochains of a topological space C • sing ( X ) – 1 non-commutative DGA. De Rham algebra of a manifold Ω • ( M ) – super-commutative DGA. 2

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Aside: A ∞ algebras Motivating example: Cohomology of a top. space H • ( X ) carries a natural A ∞ algebra structure, with m 1 = 0 , m 2 the cup product, m 3 , m 4 , · · · the (higher) Massey products on H • ( X ) .

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Aside: A ∞ algebras Motivating example: Cohomology of a top. space H • ( X ) carries a natural A ∞ algebra structure, with m 1 = 0 , m 2 the cup product, m 3 , m 4 , · · · the (higher) Massey products on H • ( X ) . Quillen, Sullivan: this A ∞ structure encodes the data of rational homotopy type of X , i.e. rational homotopy groups Q ⊗ π k ( X ) can be recovered from { m n } .

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Homotopy transfer of A ∞ algebras Homotopy transfer theorem for A ∞ algebras (Kadeishvili, Kontsevich-Soibleman) If ( V • , { m n } ) is an A ∞ algebra and V ′ ֒ → V a deformation retract of ( V, m 1 ) , then V ′ carries an A ∞ structure with n = � : ( V ′ ) ⊗ n → V ′ m ′ T where T runs over rooted trees with n leaves.

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory Homotopy transfer of A ∞ algebras Homotopy transfer theorem for A ∞ algebras (Kadeishvili, Kontsevich-Soibleman) If ( V • , { m n } ) is an A ∞ algebra and V ′ ֒ → V a deformation retract of ( V, m 1 ) , then V ′ carries an A ∞ structure with n = � : ( V ′ ) ⊗ n → V ′ m ′ T where T runs over rooted trees with n leaves. Decorations: i : V ′ ֒ p : V ։ V ′ leaf → V root − s : V • → V •− 1 edge ( k + 1) -valent vertex m k where s is a chain homotopy, m 1 s + s m 1 = id − i p . Example: V = Ω • ( M ) , d, ∧ the de Rham algebra of a Riemannian manifold ( M, g ) , V ′ = H • ( M ) de Rham cohomology realized by harmonic forms. Induced (transferred) A ∞ algebra gives Massey products.

Introduction Algebra of ”discrete forms” on the interval BV pushforward Exact discretizations in topological field theory A ∞ algebra on cochains of the interval Back to the A ∞ algebra on cochains of the interval. Explicit answer for algebra operations: � n � m n +1 ( e 01 , . . . , e 01 , e 1 , e 01 , . . . , e 01 ) = ± · B n · e 01 � �� � � �� � k k n − k (and similarly for e 1 ↔ e 0 ), where B 0 = 1 , B 1 = − 1 2 , B 2 = 1 6 , B 3 = 0 , B 4 = − 1 30 , . . . are Bernoulli numbers, e x − 1 = � x B n n ! x n . i.e. coefficients of n ≥ 0