Goal: deformation quantization of Poisson formal manifolds Open one-shifted Frobenius algebras The ⋆ -product Field-theoretic interpretation ⋆ -quantization via lattice topological field theory Theo Johnson-Freyd, Northwestern University Tuesday, 18 June 2013, String-Math Conference, SCGP These slides available at math.berkeley.edu/~theojf/slides-2013-6-18.pdf Preprint available at math.berkeley.edu/~theojf/StarQuantization.pdf Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Open one-shifted Frobenius algebras The ⋆ -product Field-theoretic interpretation Goal: deformation quantization of Poisson formal manifolds Open one-shifted Frobenius algebras Definition hFrob o 1 -algebra structure on Chains • ( R ) The ⋆ -product Deforming the differential on � Sym(Chains • ( R ) ⊗ V ) � � � Reconstructing the multiplication Field-theoretic interpretation Topological field theories of AKSZ type Quantization Factorization algebra in the large-volume limit A dictionary with field theory Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Open one-shifted Frobenius algebras The ⋆ -product Field-theoretic interpretation Definition A Poisson formal manifold is vector space V (over Q ) and a Poisson structure π on � Sym( V ). Equivalently, Taylor coefficients ... : V ∧ 2 → Sym n ( V ) satisfying, for every n : n 1 n 2 n 1 n 2 n 1 n 2 ... ... ... ... ... ... � � n 2 + 1 � n � 0 = + − n 1 n 1 + n 2 = n shuffles of the outgoing strands Remark This is an example of a properad PoisF: compositions make sense along any connected directed acyclic graph, or dag for short. Goal 2 + � 2 a 2 + . . . on Construct associative multiplication ⋆ = ⊙ + � π � Sym( V ) � � � ; all a i s bidifferential operators in Q [ π ij , ∂ k π ij , . . . ]. Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Open one-shifted Frobenius algebras Definition hFrob o The ⋆ -product 1 -algebra structure on Chains • ( R ) Field-theoretic interpretation Definition The properad Frob o 1 of open(=nonunital) one-shifted commutative Frobenius algebras has generators and = of degree 0 and = − of degree − 1, and relations = , = , = , = . The bar/cobar construction gives a cofibrant replacement hFrob o 1 freely generated by arbitrarily-complicated dags modulo symmetric group actions, with complicated differential. Example Homology H • ( S 1 , Q ) is a Frob o 1 -algebra. Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Open one-shifted Frobenius algebras Definition hFrob o The ⋆ -product 1 -algebra structure on Chains • ( R ) Field-theoretic interpretation Theorem 0 1 2 hFrob o 1 acts on C • = CellularChains • ( · · · · · · ) , lifting the action on H • , by translation-invariant operators that are quasilocal : for each generator there exists ℓ such that only cells within distance ℓ interact. The space of such actions is contractible. Proof. = 0 by symmetry, Frob o Since 1 is Koszul, giving a minimal cofibrant replacement shFrob o 1 and a map hFrob o 1 → shFrob o 1 . To build the shFrob o 1 -action (and prove contractiblity), check by hand that a short finite list of obstructions vanish; the rest of the obstructions vanish by computing the homology of the complex of translation-invariant quasilocal operators. Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Deforming the differential on � Open one-shifted Frobenius algebras Sym(Chains • ( R ) ⊗ V ) � � � The ⋆ -product Reconstructing the multiplication Field-theoretic interpretation Given PoisF-algebra V , let ∆ m , n , g : (C • ⊗ V ) ⊗ m → (C • ⊗ V ) ⊗ n be: � ∆ m , n , g = (combinatorial term) × dags Γ with m inputs, n outputs, and genus g × (Γ as generator of hFrob o 1 ) ⊗ (Γ as composition in PoisF) Extend to an m th-order differential operator on � Sym(C • ⊗ V ). Lemma The operator ∆ = � m , n , g � g + m − 1 ∆ m , n , g on � Sym(C • ⊗ V ) � � � satisfies ( ∂ + ∆) 2 = 0 , is O ( � ) , and vanishes on Sym i (C • ) ⊗ Sym i ( V ) ⊆ Sym i (C • ⊗ V ) . Corollary � � � For z ∈ Z , insertion ι z : � Sym( V ) � � � → Sym(C • ⊗ V ) , ∂ + ∆ at z is a quasiiso. Its left inverse p is unique and z-independent, and given explicitly by the Homological Perturbation Lemma. Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Deforming the differential on � Open one-shifted Frobenius algebras Sym(Chains • ( R ) ⊗ V ) � � � The ⋆ -product Reconstructing the multiplication Field-theoretic interpretation Definition The star-product ⋆ : � Sym( V ) � � � ⊗ � Sym( V ) � � � → � Sym( V ) � � � is defined modulo high powers of V , � by: ⋆ = p ◦ ⊙ ◦ ( ι z 1 ⊗ ι z 2 ) z 2 − z 1 > ℓ ; ℓ > 0 depends on the powers of V , � that you want. Main Theorem ⋆ is a well-defined associative deformation of ⊙ , and is independent of the choice of hFrob o 1 action used. It satisfies all requirements to be a universal star product. Proof. Well-definedness, associativity, and independence are formal, and use that ∆ vanishes on symmetric-times-symmetric. To check other requirements involves combinatorics of diagrams. Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Topological field theories of AKSZ type Open one-shifted Frobenius algebras Quantization The ⋆ -product Factorization algebra in the large-volume limit Field-theoretic interpretation A dictionary with field theory Definition The classical TFT of AKSZ type with target X assigns to a spacetime M the derived space of locally-constant maps M → X , called Maps( M dR , X ). If M is an oriented d -dimensional manifold and X has a symplectic form of homological degree − k , then Maps( M dR , X ) has a symplectic form of homological degree d − k . In infinite dimensions, symplectic forms do not invert. And yet: Generalization If X has a k-shifted homotopy Poisson structure, then Maps( M dR , X ) has a ( k − d ) -shifted homotopy Poisson structure. Choosing this amounts to choosing a d-shifted open homotopy Frobenius algebra structure on Chains • ( M ) at the dioperadic level. Definition Dioperads are like properads, but only use tree-level compositions. Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Topological field theories of AKSZ type Open one-shifted Frobenius algebras Quantization The ⋆ -product Factorization algebra in the large-volume limit Field-theoretic interpretation A dictionary with field theory For every oscillating integral, there is a BV complex : 0-cycles are gauge-invariant observables and 0-boundaries are Ward identities. In the classical limit, you get the derived critical locus , a dg space with ( − 1)-shifted homotopy Poisson structure. Definition A BV quantization of a ( − 1)-shifted homotopy Poisson structure is a deformation of the dg structure that matches the Poisson structure to first order. For derived critical loci, this is the same as constructing a measure. Lemma Homotopy ( − 1) -shifted Poisson = dioperadic Bar(Frob o 0 ) . BV quantization = properadic Bar(Frob o 0 ) . (Frob o 0 = open 0-shifted commutative Frobenius (pr/di)operad.) Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
Goal: deformation quantization of Poisson formal manifolds Topological field theories of AKSZ type Open one-shifted Frobenius algebras Quantization The ⋆ -product Factorization algebra in the large-volume limit Field-theoretic interpretation A dictionary with field theory Definition A factorization algebra F encodes the derived (i.e. BV–BRST) space of observables of a QFT: for every open neighborhood U in spacetime, F ( U ) is a chain complex, and for every U 1 , . . . , U n ⊆ U pairwise disjoint (to enforce Heisenberg uncertainty), there is a multiplication map � F ( U i ) → F ( U ). (+locality axioms) Fact (Francis, Lurie) Framed n-dim topological factorization algebras = E n algebras. � � � My construction is not local, so Sym(Chains • ( U ) ⊗ V ) � � � , ∂ + ∆ is not a factorization algebra. But quasilocality ⇒ locality in the “large-volume limit” = take the lattice spacing very small. Theo Johnson-Freyd, Northwestern University ⋆ -quantization via lattice topological field theory
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