-quantization via lattice topological field theory Theo - - PowerPoint PPT Presentation

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 hFrobo

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 → Symn(V ) satisfying, for every n: 0 =

• n1+n2=n

n2 + 1 n

n1

• shuffles of the
• utgoing strands

... ... n1 n2

−

... ... n1 n2

+

... ... n1 n2

Remark

This is an example of a properad PoisF: compositions make sense along any connected directed acyclic graph, or dag for short.

Goal

Construct associative multiplication ⋆ = ⊙ + π

2 + 2a2 + . . . on

• Sym(V ); all ais bidifferential operators in Q[πij, ∂kπij, . . . ].

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 hFrobo

1-algebra structure on Chains•(R)

Definition

The properad Frobo

1 of open(=nonunital) one-shifted commutative

Frobenius algebras has generators and =

• f degree 0

and = −

• f degree −1, and relations

= , = , = , = . The bar/cobar construction gives a cofibrant replacement hFrobo

1

freely generated by arbitrarily-complicated dags modulo symmetric group actions, with complicated differential.

Example

Homology H•(S1, Q) is a Frobo

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 The ⋆-product Field-theoretic interpretation Definition hFrobo

1-algebra structure on Chains•(R)

Theorem

hFrobo

1 acts on C• = CellularChains•(· · · 1 2

· · · ), 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.

Since = 0 by symmetry, Frobo

1 is Koszul, giving a minimal

cofibrant replacement shFrobo

1 and a map hFrobo 1 → shFrobo

• 1. To

build the shFrobo

1-action (and prove contractiblity), check by hand

that a short finite list of obstructions vanish; the rest of the

• bstructions 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 Open one-shifted Frobenius algebras The ⋆-product Field-theoretic interpretation Deforming the differential on Sym(Chains•(R) ⊗ V ) Reconstructing the multiplication

Given PoisF-algebra V , let ∆m,n,g : (C• ⊗V )⊗m → (C• ⊗V )⊗n be: ∆m,n,g =

• dags Γ with m inputs,

n outputs, and genus g

(combinatorial term)× × (Γ as generator of hFrobo

1) ⊗ (Γ as composition in PoisF)

Extend to an mth-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 Symi(C•) ⊗ Symi(V ) ⊆ Symi(C• ⊗V ).

Corollary

For z ∈ Z, insertion ιz : 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 Open one-shifted Frobenius algebras The ⋆-product Field-theoretic interpretation Deforming the differential on Sym(Chains•(R) ⊗ V ) Reconstructing the multiplication

Definition

The star-product ⋆ : Sym(V ) ⊗ Sym(V ) → Sym(V ) is defined modulo high powers of V , by: ⋆ = p ◦ ⊙ ◦ (ιz1 ⊗ ιz2) z2 − z1 > 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 hFrobo

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

• ther requirements involves combinatorics of diagrams.

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 Topological field theories of AKSZ type Quantization Factorization algebra in the large-volume limit 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(MdR, X). If M is an oriented d-dimensional manifold and X has a symplectic form of homological degree −k, then Maps(MdR, 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(MdR, 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 Open one-shifted Frobenius algebras The ⋆-product Field-theoretic interpretation Topological field theories of AKSZ type Quantization Factorization algebra in the large-volume limit 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(Frobo

0).

BV quantization = properadic Bar(Frobo

0).

(Frobo

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 Open one-shifted Frobenius algebras The ⋆-product Field-theoretic interpretation Topological field theories of AKSZ type Quantization Factorization algebra in the large-volume limit 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 U1, . . . , Un ⊆ U pairwise disjoint (to enforce Heisenberg uncertainty), there is a multiplication map F(Ui) → F(U). (+locality axioms)

Fact (Francis, Lurie)

Framed n-dim topological factorization algebras = En 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

Goal: deformation quantization of Poisson formal manifolds Open one-shifted Frobenius algebras The ⋆-product Field-theoretic interpretation Topological field theories of AKSZ type Quantization Factorization algebra in the large-volume limit A dictionary with field theory

Standard constructions from quantum field theory, and how they appear here: Derived space of classical fields

• Cochains•(R) ⊗ V ∗, ∂
• Algebra of quantum observables

Sym(Chains• ⊗V ), ∂ + ∆

• Renormalization scheme

choice of higher homotopies in hFrobo

1-action

effective action integrating out ∆ = (Γ ∈ hFrobo

1) ⊗ (Γ ∈ PoisF)

modes inside each interval n-point function n-fold ⋆-product Path integral Homological Perturbation Lemma

Theo Johnson-Freyd, Northwestern University ⋆-quantization via lattice topological field theory