Poisson AKSZ theories and their quantizations

Theo Johnson-Freyd

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Poisson AKSZ theories and their quantizations

Theo Johnson-Freyd

Mathematical Physics · Submitted 22 Jul 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms.

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We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads. The quantization problem is posed in terms of extending from dioperads to properads. We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures.	contribution 1.00	0.92	0.16	asserted 0.98
s1	Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads.	background 0.79	0.29	0.04	asserted 1.00
s2	The quantization problem is posed in terms of extending from dioperads to properads.	method 0.41	0.42	0.07	asserted 0.99
s3	We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms.	future work 0.61	0.58	0.79	speculative 0.98

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Original abstract

We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads. The quantization problem is posed in terms of extending from dioperads to properads. We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms.

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The homological perturbation lemma (see e.g. [ref] ) allows us to deform the whole deformation retraction: [equation] Then $ι̃_x$ and $p̃_x$ are quasiisomorphisms between $(V⊗^_∙(^d)) [equation]$ with the differential $∂+ Δ$ and $(V) [equation]$ with the differential $∂_V + Δ̃_x$; that they are quasiisomoprhisms is witnessed by the equations $p̃_x ∘ι̃_x = 𝕀$ and $[∂+Δ, h̃_x] = 𝕀- ι̃_x ∘p̃_x$.

On the perturbation lemma, and deformations · Classical and quantum AKSZ theories · 0.99
There are quasiisomorphisms relating our model to these, and a homotopy operad structure can be pulled through such quasiisomorphisms via homotopy transfer theory [ref] .

Coloured Koszul duality and strongly homotopy operads · Classical and quantum AKSZ theories · 0.88
There are quasiisomorphisms relating our model to these, and a homotopy operad structure can be pulled through such quasiisomorphisms via homotopy transfer theory [ref] .

Rectifying Partial Algebras Over Operads of Complexes · Classical and quantum AKSZ theories · 0.88

Cites as background 1 paper

See [ref] for details.

Chains(R) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure · A homotopy Frobenius algebra structure on Chains(M) · 0.91

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