dorsal/arxiv
View SchemaDeformation quantization of Poisson manifolds, I
| Authors | Maxim Kontsevich |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9709040 |
| URL | https://arxiv.org/abs/q-alg/9709040 |
| DOI | 10.1023/B:MATH.0000027508.00421.bf |
| Journal | Lett.Math.Phys.66:157-216,2003 |
Abstract
I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.
{
"annotation_id": "d593561d-a17c-4f6d-b005-97002c9b5ef3",
"date_created": "2026-03-02T18:01:28.417000Z",
"date_modified": "2026-03-02T18:01:28.417000Z",
"file_hash": "40f7c4ced92b220a7ae6dd3e4c305be4f8dacfcaf4bb55a27d70331b39341c7a",
"private": false,
"record": {
"abstract": "I prove that every finite-dimensional Poisson manifold X admits a canonical\ndeformation quantization. Informally, it means that the set of equivalence\nclasses of associative algebras close to the algebra of functions on X is in\none-to-one correspondence with the set of equivalence classes of Poisson\nstructures on X modulo diffeomorphisms. In fact, a more general statement is\nproven (\"Formality conjecture\"), relating the Lie superalgebra of polyvector\nfields on X and the Hochschild complex of the algebra of functions on X.\nCoefficients in explicit formulas for the deformed product can be interpreted\nas correlators in a topological open string theory, although I do not use\nexplicitly the language of functional integrals. One of corollaries is a\njustification of the orbit method in the representation theory.",
"arxiv_id": "q-alg/9709040",
"authors": [
"Maxim Kontsevich"
],
"categories": [
"q-alg",
"alg-geom",
"hep-th",
"math.AG",
"math.QA"
],
"doi": "10.1023/B:MATH.0000027508.00421.bf",
"journal_ref": "Lett.Math.Phys.66:157-216,2003",
"title": "Deformation quantization of Poisson manifolds, I",
"url": "https://arxiv.org/abs/q-alg/9709040"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "801b1977-d07b-4825-bb4b-6bbde2d38e4d",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}