| Title:
|
Poisson reduction as a coisotropic intersection (English) |
| Author:
|
Safronov, Pavel |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
1 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
87-121 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give a definition of coisotropic morphisms of shifted Poisson (i.e. $\Bbb P_n$) algebras which is a derived version of the classical notion of coisotropic submanifolds. Using this we prove that an intersection of coisotropic morphisms of shifted Poisson algebras carries a Poisson structure of shift one less. Using an interpretation of Hamiltonian spaces as coisotropic morphisms we show that the classical BRST complex computing derived Poisson reduction coincides with the complex computing coisotropic intersection. Moreover, this picture admits a quantum version using brace algebras and their modules: the quantum BRST complex is quasi-isomorphic to the complex computing tensor product of brace modules. (English) |
| Keyword:
|
Poisson reduction |
| Keyword:
|
shifted Poisson structure |
| Keyword:
|
BRST complex |
| Keyword:
|
brace algebra |
| MSC:
|
17B63 |
| MSC:
|
53D20 |
| MSC:
|
53D55 |
| idZBL:
|
Zbl 1429.17024 |
| idMR:
|
MR3912052 |
| DOI:
|
10.21136/HS.2017.04 |
| . |
| Date available:
|
2026-03-10T11:31:30Z |
| Last updated:
|
2026-03-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153395 |
| . |
| Reference:
|
[1] Behrend, K., Fantechi, B.: Gerstenhaber and Batalin–Vilkovisky structures on Lagrangian intersections.Algebra, Arithmetic and Geometry, Progr. in Math. 269 1–47 |
| Reference:
|
[2] Baranovsky, V., Ginzburg, V.: Gerstenhaber–Batalin–Vilkoviski structures on coisotropic intersections.Math. Res. Lett. 17 211–229, arxiv:0907.0037 http://arxiv.org/pdf/0907.0037 10.4310/MRL.2010.v17.n2.a2 |
| Reference:
|
[3] Calaque, D.: Lagrangian structures on mapping stacks and semi-classical TFTs.Stacks and Categories in Geometry, Topology, and Algebra, Contemp. Math. 643, arxiv:1306.3235 http://arxiv.org/pdf/1306.3235 |
| Reference:
|
[4] Calaque, D., Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted Poisson Structures and deformation quantization.J. of Topology 10 483–584, arxiv:1506.03699 http://arxiv.org/pdf/1506.03699 10.1112/topo.12012 |
| Reference:
|
[5] Calaque, D., Rossi, C.: Lectures on Duflo isomorphisms in Lie algebras and complex geometry.EMS series of Lectures in Math. 14 |
| Reference:
|
[6] Calaque, D., Willwacher, T.: Triviality of the higher formality theorem.Proc. Amer. Math. Soc. 143 5181–5193, arxiv:1310.4605 http://arxiv.org/pdf/1310.4605 10.1090/proc/12670 |
| Reference:
|
[7] Costello, K., Gwilliam, O.: Factorization algebras in quantum field theory, Volume 2.available at http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf |
| Reference:
|
[8] Dolgushev, V., Tamarkin, D., Tsygan, B.: Proof of Swiss cheese version of Deligne’s conjecture.Int. Math. Res. Not. 4666–4746, arxiv:0904.2753 http://arxiv.org/pdf/0904.2753 |
| Reference:
|
[9] Etingof, P.: Calogero-Moser systems and representation theory.Zurich Lectures |
| Reference:
|
[10] Fresse, B.: Théorie des opérades de Koszul et homologie des algèbres de Poisson.Ann. Math. Blaise Pascal 13 237–312 10.5802/ambp.219 |
| Reference:
|
[11] Getzler, E., Jones, J.: A_(∞)-algebras and the cyclic bar complex.Illinois J. Math. 34 256–283 |
| Reference:
|
[12] Gerstenhaber, M., Voronov, A.: Homotopy G-algebras and moduli space operad.Int. Math. Res. Not. 141–153, arxiv:hep-th/9409063 http://arxiv.org/pdf/hep-th/9409063 |
| Reference:
|
[13] Gerstenhaber, M., Voronov, A.: Higher operations on the Hochschild complex.Func. Anal. and its App. 29 1–5 |
| Reference:
|
[14] Ginot, G.: Notes on factorization algebras, factorization homology and applications.Mathematical aspects of quantum field theories, Springer, 429–552, arxiv:1307.5213 http://arxiv.org/pdf/1307.5213 |
| Reference:
|
[15] Joyce, D., Safronov, P.: A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes.arxiv:1506.04024 http://arxiv.org/pdf/1506.04024 |
| Reference:
|
[16] Kontsevich, M.: Operads and Motives in Deformation Quantization.Lett. Math. Phys. 48 35–72, arxiv:math/9904055 http://arxiv.org/pdf/math/9904055 10.1023/A:1007555725247 |
| Reference:
|
[17] Kostant, B., Sternberg, S.: Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras.Ann. of Physics 176 49–113 10.1016/0003-4916(87)90178-3 |
| Reference:
|
[18] Lurie, J.: Higher Algebra.available at http://math.harvard.edu/~lurie/papers/HA.pdf |
| Reference:
|
[19] Loday, J-L., Vallette, B.: Algebraic operads.Grundlehren Math. Wiss. 346, Springer, Heidelberg |
| Reference:
|
[20] Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry.Rep. Math. Phys. 5 121–130 10.1016/0034-4877(74)90021-4 |
| Reference:
|
[21] McClure, J., Smith, J.: A solution of Deligne’s Hochschild cohomology conjecture.Contemp. Math. 293 153–193, arxiv:math/9910126 http://arxiv.org/pdf/math/9910126 10.1090/conm/293/04948 |
| Reference:
|
[22] Melani, V.: Poisson bivectors and Poisson brackets on affine derived stacks.Adv. in Math. 288 1097–1120, arxiv:1409.1863 http://arxiv.org/pdf/1409.1863 10.1016/j.aim.2015.11.008 |
| Reference:
|
[23] Melani, V., Safronov, P.: Derived coisotropic structures I: affine case.arxiv:1608.01482 http://arxiv.org/pdf/1608.01482 |
| Reference:
|
[24] Melani, V., Safronov, P.: Derived coisotropic structures II: stacks and quantization.arxiv:1704.03201 http://arxiv.org/pdf/1704.03201 |
| Reference:
|
[25] Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted symplectic structures.Publ. Math. IHES 117 271–328, arxiv:1111.3209 http://arxiv.org/pdf/1111.3209 10.1007/s10240-013-0054-1 |
| Reference:
|
[26] Safronov, P.: Quasi-Hamiltonian reduction via classical Chern–Simons theory.Adv. in Math. 287 733–773, arxiv:1311.6429 http://arxiv.org/pdf/1311.6429 |
| Reference:
|
[27] Safronov, P.: Braces and Poisson additivity.arxiv:1611.09668 http://arxiv.org/pdf/1611.09668 |
| Reference:
|
[28] Stasheff, J.: Homological reduction of constrained Poisson algebras.J. Diff. Geom. 45 221–240, arxiv:q-alg/9603021 http://arxiv.org/pdf/q-alg/9603021 |
| Reference:
|
[29] Thomas, J.: Kontsevich’s Swiss cheese conjecture.Geom. Top. 20 1–48, arxiv:1011.1635 http://arxiv.org/pdf/1011.1635 |
| Reference:
|
[30] Voronov, A.: The Swiss-cheese operad.Homotopy invariant algebraic structures, Baltimore, 1998, Contemp. Math. 239 365–373, arxiv:math/9807037 http://arxiv.org/pdf/math/9807037 10.1090/conm/239/03610 |
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