Bosonization of Feigin-Odesskii Poisson varieties.(English)Zbl 07972359
Summary: The derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Projective spaces with Feigin-Odesskii Poisson brackets are examples of such moduli spaces over complex elliptic curves [A. V. Odesskiĭ andB. L. Feigin, Funct. Anal. Appl. 23, No. 3, 207–214 (1989;Zbl 0713.17009); Transl. Math. Monogr. 185, 65–84 (1998;Zbl 0916.16014)]. By generalizing several results in our previous work [Adv. Math. 338, 991–1037 (2018;Zbl 1400.53070); Sel. Math., New Ser. 25, No. 3, Paper No. 42, 45 p. (2019;Zbl 1412.14008); J. Topol. 16, No. 4, 1389–1422 (2023;Zbl 1529.14004)], we construct a collection of auxiliary Poisson varieties equipped with Poisson morphisms to Feigin-Odesskii varieties. We call thembosonizations of Feigin-Odesskii varieties. These spaces appear as special cases of the moduli spaces ofchains, which we introduce. We show that the moduli space of chains admits a shifted Poisson structure when the base is a Calabi-Yau variety of an arbitrary dimension. Using bosonization spaces mapping to the zero loci of the Feigin-Odesskii varieties, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries. We also derive explicit formulas for the Poisson brackets on the bosonizations of the Feigin-Odesskii varieties associated with line bundles in a simplest nontrivial case.
MSC:
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 16E10 | Homological dimension in associative algebras |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16T05 | Hopf algebras and their applications |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 53D55 | Deformation quantization, star products |
| 81S10 | Geometry and quantization, symplectic methods |
Keywords:
shifted Poisson structure;moduli of complexes;Calabi-Yau curves;Feigin-Odesskii elliptic algebrasCite
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