On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

详细信息	查看全文 | 推荐本文 |

• 作者：A. Buryak^a ; ^b ; ^{a.y.buryak@uva.nl} ; ^{buryaksh@mail.ru} ; H. Posthuma^a ; ^{h.b.posthuma@uva.nl} ; S. Shadrin^a ; ^{s.shadrin@uva.nl}
• 关键词：Frobenius manifolds ; Integrable hierarchies ; Quasi-Miura transformations ; Dubrovin&ndash ; Zhang bracket ; Givental group action
• 刊名：Journal of Geometry and Physics
• 出版年：2012
• 期刊代码：29_03930440
• 类别：ph
• 出版时间：July, 2012
• 卷：62
• 期：7
• 页码：1639-1651
• 文件大小：266 K

摘要

In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy.

Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental-Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we give an alternative derivation using a deformation formula for the weak quasi-Miura transformation that relates our hierarchy of PDEs with its dispersionless limit.