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Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations

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Abstract

This series of papers is devoted to the study of Virasoro symmetries of deformations of the Principal Hierarchy associated with a semisimple Frobenius manifold. The main tool we use is a generalization of the bihamiltonian cohomology called the variational bihamiltonian cohomology. In the present paper, we give its definition and compute the associated cohomology groups that will be used in our study of Virasoro symmetries. As an application of the variational bihamiltonian cohomology theory, we classify conformal bihamiltonian structures with semisimple hydrodynamic limits.

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Acknowledgements

The authors sincerely thank the anonymous referee for very helpful suggestions to improve the presentation of the paper. This work is supported by NSFC No. 12171268, No. 11725104 and No. 11771238.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Si-Qi Liu, Zhe Wang & Youjin Zhang

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  1. Si-Qi Liu
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  2. Zhe Wang
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  3. Youjin Zhang
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Correspondence to Youjin Zhang.

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Communicated by Y. Kawahigashi.

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Liu, SQ., Wang, Z. & Zhang, Y. Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations. Commun. Math. Phys. 401, 985–1031 (2023). https://doi.org/10.1007/s00220-023-04658-0

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