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Geometric interpretation of Zhou’s explicit formula for the Witten–Kontsevich tau function

Abstract

Based on the work of Itzykson and Zuber on Kontsevich’s integrals, we give a geometric interpretation and a simple proof of Zhou’s explicit formula for the Witten–Kontsevich tau function. More precisely, we show that the numbers \(A_{m,n}^\mathrm{Zhou}\) defined by Zhou coincide with the affine coordinates for the point of the Sato Grassmannian corresponding to the Witten–Kontsevich tau function. Generating functions and new recursion relations for \(A_{m,n}^\mathrm{Zhou}\) are derived. Our formulation on matrix-valued affine coordinates and on tau functions remains valid for generic Grassmannian solutions of the KdV hierarchy. A by-product of our study indicates an interesting relation between the matrix-valued affine coordinates for the Witten–Kontsevich tau function and the V-matrices associated with the R-matrix of Witten’s 3-spin structures.

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Notes

  1. The rescaling is due to a different choice of the flow normalizations where our choice is more natural and has the advantage that all coefficients are rational; see (3.23).

  2. The standard Giambelli formula [14] says

    $$\begin{aligned} s_{(m_1,\dots , m_k |n_1,\dots ,n_k)}({\varvec{\theta }}) = \det _{1\le i,j\le k}(s_{(m_i|n_j)}({\varvec{\theta }})). \end{aligned}$$
  3. In [4], the notation for \(\rho \) is R. In this paper, the notation R is used for the R-matrix.

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Acknowledgements

We would like to thank Boris Dubrovin, Marco Bertola, and Jian Zhou for many helpful discussions and encouragements. F. B. wishes to thank John Harnad for introducing him to the subject of tau functions. D. Y. wishes to thank Youjin Zhang for his advises and helpful discussions. The work is partially supported by PRIN 2010-11 Grant “Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions” of the Italian Ministry of Universities and Researches, and by the Marie Curie IRSES project RIMMP.

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Balogh, F., Yang, D. Geometric interpretation of Zhou’s explicit formula for the Witten–Kontsevich tau function. Lett Math Phys 107, 1837–1857 (2017). https://doi.org/10.1007/s11005-017-0965-8

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  • DOI: https://doi.org/10.1007/s11005-017-0965-8

Keywords

  • Witten–Kontsevich tau function
  • Sato Grassmannian
  • Matrix-valued affine coordinates
  • R-matrix

Mathematics Subject Classification

  • 37K10
  • 53D45