Geometric interpretation of Zhou's explicit formula for the Witten-Kontsevich tau function

2014/12/14 by Ferenc Balogh, Di Yang, Balogh, Ferenc +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.1412.4419

openalex publication_date 2014/12/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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^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^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 to the R-matrix of Witten's 3-spin structures.

Citations

• Explicit Formula for Witten-Kontsevich Tau-Function

Cited by

• A Formula for Connected Bosonic \(n\)-Point Functions for the BKP Hierarchy

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