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

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 with the R -matrix of Witten’s 3-spin structures.