Nested ansatz method for Baker-Akhiezer functions

TL;DR

The paper develops a nested ansatz to express Baker-Akhiezer functions (BAFs) for $N$ variables in terms of $(N-1)$-variable BAFs, deriving NS-type, directly quantized structures from linear equations. It first validates the method for non-twisted BAFs ($a=1$), recovering the Noumi-Shiraishi expansion via explicit coefficient factorization with symmetric $q$-numbers, and then extends to the first non-trivial twisted case ($N=3$, $a=2$), where the approach yields manifest NS-like coefficients and reveals a structured, root-centered factorization. The work introduces $R$-operators to organize the nested coefficients and shows that the general $ ilde{oldsymbol{ ho}}_{m,n_1,n_2}$ obeys a polynomial in these operators divided by a simple $ ext{Denom}$, with a dequantization path to symmetric quantities. Overall, the nested ansatz provides a principled route to fix gauge ambiguities in twisted BAFs and to realize direct quantization consistent with NS/Macdonald structures, offering a framework for broader generalizations to other root systems and DIM/twisted settings.

Abstract

We explain that the logic behind the derivation of the Noumi-Shiraishi function can be applied directly to the Baker-Akhiezer function (BAF). This amounts to changing an ansatz for BAF to a nested one, where the BAF of N + 1 variables is recursively expressed as a sum over BAFs of N variables. This may be seen as a generalization of symmetrization trick from [1], but for the generally non-symmetric BAF. We demonstrate that, for usual non-twisted (a = 1) BAFs, this method correctly reproduces the Noumi-Shiraishi formula directly from linear equations, resolving the ambiguity related to non-simple roots. For the first non-trivial twisted case (N = 3, a = 2) this method also fixes this ambiguity, moreover, answers for the first few layers of coefficients are in the form of direct quantization of [1].