Baker-Akhiezer Function (BAF) Overview

• The Baker-Akhiezer Function (BAF) is a highly structured multivariable function central to quantum integrable systems, algebraic geometry, and representation theory.
• Its construction via the Noumi-Shiraishi hypergeometric series unifies symmetric Macdonald polynomials with explicit determinantal and recursive formulations under various deformations.
• BAF plays a pivotal role in AGT correspondences and gauge theory, facilitating insights into operator eigenfunctions, dualities, and elliptic deformations.

The Baker-Akhiezer Function (BAF) is a highly structured multivariable function central to the theory of integrable quantum many-body systems, algebraic geometry, and representation theory. In its modern quantum incarnation—especially in connection to the Ruijsenaars-Schneider and Macdonald operators—the BAF is most transparently realized as a specialization or reduction of the Noumi-Shiraishi (NS) hypergeometric-type power series, which simultaneously interpolates between symmetric Macdonald polynomials and the BA function. Its construction, analytic properties, operator eigenfunction structure, and deformations (twisted, elliptic, and double-elliptic) are deeply entangled with the Ding–Iohara–Miki (DIM) algebra, AGT correspondences, and the representation theory of elliptic quantum algebras.

1. Foundational Definitions and Explicit Formulations

The archetype of the quantum BAF is the nonsymmetric eigenfunction of the trigonometric Ruijsenaars-Schneider operator, constructed as a quasi-polynomial in $N$ variables, parameterized by a set of complex spectral parameters. Let $N$ denote the number of variables, with $x_i = q^{z_i}$ and spectral data $\lambda = (\lambda_1, \dots, \lambda_N)$. The BAF admits the NS power-series representation: $$\Psi^{\text{BAF}}_m(\mathbf z, \lambda) = q^{\mathbf z \cdot \lambda} \, t^{\mathbf z \cdot \rho} \!\! \sum_{\{k_{ij} \ge0\}} \!\! \psi(\lambda, \{k_{ij}\}; q, t) \prod_{i<j} (q^{z_j - z_i})^{k_{ij}}$$ where $\rho$ is the $A_{N-1}$ Weyl vector, and the sum extends over all strictly upper-triangular collections of nonnegative integers $k_{ij}$. The coefficient $\psi(\lambda, \{k_{ij}\}; q, t)$ encodes explicit root system data and is a finite product of rational functions of $q, t$, and the $\lambda_i$ (Mironov et al., 2024).

In the "BA regime," $t = q^{-m}$ with $m \in \mathbb{Z}_{\ge0}$, or for general twisted cases (see $a>1$ below), the BAF becomes a finite quasi-polynomial, and, up to normalization, is uniquely characterized by translation (or "periodicity") constraints: $$\Psi^{[N]}_m(\ldots, z_k + j, \ldots; \lambda) = \Psi^{[N]}_m(\ldots, z_k, \ldots; \lambda), \quad j = 1, \ldots, m$$ for each $k$ (Mironov et al., 2024, Mironov et al., 24 Jan 2026). This system of linear equations has a unique (up to scaling) solution, and Cramer's rule provides a determinantal formula for the expansion coefficients (Mironov et al., 2 Apr 2025).

The BAF can also be obtained recursively via a nested ansatz: $$\Psi^{[N]}_m(x_1, \ldots, x_N \mid \lambda_1, \ldots, \lambda_N) = x_1^{\lambda_1+\frac{m}{2}(N-1)} \sum_{k_{1,i}=0}^m \tilde\psi_{m;\{k_{1,i}\}}(\lambda) \, x_1^{\sum_{i=2}^N k_{1,i}} \Psi^{[N-1]}_m(\ldots)$$ resolving non-simple root ambiguities and producing a fully gauge-fixed explicit expansion (Mironov et al., 24 Jan 2026).

2. Triad Structure: BAF, Macdonald Polynomials, and Noumi-Shiraishi Series

The NS power series defines a "triad" unifying the BAF, Macdonald polynomials, and itself:

• BAF: The specialization $t = q^{-m}$ yields the BAF as a truncating sum (non-symmetric, subject to periodicity constraints).
• Macdonald Polynomials: Specializing the spectral parameters $\lambda$ to partition data and related loci ($\lambda = \mu - \rho \log_t q$) collapses the series to symmetric Macdonald polynomials $P_\mu(x;q,t)$ (Mironov et al., 2024, Mironov et al., 2024).
• NS Function: For generic $(q,t)$ and $\lambda \in \mathbb{C}^N$, one has a non-symmetric function that interpolates between these extremes.

This triad is algebraically unified via the representation theory of the DIM algebra and can be connected to physical models (e.g., wavefunctions of quantum integrable systems) and to gauge theory partition functions in the AGT and "network models" context (Mironov et al., 10 Mar 2025, Mironov et al., 2024).

3. Operator Structure and Eigenvalue Problems

The BAF is a common eigenfunction for families of commuting quantum difference operators:

• Trigonometric Case: The trigonometric Ruijsenaars-Macdonald operator,

$$\widehat H_{\rm MR} = \sum_{i=1}^N \prod_{j \ne i} \frac{t x_i - x_j}{x_i - x_j} T_{q,x_i}$$

where $T_{q,x_i}$ is the $q$-shift operator. The BAF $\Psi_m$ satisfies

$$\widehat H_{\rm MR} \Psi_m = \left( \sum_{i=1}^N q^{\lambda_i} \right) \Psi_m$$

(Mironov et al., 2024).

• Elliptic and Bi-elliptic Cases: In the elliptic setting, the BAF arises as an eigenfunction of the elliptic lift of these operators, including Ruijsenaars-Schneider (eRS) and Koroteev-Shakirov (KS) type Hamiltonians. The elliptic operators involve theta-functions and additional elliptic moduli (Mironov et al., 2021, Awata et al., 2020, Mironov et al., 2024). In the double-elliptic (Dell) case, the elliptic BAF is conjectured to be a solution to non-stationary Dell-type difference equations (Awata et al., 2020).

In particular, the duality between $x$ (coordinate) and $\lambda$ or $y$ (spectral) variables is reflected algebraically: the BAF is simultaneously an eigenfunction of two commuting sets of operators (dual or bispectral property) (Mironov et al., 2021, Mironov et al., 10 Mar 2025).

4. Twisted Generalizations and Determinantal Solutions

For integer rays $a>1$ in the DIM algebra context, one obtains twisted BAFs, which are solutions to overdetermined linear systems ("Chalykh equations"): $$\sum_{k=0}^m M^{(a)}_{l,k} u^{(a)}_{m,k} = 0, \qquad l = 1, \ldots, am$$ where $M^{(a)}$ is a matrix built from cyclic and $q$-shifted parameters, and the solution admits a Cramer's rule expression (Mironov et al., 2 Apr 2025). In these cases, the determinant minors undergo abrupt phase changes as the degree index varies, leading to partial (but not full) factorization (Mironov et al., 2 Apr 2025). The nested ansatz method uniquely resolves ambiguities in the expansion for $a>1$ (e.g., $N=3$, $a=2$), providing direct-quantized forms for the coefficients in the expansion (Mironov et al., 24 Jan 2026).

5. Elliptic and Bi-Elliptic Deformations

Elliptic deformation of the BAF is achieved by replacing $q$-Pochhammer symbols in the NS series with elliptic analogues (theta or elliptic Gamma functions), yielding an ELS (elliptic-lift Shiraishi) function. The associated operator theory moves to elliptic Ruijsenaars-Schneider-type difference operators (Mironov et al., 2021, Saito, 2013). This construction admits an additional deformation parameter (second elliptic nome), essential for describing generic eigenfunctions of double-elliptic quantum integrable systems (Awata et al., 2020, Mironov et al., 10 Mar 2025).

At specific loci, the elliptic BAF truncates to elliptic analogues of Macdonald and BA polynomials, preserving certain dualities and symmetries under specialization, but loses properties like full self-duality or simple orthogonality, as observed in the non-elliptic case (Awata et al., 2020, Mironov et al., 2024).

6. Algebraic and Physical Contexts: DIM Algebra, AGT Correspondence, and Gauge Theory

The deep appearance of the BAF is intertwined with the representation theory of the (elliptic) Ding–Iohara–Miki algebra (DIM), whose Drinfeld current intertwiners encode the combinatorics of the NS/BAF expansion (Mironov et al., 10 Mar 2025, Saito, 2013). The vertex operator and screening formalism naturally produces both symmetric and non-symmetric (BA) eigenfunctions as correlation functions or matrix elements. The BAF is realized as the partition function of surface or orbifold defects in 5d and 6d supersymmetric gauge theories, particularly in circular quiver settings (Mironov et al., 29 Dec 2025, Awata et al., 2020). In these contexts, the BAF encodes the defect partition function, is expressible as a sum over colored Young diagrams with Nekrasov factors, and precisely matches dual conformal blocks calculated via Dotsenko–Fateev integrals.

The BAF and its deformations are also essential objects in the context of AGT-type correspondences, with the functions providing the bridge between free-field (screening) constructions of conformal blocks and explicit gauge-theory instanton partition functions (via surface defect insertions) (Mironov et al., 29 Dec 2025).

7. Analytic Properties, Dualities, and Structural Symmetries

The BAF, through its NS/ELS origin, enjoys rich dualities and symmetries:

• Bispectral duality: Interchange between coordinate and spectral variables.
• Poincaré duality: $t \leftrightarrow q/t$, exact in the non-elliptic limit, but broken in the elliptic case (Mironov et al., 2024).
• Spectral (P-Q) duality: Exchange between $x, p$ and $y, w$ in the bi-elliptic (Dell) lift (Awata et al., 2020).
• Branch structure and phase transitions: Determinantal formulas for the twisted BAF display abrupt changes ("jumps") as expansion indices cross certain loci (Mironov et al., 2 Apr 2025).

Symmetric and bi-orthogonal scalar products exist in the classical theory (e.g., Macdonald power-sum product and contour product), but in the deformed (elliptic) theory, only bi-orthogonality persists, and no single orthonormal basis survives (Mironov et al., 2024, Mironov et al., 2021). Relations to explicit integrations (Dotsenko–Fateev integrals), analytic continuation, and various limiting processes (classical, perturbative, Nekrasov–Shatashvili) further enrich the analytic landscape.

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Key references: (Mironov et al., 2024, Mironov et al., 24 Jan 2026, Mironov et al., 2 Apr 2025, Mironov et al., 10 Mar 2025, Mironov et al., 2024, Mironov et al., 2021, Mironov et al., 29 Dec 2025, Awata et al., 2020, Saito, 2013).