[Paper Review] Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator

Giovanni Felder, L. Stevens+1 more|ArXiv.org|Mar 6, 2002

Algebraic structures and combinatorial models14 references3 citations

TL;DR

This paper derives explicit formulas for the modular S-transformation action on conformal blocks of the $\mathfrak{sl}_2$ WZW model on elliptic curves using elliptic hypergeometric integrals. It establishes a direct link between the matrix elements of the modular group and values of $A_1$-Macdonald polynomials at roots of unity, revealing that the shift operator governs the recursive structure of these polynomials via Stokes' theorem applied to the integrals. The key result is a closed-form expression for the S-matrix in terms of renormalized trace functions of quantum group intertwiners.

ABSTRACT

We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl_2 type.

Motivation & Objective

• To compute the projective action of the modular group $\mathrm{SL}(2,\mathbb{Z})$ on the space of conformal blocks for the $\mathfrak{sl}_2$ WZW model on elliptic curves.
• To relate the matrix elements of the $S$-transformation $\tau \mapsto -1/\tau$ to values of $A_1$-Macdonald polynomials evaluated at roots of unity.
• To provide an explicit basis of elliptic hypergeometric integral solutions for which the modular representation matrices are computed directly, not just up to conjugacy.
• To clarify the role of the shift operator in the recursive construction of Macdonald polynomials via the Stokes theorem applied to the integral representations.

Proposed method

• The space of conformal blocks is identified with solutions to the KZB heat equation on the torus, characterized by periodicity, quasi-periodicity, and holonomy conditions.
• Elliptic hypergeometric integrals are used as explicit integral representations of the conformal blocks, providing a basis for the solution space.
• The modular $S$-transformation is computed by analyzing the monodromy of these integral solutions under covering transformations of the universal cover of the moduli space of elliptic curves.
• The matrix elements of the $S$-transformation are expressed in terms of renormalized trace functions $\Psi^{(k)}(q^{-1}, \nu, \mu)$ associated with quantum group intertwiners of $U_q(\mathfrak{sl}_2)$ at roots of unity.
• A recursive procedure based on Stokes' theorem is applied to the integrals, which reproduces the action of the shift operator on Macdonald polynomials.
• The identification of $f_{m,n}^{(k)}$ with $\Psi^{(k)}$ functions allows the derivation of a closed-form expression for the $S$-matrix in terms of Macdonald polynomial values at roots of unity.

Experimental results

Research questions

• RQ1 How does the modular group $\mathrm{SL}(2,\mathbb{Z})$ act projectively on the space of conformal blocks for the $\mathfrak{sl}_2$ WZW model on elliptic curves?
• RQ2 What is the precise relation between the matrix elements of the $S$-transformation and values of $A_1$-Macdonald polynomials at roots of unity?
• RQ3 How does the shift operator emerge from the recursive structure of elliptic hypergeometric integrals via Stokes' theorem?
• RQ4 Can the $S$-matrix be computed explicitly in a canonical basis of integral solutions, rather than up to conjugacy?
• RQ5 What is the connection between the trace functions of quantum group intertwiners and the modular representation of conformal blocks?

Key findings

• The matrix elements of the $S$-transformation are given explicitly by a formula involving $\Psi^{(k)}$ functions, which are renormalized traces of $U_q(\mathfrak{sl}_2)$ intertwiners at roots of unity.
• The $S$-matrix is expressed as a sum over $m$ with coefficients involving $q^{pn - km - k(k+1)}$, binomial coefficients $\begin{bmatrix}p\ k\end{bmatrix}^{-1}$, and differences of $\Psi^{(k)}$ functions.
• The shift operator acts on $\Psi^{(k)}(q^{-1}, m, n)$ as $D \Psi^{(k)} = q^{-k-1} \Psi^{(k+1)}$, establishing a recursive structure identical to that of Macdonald polynomials.
• The values of $A_1$-Macdonald polynomials at roots of unity are shown to coincide with specific combinations of $\Psi^{(k)}$ functions, confirming a special case of Kirillov’s theorem.
• The recursive construction of Macdonald polynomials from Schur functions via the shift operator is realized geometrically through the application of Stokes’ theorem to the elliptic hypergeometric integrals.
• The $S$-matrix formula is derived via an explicit identification of the integral basis $f_{m,n}^{(k)}$ with the renormalized trace functions $\Psi^{(k)}$, leading to a closed-form expression for the modular transformation.