[Paper Review] CFT approach to constraint operators for ($β$-deformed) hermitian one-matrix models

Rui Wang, Chun-Hong Zhang+2 more|arXiv (Cornell University)|Mar 28, 2022

Algebraic structures and combinatorial models4 citations

TL;DR

This paper develops a conformal field theory (CFT) approach to derive constraint operators for (β-deformed) hermitian one-matrix models using Heisenberg algebra generators. It constructs second-order total derivative operators associated with these constraints, proves a long-standing conjecture on Jack polynomial averages, and establishes connections between constraint operators and W-representations, enabling character expansions and superintegrability in Gaussian and β-deformed matrix models.

ABSTRACT

Since the ($β$-deformed) hermitian one-matrix models can be represented as the integrated conformal field theory (CFT) expectation values, we construct the operators in terms of the generators of the Heisenberg algebra such that the constraints can be derived by inserting the constructed operators into the integrated expectation values. We also obtain the second order total derivative operators associating with the derived constraint operators and analyze their properties. We explore the intrinsic connection between the derived constraint operators and $W$-representations of some matrix models. For the Gaussian hermitian one-matrix model in the external field and $β$-deformed $N imes N$ complex matrix model, we investigate the superintegrability and derive the corresponding character expansions from their $W$-representations. Moreover a conjectured formula for the averages of Jack polynomials in the literature is proved.

Motivation & Objective

• To develop a CFT-based framework for deriving constraint operators in (β-deformed) hermitian one-matrix models.
• To establish a connection between the derived constraint operators and W-representations of matrix models.
• To investigate superintegrability and character expansions via W-representations in Gaussian and β-deformed matrix models.
• To construct and analyze second-order total derivative operators associated with the constraint operators.
• To prove a conjectured formula for averages of Jack polynomials in the literature.

Proposed method

• Construct operators using generators of the Heisenberg algebra to derive constraints in integrated CFT expectation values.
• Derive second-order total derivative operators (denoted as ̄Wₙ) with respect to integration variables from the constraint operators.
• Apply rescaling variable transformations to link constraint operators to W-representations of Gaussian hermitian and N×N complex matrix models.
• Use W-representations to directly derive character expansions in terms of Schur functions and Jack polynomials.
• Construct extended operators Hₙ as combinations of Euler operators and ̄Wₙ, generalizing known operators like Laplace-Beltrami and Lassalle operators.
• Employ the Cauchy identity and properties of Jack polynomials to prove the conjectured average formula (3.77).

Experimental results

Research questions

• RQ1 How can constraint operators for (β-deformed) hermitian one-matrix models be systematically derived using CFT and Heisenberg algebra generators?
• RQ2 What is the intrinsic connection between the derived constraint operators and W-representations of matrix models?
• RQ3 How do the second-order total derivative operators ̄Wₙ relate to the eigenfunctions of many-body systems such as the A_{N-1}-Calogero model?
• RQ4 Can the character expansions of matrix models be directly derived from their W-representations?
• RQ5 Is the conjectured formula for averages of Jack polynomials in Ref. [54] valid, and can it be rigorously proven?

Key findings

• The paper proves the conjectured formula for averages of Jack polynomials: ⟨J_λ{p_k = ∑z_i^k}⟩ = J_λ{p_k = N} J_λ{p_k = N + β⁻¹ - 1} / J_λ{p_k = β⁻¹ δ_{k,1}}.
• The partition function of the β-deformed N×N complex matrix model is expressed as a sum over Jack polynomials with coefficients involving ratios of Jack polynomials evaluated at specific parameters.
• Two constraint operators in the β-deformed model are shown to be associated with W-representations of the Gaussian hermitian one-matrix model in an external field and the N×N complex matrix model.
• The ground state eigenfunction of the A_{N-1}-Calogero model (a power of the Vandermonde determinant) is annihilated by the total derivative operators ̄Wₙ.
• The extended operators Hₙ include known operators such as the Laplace-Beltrami and Lassalle operators (with exchange operator) as special cases.
• Superintegrability is derived directly from W-representations for the Gaussian hermitian one-matrix model in an external field and the β-deformed N×N complex matrix model.