[Paper Review] Supersymmetric Analogs of the Gordon-Andrews Identities, and Related TBA Systems

Ezer Melzer|ArXiv.org|Dec 18, 1994

Algebraic structures and combinatorial models4 references22 citations

TL;DR

This paper proposes supersymmetric analogs of the Gordon-Andrews identities for the N=1 superconformal minimal models SM(2,4k), presenting two distinct fermionic forms for their characters. Using thermodynamic Bethe Ansatz (TBA) systems, the authors identify these forms with perturbations by the $φ_{1,3}^{\text{top}}$ and $φ_{1,5}^{\text{bot}}$ fields, establishing a correspondence between TBA systems and perturbed super-CFTs, with explicit central charges and conformal dimensions derived for key cases.

ABSTRACT

The Gordon-Andrews identities, which generalize the Rogers-Ramanujan-Schur identities, provide product and fermionic forms for the characters of the minimal conformal field theories (CFTs) M(2,2k+1). We discuss/conjecture identities of a similar type, providing two different fermionic forms for the characters of the models SM(2,4k) in the minimal series of N=1 super-CFTs. These two forms are related to two families of thermodynamic Bethe Ansatz (TBA) systems, which are argued to be associated with the $\hatϕ_{1,3}^{ m top}$- and $\hatϕ_{1,5}^{ m bot}$-perturbations of the models SM(2,4k). Certain other q-series identities and TBA systems are also discussed, as well as a possible representation-theoretical consequence of our results, based on Andrews's generalization of the Gollnitz-Gordon theorem.

Motivation & Objective

• To extend the Gordon-Andrews identities—originally for non-supersymmetric minimal CFTs M(2,2k+1)—to their N=1 superconformal analogs, specifically for the SM(2,4k) models.
• To derive two distinct fermionic forms for the characters of SM(2,4k) super-CFTs, analogous to the fermionic sum forms in the non-supersymmetric case.
• To associate these fermionic forms with two families of thermodynamic Bethe Ansatz (TBA) systems, identifying them with specific perturbations of the super-CFTs.
• To establish a correspondence between the TBA systems and perturbations by the $φ_{1,3}^{\text{top}}$ and $φ_{1,5}^{\text{bot}}$ fields in SM(2,4k), linking TBA structures to physical field theory perturbations.
• To explore the representation-theoretic implications of these identities, particularly through Andrews’s generalization of the Göllnitz-Gordon theorem.

Proposed method

• The authors derive two fermionic sum forms for the characters of SM(2,4k) super-CFTs, analogous to the Gordon-Andrews identities, using q-series identities and generalized partition functions.
• They construct TBA systems associated with the characters, identifying them with the thermodynamic limit of integrable quantum field theories perturbed by specific primary fields.
• The TBA systems are linked to the $φ_{1,3}^{\text{top}}$ and $φ_{1,5}^{\text{bot}}$ perturbations of SM(2,4k) via the effective central charge and conformal dimension of the perturbing field.
• The central charge and conformal dimension of the perturbed CFT are computed using the periodicity of the TBA system and the three-point function condition, distinguishing cases based on whether the three-point function of the perturbing field with the minimal model fields vanishes.
• The method involves identifying TBA systems with known minimal models: for example, $(T_1 \diamond T_k)_1$ is shown to correspond to ${\cal M}(k+2,2k+2)$ for odd $k$ and ${\cal M}(k+1,2k+4)$ for even $k$, with matching central charges and dimensions.
• The framework is validated by matching the TBA solutions to known finite-volume ground state energies from conformal perturbation theory, particularly in special cases like $k=2$ and $n=1$.

Experimental results

Research questions

• RQ1 Can the Gordon-Andrews identities, which relate product and fermionic forms for characters of M(2,2k+1), be generalized to the N=1 superconformal minimal models SM(2,4k)?
• RQ2 What are the two distinct fermionic sum forms for the characters of SM(2,4k), and how do they relate to different TBA systems?
• RQ3 Which perturbations of SM(2,4k) correspond to the two families of TBA systems identified in the paper, and how are their central charges and conformal dimensions determined?
• RQ4 How can the TBA systems for SM(2,4k) be matched to known minimal models via the effective central charge and perturbing field analysis?
• RQ5 What is the representation-theoretic significance of the proposed identities, particularly in relation to Andrews’s generalization of the Göllnitz-Gordon theorem?

Key findings

• The paper proposes two fermionic sum forms for the characters of SM(2,4k), analogous to the Gordon-Andrews identities, with one form corresponding to the $φ_{1,3}^{\text{top}}$-perturbation and the other to the $φ_{1,5}^{\text{bot}}$-perturbation.
• For the case $(T_1 \diamond T_1)_1$, the effective central charge is $\tilde{c} = 1$, corresponding to the free Dirac point on the c=1 Gaussian line, with a perturbing field of dimension $\Delta_p = \frac{1}{2}$.
• The system $(T_1 \diamond T_2)_1$ is identified with ${\cal M}(3,8) + \phi_{1,5}$, and is equivalent to the folded TBA system for ${\cal SM}(2,8) + \hat{\phi}_{1,3}^{\text{top}}$, confirming the correspondence between TBA systems and perturbed super-CFTs.
• For $(T_1 \diamond T_k)_1$ with odd $k$, the effective central charge is $\tilde{c} = 1 - \frac{6}{2(k+1)(k+2)}$, identifying the unperturbed CFT as ${\cal M}(k+2,2k+2)$, and the perturbing field as $\phi_{2,1}^{(k+2,2k+2)}$.
• For $(T_1 \diamond T_k)_2$ with odd $k \geq 3$, the effective central charge is $\tilde{c} = \frac{3}{5} + \left(1 - \frac{6}{k(k+2)}\right)$, identifying the unperturbed CFT as ${\cal M}(3,5) \otimes {\cal M}(k,k+2)$, with a perturbing field of dimension $\Delta_p = \frac{2k+1}{2(k+2)}$.
• The TBA system $(A_2 \diamond T_2)_1$ is identified as corresponding to the most relevant perturbation of the second model in the N=2 minimal series, and is shown to yield the fermionic forms for ${\cal SM}(3,7)$ when expanded via the identity in [3].