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On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions

Nikolay Borozenets, Eric T. Mortenson

TL;DR

The paper investigates string functions attached to admissible highest weight representations of the Kac–Moody algebra $A_1^{(1)}$, linking them to generalized parafermionic theories and mock modular phenomena. It develops a unified framework based on Hecke-type double sums and Appell functions to express $1/2$-level string functions in terms of Ramanujan’s mock theta functions and theta-corrected pieces, and proves mixed mock modular transformations on the full modular group. For negative admissible levels, it proves that string functions admit false theta expansions, with explicit treatments of the $(-1/2)$ and $(-2/3)$ cases, supported by a detailed structural theory (including recurrences and symmetric double-sum evaluations). Together, these results provide new modular- and theta-function–based representations of string functions, yielding concrete identities, recurrence relations, and expansions with potential applications in representation theory and mathematical physics.

Abstract

Kac and Wakimoto introduced the admissible highest weight representations in order to classify all modular invariant representations of the Kac--Moody algebras. For the Kac--Moody algebra $A_1^{(1)}$ the string functions of admissible representations are allowed to have certain rational levels and were realized by Ahn, Chung, and Tye as the characters of the generalized Fateev--Zamolodchikov parafermionic theories. For the $1/2$-level string functions, we present their mixed mock modular properties as well as elegant mock theta conjecture-like identities involving two mock theta functions from Ramanujan's Lost Notebook. In addition, we demonstrate that the negative level string functions can be evaluated in terms of false theta functions.

On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions

TL;DR

The paper investigates string functions attached to admissible highest weight representations of the Kac–Moody algebra , linking them to generalized parafermionic theories and mock modular phenomena. It develops a unified framework based on Hecke-type double sums and Appell functions to express -level string functions in terms of Ramanujan’s mock theta functions and theta-corrected pieces, and proves mixed mock modular transformations on the full modular group. For negative admissible levels, it proves that string functions admit false theta expansions, with explicit treatments of the and cases, supported by a detailed structural theory (including recurrences and symmetric double-sum evaluations). Together, these results provide new modular- and theta-function–based representations of string functions, yielding concrete identities, recurrence relations, and expansions with potential applications in representation theory and mathematical physics.

Abstract

Kac and Wakimoto introduced the admissible highest weight representations in order to classify all modular invariant representations of the Kac--Moody algebras. For the Kac--Moody algebra the string functions of admissible representations are allowed to have certain rational levels and were realized by Ahn, Chung, and Tye as the characters of the generalized Fateev--Zamolodchikov parafermionic theories. For the -level string functions, we present their mixed mock modular properties as well as elegant mock theta conjecture-like identities involving two mock theta functions from Ramanujan's Lost Notebook. In addition, we demonstrate that the negative level string functions can be evaluated in terms of false theta functions.
Paper Structure (24 sections, 32 theorems, 251 equations)

This paper contains 24 sections, 32 theorems, 251 equations.

Table of Contents

  1. The introduction
  2. String functions of admissible highest weight representations
  3. Computation of string functions
  4. String functions as the characters of generalized parafermionic theories
  5. String functions in Hecke-type double-sum form
  6. Mock theta functions and Appell functions
  7. Hecke-type double-sums and Appell functions
  8. Hecke-type double-sums and false theta functions
  9. Main results
  10. The $1/2$-level string functions
  11. The string functions of negative admissible level
  12. Overview of the paper
  13. Properties of theta functions, Appell functions, and Hecke-type double-sums
  14. Technical results for theta functions
  15. The Hecke-type double-sum form of the string functions
  16. ...and 9 more sections

Key Result

Proposition 1.2

Let $p'\geq 2$, $p\geq 1$ be coprime integers, $0\leq \ell \leq p'-2$ and $m\in 2\mathbb{Z} +\ell$. We have

Theorems & Definitions (54)

  • Definition 1.1
  • Proposition 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • ...and 44 more