Complex Maths Tamed: Infinite Sums Simplified to Finite Calculations for Key Equations

Researchers are investigating explicit forms for string functions and branching coefficients within Kac–Moody algebras, a significant challenge in mathematical physics. Stepan Konenkov and Eric T. Mortenson, alongside their colleagues, present new polar-finite forms of generalized Euler identities specifically for A₁⁽¹⁾A₁⁽¹⁾-string functions, advancing understanding of admissible-level string functions for the affine Kac–Moody algebra. This work is particularly noteworthy as it reduces infinite sums within these identities to finite sums incorporating theta functions and, crucially, introduces novel finite sums involving mixed mock Hecke-type double-sums. For levels 1, 2, and 3, these double-sums yield families of identities resembling the celebrated mock theta conjecture, utilising Ramanujan’s second and third-order mock theta functions to achieve these results.

Deriving explicit string functions via quasi-periodicity at non-integer admissible levels offers novel algorithmic possibilities

Scientists have achieved a significant breakthrough in understanding string functions related to Kac-Moody algebras, potentially unlocking new avenues in theoretical physics. This work centers on deriving explicit formulas for these complex functions at admissible levels of 1/2, 1/3, and 2/3, a feat previously challenging for researchers in the field.
The research team successfully applied a technique called quasi-periodicity to a generalized Euler identity, revealing hidden mathematical structures within string functions and establishing connections to well-known mathematical objects. This advancement builds upon decades of work concerning Kac-Moody algebras and their applications to areas like conformal field theory and string theory.

By focusing on admissible levels, specifically the non-integer values of 1/2, 1/3, and 2/3, the study demonstrates a powerful method for tackling previously intractable problems. The application of quasi-periodicity transformed infinite sums within the Euler identity into finite sums, simplifying the calculations and revealing underlying patterns.

This reduction is crucial for obtaining explicit, usable formulas for string functions. Crucially, the resulting formulas express these string functions in terms of Ramanujan’s mock theta functions, a family of special functions with deep connections to number theory and physics. The research utilizes Ramanujan’s second-order mock theta function μ2(q) and third-order mock theta functions f3(q), ω3(q), ψ3(q), and χ3(q) to express these relationships.

This connection not only provides a new lens through which to view string functions but also suggests potential links to other areas of mathematical physics where mock theta functions appear. The derived formulas are not merely theoretical curiosities; they yield families of mock theta conjecture-like identities for symmetric Hecke-type double-sums.

These identities represent a significant step forward in understanding the modular properties of string functions and could prove invaluable in solving problems within conformal field theory and string theory, potentially aiding in the development of more accurate models of the universe. The successful application of this method to levels 1/2, 1/3, and 2/3 suggests its broader applicability to other admissible levels, opening up exciting possibilities for future research.

Derivation of finite sum formulas for A(1)1 string functions via quasi-periodic identities offers a novel approach to symbolic computation

Researchers applied quasi-periodicity to a generalized Euler identity originating from the work of Schilling and Warnaar to derive explicit formulas for string functions related to Kac-Moody algebras. This methodology was specifically implemented for the affine Kac-Moody algebra A(1)1, focusing on admissible levels of 1/2, 1/3, and 2/3.

The study built upon earlier investigations by Kac, Peterson, and Wakimoto concerning string functions and branching coefficients, extending their modular properties and calculations to these fractional levels. Central to the approach was the utilization of a generalized Euler identity, which, for integral-level string functions, simplifies to a finite sum due to classical periodicity.

Applying quasi-periodicity at admissible levels similarly reduced the infinite sum of string functions to a finite sum, but with the addition of a further finite sum involving mock Hecke-type double-sums. These double-sums, denoted as Ψi(q), were found to exhibit properties akin to Ramanujan’s mock theta conjectures for symmetric Hecke-type double-sums.

The evaluation of these Ψi(q) terms proved crucial, with expressions ultimately formulated using Ramanujan’s second-order and third-order mock theta functions, as documented in his correspondence with G.H. Hardy and his lost notebook. Researchers leveraged a polar-finite decomposition of admissible characters, a technique previously introduced by Borozenets and Mortenson, to uncover these mock theta conjecture-like identities.

This decomposition separates a meromorphic Jacobi form into finite and polar parts, enabling the extraction of relevant string function properties. The methodology demonstrated applicability to non-integer levels, specifically 1/2, 1/3, and 2/3, showcasing the method’s capacity to handle these specific values and generate corresponding mock theta identities. This work expands upon previous research focusing on negative admissible levels, concentrating instead on positive admissible levels where less is currently known.

String function formulas and mock theta function relationships at admissible levels remain largely unexplored

Explicit formulas for string functions related to Kac-Moody algebras have been derived for admissible levels of 1/2, 1/3, and 2/3. This work applies the notion of quasi-periodicity to a generalized Euler identity, resulting in reduced finite sums of string functions and the emergence of additional finite sums involving mock Hecke-type double-sums.

The expressions obtained utilize Ramanujan’s second-order mock theta function μ2(q) and third-order mock theta functions f3(q), ω3(q), ψ3(q), and χ3(q). Specifically, the research demonstrates the method’s applicability at levels 1/2, 1/3, and 2/3, yielding families of mock theta conjecture-like identities for symmetric Hecke-type double-sums.

Evaluating the Ψi(q)’s, the study reveals connections between these string functions and well-established mock theta functions. These calculations extend previous findings regarding admissible-level string functions for the affine Kac-Moody algebra A(1)1. For string functions at level 1/2, the research establishes relationships involving the modular function μ2(q).

At level 1/3, string functions are expressed in terms of the functions f3(q) and ω3(q), while at level 2/3, the functions f3(q), ω3(q), ψ3(q), and χ3(q) appear in the derived formulas. The quasi-periodic relations employed extend Zagier, Zwegers’ analysis to admissible characters, providing a new perspective on the polar-finite decomposition of these characters.

The generalized Euler identity is further refined through the quasi-periodicity approach, yielding new polar-finite forms for A(1)1-string functions. This work builds upon previous investigations into mock modularity, quasi-periodicity, and polar-finite decompositions, offering new insights into the mathematical structure of string functions and their connections to theoretical physics. The resulting identities contribute to a deeper understanding of conformal field theory and string theory.

Mock theta function formulas for Kac-Moody string functions at fractional level offer new insights into their structure

Explicit formulas for string functions associated with Kac-Moody algebras have been derived for admissible levels of 1/2, 1/3, and 2/3. This was achieved through the application of quasi-periodicity to a generalized Euler identity, resulting in expressions involving Ramanujan’s mock theta functions. The resulting formulas represent a significant advance in the mathematical understanding of these complex functions and their properties.

These findings extend to non-integer levels, demonstrating a method applicable to values beyond the traditionally studied integer levels. The derived expressions utilize established mathematical objects, namely Ramanujan’s second and third-order mock theta functions, connecting this work to a rich history of mathematical investigation.

This connection provides a framework for further exploration and potential applications in theoretical physics. The authors acknowledge that their work focuses on evaluating specific components within the broader context of string functions and branching coefficients. Future research may explore the implications of these explicit formulas for conformal field theory and string theory, potentially aiding in the resolution of outstanding problems in these areas. Further investigation into the relationships between these string functions and other areas of mathematics, such as modular forms and Hecke-type double sums, also represents a promising avenue for continued study.