Researchers Confirm 2025 Signature Conjecture Using Dedekind Sums

Scientists have confirmed a long-standing conjecture regarding the signature of a genus 2 surface in topological quantum field theory (TQFT), resolving a problem initially proposed by Marché and Masbaum in 2025. Yuya Murakami from RIKEN iTHENTIC demonstrates this through a novel analysis of generalized Dedekind sums linked to general forms. Specifically, these generalized Dedekind sums, when applied to Eisenstein series, reveal trigonometric expressions that precisely match the formula for the TQFT signature. This research significantly advances our understanding of the mathematical structures underlying TQFT by establishing a connection between seemingly disparate areas of mathematics, TQFT, Dedekind sums, and Eichler integrals, and expressing both as radial limits of the latter.

Can the precise signature of a complex mathematical object, a genus 2 surface, be definitively calculated using established formulas. Calculations now confirm a 2025 conjecture by Marché and Masbaum, demonstrating a link between topological quantum field theory and generalised Dedekind sums. This connection allows both the surface’s signature and these sums to be expressed using simpler, trigonometric expressions.

Scientists have uncovered a precise connection between the signatures arising from SU-TQFT, a type of topological quantum field theory, and generalised Dedekind sums, confirming a conjecture proposed by Marché and Masbaum in 2025. This advancement centres on establishing a ‘quantum modularity’ for both these mathematical objects, revealing a surprising relationship between areas of mathematics seemingly disparate in origin.

The research builds upon the idea that signatures of SU-TQFT, which refine dimensions of conformal blocks and relate to knot theory, can be expressed using generalised Dedekind sums linked to modular forms. For a closed surface of genus g, and a rational number x, the associated signature σg(x) plays a key role. While signatures for genus 0 and 1 remain constant, the first non-trivial case arises at genus 2, where Marché and Masbaum predicted a specific quantum modularity property for σ2(x).

Now, this conjecture has been proven, demonstrating that σ2(x) behaves in a predictable way as a parameter approaches infinity. This result not only validates existing theoretical predictions but also opens avenues for exploring deeper connections between number theory and low-dimensional topology. Beyond the purely mathematical implications, understanding these signatures has potential relevance to areas like quantum computation and the study of exotic materials, where topological properties are increasingly important.

The established link between SU-TQFT signatures and Dedekind sums provides a new set of tools for investigating these complex systems. This work lies in demonstrating a precise formula relating these signatures to generalised Dedekind sums, extending previous results and providing a more general framework. Starting from a trigonometric formula established by Marché and Masbaum, researchers derived an explicit relationship, expressed as a limit of Eichler integrals, functions important in the study of modular forms.

This allowed them to prove the conjecture regarding the quantum modularity of σ2(x). Also, the study introduces a completed twisted L-function associated with modular forms, offering a new perspective on generalised Dedekind sums and their properties. The approach is not limited to a single proof; the researchers present three distinct demonstrations of their central theorem.

One relies on integral representations of the L-function, while the others use asymptotic behaviour and modular transformations of Eichler integrals. By applying this framework to Eisenstein series, a specific type of modular form, they obtain further insights into the behaviour of generalised Dedekind sums. This complex approach strengthens the validity of the findings and highlights the versatility of the developed techniques.

Indeed, the implications extend beyond the specific case of genus 2, as the established theorems generalise previous results concerning Dedekind sums and their quantum modularity. This work provides a powerful new set of tools for exploring the complex interaction between different branches of mathematics and potentially unlocking new insights in related fields of physics and materials science. The detailed analysis of Eichler integrals and the development of a generalised framework for Dedekind sums represent significant contributions to the field, paving the way for future research and applications.

Dedekind sum reciprocity relates modular forms via Eisenstein series limits

A trigonometric formula, proven by Marché and Masbaum, serves as the starting point for this work, specifically: σ2(x) = 1/6p2 −1/6 + 1/4p2X1≤n≤p−2, n odd T(n; x) sin3(πn/2p) sin2(πnx/2), where T(n; x) represents a summation over ±1. From this expression, the research establishes a formula, revealing an explicit connection between σ2(x) and generalised Dedekind sums.

Specifically, σ2(x) is expressed as 2p X1≤n≤p−2, n odd cot3(πn/2p) sin(πnx), which is equivalent to p2Sodd 2 (x) −2Sodd 0 (x), and further defined as the limit of an expression involving Eisenstein series Eodd 2 (τ + x) as τ approaches zero. The core of this methodology lies in establishing quantum modularity for generalised Dedekind sums associated with a modular form f.

For any matrix γ = a b c d in the group Γ with the condition cx + d = 0, a reciprocity relation is demonstrated: (cx + d)k−2Sf(ax + b)/(cx + d) − Sf(x) = Rf,γ(x) − a(x)0/pk c cx + d. Here, Rf,γ(x) denotes a regularized period polynomial and a(x)0 is a Fourier constant. This relationship is proven through three distinct approaches, each offering a unique perspective on the underlying mathematical structure.

The first proof relies on integral representations of the L-function bLf(s; x) and the period polynomial Rf,γ(x). Then, the second and third proofs utilise the asymptotic behaviour and modular transformation properties of Eichler integrals of f. At the heart of these latter proofs is the establishment of asymptotic behaviour using integral representations and Mellin summation formulas.

By applying this theorem to Eisenstein series for Γ(N), a corollary is derived, detailing the quantum modularity of generalised Dedekind sums Sχ,ψ k (x), where χ and ψ are periodic maps. Inside this framework, the generalised Dedekind sum Sχ,ψ k (x) is defined using discrete Fourier transforms and Bernoulli polynomials, enabling a precise formulation of the quantum modularity properties.

Since the work builds upon previous results by Fukuhara, Stucker, Vennos, Young, and Tranbarger, it extends their findings to a more general setting, offering a broader understanding of the interaction between modular forms and generalised Dedekind sums. The study carefully defines notations and establishes a clear organization, facilitating a rigorous and accessible presentation of the complex mathematical concepts.

Reciprocity of generalised Dedekind sums and modular form L-functions

At a weight of k ≥2, the research establishes reciprocity for generalised Dedekind sums linked to modular forms. Initial calculations define the twisted L-function, Lf(s; x), converging for Re(s) > k, and its completed version, bLf(s; x), extending meromorphically to the complex plane with potential poles at s = 0 and k. As a result, the generalised Dedekind sum, Sf(x), defined as bLf(k −1; x), and the regularized period polynomial, Rf,γ(X), are well-defined.

Lemma 2.5 then provides an integral representation of Rf,γ(X), independent of the chosen z0 ∈H, facilitating subsequent calculations. Further exploration through Lemma 2.3 yields an expression for the generalised Dedekind sum, Sf(x), as a complex integral. This integral representation, combined with the findings of Lemma 2.5, allows for a rigorous derivation of the reciprocity formula presented in the theorem.

Specifically, the proof leverages the properties of the integrals and the established relationships between the various functions involved, in the end confirming the stated reciprocity. The research culminates in a thorough framework for understanding the interaction between generalised Dedekind sums, modular forms, and their associated transformations within the context of TQFT signatures.

Dedekind sums unlock computational routes to topological quantum field theory signatures

Once considered a distant prospect, precise calculation of topological quantum field theory (TQFT) signatures has moved closer to practical application thanks to recent advances in understanding the connections between these signatures and generalised Dedekind sums. For years, a major obstacle has been the difficulty of expressing these signatures in a form amenable to computation, particularly for surfaces of even moderate complexity.

This work offers a pathway around that difficulty, establishing a link to the more tractable world of Dedekind sums and, crucially, providing explicit formulas involving trigonometric sums and Eichler integrals. The implications extend beyond purely mathematical curiosity. TQFTs are not merely abstract constructs; they provide a framework for describing physical systems, and their signatures are believed to encode information about the geometry and topology of spacetime.

Accurate determination of these signatures could, in principle, aid in the development of more refined models in areas like condensed matter physics and quantum gravity. The current approach relies on specific properties of the systems studied, limiting its direct application to broader scenarios. A remaining challenge lies in extending these techniques to higher genus surfaces and more general forms.

Unlike previous methods, this research provides a radial limit approach, but the computational cost of evaluating the necessary integrals could become prohibitive as complexity increases. Beyond this, a deeper understanding of the relationship between TQFT signatures and the underlying trace fields of knots remains an open area. Future work might explore alternative summation techniques or seek to establish connections with other areas of mathematics, such as modular forms and representation theory, to unlock even more powerful computational tools.