Dynkin diagrams, generalized Nahm sums and 2d CFTs

Abstract: A folklore conjecture states that the Nahm sum associated with a pair of Dynkin diagrams of type $ADET$ is a modular function. In this paper, we extend this conjecture to Dynkin diagrams of type $ABCDEFGT$ in the context of generalized Nahm sums. The modular Nahm sums are closely related to the characters of 2d rational conformal field theories. In this work, we identify many specific generalized Nahm sums with characters of some well-studied 2d CFTs. For example, we find that the generalized Nahm sums associated with $(T_1, C_r)$ and $(T_1,D_r)$ correspond to the supersymmetric Virasoro minimal models $\mathrm{SM}(4r+6, 4)$ and $\mathrm{SM}(8r+4, 2)$, respectively.

• The paper establishes a novel generalization of Nahm sums by extending the modularity conjecture to pairs of Dynkin diagrams from types A–G and T.
• It employs symmetrizable Cartan matrices and diagonal matrices encoding root-length data to systematically relate q-series identities with RCFT characters.
• The findings provide a unified framework that connects combinatorial identities, modular forms, and quantum topology with rational 2D conformal field theory.

Generalized Nahm Sums, Dynkin Diagrams, and Rational 2D Conformal Field Theories

Introduction and Context

This paper provides a comprehensive study of generalized Nahm sums associated with pairs of Dynkin diagrams of types $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $T$, extending the classical conjecture relating Nahm sums and modularity beyond the traditional $ADET$ classification. The motivation originates from the observed relationships between Nahm sums, modular forms, and characters of rational two-dimensional CFTs (RCFTs), notably in the context of identities such as those of Rogers–Ramanujan and their connections to representation-theoretical and combinatorial physics.

The authors explicitly propose and test a generalization of the so-called folklore "Nahm sums modularity conjecture" to encompass all simply- and non-simply-laced types $ABCDEFGT$. The approach involves formulating and analyzing generalized Nahm sums using symmetrizable Cartan matrices, thereby unifying prior results and conjectures in CFT, modular $B$0-series, and quantum topology.

Nahm Sums, Modularity, and Dynkin Diagram Pairs

A Nahm sum, for data $B$1 with $B$2 a positive-definite matrix, $B$3 a rational vector, and $B$4 a rational scalar, is a $B$5-hypergeometric series:

$B$6

where $B$7 and $B$8 is the standard $B$9-Pochhammer symbol. In the rational CFT context, such sums manifest as fermionic representations for CFT characters, and their modularity properties encode deep combinatorial and arithmetic information.

The classical folklore conjecture asserts that the Nahm sum associated to a pair $C$0 of Dynkin diagrams of types $C$1, $C$2, $C$3, or $C$4 yields a modular function when $C$5 is chosen as the Cartan matrix combination $C$6 with appropriate $C$7 and $C$8. This conjecture is both physically and mathematically motivated by the correspondence between affine algebras, coset constructions in CFT, and integrable lattice models.

This work extends the conjecture for Nahm sums to all types $C$9 by considering symmetrizable Cartan matrices, and introduces a new quadruple $D$0 where $D$1 is a diagonal matrix encoding root length data, reflecting the possible presence of non-simply-laced diagrams.

Generalized Nahm Sums and New Modularity Conjecture

The generalization is framed as follows: For any pair of Dynkin diagrams $D$2, $D$3 (of type $D$4), with Cartan matrices $D$5, $D$6 and diagonal matrices $D$7, $D$8 (where diagonal elements encode simple root length ratios), set:

• $D$9,
• $E$0,
• $E$1, with central charge $E$2 ($E$3 denotes the Coxeter number).

The main conjecture is that, for this quadruple and $E$4, the associated generalized Nahm sum

$E$5

is modular. This extends prior conjectures and, for many cases, is proven or supported by identification with explicit $E$6-series identities or correspondence with CFT characters.

The connection to modular forms is crucial, as the modularity of Nahm sums is closely tied to their interpretation as RCFT characters with positive, integral Fourier coefficients and central charge as above.

Explicit Identifications and New Correspondences

A substantial part of the paper is devoted to the explicit identification of the generalized Nahm sums for many pairs $E$7 with characters of well-understood 2D CFTs. Key examples include:

• For $E$8, the sum corresponds to the supersymmetric Virasoro minimal model $E$9.
• For $F$0, the correspondence is with the supersymmetric Virasoro minimal model $F$1.
• The well-known Rogers–Ramanujan/Andrews–Gordon type combinations are recovered as special cases.

Each instance involves a careful matching between the modular Nahm sum and an explicitly known sum of CFT characters (often in both Neveu–Schwarz and Ramond sectors), sometimes yielding new or "non-diagonal" modular invariants. The identifications are justified via detailed calculation (to high order in $F$2-expansions) and by exploiting explicit product formulas in the literature.

The paper also demonstrates that generalized Nahm sums reduce to ordinary Nahm sums under rank lifts, showing that the generalization provides a vast new family of modular candidates.

Theoretical Implications and Structures

The results have direct implications for several areas:

• Modular forms and $F$3-series: The proposed generalization dramatically enlarges the catalogue of modular $F$4-series arising from representation-theoretic or quantum field-theoretic structures. The explicit identifications provide a unifying framework and offer guidance on patterns of modularity, linear relations among characters, and the construction of explicit modular invariants.
• Conformal field theory: These results explicitly identify which generalized Nahm sums yield CFT characters for a broad class of RCFTs, including beyond the simply-laced series. This has consequences for the construction and classification of rational CFTs, dual coset descriptions, and GKO-like correspondences.
• Representation theory and quantum topology: By connecting these Nahm sums with graded dimensions of principal subspaces of (twisted) affine Kac–Moody algebra modules and with quantum invariants of knots and 3D TQFTs, the work offers new tools and classification frameworks, opening the door to a systematic study of modularity in these contexts.
• Arithmetic and algebraic $F$5-theory: The structure of the Nahm equation, its solutions in the Bloch group, and the relation to modularity further connect to deep aspects of algebraic $F$6-theory, specifically through the torsion elements associated with these equations.

Despite the breadth of validity, counterexamples exist to the general "duality" conjecture for modular triples/quadruples, but the proposed construction often respects duality when $F$7, providing a systematic structure.

Detailed Analysis of Low Rank Cases and Explicit Families

The paper provides a comprehensive classification of known modular generalized Nahm sums for pairs of low-rank diagrams, verifying modularity in all but a handful of cases. For each, concrete CFT identifications are provided, and for exceptional or "folded" diagrams (e.g., $F$8, $F$9 as foldings of $G$0, $G$1), clear relationships between the corresponding characters are established.

Significant analytic, combinatorial, and modular identities (such as Warnaar's and Kanade–Russell conjectures) are incorporated to confirm modularity and express Nahm sums in terms of rational CFT spectra, further reinforcing the theoretical conjecture.

Future Directions

This framework indicates numerous directions for further research:

• Classification and proof of modularity for all pairs in the $G$2 regime, including higher-rank and as-yet unclassified cases.
• Detailed study of the consequences for the theory of vertex operator algebras, including connections to the classification of modular tensor categories.
• Application to the construction of new quantum knot invariants and advances in arithmetic $G$3-theory motivated by the modularity and Bloch group structure of the Nahm sums.
• Systematic investigation of folding, conformal embeddings, and their impact on modular invariants and sub-CFT structure.

The connections exhibited between combinatorial identities, modular functions, Lie theoretic data, and CFT characters suggest a unifying structure underlying a significant portion of algebraic and quantum mathematics.

Conclusion

This work systematically extends the landscape of modular $G$4-series—originating with the classical Nahm sums—by leveraging the structure of all affine Dynkin diagrams, incorporating root-length data, and aligning the resulting generalized Nahm sums with characters of rational 2D CFTs. The explicit identifications and accompanying conjecture chart new territory for the interface of representation theory, combinatorics, and quantum field theory, and provide a canonical direction for future research in algebra, number theory, and quantum topology.

Reference: "Dynkin diagrams, generalized Nahm sums and 2d CFTs" (2604.00847)