Modularity of generalized Nahm sums for Dynkin pairs (X,Y) of type ABCDEFGT

Prove that for any pair of Dynkin diagrams X and Y of types A, B, C, D, E, F, G, or T, with Cartan matrices C(X) and C(Y), and with D(X) the diagonal matrix of squared simple-root length ratios, the generalized Nahm sum f_{A(X,Y),0,C(X,Y),D(X,Y)}(τ) defined by A(X,Y)=C(X)⊗C(Y)^{-1}, D(X,Y)=D(X)⊗D(Y), and C(X,Y)=−c(X,Y)/24 where c(X,Y)=tr(D(X,Y))·h(X)/(h(X)+h(Y)) (h(·) the Coxeter number), is a modular function; equivalently, establish that the quadruple (A(X,Y),0,C(X,Y),D(X,Y)) is modular for all such X and Y.

Background

The paper introduces generalized Nahm sums f_{A,B,C,D}(τ) for symmetrizable matrices following Mizuno, and proposes a broad extension of the folklore ADE/T modularity phenomenon to all simple Dynkin types including non-simply-laced cases and tadpoles. The construction uses Kronecker products of Cartan and diagonal root-length matrices, together with a central charge formula motivated by 2d rational CFTs.

This conjecture unifies and extends many known modular cases (e.g., Andrews–Gordon, Bressoud, Warnaar identities, and various low-rank examples) and would generate infinitely many new modular Nahm sums (ordinary or generalized).