Simply-laced quantum connections generalising KZ

We construct a new family of flat connections generalising the KZ connection, the Casimir connection and the dynamical connection. These new connections are attached to simply-laced graphs, and are obtained via quantisation of time-dependent Hamiltonian systems controlling the isomonodromic deformations of meromorphic connections on the sphere.

💡 Research Summary

The paper introduces a broad new class of flat quantum connections—called simply‑laced quantum connections (SLQC)—that simultaneously generalize the Knizhnik‑Zamolodchikov (KZ) connection, the Casimir (DMT) connection, and the Felder‑Markov‑Tarasov‑Varchenko (FMTV) connection. The construction is rooted in the quantisation of a family of non‑autonomous Hamiltonian systems known as simply‑laced isomonodromy systems (SLIMS), which were recently introduced as a unifying framework for isomonodromic deformations associated with complete k‑partite graphs and their “splayed” versions.

The authors begin by recalling that the classical KZ equations arise as a quantisation of the Schlesinger isomonodromy system. They then describe how SLIMS extend this picture: given a finite set J of “parts” and a surjection π: I → J, one obtains a complete k‑partite graph G on the vertex set I, together with a splayed graph eG on the parts J. A “reading” a: J → ℂ∪{∞} assigns distinct complex parameters (or ∞) to the parts, which become the regular and irregular times of the deformation problem. The base space B is the product over parts of configuration spaces of the associated times, while the fibre M = Rep(G,V) consists of off‑diagonal linear maps between the vector spaces V_i attached to each vertex. A symplectic form ω_a on M is defined using the scalar factors φ_{ij} = (a_i−a_j)^{−1}.

The isomonodromic deformation problem for a meromorphic connection on the Riemann sphere (with simple poles at the regular times and an irregular singularity at infinity) translates into a time‑dependent Hamiltonian system H_i : F_a → ℂ, where F_a = M × B. The Hamiltonians are obtained from a 1‑form Θ (equation (4) in the paper) that is built from traces of products of the off‑diagonal matrices and the diagonal part of the irregular term. Importantly, the H_i’s Poisson‑commute and satisfy the strong flatness condition {H_i, H_j}=0 together with ∂{t_j}H_i−∂{t_i}H_j=0.

To quantise, the authors replace the commutative polynomial algebra O(M) ≅ Sym(M^*) with the Weyl algebra A = 𝔚(M) and then apply the Rees construction to obtain a filtered deformation 𝔟A depending on a formal parameter h. Classical potentials—linear combinations of oriented cycles in the graph—are lifted to quantum potentials, i.e. elements of A (or 𝔟A) obtained by the same trace procedure but now interpreted as non‑commutative operators. The quantum Hamiltonians 𝔟H_i are defined by inserting the quantum potentials into the same trace formulas, yielding a family of h‑dependent operators on the trivial bundle 𝔟A × B → B.

The main theorem (Theorem 5.1) proves that the universal simply‑laced quantum connection \