Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices

We discuss quantum dynamical elliptic R-matrices related to arbitrary complex simple Lie group G. They generalize the known vertex and dynamical R-matrices and play an intermediate role between these two types. The R-matrices are defined by the corresponding characteristic classes describing the underlying vector bundles. The latter are related to elements of the center Z(G) of G. While the known dynamical R-matrices are related to the bundles with trivial characteristic classes, the Baxter-Belavin-Drinfeld-Sklyanin vertex R-matrix corresponds to the generator of the center Z_N of SL(N). We construct the R-matrices related to SL(N)- bundles with an arbitrary characteristic class explicitly and discuss the corresponding IRF models.

💡 Research Summary

The paper investigates a broad family of quantum dynamical elliptic R‑matrices associated with an arbitrary complex simple Lie group G, focusing in particular on the special linear group SL(N). The authors start by recalling the two well‑known classes of elliptic R‑matrices: the vertex‑type (Baxter‑Belavin‑Drinfeld‑Sklyanin, BBDS) matrices, which are tied to non‑trivial elements of the centre Z(G), and the dynamical‑type (Felder) matrices, which correspond to the trivial characteristic class of the underlying vector bundle. In the case of SL(N) the centre is the cyclic group Z_N; the generator of Z_N produces the BBDS vertex matrix, while the identity element yields the Felder dynamical matrix.

The central observation of the work is that the characteristic class of an SL(N)‑bundle is in one‑to‑one correspondence with an element ζ ∈ Z_N. A non‑trivial characteristic class therefore encodes a “twist’’ of the bundle, i.e. a modification of the transition functions by a phase factor determined by ζ. The authors construct a family of R‑matrices R^{(ζ)}(u,λ) that depend explicitly on this twist. The construction proceeds by multiplying the BBDS vertex matrix by a scalar phase χ_ζ(u)=exp(2πi k u/N) (where ζ=e^{2πik/N}) and by inserting a dynamical twist operator Φ_ζ(λ) that acts on the Felder dynamical matrix. The resulting formula

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