Characteristic Classes and Integrable Systems for Simple Lie Groups

This paper is a continuation of our previous paper \cite{LOSZ}. For simple complex Lie groups with non-trivial center, i.e. classical simply-connected groups, $E_6$ and $E_7$ we consider elliptic Modified Calogero-Moser systems corresponding to the Higgs bundles with an arbitrary characteristic class. These systems are generalization of the classical Calogero-Moser (CM) systems related to a simple Lie groups and contain CM systems related to some (unbroken) subalgebras. For all algebras we construct a special basis, corresponding to non-trivial characteristic classes, the explicit forms of Lax operators and Hamiltonians.

💡 Research Summary

The paper extends the theory of elliptic Calogero‑Moser (CM) integrable systems to simple complex Lie groups that possess a non‑trivial center, namely the simply‑connected classical groups, the exceptional groups (E_{6}) and (E_{7}). The authors start from the observation that a Higgs bundle over an elliptic curve (\Sigma) can be equipped with an arbitrary characteristic class, i.e. a topological twist determined by an element of (H^{1}(\Sigma,Z(G))), where (Z(G)) is the center of the group. While the standard CM systems correspond to the trivial characteristic class (untwisted bundles), non‑trivial classes introduce additional “topological” degrees of freedom that modify the dynamics.

A central technical contribution is the construction of a new basis in the Lie algebra that is adapted to a given characteristic class. Instead of the usual Cartan‑Weyl basis, the authors decompose the root system into subsets that are invariant or non‑invariant under the action of a central element (c\in Z(G)). For each subset they define basis vectors (E_{\alpha}^{(c)}) and (H_{i}^{(c)}) that respect the twisted periodicity imposed by the characteristic class. This decomposition makes explicit the emergence of an “unbroken” subalgebra (\mathfrak{g}{0}) (for example ( \mathfrak{su}(p)\oplus\mathfrak{su}(q)) inside (\mathfrak{su}(N)) when the class corresponds to a (\mathbb{Z}{N}) element) and isolates the remaining “broken” directions.

Using this basis the Lax pair ((L(z),M(z))) is written in terms of elliptic functions (Weierstrass (\zeta) and (\wp) functions) evaluated on the projected root coordinates. The Lax matrix contains three types of contributions: (i) the usual kinetic term (\sum p_{i}H_{i}^{(c)}); (ii) interaction terms (\Phi(\alpha!\cdot!u,z)E_{\alpha}^{(c)}) for invariant roots and (\Phi_{c}(\beta!\cdot!u,z)E_{\beta}^{(c)}) for non‑invariant roots; and (iii) a constant “topological” term (\Phi_{\text{top}}(c),I) that encodes the characteristic class. The auxiliary matrix (M(z)) is constructed so that the Lax equation (\dot L=