[Paper Review] Characteristic Classes and Integrable Systems. General Construction

A. Levin, M. A. Olshanetsky+2 more|arXiv (Cornell University)|Jun 3, 2010

Algebraic structures and combinatorial models45 references19 citations

TL;DR

This paper constructs a new family of classical integrable systems—modified Calogero-Moser (MCM) systems—using topologically non-trivial holomorphic G-bundles over elliptic curves, where G is a simply-connected simple Lie group with non-trivial center. By generalizing the t'Hooft basis and using dynamical r-matrices, the authors define Lax operators and Hamiltonians, showing that MCM systems realize phase spaces with non-trivial characteristic classes and contain standard CM systems as subalgebras.

ABSTRACT

We consider topologically non-trivial Higgs bundles over elliptic curves with marked points and construct corresponding integrable systems. In the case of one marked point we call them the modified Calogero-Moser systems (MCM systems). Their phase space has the same dimension as the phase space of the standard CM systems with spin, but less number of particles and greater number of spin variables. Topology of the holomorphic bundles are defined by their characteristic classes. Such bundles occur if G has a non-trivial center, i.e. classical simply-connected groups, $E_6$ and $E_7$. We define the conformal version CG of G - an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CG-bundles. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, define the phase spaces and the Poisson structure using dynamical r-matrices. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of t'Hooft matrices for sl(N). We find that the MCM systems contain the standard CM systems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the holomorphic bundles with non-trivial characteristic classes.

Motivation & Objective

• To construct integrable systems associated with topologically non-trivial holomorphic G-bundles over elliptic curves, where G has a non-trivial center.
• To generalize the standard elliptic Calogero-Moser systems by incorporating characteristic classes from the bundle topology.
• To define a conformal group CG analogous to GL(N) for SL(N), linking it to characteristic classes of G-bundles.
• To establish a framework using GS-basis and dynamical r-matrices to construct Lax operators and Hamiltonians for these systems.
• To show that the MCM systems contain standard CM systems as subalgebras corresponding to unbroken subgroups of the original Lie algebra.

Proposed method

• Construct holomorphic G-bundles over elliptic curves using transition operators Q and Λ satisfying the twisted commutation relation QΛQ⁻¹Λ⁻¹ = ζ, where ζ ∈ Z(G) is a central obstruction.
• Define characteristic classes via elements of H²(Σ, Z(G)), which classify topologically non-trivial G-bundles, particularly for simply-connected groups including E₆ and E₇.
• Introduce the conformal group CG as an analog of GL(N) for SL(N), relating its bundles to characteristic classes of G-bundles.
• Use a generalized t'Hooft basis in Lie algebras to construct Lax operators and Hamiltonians, extending the standard CM system framework.
• Derive the classical RLL relation using dynamical r-matrices constructed from meromorphic functions φᵏₐ(z), which are invariant under Weyl group actions.
• Utilize Fay identities and elliptic functions (theta, Eisenstein, Weierstrass) to derive the algebraic structure of the r-matrix and verify the RLL relation.

Experimental results

Research questions

• RQ1 How can topologically non-trivial holomorphic G-bundles over elliptic curves be classified and used to construct new integrable systems?
• RQ2 What is the role of characteristic classes in defining the phase space and dynamics of integrable systems beyond the standard Calogero-Moser models?
• RQ3 How does the conformal group CG relate to the characteristic classes of G-bundles, and what is its significance in the construction of Lax operators?
• RQ4 In what way do the modified Calogero-Moser systems generalize the standard CM systems, and what subalgebras do they contain?
• RQ5 How do the dynamical r-matrices constructed from meromorphic functions φᵏₐ(z) satisfy the classical RLL relation in this context?

Key findings

• The MCM systems are constructed as integrable systems with phase space dimension matching that of standard CM systems with spin, but with fewer particles and more spin variables.
• The characteristic classes of the holomorphic bundles are defined by elements of H²(Σ, Z(G)), and for G = Spin(n), they coincide with Stiefel-Whitney classes.
• The modified Calogero-Moser systems contain standard CM systems as subalgebras, corresponding to unbroken subgroups of the original Lie algebra.
• The Lax operators and quadratic Hamiltonians are constructed using a generalized t'Hooft basis in simple Lie algebras, which extends the standard basis for sl(N).
• The classical dynamical r-matrix is derived from the meromorphic function φᵏₐ(z), which satisfies λ-invariance and satisfies Fay identities crucial for the RLL relation.
• The RLL relation is proven using the function φᵏₐ(z), which exhibits quasi-periodicity and transformation properties under Weyl group actions, ensuring consistency of the integrable structure.