[Paper Review] Integrability in Hamiltonian Chern-Simons theory

Anton Alekseev|ArXiv.org|Nov 12, 1993

Algebraic structures and combinatorial models4 references22 citations

TL;DR

This paper constructs an integrable model on the moduli space of flat connections with marked points using Wilson line observables in Hamiltonian Chern-Simons theory. By defining a transfer matrix from these observables, the authors extract a family of commuting Hamiltonians and show the model is gauge-equivalent to a finite XXZ spin chain, establishing a direct link between topological quantum field theory and integrable systems via quantum group representations.

ABSTRACT

We consider the moduli space of flat connections on the Riemann surface with marked points. The new efficient parametrization is suggested and used to construct an integrable model on the moduli space. A family of commuting Hamiltonians is extracted from the trace of the transfer matrix built from the Wilson line observables of the Chern-Simons theory. Our model appears to be gauge equivalent to XXZ magnetic chain with finite number of sites.

Motivation & Objective

• To construct a complete set of commuting Hamiltonians on the moduli space of flat connections with marked points in Hamiltonian Chern-Simons theory.
• To establish a connection between the observables of Chern-Simons theory and integrable systems using the inverse scattering method.
• To demonstrate that the resulting integrable model is gauge-equivalent to a finite-site XXZ spin chain.
• To provide a representation theory for the moduli algebra of flat connections with marked points, especially for generic quantum group deformation parameter q.
• To explore the structure of polarizations on the moduli space and their relation to spectral shifts and quantum group symmetries.

Proposed method

• Introduce a new parametrization of the moduli space of flat connections with marked points using link variables on a graph embedded in the Riemann surface.
• Construct a transfer matrix from Wilson line observables, using spectral parameter-dependent L-operators to generate commuting Hamiltonians.
• Utilize the R-matrix algebra for link variables to ensure commutativity of the transfer matrix, leveraging the non-ultralocal structure of the lattice Chern-Simons model.
• Establish gauge equivalence between the constructed integrable model and the XXZ spin chain by identifying the same R-matrix algebra and Hamiltonian structure.
• Define the moduli algebra as a quotient of a larger algebra by an ideal corresponding to quantum trace conditions, ensuring consistency with topological invariance.
• Construct irreducible representations of the moduli algebra as spaces of q-invariants in tensor products of U_q(G)-representations and a fundamental representation of the quantum group.

Experimental results

Research questions

• RQ1 Can a complete set of commuting Hamiltonians be extracted from Wilson line observables in Hamiltonian Chern-Simons theory on a Riemann surface with marked points?
• RQ2 Is the resulting integrable model on the moduli space of flat connections gauge-equivalent to a known spin chain model, such as the XXZ chain?
• RQ3 How can the moduli algebra of flat connections with marked points be represented, especially in terms of quantum group representations for generic q?
• RQ4 What is the structure of polarizations on the moduli space when parameterized by spectral shifts, and how do they differ from those induced by complex structures?
• RQ5 What is the role of the Knizhnik-Zamolodchikov equation in relating different quantizations arising from distinct polarizations?

Key findings

• The integrable model constructed from Wilson line observables yields a family of commuting Hamiltonians via the transfer matrix formalism, providing a complete set of conserved quantities on the moduli space.
• The model is shown to be gauge-equivalent to a finite XXZ spin chain with a specific R-matrix structure derived from the Chern-Simons lattice formulation.
• Irreducible representations of the moduli algebra are realized as spaces of q-invariants in tensor products of representations of U_q(G), specifically W^e_{I_1,…,I_n} = Inv_q(I_1 ⊗ … ⊗ I_n ⊗ R^{⊗g}).
• The construction is valid for generic values of the deformation parameter q; the case of q being a root of unity requires further analysis due to non-associative structures.
• The family of commuting Hamiltonians can be extended to include spectral shifts, suggesting a broader class of polarizations on the moduli space, though their full completeness remains to be established.
• The paper hints at a new quantum connection—generalizing the KZ equation—governing the identification of Hilbert spaces across different polarizations parameterized by spectral shifts.