[Paper Review] Boundary G/G theory and topological Poisson-Lie sigma model

Fernando Falceto, Krzysztof Gawędzki|ArXiv.org|Aug 27, 2001

Black Holes and Theoretical Physics13 references4 citations

TL;DR

This paper establishes a covariant equivalence between the boundary $G/G$ gauged Wess-Zumino-Witten model and the topological Poisson-Lie sigma model with the dual Poisson-Lie group $G^*$ as the target. By restricting the target space of the $G/G$ theory and incorporating boundary terms, the authors show that the phase space on a strip is isomorphic to the Heisenberg double of $G$, providing a classical realization of quantum group structures via deformation quantization.

ABSTRACT

We study a boundary version of the gauged WZW model with a Poisson-Lie group G as the target. The Poisson-Lie structure of G is used to define the Wess-Zumino term of the action on surfaces with boundary. We clarify the relation of the model to the topological Poisson sigma model with the dual Poisson-Lie group G^* as the target and show that the phase space of the theory on a strip is essentially the Heisenberg double of G introduced by Semenov-Tian-Shansky

Motivation & Objective

• To establish a covariant formulation of the classical equivalence between the $G/G$ gauged WZW model and the topological Poisson-Lie sigma model on surfaces with boundary.
• To clarify how boundary conditions and target space restrictions in the $G/G$ theory arise naturally from the requirement of equivalence to the Poisson-Lie sigma model.
• To identify the classical phase space of the theory on a strip as a version of the Heisenberg double of the Poisson-Lie group $G$, generalizing the cotangent bundle $T^*G$.
• To explore the implications for quantization, particularly the emergence of the quantum group algebra $\mathcal{U}_q(\mathfrak{g})$ as the algebra of boundary states.

Proposed method

• Use of Poisson-Lie group structures on $G$ and its dual $G^*$ to define the Wess-Zumino term in the $G/G$ model on surfaces with boundary.
• Restriction of the $G/G$ target space to an open subset of $G$ and quotient of $G^*$ by a discrete subgroup to achieve precise classical equivalence with the Poisson-Lie sigma model.
• Derivation of the canonical structure on a cylinder and a strip using symplectic forms on the Heisenberg double, with explicit parametrization via $h = h_\pi h_0^{-1}$ and $p = h_0 \tau h_0^{-1}$.
• Application of deformation quantization to the phase space, identifying the algebra of boundary states with $\mathcal{U}_q(\mathfrak{g})$ for generic $q$, and with $\widetilde{\mathcal{U}}_q(\mathfrak{g})$ for $q$ a root of unity.
• Use of duality between $\mathcal{H} = \mathcal{A}(G)$ and $\mathcal{H}^* = \prod_\lambda \text{End}(V_\lambda)$ to relate classical functions on $T^*G$ to quantum group structures.
• Construction of a non-degenerate bilinear form on the finite-dimensional quotient $\widetilde{\mathcal{U}}_q(\mathfrak{g})$ using modular matrices, enabling a full topological quantum field theory structure.

Experimental results

Research questions

• RQ1 How can the $G/G$ gauged WZW model be consistently formulated on surfaces with boundary while preserving equivalence to the topological Poisson-Lie sigma model?
• RQ2 What is the precise classical phase space of the $G/G$ theory on a strip, and how does it relate to the Heisenberg double of $G$?
• RQ3 How does the quantization of the boundary theory lead to the quantum group algebra $\mathcal{U}_q(\mathfrak{g})$?
• RQ4 Why does the $G/G$ theory on a strip yield an infinite-dimensional space of boundary states despite its topological nature?
• RQ5 Under what conditions does the quantum boundary algebra become finite-dimensional and unitary, enabling a true topological field theory?

Key findings

• The phase space of the $G/G$ theory on a strip is isomorphic to the Heisenberg double of the Poisson-Lie group $G$, as defined by Semenov-Tian-Shansky, providing a Poisson-Lie generalization of $T^*G$.
• The classical action of the $G/G$ model on a surface with boundary is uniquely fixed by requiring equivalence to the topological Poisson-Lie sigma model with $G^*$ as the target, up to a discrete quotient of $G^*$.
• The algebra of boundary states is isomorphic to $\mathcal{U}_q(\mathfrak{g})$ for generic $q$, arising from deformation quantization of the Heisenberg double, and carries a natural non-commutative algebra structure.
• For $q$ a root of unity (integer $k$), the algebra of boundary states reduces to a finite-dimensional quotient $\widetilde{\mathcal{U}}_q(\mathfrak{g})$, which supports a non-degenerate bilinear form via modular matrices, enabling a genuine topological quantum field theory.
• The theory lacks unitarity in Minkowski signature due to a non-real action, but this is resolved in the quantum regime when $q$ is a root of unity, yielding a consistent TQFT.
• The duality between the commutative algebra $\mathcal{A}(G)$ of functions on $G$ and the non-commutative algebra $\mathcal{H}^*_q = \prod_\lambda \text{End}(V^q_\lambda)$ of quantum group representations is preserved under quantization, with products and coproducts interchanged.