Lambda Models From Chern-Simons Theories

TL;DR

This paper establishes a precise, strong connection between superstring lambda models and double Chern-Simons theories, revealing how integrable structures and non-ultralocality issues are resolved through symplectic reduction.

Contribution

It extends previous work by rigorously linking lambda models to Chern-Simons theories via symplectic reduction, clarifying their algebraic structures and spectral parameter dependence.

Findings

The Maillet algebra emerges as a boundary current algebra after reduction.

Poisson algebra of Wilson loops generalizes Goldman bracket with spectral parameter.

Non-ultralocality issues in lambda models are resolved through Chern-Simons symplectic reduction.

Abstract

In this paper we refine and extend the results of arXiv:1701.04138, where a connection between the $AdS_{5}\times S^{5}$ superstring lambda model on $S^{1}=\partial D$ and a double Chern-Simons (CS) theory on $D$ based on the Lie superalgebra $\mathfrak{psu}(2,2|4)$ was suggested, after introduction of the spectral parameter $z$. The relation between both theories mimics the well-known CS/WZW symplectic reduction equivalence but is non-chiral in nature. All the statements are now valid in the strong sense, i.e. valid on the whole phase space, making the connection between both theories precise. By constructing a $z$-dependent gauge field in the 2+1 Hamiltonian CS theory it is shown that: i) by performing a symplectic reduction of the CS theory the Maillet algebra satisfied by the extended Lax connection of the lambda model emerges as a boundary current algebra and ii) the Poisson algebra of the supertraces of $z$-dependent Wilson loops in the CS theory obey some sort of spectral parameter generalization of the Goldman bracket. The latter algebra is interpreted as the precursor of the (ambiguous) lambda model monodromy matrix Poisson algebra prior to the symplectic reduction. As a consequence, the problematic non-ultralocality of lambda models is avoided (for any value of the deformation parameter $\lambda \subset [0,1]$), showing how the lambda model classical integrable structure can be understood as a byproduct of the symplectic reduction process of the $z$-dependent CS theory.