Lambda models from Chern-Simons theories

Abstract

In this paper we refine and extend the results of [1], where a connection between the AdS 5 × S 5 superstring lambda model on S 1 = ∂D and a double Chern-Simons (CS) theory on D based on the Lie superalgebra 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 λ ⊂ [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.

Holomorphic Chern-Simons theory and lambda models: PCM case

“…As a consequence, the expressions (2.17) and (2.19) are valid in the strong sense. At this point, we can anticipate that no Hamiltonian extension of the Lax connection will be required in contrast to the lambda models on (semi)-symmetric spaces [23].…”

“…Because of the Lax connection is strongly flat, the relations (2.14) are valid off-shell and in terms of the variables Ψ(z ± ), the action (2.1) takes the form [23]…”

“…It is important to emphasized that this is done without fixing the gauge symmetry of the CS theory completely (as in [29,30]), but rather from a symplectic reduction perspective (as in [22,23,35]). The main result is that in the holomorphic CS theory case, the SR works as a localization mechanism that eliminates the spectral parameter from the reduced CS theory phase space, which is identified as being equivalent to the lambda model.…”

“…iii) The form of the Hamiltonian when expressed in terms of the components of the Lax connection L [22,23],…”

“…By considering the addition of two CS actions of opposite levels defined on a solid cylinder, one for each pole z ± , it is possible to recover (1.1), (1.2), (1.3) and the lambda deformed PCM action functional as well. This more traditional approach is considered in [22,23], where all the results are presented. The major drawback of this bottom-up approach, is that it is not clear how to include the spectral parameter z in the double CS theory action functional from the very beginning and hence only works partially.…”