A unifying 2D action for integrable $$\sigma $$-models from 4D Chern–Simons theory

Abstract

In the approach recently proposed by K. Costello and M. Yamazaki, which is based on a four-dimensional variant of Chern-Simons theory, we derive a simple and unifying two-dimensional form for the action of many integrable σ-models which are known to admit descriptions as affine Gaudin models. This includes both the Yang-Baxter deformation and the λ-deformation of the principal chiral model. We also give an interpretation of Poisson-Lie T -duality in this setting and derive the action of the E-model. The four-dimensional actionLet G C be a complex semisimple Lie group with Lie algebra g C , on which we fix a choice of non-degenerate invariant symmetric bilinear form ·, · : g C × g C → C.Let CP 1 := C ∪ {∞} denote the Riemann sphere. We shall fix a choice of global holomorphic coordinate z on C ⊂ CP 1 .2.1. Bulk and boundary equations of motion. Consider the action (1.2) where ω is a meromorphic 1-form on CP 1 and the Chern-Simons 3-form for the 1-form A = A σ dσ + A τ dτ + Azdz is given by

Holomorphic Chern-Simons theory and lambda models: PCM case

“…A similar expression was found in [30] to be valid as well for a wide range of other integrable deformations of string sigma models, after fixing a particular form of the CS gauge field (roughly) enforcing the condition A z = 0 and imposing the so-called archipelago conditions on an analogue to the field F defined above. Here, the CS theory naturally localize at p, as a consequence of the SR.…”

“…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.…”

“…where Σ = R × S 1 is the closed string world-sheet manifold, ω is as defined in (2.50), CS(B) is the CS three-form for the gauge field B and CP 1 is the spectral space introduced above. Under certain circumstances, as considered in [30], the action is real.…”

“…It is the purpose of this note to work out the approach (II) in detail and to show how (1.1), (1.2), (1.3), the lambda deformed PCM action functional and its classical integrability properties can be recovered from the SR of a holomorphic CS theory defined on Σ × CP 1 . We emphasize that in this setup the gauge is not fixed completely but only partially, in contrast to [29,30], where the CS gauge symmetry is fixed in totality. In this regard our results are complementary.…”

“…The action functional associated to the symplectic form (1.7) is the four-dimensional holomorphic CS theory first presented in [24,25] and subsequently thoroughly studied in a series of papers [1,[26][27][28]. This top-down approach introduce successfully the spectral parameter z into the CS action functional from first principles and everything points towards it is the correct way to do so, as a wide range of known and even new integrable field theories can be described in this way [1,30], not to mention several lattice integrable models as well. In this approach, and at least for the explicit example to be considered in this note, the SR restricts the degrees of freedom of the CS theory from M = S 1 × CP 1 to M = S 1 × p inducing a localization mechanism and, as a consequence, the information of the reduced theory phase space is determined by the restriction of part of the original CS gauge field to the set of poles p in the spectral manifold, that is identified with the Lax connection of the lambda model in a natural way.…”