Abstract
We consider several classes of σ-models (on groups and symmetric spaces, η-models, ⋋-models) with local couplings that may depend on the 2d coordinates, e.g. on time τ . We observe that (i) starting with a classically integrable 2d σ-model, (ii) formally promoting its couplings hα to functions hα(τ ) of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that hα(τ ) must solve the 1-loop RG equations of the original theory with τ interpreted as RG time. This provides a novel example of an ‘integrability-RG flow’ connection. The existence of a Lax connection suggests that these time-dependent σ-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such σ-models with D-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a (D + 2)-dimensional conformal σ-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.
Highlights
We observe that (i) starting with a classically integrable 2d σ-model, (ii) formally promoting its couplings hα to functions hα(τ ) of 2d time, and (iii) demanding that the resulting time-dependent model admits a Lax connection implies that hα(τ ) must solve the 1-loop RG equations of the original theory with τ interpreted as RG time
As we have found above on several examples, if the couplings of an integrable σ-model are promoted to functions of time that solve the 1-loop RG equations, hα → hα(τ ), ∂τ hα = βα(h), the Lax connection of the original model L(hα) admits a natural generalization to a classical Lax connection for the time-dependent model L(hα(τ ))
As we discussed in the Introduction, such time-dependent models can be naturally embedded into string theory by starting with a Weyl-invariant σ-model (1.3) with two extra ‘null’ directions (u, v) and fixing a l.c. gauge u = τ
Summary
We shall discuss how, starting from a classically integrable model There, for each model, we give the original Lagrangian, original Lax pair, 1-loop RG equation and its solution, and the expression for the function z(w; τ, σ) that generalizes the original spectral parameter in such a way that (2.3) gives a Lax pair for the time-dependent theory (2.2).. One could remove square roots by redefining the spectral parameter (or more formally moving to an appropriate Riemann surface on which the Lax connection is meromorphic) This is not possible since the positions of the branch cuts depend on (τ, σ) and one cannot redefine the spectral parameter in a way depending on (τ, σ) without changing the equations of motion encoded in the zero curvature condition. An equivalent expression for the Lax connection of this ‘local’ PCM (or symmetric space σ-model, see table 1) was originally found in [32] where the dependence of the analog of the spectral parameter on the functions f +(ξ+) and f −(ξ−) was discovered.
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