Differential equations and integrable models: the SU(3) case

Abstract

We exhibit a relationship between the massless a Ž2. integrable quantum field theory and a 2 certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schrodinger equation. This forms part of a more general correspondence involving A -related Bethe ansatz systems and third-order differential equations. IntroductionA curious connection between certain integrable quantum field theories and the w x theory of the Schrodinger equation has been the subject of some recent work 1-4 . In this paper we extend these results by establishing a link between functional relations for Ž w x. A -related Bethe ansatz systems see, for example, Refs. 5,6 and third-order differen-2 tial equations. Most of our analysis concerns a certain specialisation of the model, a w x particularly symmetric case that can also be related to the dilute A-model of 7 . 0550-3213r00r$ -see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž . PII: S 0 5 5 0 -3 2 1 3 9 9 0 0 7 9 1 -9 Dorey, R. Tateo r Nuclear Physics B 571 PM 2000 583-606 584 w x In the cases studied in 1-4 , the most general differential equation was a radial Schrodinger problem with 'angular momentum' l and homogeneous potential x 2 M , Ž . initially defined on the positive real axis x g 0,`:The relevant integrable quantum field theories were the massless twisted sine-Gordon models or, equivalently, the twisted XXZr6-vertex models in their thermodynamic limits, and their reductions. It is worth noting that these models are all related to the Lie Ž . w x algebra A . Spectral functions associated with 1.1 satisfy functional relations 8-11 , 1 and these were mapped into functional equations appearing in the context of integrable w x quantum field theory in 1-4 . We will follow a similar strategy here, taking a simple third-order ordinary differential equation as our starting-point and showing that the Stokes multipliers and certain spectral functions for this equation together satisfy relations which are essentially the analogues, for the Bethe ansatz systems treated in w x 6,7 , of the T-Q systems which arise in the context of the integrable quantum field w x theories related to A 12,13 . This is the subject of Section 2, while in Section 3 we 1 borrow some other ideas from integrable quantum field theory in order to derive a non-linear integral equation for the spectral functions, an equation which is put to the test in a simple example in Section 4. Duality properties are discussed in Section 5, Ž . allowing us to find the equivalent of the angular-momentum term in 1.1 for the third-order equation. Connections with various perturbed conformal field theories are discussed and tested in Section 6. Finally, Section 7 discusses the most general A -related BA equations that arise in this context, and Section 8 contains our conclu-2 sions. The differential equationWe begin with the following third-order ordinary differential equation:3 M Ž . and initially restrict ourselves to purely homoge...

New families of flows between two-dimensional conformal field theories

“…2 model was conjectured [25] ‡ , and checked to describe the massive φ 12 , φ 21 and φ 15 perturbations of minimal models on the imposition of a suitably-chosen twist. (The connection between the a (2) 2 model and these perturbations can be understood through quantum group reduction [30][31][32].)…”

“…As mentioned above, φ 21 is odd, so only the even coefficients C 2n are nonzero for this case. for some value of y. Periodicity arguments suggest that this will be a series in r 6/(1+ξ) , and if ξ is chosen according to (2.5) or (2.6) then y = 2 − 2h UV and the perturbative expansion (4.10) is matched, with h UV the conformal dimension of either φ 21 or φ 15 , and all odd terms zero for the φ 21 case [25]. The irregular bulk term B(r) must be subtracted before c pert can be compared with c eff .…”

“…The previously-studied sequence (1.1), (1.2) is picked out by the monotonicity of the function c eff as a function of r. In all other cases, somewhat to our surprise, we found that the effective central charge undergoes a number of oscillations as it interpolates between its short and long distance limits. Our approach differs from the TBA method, and is much closer to that of the papers [20][21][22][23][24][25][26], in that a single nonlinear integral equation (NLIE) is proposed to describe infinitely-many different perturbed conformal field theories, each being picked out by an appropriate choice of certain parameters. For massless flows, the one previous example of such an equation was found by Al.…”

“…The parameter ξ, related to the a (2) 2 coupling γ as ξ = γ/(2π − γ), is the analogue of the sine-Gordon parameter ζ mentioned earlier, while α corresponds to the twist. To obtain massive φ 12 , φ 21 and φ 15 perturbations of a minimal model M p,q , the values of ξ and α must be chosen as follows [25]:…”

Massless flows between minimal W models

“…2 [42], with appropriate tuning of the free parameters. Thus by level-rank duality the models W A p−2,p p−3 have up to two extra massive perturbations in addition to Φ adj whose groundstate energy can be described in terms of a nonlinear integral equation.…”