ABCD and ODEs

Patrick Dorey, Clare Dunning, Davide Masoero, Junji Suzuki and Roberto Tateo

Abstract We outline a relationship between conformal field theories and spectral problems of ordinary differential equations, and discuss its generalization to models related to classical Lie algebras.

1 Introduction

The ODE/IM correspondence [8, 4, 20, 13] has established a link between two dimensional conformal field theory (CFT) and generalised spectral problems in ordinary differential and pseudo-differential equations. It is based on an equivalence between transfer matrix eigenvalues [1, 2] and Baxter Q-functions in integrable models (IMs), and spectral determinants [19, 22] of ordinary differential equations (ODEs).

In statistical mechanics, the transfer matrix and its largest eigenvalue—denoted by T in the following—are central objects. For example, consider the six-vertex model defined on a square lattice with N columns and N rows; T can be written in terms of an auxiliary entire function Q through the so-called Baxter T Q relation.

Patrick Dorey

Dept. of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK, e-mail: p.e.dorey@durham.ac.uk

Clare Dunning

IMSAS, University of Kent, Canterbury CT2 7NF, UK, e-mail: t.c.dunning@kent.ac.uk

Davide Masoero

SISSA, via Beirut 2-4, 34014 Trieste, Italy, e-mail: masoero@sissa.it

Junji Suzuki

Department of Physics, Shizuoka University, Ohya 836, SURUGA, Shizuoka, Japan, e-mail: sjsuzuk@ipc.shizuoka.ac.jp

Roberto Tateo

Dip. di Fisica Teorica and INFN, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy, e-mail: tateo@to.infn.it

Up to an overall constant, Q is completely determined by the knowledge of the positions of its zeros, the Bethe roots, which are constrained by the Bethe ansatz equations (BAE). Subject to some qualitative information on the positions of the Bethe roots, easily deduced by studying systems with small size, the Bethe ansatz leads to a unique set of ground-state roots. In the N → ∞ limit the free energy per site f is simply related to T by

In [1, 2], Bazhanov, Lukyanov and Zamolodchikov showed how to adapt the same techniques directly to the conformal field theory (CFT) limit of the six-vertex model. In this setting, we consider the conformal field theory with Virasoro central charge c = 1 corresponding to the continuum limit of the six-vertex model, defined on an infinitely-long strip with twisted boundary conditions along the finite size direction. The largest transfer matrix eigenvalue T depends on three independent parameters: the (rescaled) spectral parameter ν, the anisotropy η and the twist φ. Defining E, M, l, ω, Ω through the following relations

The Baxter function Q for this largest eigenvalue is fixed by demanding entirety of both T and Q, and reality, positivity and ‘extreme packing’ for l > −1/2 of the set {Ei } of zeros of Q. The BAE follow from the entirety of T and Q via

Surprisingly, (3) and (4) also emerge from an apparently unrelated context: the study of particular spectral problems for the following differential equation

with x and E possibly complex. To see the emergence of (4) from (5), we start from the unique solution ψ (x, E, l) of (5) on the punctured complex plane x C \ {0} which has the asymptotic

solution is entire in E and x. From ψ we introduce a family of solutions to (5) using the ‘Sibuya trick’ (also known as ‘Symanzik rescaling’):

In (7), k takes integer values; any pair {ψk , ψk+1} constitutes a basis of solutions to (5). An alternative way to characterize a solution to (5) is through its behaviour near the origin x = 0. The indicial equation is

and χ −(x, E) = χ (x, E, −l − 1), which transform trivially under Symanzik rescaling as

where the coefficients has been fixed by consistency among (11), (12) and (10) and

Here W [f, g] = f ddgx − g dfdx denotes the Wronskian of f and g. Taking the ratio (11)/(12) evaluated at a zero E = Ei of Q leads immediately to the Bethe ansatz equations (4) without the need to introduce the T Q relation, though in this case it can be done very easily (see, for example the recent ODE/IM review article [13]). Correspondingly, χ becomes subdominant at x → ∞ on the positive real axis: χ (x, Ei , l) ψ (x, Ei , l). The motivation of dealing with χ , instead of ψ (6), is two-fold. Firstly, χ can be obtained by applying the powerful and numerically efficient iterative method proposed by Cheng many years ago [7] in the context of Regge pole theory, and applied to spectral problems of this sort in [11]. To this end we introduce the linear operator L, defined through its formal action

So for any polynomial P (x) of x ,

and the basic differential equation (5), with the boundary conditions (9) at the origin, is equivalent to

Equation (16) is solvable by iteration and it allows the predictions of the ODE/IM correspondence to be checked with very high precision.

The initial results of [8, 4, 20] connected conformal field theories associated with the Lie algebra A1 to (second-order) ordinary differential equations. The generalisation to An−1-models was established in [21, 9] but it was only recently [12] that the ODE/IM correspondence was generalised to the remaining classical Lie algebras Bn, Cn and Dn. Our attempts to derive generalised T Q relations from the proposed set of pseudo-differential equations were unsuccessful, but a series of well-motivated conjectures led us directly to the BAE, allowing us to establish the relationship between BAE and pseudo-differential equation parameters. Moreover, while the numerics to calculate the analogs of the functions ψ turned out to be very costly in CPU time, the generalisation of Cheng’s method proved very efficient and allowed very high precision tests to be performed. This is our second main reason to deal with solutions defined through the behaviour about x = 0, rather than x = ∞.

2 Bethe Ansatz for Classical Lie Algebras

For any classical Lie algebra g, conformal field theory Bethe ansatz equations depending on a set of rank(g) twist parameters γ ={γa } can be written in a compact form as

where Q(a)k (E, γ ) = Q(a)(Ωk E, γ ), and the numbers Ei(a) are the (in general complex) zeros of the functions Q(a). In (17) the indices a and b label the simple roots of the Lie algebra g, and

where the α’s are the simple roots of g. The constant Ω = exp(i h2πμ ) is a pure phase, μ is a positive real number and h is the dual Coxeter number.

It turns out that the Bethe ansatz roots generally split into multiplets (strings) with approximately equal modulus |Ei(a)|. The ground state of the model corresponds to a configuration of roots containing only multiplets with a common dimension da =