Completing the towers of local charges in involution and their hierarchies in integrable sigma-models with twist function: type-A commutativity, cyclotomic one-parameter families, and the type-D Pfaffian charge

Integrable two-dimensional sigma-models such as the principal chiral model, its deformations, and sigma-models on symmetric spaces and other Z_T-cosets are Hamiltonian field theories whose Lax matrix has a Maillet r/s Poisson bracket with twist function. Lacroix, Magro and Vicedo (JHEP 2017) attached to each regular zero of the twist function an infinite tower of local conserved charges in involution, with degrees given by one plus the exponents of the untwisted affine Kac-Moody algebra, and proved that the Hamiltonian flows of these charges define mutually commuting (zero-curvature) hierarchy connections — but only for underlying Lie algebras of types B, C and D. For type A they verified commutativity only at low degrees and conjectured it in general; they also left open the existence of one-parameter families of commuting charges at cyclotomic zeros and the construction of the Pfaffian-type charge in type D. No proof, refutation, or construction resolving any of these gaps is known in the literature to date, so the completion problem remains open.

Background

A two-dimensional integrable sigma-model is a field theory on a worldsheet with coordinates $(x,t)$ whose prototype is the principal chiral model (PCM): the field is a group-valued map $g(x,t)$ into a real Lie group $G_0$ and the basic currents are $j_\pm = g^{-1}\partial_\pm g$. Sigma-models on symmetric spaces, more generally on $\mathbb{Z}_T$-cosets $G/H$ graded by an automorphism $\sigma$ of order $T$, and integrable Yang–Baxter- and $\lambda$-deformations of these models all share the same Hamiltonian structure: they possess a Lax matrix $\mathcal{L}(\lambda,x)$, a matrix-valued function of $x$ depending meromorphically on a complex spectral parameter $\lambda$, whose Poisson bracket is of Maillet's $r/s$-form. In this bracket the coefficient of the derivative $\delta'(x-y)$ of the delta distribution is not proportional to the coefficient of $\delta(x-y)$; this feature, called non-ultralocality, obstructs the standard quantisation routes and is characteristic of sigma-models. Semenov-Tian-Shansky showed that the $r$- and $s$-matrices combine into a single solution $\mathcal{R}_{12}(\lambda,\mu)$ of the classical Yang–Baxter equation for the loop algebra of the complexified Lie algebra $\mathfrak{g}$, related to the standard rational solution by a twist function $\varphi(\lambda)$, a rational function of $\lambda$ that encodes the pole structure of the model and characterises it.

A zero $\lambda_0$ of $\varphi(\lambda)$ is called regular if $\varphi(\lambda)\mathcal{L}(\lambda,x)$ is regular at $\lambda_0$. Lacroix, Magro and Vicedo (2017) showed that to each regular zero one can attach an infinite tower of local conserved charges $\mathcal{Q}^{\lambda_0}_n$, indexed by a set $\mathcal{E}_{\lambda_0}$ of degrees with $n-1$ an exponent of the untwisted affine Kac–Moody algebra $\widehat{\mathfrak{g}}$ associated with $\mathfrak{g}$ (an affine Kac–Moody algebra is the central extension of the loop algebra of $\mathfrak{g}$; its exponents are obtained from the exponents of the finite algebra, i.e. of the corresponding finite Weyl group, by adding arbitrary non-negative multiples of the Coxeter number $h$, counting repetitions by the rank; for instance the finite exponents are $1,\dots,d-1$ for $\mathfrak{sl}(d)$, the odd integers below the Coxeter number for types B and C, and those odd integers together with one additional exponent in type D (the exponent related to the Pfaffian invariant, see below)). The charge densities are polynomials in the fields of the Lax matrix, obtained from traces of powers of $\varphi(\lambda)\mathcal{L}(\lambda,x)$ evaluated at $\lambda_0$ — more generally from the symmetric ad-invariant tensors of Evans, Hassan, MacKay and Mountain (EHMM tensors), whose degrees reproduce the affine exponents and whose key algebraic property guarantees involution. This generalised the construction of Evans, Hassan, MacKay and Mountain for the PCM (1999) and of Evans and Mountain for symmetric spaces (2000).

Two refinements matter. First, when $T>1$ the origin $\lambda=0$ (and infinity) is distinguished: it is fixed by the cyclotomic action $\lambda\mapsto\omega\lambda$, $\omega=e^{2\pi i/T}$, under which the Lax matrix and twist function are equivariant. Zeros are then called cyclotomic (the origin) or non-cyclotomic, and at a cyclotomic zero some exponents 'drop': the corresponding degree is excluded when a certain residue $r_n$ equals $T-1$. Second, for $\mathfrak{g}$ of types B, C and D at a non-cyclotomic zero there is in fact a one-parameter family $\mathcal{Q}^{\lambda_0}_{2n}(\xi)$ of commuting charges, $\xi\in\mathbb{R}$, built from a generating function following Evans–Hassan–MacKay–Mountain; the charges usually cited are the special case $\xi=0$.

Finally, the Hamiltonian flow of each charge admits a zero-curvature (Lax) representation: there is a matrix $\mathcal{M}^{\lambda_0}_n(\lambda,x)$ such that $\{\mathcal{Q}^{\lambda_0}_n,\mathcal{L}(\lambda,x)\}-\partial_x\mathcal{M}^{\lambda_0}_n(\lambda,x)+[\mathcal{M}^{\lambda_0}_n(\lambda,x),\mathcal{L}(\lambda,x)]=0$, and the flows define an integrable hierarchy precisely when the connections $\nabla^{\lambda_0}_n=\{\mathcal{Q}^{\lambda_0}_n,\cdot\}+\mathcal{M}^{\lambda_0}_n(\lambda,x)$ mutually commute, i.e. satisfy pairwise zero-curvature equations. Lacroix–Magro–Vicedo proved these commutativity statements (their Theorem 6.3) in full for $\mathfrak{g}$ of types B, C and D, and for all types when the two zeros are distinct. When infinity is a regular zero, the source establishes the corresponding statements only in a weak sense: the equalities hold after setting to zero a certain field $\mathcal{C}(x)$, which in the $\mathbb{Z}_T$-coset sigma-models coincides with the gauge constraint. For $\mathfrak{g}=\mathfrak{sl}(d,\mathbb{C})$ of type A, where the invariant tensors are mixed traces rather than powers of a single trace, the proof breaks down: commutativity was verified by explicit computation for degrees $n,m\le 7$ at a non-cyclotomic zero in the diagonal case $\lambda_0=\mu_0$, and for degrees $n,m\le 8$ with $T=2,\dots,6$ at the cyclotomic zero $\lambda_0=\mu_0=0$ when $\sigma$ is non-inner (for types B, C, D and for type A with $\sigma$ inner the cyclotomic case is proved). In the same framework two charges are missing altogether: the one-parameter families at cyclotomic zeros, expected but not constructed, and the charge associated with the Pfaffian invariant of $\mathfrak{so}(2N)$ in type D, which corresponds to the exponent missing from the trace construction and is not of trace type. The whole construction was recast by Vicedo in the language of dihedral affine Gaudin models — classical analogues of the quantum Gaudin models associated with affine Kac–Moody algebras, whose finite-dimensional counterparts have a commuting Gaudin algebra generated in degrees given by exponents plus one (Feigin–Frenkel–Reshetikhin 1994); the regularity of zeros translates into a condition on the Gaudin levels.

Problem Statement

Within the framework of $r/s$-systems with twist function (equivalently, dihedral affine Gaudin models) of Lacroix–Magro–Vicedo (2017), complete the construction of the towers of local charges in involution and their integrable hierarchies at the framework's natural generality — all classical Lie algebra types, cyclotomic and non-cyclotomic regular zeros — by resolving the following cases explicitly left open there.

(i) Prove, for $\mathfrak{g}=\mathfrak{sl}(d,\mathbb{C})$ of type A, that the hierarchy connections $\nabla^{\lambda_0}_n=\{\mathcal{Q}^{\lambda_0}_n,\cdot\}+\mathcal{M}^{\lambda_0}_n(\lambda,x)$ mutually commute in the case of Theorem 6.3 of the source left unproven there: at a non-cyclotomic regular zero $\lambda_0$, the diagonal case $\lambda_0=\mu_0$, i.e. the zero-curvature equation

$\{\mathcal{Q}^{\lambda_0}_n,\mathcal{M}^{\lambda_0}_m\}-\{\mathcal{Q}^{\lambda_0}_m,\mathcal{M}^{\lambda_0}_n\}+[\mathcal{M}^{\lambda_0}_n,\mathcal{M}^{\lambda_0}_m]=0$

for all $n,m\in\mathcal{E}_{\lambda_0}$, removing the restriction to the degrees $n,m\le 7$ up to which it was verified by explicit computation. Pairs of distinct regular zeros, including the case where one zero is non-cyclotomic and the other is the origin, are already proved in the source for all Lie types and are recalled here only as context. The corresponding diagonal statement at $\lambda_0=\mu_0=\infty$, when infinity is a regular zero, is part of the same question in the weak sense of the source (equalities holding after setting the constraint field $\mathcal{C}(x)$ to zero), since the source's method extends there by inversion of the spectral parameter.

(ii) Prove the same zero-curvature statement at the cyclotomic zero $\lambda_0=\mu_0=0$ for $\mathfrak{g}=\mathfrak{sl}(d,\mathbb{C})$ with the automorphism $\sigma$ non-inner, for all degrees $n,m\in\mathcal{E}_0$ and all orders $T\ge 2$ of the cyclotomy, removing the restriction to $n,m\le 8$ and $T=2,\dots,6$ up to which it was verified.

(iii) Construct one-parameter families $\mathcal{Q}^0_n(\xi)$, $\xi\in\mathbb{R}$, of local charges at a cyclotomic regular zero — expected in the source to arise as the first non-zero coefficient of a suitable $\xi$-dependent generating function around $\lambda=0$ — for $\mathfrak{g}$ of types B, C, D and for $\mathfrak{g}$ of type A with $\sigma$ inner, and prove that the charges within each fixed-$\xi$ family are in involution; establishing the associated hierarchy (which is proven only for $\xi=0$, even in the non-cyclotomic case) is a natural companion question and is not required for a complete answer.

(iv) Construct, within the same framework, the local charge(s) in involution corresponding to the Pfaffian invariant of $\mathfrak{g}=\mathfrak{so}(2N)$ in type D, i.e. the charge(s) of the degree(s) missing from the trace construction (related to the rank modulo the Coxeter number), and prove their involution with the full tower; establishing their hierarchy flows is likewise a natural companion question and is not required for a complete answer.

A complete answer establishes (or refutes, with explicit counterexamples) each of (i)–(iv) at this generality; items (i)–(iv) are independent open problems, so resolving any one of them in full is a legitimate answer to that component, and resolving an explicitly listed sub-case within one of them is a legitimate partial resolution of its component, while the completion problem remains open until all four items are settled.

References

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