Momentum-conserving type-II defects for the exceptional simply-laced affine Toda field theories $e_6^{(1)}$, $e_7^{(1)}$, $e_8^{(1)}$ and their defect Lax pairs

Bristow and Bowcock obtained momentum-conserving type-II integrable defects for the affine Toda field theories based on the classical series $a_n^{(1)}$, $b_n^{(1)}$, $c_n^{(1)}$ and $d_n^{(1)}$, but the exceptional simply-laced series $e_6^{(1)}$, $e_7^{(1)}$, $e_8^{(1)}$ resisted every attempted construction: no splitting of the root space into the 1-space and 2-space required by their framework is known, and no proof excludes one. In addition, the classical integrability of these multi-field defects is established by a defect Lax pair (zero-curvature representation) only for two low-rank cases, the Tzitzéica model (the $a_2^{(2)}$ affine Toda theory) and $d_4^{(1)}$. The problem asks for explicit momentum-conserving type-II defects for the three exceptional theories — or a proof that none exist — together with defect Lax matrices establishing their integrability, which would complete the classification of integrable defects over all affine Toda field theories.

Background

Affine Toda field theory (ATFT) is a family of integrable relativistic field theories in two dimensions. The fields $u=(u_1,\dots,u_r)$ take values in the real vector space spanned by the simple roots $\alpha_1,\dots,\alpha_r$ of a simple Lie algebra, and the Lagrangian is $\mathcal{L}=\frac12\,\partial_\mu u\cdot\partial^\mu u-U$ with $U=(m^2/\beta^2)\,\sum_{i=0}^{r} n_i\, e^{\beta\,\alpha_i\cdot u}$, where the affine root $\alpha_0$ extends the Dynkin diagram to the corresponding affine Kac–Moody algebra, the integers $n_i$ are the Kac labels, and $\beta$ is the coupling. In the simply-laced cases — $a_n^{(1)}$, $d_n^{(1)}$, $e_6^{(1)}$, $e_7^{(1)}$, $e_8^{(1)}$, named by their Dynkin diagrams — all roots have equal length, and the theories support classical solitons and are classically integrable in the Liouville sense: they possess a Lax pair, that is, a pair of matrices depending on an auxiliary spectral parameter whose zero-curvature condition is equivalent to the field equations and generates an infinite tower of conserved charges.

An integrable defect is a local modification of a field theory on a line (say $x=0$) that joins two copies of the theory, one on each side, while preserving the conserved quantities that make the bulk theory solvable; physically, solitons pass through the defect with a delay and possibly an internal change rather than dispersing into radiation. Type-I defects, relating a single field on each side and generalizing the Bäcklund transformations (classical maps taking one solution of an integrable equation to another) of the sine-Gordon model, are only available for single-field or very special multi-field theories. Type-II defects were introduced for multi-field theories: the sewing of the two bulk Lagrangians at the defect line involves an auxiliary pair of fields and is governed by two functions $D(u,\tilde u)$ and $\bar D(u,\tilde u)$ built from the fields $u$ and $\tilde u$ on the two sides. Energy conservation holds automatically in this framework, but momentum conservation imposes a nontrivial algebraic identity on $D$ and $\bar D$, which is the real obstruction.

Bristow and Bowcock (2017) showed that momentum-conserving type-II defects exist for the ATFTs based on the classical algebras $a_n^{(1)}$, $b_n^{(1)}$, $c_n^{(1)}$ and $d_n^{(1)}$. Their construction splits the root space into two subspaces — a 1-space and a 2-space — whose associated exponential sums enter the defect potentials $D$ and $\bar D$ in inequivalent ways, and it fixes constant matrices appearing in the sewing (among them a matrix $W$ with $W^2=-I$ and, in principle, an antisymmetric matrix $A$) together with the auxiliary-field sector. Robertson (2014) established a folding map that derives defects in the non-simply-laced theories from defects in their simply-laced parents, so the $a_n^{(1)}$ and $d_n^{(1)}$ constructions also deliver the folded cases. Two gaps remained. First, for the exceptional algebras $e_6$, $e_7$, $e_8$ no appropriate splitting — with or without a nonzero $A$ or additional paired auxiliary fields — was found. Second, integrability of the new multi-field defects was supported by momentum conservation, soliton transit and a Bäcklund interpretation, but not by a Lax pair; Bristow (2017) constructed defect Lax matrices satisfying zero curvature only for the Tzitzéica model ($a_2^{(2)}$) and for $d_4^{(1)}$.

Problem Statement

For each of the three exceptional simply-laced affine Toda field theories $e_6^{(1)}$, $e_7^{(1)}$ and $e_8^{(1)}$, construct a momentum-conserving type-II defect within the Bristow–Bowcock framework — that is, specify the splitting of the root space into a 1-space and a 2-space, the constant matrices and auxiliary-field assignments entering the sewing (allowing a nonzero antisymmetric matrix $A$ and nontrivial paired auxiliary fields if required), and the defect potentials $D$ and $\bar D$ as explicit sums of exponentials satisfying the momentum-conservation identity — or prove that no such defect exists. For every defect so constructed, establish its classical integrability by exhibiting defect Lax matrices: a zero-curvature representation whose bulk limits reproduce the standard Lax pairs of the theory and whose sewing across the defect line reproduces the defect conditions, on par with the Tzitzéica and $d_4^{(1)}$ cases. A systematic method for finding the splitting, $A$-matrix and auxiliary sector is welcomed as the vehicle, but the deliverable is the three constructions (or their impossibility) together with their Lax pairs; via the folding map of Robertson (2014) this completes the classification of integrable defects over all affine Toda field theories.

References

1. Rebecca Bristow and Peter Bowcock, 'Momentum conserving defects in affine Toda field theories', Journal of High Energy Physics 2017 (2017) 153, doi:10.1007/JHEP05(2017)153, arXiv:1612.03002, https://arxiv.org/abs/1612.03002
2. Rebecca Bristow, 'Integrability of generalised type II defects in affine Toda field theory', Journal of High Energy Physics 2017 (2017) 067, doi:10.1007/JHEP11(2017)067, arXiv:1709.03927, https://arxiv.org/abs/1709.03927
3. Edward Corrigan and Cristina Zambon, 'Type II defects revisited', Journal of High Energy Physics 2018 (2018) 019, doi:10.1007/JHEP09(2018)019, arXiv:1807.05747, https://arxiv.org/abs/1807.05747
4. Silvia Penati and Davide Polvara, 'Quantum anomalies in A_r^(1) Toda theories with defects', Journal of High Energy Physics 2019 (2019) 062, doi:10.1007/JHEP06(2019)062, arXiv:1902.10690, https://arxiv.org/abs/1902.10690
5. Patrick Dorey and Davide Polvara, 'Tree level integrability in 2d quantum field theories and affine Toda models', Journal of High Energy Physics 2022 (2022) 199, doi:10.1007/JHEP02(2022)199, arXiv:2111.02210, https://arxiv.org/abs/2111.02210
6. Edward Corrigan and Cristina Zambon, 'Adding integrable defects to the Boussinesq equation', Journal of Physics A: Mathematical and Theoretical 56 (38) (2023) 385701, doi:10.1088/1751-8121/aceec9, arXiv:2306.03772, https://arxiv.org/abs/2306.03772
7. Craig Robertson, 'Folding defect affine Toda field theories', Journal of Physics A: Mathematical and Theoretical 47 (18) (2014) 185201, doi:10.1088/1751-8113/47/18/185201, arXiv:1304.3129, https://arxiv.org/abs/1304.3129