Native finitization of Bressoud $(k-1)$-restricted paths for $\mathbb{Z}_k$ parafermionic characters: a canonical length assignment

The states of the irreducible modules of the $\mathbb{Z}_k$ parafermionic conformal field theories have two combinatorial descriptions: Andrews-Baxter-Forrester (ABF) restricted solid-on-solid (RSOS) paths in regime II, which carry an intrinsic length and thereby finitize the characters as polynomials, and Bressoud $(k-1)$-restricted lattice paths, the path avatar of the quasi-particle (multiple-partition) basis. Jacob and Mathieu (2007) proved a weight-preserving bijection between the two path families and used it to derive finitized parafermionic characters - but only through the RSOS side, remarking that Bressoud paths "do not embody a natural notion of length". The problem is to construct a native, canonical finitization on the Bressoud-path (multiple-partition) side itself: an intrinsic finite-length truncation defined without importing the RSOS bijection, whose weighted generating functions are polynomials that agree with the RSOS finitized characters under the established correspondence and converge, as the truncation parameter grows, to the full $\mathbb{Z}_k$ parafermionic characters. No such construction is known in the literature.

Background

The $\mathbb{Z}_k$ parafermionic conformal field theories are two-dimensional conformal field theories with a discrete $\mathbb{Z}_k$ symmetry, introduced by Zamolodchikov and Fateev through non-local (parafermionic) currents of fractional spin; they describe the continuum limit of regime II of the Andrews-Baxter-Forrester (ABF) restricted solid-on-solid lattice model. In the ABF model a configuration assigns heights $\sigma_i\in\{0,1,\dots,k\}$ to the sites of a one-dimensional chain, adjacent heights differing by exactly one unit; plotting the heights and joining adjacent points turns each configuration into a lattice path (an "RSOS path"). The one-dimensional configuration sum - the sum of $q^{\mathrm{weight}}$ over all such paths with fixed endpoints - reproduces, in the infinite-length limit, the characters (graded partition functions of the module basis states) of an associated conformal field theory: the unitary minimal models $\mathcal{M}(k+1,k+2)$ in regime III and the $\mathbb{Z}_k$ parafermionic theory in regime II, the two regimes being related, up to an overall factor, by the substitution $q\to q^{-1}$.

A finitized character is a polynomial truncation of an infinite character such that the character is recovered in a limiting process. For the minimal models such polynomials first arose from corner-transfer-matrix (CTM) sums - finite-lattice quantities - and Warnaar's "fermionic solution" of the ABF model expressed them as finite multiple sums over occupation numbers $m_j$ of quasi-particle species with $q$-binomial coefficients $\begin{bmatrix}p_j+m_j\\m_j\end{bmatrix}_q$ (a $q$-binomial is the generating function of partitions inside a $p_j\times m_j$ rectangle), proving the finitized (Melzer-type) polynomial identities for the minimal-model characters.

Independently, the states of the $\mathbb{Z}_k$ parafermionic modules possess a quasi-particle basis: ordered strings of modes of the $k-1$ parafermionic fields $\psi^{(j)}$ of conformal dimension $j(k-j)/k$. The integral parts of the mode labels of the $j$-th field form an ordered partition $\lambda^{(j)}=(\lambda^{(j)}_1\ge\lambda^{(j)}_2\ge\cdots)$ of length $m_j$, and the $k-1$ partitions together (a "multiple partition") satisfy gap conditions $\lambda^{(j)}_l\ge\lambda^{(j)}_{l+1}+2j$ plus lower bounds depending on the higher-charge lengths $m_{j+1},\dots,m_{k-1}$. These multiple partitions are in bijection with Bressoud's $(k-1)$-restricted lattice paths: paths confined to the strip $0\le y\le k-1$ built from up, down, and (at height zero only) horizontal steps, weighted by the sum of the $x$-coordinates of their peaks, each peak carrying a "charge" $1\le j\le k-1$ equal to the height of the peak (so that a peak of charge $j$ has minimal $x$-coordinate $j$); these are the paths behind Bressoud's lattice-path proof of Gordon-type Rogers-Ramanujan identities. The conformal dimension of the parafermionic state corresponding to such a path is its weight minus a fractional correction $m'^2/k$, where $m'=\sum_j j\,m_j$ is the total quasi-particle charge (in the module of highest-weight charge $\ell$ the correction is $m'(m'+\ell)/k$), which accounts for the fractional part of the parafermionic mode numbers.

Jacob and Mathieu (2007) proved a simple weight-preserving bijection between the two path families, so that an RSOS path in regime II corresponds directly to a parafermionic state in the quasi-particle basis, and they used it to derive the finitized characters on the RSOS side. The essential asymmetry is this: an RSOS path of fixed length is a finite object, and fixing the length truncates the configuration sum to a polynomial. Fixing the length amounts to fixing the total quasi-particle charge $m'=\sum_j j\,m_j$ (the geometric length of a connected RSOS path is twice its charge content), and the finite length then bounds how far each peak of charge $j<k$ can be displaced from its position in the minimal-weight configuration - the maximal displacement being the window $p_j=2m_{j+1}+4m_{j+2}+\cdots+2(k-j)m_k$ - which is what turns the binomial configuration-counting factors into $q$-binomials and makes the finitized character a polynomial. On the Bressoud side, by contrast, a path has no intrinsic length: horizontal steps at height zero can extend it indefinitely, so every Bressoud path is naturally an infinite object completed by a tail, and its weight is organized by peak positions rather than by a length count. The paper's finitized vacuum character

$$\chi^{(m')}_0(q)=\sum_{\substack{m_1,\dots,m_k\ge0\\ m_1+2m_2+\cdots+km_k=m'}}q^{h_{\mathrm{mwc}}}\prod_{j=1}^{k-1}\begin{bmatrix}p_j+m_j\\m_j\end{bmatrix}_q$$

(with $h_{\mathrm{mwc}}$ the conformal dimension of the minimal-weight configuration for the given charge content, and $m_k$ the number of charge-$k$ clusters that the Jacob-Mathieu construction inserts to render the corresponding RSOS path connected - a number fixed by the lower-charge data through their connectivity criterion, and one that cancels out of $h_{\mathrm{mwc}}$) is therefore constructed through the RSOS path length, and the paper concludes by stressing that the Bressoud paths "do not embody a natural notion of length". The problem raised by this limitation is to reverse the asymmetry: to build the finitization natively on the Bressoud-path (equivalently, multiple-partition) side.

Problem Statement

Construct a native finitization of the Bressoud $(k-1)$-restricted lattice-path description of the $\mathbb{Z}_k$ parafermionic characters - equivalently, of the multiple-partition (quasi-particle) basis - that is, a canonical length assignment defined intrinsically on the Bressoud side. The single common objective is an intrinsic finite-length structure on Bressoud paths that reproduces the known finitized parafermionic characters; a complete answer consists of the following jointly required, mutually supporting results, each with proof:

1. (Intrinsic definition) A rule, expressed purely in the language of Bressoud paths or of the multiple partitions $\lambda^{(j)}$ (peak positions, peak charges, gap conditions) with no reference to RSOS configuration sums and no recourse to the Jacob-Mathieu bijection, that assigns to each non-negative integer $m'$ a finite family of paths (or truncated multiple-partition sums), the parameter $m'$ playing the role of a length.

2. (Finitized characters) For each irreducible $\mathbb{Z}_k$ module in the sense of the source paper (the vacuum module and the modules of highest-weight charge $\ell=1,\dots,k-1$, each decomposed into its fixed-charge sectors $m'\equiv r \pmod{k}$), the weighted $q$-generating function of the corresponding finite family - the standard Bressoud weight (sum of peak positions) together with the appropriate fractional-dimension correction of the parafermionic modes - is, up to the fixed ground-state monomial $q^{h^{(r)}_\ell}$ of the sector (where $h^{(r)}_\ell=h_\ell+r(k-\ell-r)/k$ and $h_\ell=\ell(k-\ell)/(2k(k+2))$ is the conformal dimension of the highest-weight state, so that in the vacuum module $h^{(r)}_0=r(k-r)/k$), a polynomial in $q$ with non-negative integer coefficients.

3. (Correct limit) As $m'\to\infty$ these finitized generating functions converge coefficient-wise to the corresponding irreducible $\mathbb{Z}_k$ parafermionic characters.

4. (Compatibility with the RSOS side) The finitized polynomials agree with the RSOS-side finitized characters of Jacob-Mathieu (their Sections 3-4) up to an explicit weight-preserving correspondence - equivalently, with the $q\to q^{-1}$ duals of the finitized $\mathcal{M}(k+1,k+2)$ characters - so that the two finitizations coincide under the already-established path bijection.

References

1. P. Jacob and P. Mathieu, "Paths for $\mathbb{Z}_k$ parafermionic models", Letters in Mathematical Physics 81 (2007) 211-226. DOI: 10.1007/s11005-007-0182-y; arXiv:hep-th/0703010; https://arxiv.org/abs/hep-th/0703010
2. P. Mathieu, "Paths and partitions: Combinatorial descriptions of the parafermionic states", Journal of Mathematical Physics 50 (2009) 095210. DOI: 10.1063/1.3157921; arXiv:0903.3550; https://arxiv.org/abs/0903.3550
3. O. Blondeau-Fournier, P. Mathieu and T. A. Welsh, "A bijection between paths for the $\mathcal{M}(p,2p+1)$ minimal model Virasoro characters", Annales Henri Poincare 11 (2010) 101-125. DOI: 10.1007/s00023-010-0031-x; arXiv:0912.0427; https://arxiv.org/abs/0912.0427
4. P. Jacob and P. Mathieu, "A new path description for the $\mathcal{M}(k+1,2k+3)$ models and the dual $\mathbb{Z}_k$ graded parafermions", Journal of Statistical Mechanics (2007) P11005. DOI: 10.1088/1742-5468/2007/11/P11005; arXiv:0709.1658; https://arxiv.org/abs/0709.1658
5. P. Jacob and P. Mathieu, "Particles in RSOS paths", Journal of Physics A: Mathematical and Theoretical 42 (2009) 122001. DOI: 10.1088/1751-8113/42/12/122001; arXiv:0901.1814; https://arxiv.org/abs/0901.1814
6. P. Jacob and P. Mathieu, "Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence", Discrete Mathematics 309 (2009) 878-886. DOI: 10.1016/j.disc.2008.01.029; arXiv:math/0609001; https://arxiv.org/abs/math/0609001
7. J. Lamy-Poirier and P. Mathieu, "Path representation of su(2)_k states I: Operators and particles for k=1,2", Nuclear Physics B 845 (2011) 257-296. DOI: 10.1016/j.nuclphysb.2010.11.019; arXiv:1010.2462; https://arxiv.org/abs/1010.2462
8. J. Lamy-Poirier and P. Mathieu, "Path representation of su(2)_k states II: Operator construction of the fermionic character and spin-1/2-RSOS factorization", Nuclear Physics B 847 (2011) 247-273. DOI: 10.1016/j.nuclphysb.2011.01.027; arXiv:1012.4722; https://arxiv.org/abs/1012.4722
9. S. O. Warnaar, "Fermionic solution of the Andrews-Baxter-Forrester model. I. Unification of TBA and CTM methods", Journal of Statistical Physics 82 (1996) 657-685. DOI: 10.1007/BF02179790; arXiv:hep-th/9501134; https://arxiv.org/abs/hep-th/9501134
10. S. O. Warnaar, "Fermionic solution of the Andrews-Baxter-Forrester model. II. Proof of Melzer's polynomial identities", Journal of Statistical Physics 84 (1996) 49-83. DOI: 10.1007/BF02179577; arXiv:hep-th/9508079; https://arxiv.org/abs/hep-th/9508079
11. J.-F. Fortin, P. Mathieu and S. O. Warnaar, "Characters of graded parafermion conformal field theory", Advances in Theoretical and Mathematical Physics 11 (2007) 945-989. DOI: 10.4310/ATMP.2007.v11.n6.a2; arXiv:hep-th/0602248; https://arxiv.org/abs/hep-th/0602248
12. G. Z. Fehér, P. A. Pearce and A. Vittorini-Orgeas, "One-dimensional sums and finitized characters of $2\times2$ fused RSOS models", Journal of Statistical Mechanics (2018) 073104. DOI: 10.1088/1742-5468/aace1d; arXiv:1804.03332; https://arxiv.org/abs/1804.03332
13. M. Butorac, S. Kožić and M. Primc, "Parafermionic bases of standard modules for affine Lie algebras", Mathematische Zeitschrift 298 (2021) 1003-1032. DOI: 10.1007/s00209-020-02639-w; arXiv:2002.00435; https://arxiv.org/abs/2002.00435
14. D. M. Bressoud, "Lattice paths and the Rogers-Ramanujan identities", in Number Theory, Madras 1987, Lecture Notes in Mathematics 1395, Springer, Berlin (1989), pp. 140-172. DOI: 10.1007/bfb0086403; https://link.springer.com/chapter/10.1007/bfb0086403
15. A. B. Zamolodchikov and V. A. Fateev, "Parafermionic currents in two-dimensional conformal quantum field theory and self-dual critical points in Z(N) invariant statistical systems", Soviet Physics JETP 62 (1985) 215-225 [Russian original: Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 89 (1985) 380-399]; https://inspirehep.net/literature/219875