Singular parameter regimes and localisation hypotheses in the asymmetric six-vertex model construction of quantum cohomology of Grassmannians

Korff (Commun. Math. Phys. 381:591-640, 2021) constructed 'cylindric Hecke characters' as partition functions of the asymmetric six-vertex model on a cylinder and proved that their restriction coproduct in the Grothendieck ring of type A Iwahori-Hecke algebras has the 3-point genus-0 Gromov-Witten invariants of Grassmannians as structure constants, the link being a Bethe ansatz whose vanishing ideal reproduces the Siebert-Tian presentation of small quantum cohomology. Every identification in this chain is proved only after adjoining an n-th root $q^{1/n}$ of the twist parameter, inverting the Vandermonde $\Delta$ of the Bethe roots, and inverting $\Delta_{a,b}=\prod_{i,j}(a y_i+b y_j)$; moreover the asymmetric six-vertex R-matrices degenerate at special parameter values (for instance $a=-b$, where a Boltzmann weight vanishes and the deleted locus of $\Delta_{a,b}$ is met), with commutativity of transfer matrices currently rescued only through a factorisation into five-vertex models. The objective is to give a regular, localisation-free treatment of this correspondence over the whole parameter space: for each of the three localisation hypotheses, either remove it (proving the ideal equality and ring isomorphism, and the completeness of the Bethe ansatz, in the unlocalised rings) or prove it essential by explicit counterexample; and classify the singular parameter loci of the R-matrix and of the row weights, extending the construction to each locus or proving obstruction. This appears to be untreated in the literature.

Background

The six-vertex model ('ice model') is an exactly solved two-dimensional lattice model of statistical mechanics: each vertex of a square lattice has four incident edges carrying arrow (spin) variables with exactly two arrows in and two out, giving six allowed local configurations weighted by Boltzmann weights $\omega_1,\dots,\omega_6$. The row-to-row transfer matrix builds the model's partition functions. In the classical symmetric six-vertex model (related to the XXZ spin chain) the weights are invariant under simultaneously flipping all arrows; Korff's paper works with the lesser-studied asymmetric six-vertex model, in which this symmetry is broken. The R-matrix $R(\omega)$ acting on two neighbouring two-dimensional spin spaces satisfies the Yang-Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$; the entries $A,B,C,D$ of the monodromy matrix (a product of R-matrices along one lattice row) generate the Yang-Baxter algebra, whose defining relations are the RTT-relations $R_{12}T_1T_2=T_2T_1R_{12}$. In the Baxter framework used in the source ((4.6)-(4.7)), the construction of $R(\omega)$ localises at the weights $\omega_1,\omega_3$ and imposes the matching conditions $\Delta(\omega)=\Delta(\omega')$ and $\Gamma(\omega)=\Gamma(\omega')$ between the two quadrics $\Delta(\omega)=\frac{\omega_1\omega_2+\omega_3\omega_4-\omega_5\omega_6}{2\omega_1\omega_3}$, $\Gamma(\omega)=\frac{\omega_2\omega_4}{\omega_1\omega_3}$ attached to the rows. A particular four-parameter polynomial solution, the intertwiner $R(x_i,x_j;a,a',b,b')$ of Proposition 4.15 (source (4.30)-(4.31)), has weights $\omega_1=b x_i+a' x_j$, $\omega_2=a x_i+b' x_j$, $\omega_3=-a x_i+a' x_j$, $\omega_4=b x_i-b' x_j$, $\omega_5=(a'+b')x_j$, $\omega_6=(a+b)x_i$ with a spectral parameter $x_i$ attached to lattice row $i$; it satisfies the Yang-Baxter relation (4.29) against the monodromy matrices of the single-row weights introduced next, but it is a different object from them and is not specialised to them: since $\omega_1-\omega_3=(a+b)x_i=\omega_6$ in (4.30)-(4.31), the condition $\omega_1=\omega_3$ there forces $\omega_6=0$. On a cylinder of circumference $n$ with quasi-periodic (twist $q$) boundary conditions and $k$ down-spins, the transfer matrix is diagonalised by the algebraic Bethe ansatz: eigenvectors are produced by creation operators applied to a reference 'pseudovacuum' state, with the $k$ auxiliary variables ('Bethe roots') constrained by polynomial Bethe ansatz equations. For the single-row Boltzmann weights (source (4.27)) $\omega_1=\omega_3=\omega_5=1$, $\omega_2=a x_i$, $\omega_4=b x_i$, $\omega_6=(a+b)x_i$ - a two-parameter $(a,b)$-deformation of the free fermion point of the symmetric model, defining the transfer matrices $\tau(x;a,b)$ used throughout the construction - these equations reduce to $\xi_i^n+(-1)^k q=0$, and the Bethe ansatz is complete, i.e. yields all eigenvalues (Proposition 4.17 of the source).

On the geometric side, the small quantum cohomology ring $qH^*(\mathrm{Gr}_k(\mathbb{C}^n))$ of the Grassmannian of $k$-planes in $\mathbb{C}^n$ is a one-parameter deformation of the ordinary cohomology ring whose structure constants are the 3-point genus-0 Gromov-Witten invariants - intersection numbers counting rational curves in the Grassmannian meeting three Schubert varieties, and the curve-counting generalisation of Littlewood-Richardson coefficients. The Siebert-Tian presentation realises it as $\mathbb{Z}[q,e_1,\dots,e_k]/I_{k,n}$ with $I_{k,n}=\langle h_{n+1-k},\dots,h_{n-1},\,h_n+(-1)^k q\rangle$, where $e_i$ and $h_i$ are elementary and complete symmetric polynomials (geometrically, Chern classes of the tautological and quotient bundles).

Korff's main theorem connects the two worlds. Let $H_m(t)$ be the Iwahori-Hecke algebra of type A, a deformation of the group algebra of the symmetric group with deformation parameter $t$, and let $R(t)=\bigoplus_{m\ge0}R_m(t)$ be the Grothendieck ring of its finite-dimensional modules, a Hopf algebra whose coproduct is the restriction functor. Theorem 1.1 constructs, inside $R(t)$, a positive $\mathbb{Z}$-sub-coalgebra spanned by cylindric Hecke characters $\chi_t^{\lambda[d]}$ - virtual characters (integer combinations of irreducible Hecke characters) indexed by a partition $\lambda$ fitting a $k\times(n-k)$ rectangle and a cylindric winding number $d\ge0$ - whose coproduct expansion has the 3-point genus-0 Gromov-Witten invariants $C^{\lambda,d}_{\mu\nu}\in\mathbb{Z}_{\ge0}$ of $\mathrm{Gr}_k(\mathbb{C}^n)$ as structure constants. The characters are computed as partition functions of the asymmetric six-vertex model on the cylinder, and the bridge is the Bethe ansatz: writing $\Lambda^k=\mathbb{C}[y_1,\dots,y_k]^{S_k}$ for the ring of symmetric polynomials in the Bethe-root variables, the vanishing ideal of the Bethe-root set coincides with the Siebert-Tian ideal $I_{k,n}$, so the (localised) coordinate ring of Bethe roots is a two-parameter $(a,b)$-extension of quantum cohomology, and the transfer matrix acts as multiplication by $t$-deformed Chern classes.

The linchpin identifications are, however, proved only after localisation - that is, after passing to a larger ring in which selected elements are formally inverted; statements proved after localisation need not descend, because ideals can strictly grow under localisation, and the inverted elements encode precisely the degenerate situations where the standard arguments break. Three such hypotheses are used: (i) an $n$-th root $q^{1/n}$ is adjoined, since the Bethe roots $\xi_i$ are powers of $q^{1/n}$ (Lemma 4.18); (ii) the Vandermonde $\Delta=\prod_{i\ne j}(y_i-y_j)$ is inverted, which imposes that the $k$ Bethe roots are mutually distinct (Lemmas 4.19 and 4.23 work in $\Lambda^k[q^{\pm 1/n},\Delta^{-1}]$); and (iii) in the $(a,b)$-extended family the element $\Delta_{a,b}=\prod_{i,j=1}^k(a y_i+b y_j)$ is inverted: the converse implication in Lemma 4.22 (polynomiality identities implying the Bethe equations) requires $\Delta_{a,b}$ invertible, and the ideal equality $J_{k,n}[\Delta_{a,b}^{-1}]=(I_{k,n}\otimes\mathbb{C}[a,b])[\Delta_{a,b}^{-1}]$ and the resulting ring isomorphism are stated only for the localised rings (Lemma 4.23). The structure constants are nonnegative integers, but the proof of the coproduct expansion passes through the Bertram-Vafa-Intriligator summation formula, which carries a squared Vandermonde in the denominator. In parallel, some asymmetric R-matrices in the Yang-Baxter relation degenerate for special parameter choices $(a,a',b,b')$ (Remark 4.16); commutativity of the corresponding transfer matrices is nevertheless guaranteed by a factorisation of the $(a,b)$-transfer matrix into transfer matrices of two five-vertex models (the vicious and osculating walker models of Korff, Lett. Math. Phys. 104 (2014)), but no systematic analysis of these singular regimes is given. Notably, the localisation (iii) and the singular regimes are closely related but not identical: $\Delta_{a,b}$ contains the factor $(a+b)^k\prod_i y_i$ (from the terms with $i=j$, where $a y_i+b y_i=(a+b)y_i$), so its deleted locus includes the singular parameter locus $a=-b$, where the Boltzmann weight $\omega_6=(a+b)x_i$ vanishes identically and the model degenerates; but the factors with $i\ne j$ vanish also on root configurations with $a y_i=-b y_j$ at generic parameters (for instance $a=b=1$, $y_2=-y_1$ with all roots pairwise distinct), which are neither parameter singularities nor coincident-root configurations, while $\Delta$ vanishes only on coincident roots and never on parameter loci.

Problem Statement

Establish (or precisely delimit) a regular, localisation-free version of Korff's correspondence between the asymmetric six-vertex model and the small quantum cohomology of Grassmannians, valid over the entire parameter space. Concretely, with the notation of the source paper (transfer matrices $\tau(x;a,b)$ with weights $\omega_1=\omega_3=\omega_5=1$, $\omega_2=a x$, $\omega_4=b x$, $\omega_6=(a+b)x$ on a cylinder of circumference $n$ with $k$ down-spins and twist $q$; Bethe ansatz equations $\xi_i^n+(-1)^k q=0$; symmetric polynomial ring $\Lambda^k=\mathbb{C}[y_1,\dots,y_k]^{S_k}$; vanishing ideal $I_{k,n}=\langle h_{n+1-k},\dots,h_{n-1},h_n+(-1)^k q\rangle$ and its $(a,b)$-deformation $J_{k,n}$, the ideal generated by $f_0=h_n(y;a,b)-q(-1)^k b^{n-k}((-a)^k-b^k)$ and $f_i=h_{i+n}(y;a,b)+q(-1)^k b^n h_i(y;a,b)$ for $i=1,\dots,k-1$, with the deformed complete symmetric functions defined by the generating function $\prod_{i=1}^k\frac{1+a x y_i}{1-b x y_i}=\sum_{r\ge0}x^r h_r(y;a,b)$):

The two questions below arise from source results proved only under the named localisations (Proposition 4.17 and Lemmas 4.18-4.23) and from Remark 4.16; they are untreated in the source.

(1) Localisation hypotheses. For each of the three hypotheses - adjoining $q^{1/n}$, inverting the Vandermonde $\Delta=\prod_{i\ne j}(y_i-y_j)$, and inverting $\Delta_{a,b}=\prod_{i,j=1}^k(a y_i+b y_j)$ - either prove the corresponding statement without it, or prove the localisation is essential by explicit counterexample. The statements in question are: the ideal equality $J_{k,n}=(I_{k,n}\otimes\mathbb{C}[a,b])$ and the ring isomorphism of the Bethe quotient with the two-parameter extension of $qH^*(\mathrm{Gr}_k(\mathbb{C}^n))$ (Lemma 4.23, currently proved only after inverting $\Delta$ and $\Delta_{a,b}$); the converse implication from the polynomiality identities to the Bethe ansatz equations (Lemma 4.22, currently requiring $\Delta_{a,b}$ invertible); and the completeness and diagonalisability of the Bethe ansatz (Proposition 4.17, currently with the Bethe roots assumed distinct and taken as powers of $q^{1/n}$). In this item, removing the hypothesis that $q^{1/n}$ is adjoined means establishing the statements over rings of Laurent polynomials in $q$ - $\mathbb{C}[q^{\pm1}]$, or $\mathbb{Z}[q^{\pm1}]$ where integrality of the structure constants is claimed - following the source's Laurent convention in $\Lambda^k[q^{\pm1/n},\Delta^{-1}]$, rather than over a field extension in which an $n$-th root of $q$ exists automatically; inverting $q$ itself is permitted and does not count as a forbidden discriminant-type inversion, which here refers to inverting $\Delta$ or $\Delta_{a,b}$. For each hypothesis that turns out to be removable, a positive answer should in particular yield the coproduct expansion of cylindric Hecke characters with its Gromov-Witten structure constants - and their integrality - without passing through the localised Bertram-Vafa-Intriligator formula with its squared-Vandermonde denominators.

(2) Singular parameter regimes. Classify the parameter values $(a,a',b,b')$ at which the four-parameter intertwiner $R(x_i,x_j;a,a',b,b')$ of Proposition 4.15 (weights $\omega_1=b x_i+a' x_j$, $\omega_2=a x_i+b' x_j$, $\omega_3=-a x_i+a' x_j$, $\omega_4=b x_i-b' x_j$, $\omega_5=(a'+b')x_j$, $\omega_6=(a+b)x_i$) degenerates: vanishing Boltzmann weights, non-invertible R-matrix, or failure of the Baxter-framework genericity conditions $\Delta(\omega)=\Delta(\omega')$, $\Gamma(\omega)=\Gamma(\omega')$ of the source's (4.6)-(4.7) under which its R-matrices are constructed. (The RTT-relations of Corollary 4.3 are polynomial identities in the weights and therefore do not by themselves define a singular locus; the framework-dependent genericity conditions replace 'breakdown of the RTT-relations' as the degeneracy criterion.) The construction of cylindric Hecke characters depends only on the two-parameter single-row family (4.27), so the extension requirement is scoped to that family's singular loci, which include $a=-b$ - where $\omega_6=(a+b)x$ vanishes identically and the deleted locus of $\Delta_{a,b}$ is met - and the five-vertex endpoints $a=0$, $b=0$: for each such locus, determine which components of the construction (commutativity of transfer matrices, Yang-Baxter algebra, Bethe ansatz, quotient-ring identification) survive at that locus, and for each component that fails, exhibit an explicit counterexample identifying the obstruction.

References

1. Christian Korff, Cylindric Hecke Characters and Gromov-Witten Invariants via the Asymmetric Six-Vertex Model, Communications in Mathematical Physics 381(2):591-640, 2021. DOI: 10.1007/s00220-020-03906-x, arXiv:1906.02565. https://doi.org/10.1007/s00220-020-03906-x (source paper)
2. Christian Korff, Quantum Cohomology via Vicious and Osculating Walkers, Letters in Mathematical Physics 104:771-810, 2014. DOI: 10.1007/s11005-014-0685-2, arXiv:1204.4109. https://doi.org/10.1007/s11005-014-0685-2
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