Categorification and mirror symmetry for Grassmannians

TL;DR

The paper builds a cohesive framework linking the cluster structure of the Grassmannian coordinate ring $\mathbb{C}[Gr(k,n)]$ with a Frobenius $2$-Calabi–Yau categorification $\operatorname{CM}C$, and introduces two generalized cluster characters, $\mathcal{P}^T_M$ and $\mathcal{F}^T_M$, associated to a cluster tilting object $T$. By employing a weight map between Grothendieck groups and deploying network/cluster charts, the authors relate these invariants to Fu–Keller’s cluster character and connect leading exponents to $g$-vectors and the invariant $\boldsymbol{\kappa}(T,M)$, thereby describing Newton–Okounkov bodies and their polyhedral structure. They demonstrate that the GV-monoid is rational polyhedral and that its tropicalisation recovers the Marsh–Rietsch-Williams mirror–symmetry cone via the superpotential $W$, establishing a categorical Grassmannian mirror symmetry in the RW sense and extending to positroid subvarieties. The work also clarifies the role of weight modules, adjunctions, and syzygies in relating endomorphism algebras $A$ to necklace algebras $B$, and it provides a robust framework connecting classical dimer model combinatorics with modern cluster theory and mirror symmetry.

Abstract

The homogeneous coordinate ring $\mathbb{C}[\operatorname{Gr}(k,n)]$ of the Grassmannian is a cluster algebra, with an additive categorification $\operatorname{CM}C$. Thus every $M\in\operatorname{CM}C$ has a cluster character $Ψ_M\in\mathbb{C}[\operatorname{Gr}(k,n)]$. For any cluster tilting object $T$, with $A=\operatorname{End}(T)^{\mathrm{op}}$, we define two new cluster characters, a generalised partition function $\mathcal{P}^T_M\in\mathbb{C}[K(\operatorname{CM}A)]$, whose leading exponent is $g$-vector/index of $M$, and a generalised flow polynomial $\mathcal{F}^T_M\in\mathbb{C}[K(\operatorname{fd}A)]$, whose leading exponent is $\boldsymbolκ(T,M)$, an invariant introduced in earlier paper. These (formal) polynomials are related by applying a map $\operatorname{wt}\colon K(\operatorname{CM}A)\to K(\operatorname{fd}A)$ to their exponents. In the $\mathbb{X}$-cluster chart corresponding to $T$, the function $Ψ_M$ becomes $\mathcal{F}^T_M$. Further more when $T$ mutates, $\mathcal{F}^T_M$ undergoes $\mathbb{X}$-mutation and $\boldsymbolκ(T,M)$ undergoes tropical $\mathbb{A}$-mutation. We show that the monoid of $g$-vectors is given by a rational polyhedral cone, which can be described, following Rietsch-Williams' mirror symmetry strategy, by tropicalisation of the Marsh-Reitsch superpotential~$W$ and, from that, by module-theoretic inequalities. In the process, the NO-body of Rietsch--Williams can be described in terms of $\boldsymbolκ(T,M)$. This leads to a categorical incarnation of Grassmannian mirror symmetry, in the sense of Rietsch-Williams. Some of the machinery we develop works in a greater generality, which is relevant to the positroid subvarieties of $\operatorname{Gr}(k,n)$.

Figures (11)

• Figure 2.1: Crossings and their corresponding alignments
• Figure 7.1: Plabic graph, with strand diagram, and (opposite) dual quiver
• Figure 7.2: Left target labelling and the Gabriel quiver of $\operatorname{End}_B(T)^{\mathrm{op}}$
• Figure 7.3: Some matchings and their boundary values
• Figure 7.4: Cohomological flows $\mathfrak{m}_*-\mathfrak{m}$ and their weight vectors

Theorems & Definitions (266)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5
• Lemma 2.6
• proof