Calabi-Yau techniques for Namikawa-Weyl groups

TL;DR

The paper tackles computing Namikawa-Weyl groups $W$ for symplectic singularities arising from Calabi–Yau-2 algebras, with a focus on quiver varieties that admit symplectic resolutions. It develops a local-to-global strategy using an $A_ obreak ext{$ ext{∞}$}$-functor to transfer local Kleinian data to the global setting and reads off the monodromy from Dynkin automorphisms, aligning with Namikawa’s semi-explicit framework. A central contribution is the explicit decomposition of $W$ into products over codimension-2 leaves, together with a canonical decomposition formula for dimension vectors, and a stability-preserving functor that clarifies how local and global representations interact. The results recover Wu’s computations in targeted cases and generalize to Calabi–Yau-2 algebras beyond quiver varieties, providing a robust toolkit for Poisson deformation theory and resolution geometry in this class of singularities.

Abstract

The field of symplectic singularities aims to build a 21st century Lie theory. A key development is the Namikawa-Weyl group, which generalizes the classical Weyl group of Lie algebras. Another cornerstone is the integration of categorical Calabi-Yau techniques, capturing the rich algebraic structure of these singularities. In this paper, we develop a systematic strategy to determine Namikawa-Weyl groups for representation spaces of Calabi-Yau-2 algebras, leveraging local-to-global functors, symmetry analysis, and $ A_{\infty} $-methods. Applying this approach to quiver varieties, we carry out the detailed calculations and recover Yaochen Wu's result.

Figures (13)

• Figure 4.1: This figure depicts the possible symmetries among roots in the local quiver. Long equality signs in the form of double strokes have been used to indicate equality among roots. For better readability, the arrows of the quiver have been grayed out. For brevity, we have depicted similar situations only once. For instance, in the $D_4$ case it is equally possible that $β_2 = β_3$ instead of $β_1 = β_2$.
• Figure A.1: This figure depicts the double quiver $\overline{Q}$ which we study as an example.
• Figure A.2: This figure depicts the local quiver associated to the decomposition $α = e_1 + e_2 + e_2 + e_3$. Evidently, the local quiver is isomorphic to the Kleinian $A_3$ quiver.
• Figure A.3: This figure depicts the representations in the leaf $L$. They are direct sums of representations of dimension vectors $e_1$, $e_2$, $e_2$ and $e_3$.
• Figure A.4: This figure depicts two of the three one-parameter families of $θ$-stable representations which lie above the codimension-2 leaf $L$. It is easily verified that the depicted representations are indeed all non-isomorphic and semisimplify to the representation given by $S_2 = (κ_1, κ_1^*)$ and $S_3 = (κ_2, κ_2^*)$. The thick lines visualize the $A_3$ Dynkin diagram. The two one-parameter families precisely correspond to the two outer vertices of the Dynkin diagram.

Theorems & Definitions (46)

• Theorem 2.1: namikawa-II
• Theorem 2.2
• Theorem 2.3: bellamy-schedler
• Theorem 2.4: cb-geometry-moment
• Theorem 2.5: bellamy-schedler
• Theorem 2.6: bellamy-schedler
• Definition 2.1
• Theorem 2.7: bellamy-schedler
• Definition 2.2
• Theorem 2.8: bellamy-schedler