Minimal flop paths and the arrangement-complement action

Use the complete-local isolated-cDV smooth setting of Complete-local smooth threefold flops and their slice data, the finite simplicial arrangement of Generic slicing identifies flop stability walls with restricted ADE walls, and the categories of Chambers label smooth minimal models and their NCCR categories. Positive paths of minimum edge length are geodesics as in Geodesic chamber paths. Any two such paths with the same endpoints induce naturally isomorphic compositions of the geometric flop functors. Here naturally isomorphic is used in the sense of Natural isomorphism.

The Deligne groupoid is generated by directed simple wall crossings, with relations identifying all positive minimum paths with the same endpoints, and formal inverses adjoined. Write $\mathcal{G}$ for this groupoid and $U={\mathbb{C}}^n\setminus {\mathcal{H}}_{\mathrm{\Lambda },\mathbb{C}}$ for the complexified arrangement complement. For a chamber $C$, choose a basepoint $x_C\in C\subset U$ and fix a group isomorphism ${\iota }_C:{\pi }_1(U,x_C)\stackrel{\sim }{\to }{\mathrm{Hom}}_{\mathcal{G}}(C,C)$ furnished by the cited vertex-group theorem. Here the fundamental group consists of based loop classes with the multiplication convention of Based loops and the fundamental group; ${\iota }_C$ is chosen as a group isomorphism for this convention, not asserted to be a literal equality or a canonical identification.

The flop and mutation equivalences give a groupoid representation in exact equivalences modulo natural isomorphism. If $F_C$ denotes its restriction to ${\mathrm{Hom}}_{\mathcal{G}}(C,C)$, the induced action of the based fundamental group on the corresponding derived category is ${\rho }_C=F_C\circ {\iota }_C$. For the smooth complete-local setting this action is faithful.

Import Hirano--Wemyss Definition 2.7 and Theorem 2.8, printed p.5, for the Deligne groupoid and the isomorphism type of its vertex group. Corollary 2.12 and its proof, printed p.6, transport a faithful groupoid representation through such an isomorphism to give an injective fundamental-group homomorphism. The based-loop definition is Based loops and the fundamental group. The choices of $x_C$ and ${\iota }_C$ in the Statement specify the transport used here; no canonical choice is claimed.

Import Theorem 1.4(1), printed pp.2--3 (Theorem 6.9(1), pp.18--19), for the geometric minimum-chain relation; and section 6.1, Definition 6.4, Theorem 6.5 and Corollary 6.6, printed pp.17--18, for the faithful mutation representation and group action. Their section 4.1, Remark 4.4, identifies mutation with inverse flop; inversion with matched endpoints accounts for that convention. Theorem 6.9 explicitly establishes the geometric positive-path relation, so it is not inferred by incorrectly identifying a reverse positive wall crossing with the formal inverse of the forward arrow. These are imported results; their torsion-pair and faithfulness proofs are not reconstructed.