Recursive monodromy conjecture for irreducible discriminant slices

From papers

Let Xζ+X_{\zeta_{+}} be any Calabi–Yau threefold constructed by the GLSM flop procedure, with coordinates z1,,zrz_{1},\ldots,z_{r} and divisors D^α1,α\widehat D_{\alpha-1,\alpha}. Assume that each intersection ΔD^α1,α\Delta\cap\widehat D_{\alpha-1,\alpha} is irreducible. Let n(α)n^{(\alpha)} be the corresponding sums of multi-degrees, let LαL_{\alpha} be the large-volume monodromies, and let MαM_{\alpha} denote the successive discriminant monodromies. Recursive monodromy conjecture. The monodromy M0M_{0} has the recursive form

Mr:=TOXζ+,Mα=(Mα+1Lα+1)n(α+1)Lα+1n(α+1),α=0,,r1,M_{r}:=T_{\mathcal O_{X_{\zeta_{+}}}},\qquad M_{\alpha}=(M_{\alpha+1}L_{\alpha+1})^{n^{(\alpha+1)}}L_{\alpha+1}^{-n^{(\alpha+1)}},\qquad \alpha=0,\ldots,r-1,

and at every stage

(MαLα)n(α)=(LαMα)n(α),α=1,,r.(M_{\alpha}L_{\alpha})^{n^{(\alpha)}}=(L_{\alpha}M_{\alpha})^{n^{(\alpha)}},\qquad \alpha=1,\ldots,r.

Moreover, these forms are independent of the labeling of z1,,zrz_{1},\ldots,z_{r}. The conjecture proposes a uniform recursive description of monodromy for all such GLSM constructions under the stated irreducibility hypothesis.

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Sources & referencesView supporting material

Primary source

Ban Lin and Mauricio Romo, “Monodromy of Calabi-Yau threefold flops via grade restriction rule and their quantum Kahler moduli”, arXiv:2605.18514 (2026).

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