Okounkov's monodromy formula for wall operators

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Let M(z)M(z) be a rational operator defining a quantum difference connection, and let Mon(z)\mathsf{Mon}(z) denote its monodromy. For an integral cocharacter or wall parameter w\mathsf{w}, define

Monw=lim⁡q→0Mon(a,zqw).\mathsf{Mon}_\mathsf{w}=\lim_{q\to 0}\mathsf{Mon}(a,zq^\mathsf{w}).

Okounkov's wall-operator conjecture. The wall operator is given by

Bw(zshifted)=Monw⋅Monw+ε−1.\mathsf{B}_\mathsf{w}(z_\text{shifted})=\mathsf{Mon}_\mathsf{w}\cdot\mathsf{Mon}_{\mathsf{w}+\varepsilon}^{-1}.

This conjecture proposes that individual wall-crossing operators can be reconstructed from asymptotic limits of the monodromy of the quantum difference equation. The source gives no resolution status.

References

Primary source

Iakov Kononov, “Elliptic stable envelopes and 3d mirror symmetry”, arXiv:2106.09888 (2021).

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