Conjectural duality for intertwining operators of quasi-lisse vertex superalgebras

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Let VV be a quasi-lisse vertex superalgebra. Let W1W_1 and W2W_2 be finitely strongly generated VV-modules, let W0W_0 and W3W_3 be ordinary gg-twisted modules, let W4W_4 and W6W_6 be weak gg-twisted modules, and let W5W_5 be a weak VV-module. Let Y1,Y2,Y3,Y4,Y5,Y6 \mathcal{Y}_1, \mathcal{Y}_2, \mathcal{Y}_3, \mathcal{Y}_4, \mathcal{Y}_5, \mathcal{Y}_6 be intertwining operators of types

(W0′W1  W4),(W4W2  W3),(W5W1  W2),(W0′W5  W3),(W0′W2  W6),(W6W1  W3),\binom{W_0'}{W_1\;W_4},\binom{W_4}{W_2\;W_3},\binom{W_5}{W_1\;W_2},\binom{W_0'}{W_5\;W_3},\binom{W_0'}{W_2\;W_6},\binom{W_6}{W_1\;W_3},

respectively. For w1∈W1w_1\in W_1, w2∈W2w_2\in W_2, w3∈W3w_3\in W_3, and w0∈W0′w_0\in W_0', the relevant products and iterates are required to be interpreted as matrix coefficients of intertwining operators.

Duality conjecture. There exists a maximally extended multivalued analytic function f(z1,z2;w1,w2,w3,w4)f(z_1,z_2;w_1,w_2,w_3,w_4) on

{(z1,z2)∣zi≠0,  zi≠zj,  i≠j}\{(z_1,z_2)\mid z_i\neq 0,\;z_i\neq z_j,\;i\neq j\}

such that the three series

⟨w0,Y(w1,z1)Y(w2,z2)w3⟩,\langle w_0,\mathcal{Y}(w_1,z_1)\mathcal{Y}(w_2,z_2)w_3\rangle, ⟨w0,Y4(Y3(w1,z1−z2)w2,z2)w3⟩,\langle w_0,\mathcal{Y}_4(\mathcal{Y}_3(w_1,z_1-z_2)w_2,z_2)w_3\rangle, (−1)∣w1∣∣w2∣⟨w0,Y5(w2,z2)Y6(w1,z1)w3⟩(-1)^{|w_1||w_2|}\langle w_0,\mathcal{Y}_5(w_2,z_2)\mathcal{Y}_6(w_1,z_1)w_3\rangle

are absolutely convergent in the respective regions ∣z1∣>∣z2∣>0|z_1|>|z_2|>0, ∣z2∣>∣z1−z2∣>0|z_2|>|z_1-z_2|>0, and ∣z2∣>∣z1∣>0|z_2|>|z_1|>0, and converge there to branches of f(z1,z2;w1,w2,w3,w4)f(z_1,z_2;w_1,w_2,w_3,w_4).

This is a multivalued analytic formulation of duality for intertwining operators, extending associativity and commutativity-type analytic continuation to the quasi-lisse setting. The supplied text gives no resolution status, so the assertion is recorded as open.

References

Primary source

Hao Li, “Quasi-lisse vertex (super)algebras”, arXiv:2308.04993 (2025).

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