Elliptic corner VOA correlation conjecture for elliptic Macdonald polynomials

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Let m,n∈Z≥1m,n\in\mathbb{Z}^{\geq 1} with m≤nm\leq n, and let λ∈Par⁡(m)\lambda\in\operatorname{Par}(m). Let ψ~λ(q,ξ)\widetilde{\psi}^{(q,\xi)}_{\lambda} be the specialization map that sends a symmetric function in mm variables to the variables arranged in ℓ(λ)\ell(\lambda) geometric strings with ratios q−1q^{-1} and successive string parameters 1,ξ,…,ξℓ(λ)−11,\xi,\ldots,\xi^{\ell(\lambda)-1}. Let f11(3n)f^{(3^n)}_{11} and T~1(3n)\widetilde{T}^{(3^n)}_1 denote the elliptic corner-VOA structure function and current, respectively, and specialize q1=qq_1=q, q2=q−1tq_2=q^{-1}t, and q3=t−1q_3=t^{-1}. Elliptic corner VOA correlation conjecture. For each such m,n,λm,n,\lambda, there exists Nλ(z1,…,zm)∈Q(q,t)(z1,…,zm)\mathcal{N}_{\lambda}(z_1,\dots,z_m)\in\mathbb{Q}(q,t)(z_1,\dots,z_m) such that the specialized limit

lim⁡ξ→t(ψ~λ(q,ξ)∘∣q1=q,\q2=q−1t,\q3=t−1)(Nλ(z1,…,zm;p)∏1≤i<j≤mf11(3n)(zjzi;p)⟨0∣T~1(3n)(z1;p)⋯T~1(3n)(zm;p)∣0⟩)\lim_{\xi\to t}\left(\widetilde{\psi}^{(q,\xi)}_{\lambda}\circ\left.\right|_{\substack{q_1=q,\q_2=q^{-1}t,\q_3=t^{-1}}}\right)\left(\mathcal{N}_{\lambda}(z_1,\dots,z_m;p)\prod_{1\leq i<j\leq m}f^{(3^n)}_{11}\left(\frac{z_j}{z_i};p\right)\langle0|\widetilde{T}^{(3^n)}_1(z_1;p)\cdots\widetilde{T}^{(3^n)}_1(z_m;p)|0\rangle\right)

exists and equals Pλ(u1,…,un;q,t;p)P_{\lambda}(u_1,\dots,u_n;q,t;p). This conjecturally identifies the relevant elliptic corner-VOA correlation functions with elliptic Macdonald polynomials after the appropriate regularized specialization; the existence of the normalization and the asserted identification are not established in the supplied text.

References

Primary source

Panupong Cheewaphutthisakun, Jun'ichi Shiraishi and Keng Wiboonton, “Elliptic Deformation of the Gaiotto-Rapčák Corner VOA and the Associated Partially Symmetric Polynomials”, arXiv:2406.15860 (2024).

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