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In this note we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, $$X=T^{*}\Gr(k,n)$$. This integral representation can be used to compute the $$\hbar\to \infty$$ limit of the vertex function, where $$\hbar$$ denotes the equivariant parameter of a torus acting on $$X$$ by dilating the cotangent fibers. We show that in this limit the integral turns into the standard mirror integral representation of the $$A$$-series of the Grassmannian $$\Gr(k,n)$$ with the Laurent polynomial Landau-Ginzburg superpotential of Eguchi, Hori and Xiong.more » « less
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We introduce a Frobenius-like structure for the [Formula: see text] Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind.more » « less
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We consider the master functions associated with one irreducible integrable highest weight representation of a Kac-Moody algebra. We study the generation procedure of new critical points from a given critical point of one of these master functions. We show that all critical points of all these master functions can be generated from the critical point of the master function with no variables. In particular this means that the set of all critical points of all these master functions form a single population of critical points. We formulate a conjecture that the number of populations of critical points of master functions associated with a tensor product of irreducible integrable highest weight representations of a Kac-Moody algebra are labeled by homomorphisms to $$\C$$ of the Bethe algebra of the Gaudin model associated with this tensor product.more » « less
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Abstract We consider a pair of quiver varieties $$(X;X^{\prime})$$ related by 3D mirror symmetry, where $$X =T^*{Gr}(k,n)$$ is the cotangent bundle of the Grassmannian of $$k$$-planes of $$n$$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $$X \times X^{\prime} $$ (the mother function) whose restrictions to $$X$$ and $$X^{\prime} $$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $$X$$ and $$X^{\prime}$$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.more » « less
