Permutation-equivariant Quantum K-theory Viii. Explicit Reconstruction

In Part VII, we proved that the range LX of the big J-function in permutation-equivariant genus-0 quantum K-theory is an overruled cone, and gave its adelic characterization. Here we show that the ruling spaces are Dq-modules in Novikov’s variables, and moreover, that the whole cone LX is invariant under a large group of symmetries of LX defined in terms of q-difference operators. We employ this for the explicit reconstruction of LX from one point on it, and apply the result to toric X, when such a point is given by the q-hypergeometric function. Adelic characterization We begin where we left in Part VII: at a description of the range L ⊂ K in the space K of K(X) ⊗ Λ-value rational functions of q of the J-function of permutation-equivariant quantum K-theory of a given Kähler target space X: J := 1− q + t(q) + ∑