2024/12/26 by Koroteev, Peter, Smirnov, Andrey · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2412.19383
Let Ψ(z,a,q) a the fundamental solution matrix of the quantum difference equation of a Nakajima variety X. In this work, we prove that the operator Ψ(z,a,q) Ψ(zp,ap,qp2)-1 has no poles at the primitive complex p-th roots of unity q=ζp. As a byproduct, we show that the iterated product of the operators \bf ML(z,a,q ) from the q-difference equation on X: \bf ML (z q(p-1)L,a,q) ⋯ \bf ML (z qL,a,q) \bf ML (z ,a,q) evaluated at q=ζp has the same eigenvalues as \bf ML (zp,ap,qp). Upon a reduction of the quantum difference equation of X to the quantum differential equation over the field of finite characteristic, the above iterated product transforms into a Grothendiek-Katz p-curvature of the corresponding quantum connection whreas \bf ML (zp,ap,qp) becomes a certain Frobenius twist of that connection. In this way, we give an explicit description of the spectrum of the p-curvature of quantum connection for Nakajima varieties.