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Abstract: We introduce some multiple integrals that are expected to have the samesingularities as the singularities of the $ n$-particle contributions$\chi^{n}$ to the susceptibility of the square lattice Ising model. We findthe Fuchsian linear differential equation satisfied by these multiple integralsfor $ n=1, 2, 3, 4$ and only modulo some primes for $ n=5$ and $ 6$, thusproviding a large set of possible new singularities of the $\chi^{n}$. Wediscuss the singularity structure for these multiple integrals by solving theLandau conditions. We find that the singularities of the associated ODEsidentify up to $n= 6$ with the leading pinch Landau singularities. The secondremarkable obtained feature is that the singularities of the ODEs associatedwith the multiple integrals reduce to the singularities of the ODEs associatedwith a {\em finite number of one dimensional integrals}. Among thesingularities found, we underline the fact that the quadratic polynomialcondition $ 1+3 w +4 w^2 = 0$, that occurs in the linear differential equationof $ \chi^{3}$, actually corresponds to a remarkable property of selectedelliptic curves, namely the occurrence of complex multiplication. Theinterpretation of complex multiplication for elliptic curves as complex fixedpoints of the selected generators of the renormalization group, namelyisogenies of elliptic curves, is sketched. Most of the other singularitiesoccurring in our multiple integrals are not related to complex multiplicationsituations, suggesting an interpretation in terms of motivic mathematicalstructures beyond the theory of elliptic curves.



Autor: S. Boukraa, S. Hassani, J.-M. Maillard, N. Zenine

Fuente: https://arxiv.org/







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