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Landau singularities and singularities of holonomic integrals of the Ising class. (English) Zbl 1137.81017
In particle physics integral representations of functions describing physical processes are often used. The singularities of these integrals are associated with physical threshold in each of the physical regions in which they describe the process under consideration. The question of forecasting the location of the singularities of functions defined through integral representations is, thus, a general and interesting problem in mathematical physics [see {\it D. Kreimer}, Knots and Feynman diagrams. Cambridge Lecture Notes in Physics. 13. Cambridge: Cambridge University Press (2000; Zbl 0964.81052), (ch. 9)]. The authors consider families of multiple and simple integrals of the ’Ising class’ and linear ordinary differential equations over $\Bbb{C}(w)$ of which they are solutions. They compare the full set of singularities of these linear ODE and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, it is shown that the Landau conditions can be worked out, either to give the singularities of the correspondent linear differential equation or the singularities occurring in the integrals. Unfortunately, in the general case, the multiple integrals are too involved for the Landau conditions to be worked out.

81Q30Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
34M35Singularities, monodromy, local behavior of solutions, normal forms
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