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Résumé |
A growing body of evidence suggests that the complexity of physical integrals is
most naturally understood through geometry. Recent mathematical developments by
Kontsevich and Soibelman[arXiv:2402.07343] have illuminated the role of
exponential integrals as periods of twisted de Rham cocycles over Betti cycles,
offering a structured approach to address this problem in a wide range of
settings. In this talk, I will first introduce the key tools underlying this
structure and then apply them to show how families of physically relevant
integrals, ranging from holomorphic exponentials to logarithmic multivalued
functions, can be reformulated within this language. For holomorphic exponents, I
will present an explicit decomposition of a family of integrals into thimbles
expansion together with a detailed analysis of the wall-crossing structure behind
the analytic continuation of its relevant parameter. Finally, I will discuss the
generalization to multivalued functions, which provides the appropriate framework
for describing Feynman integrals in special representations.
In this context, the thimble decomposition is expected to match the decomposition
into Master Integrals, while the study of the wall-crossing structure yields a
precise count of independent Master Integrals (or periods), circumventing
complications arising from Stokes phenomena |
