We give a brief introduction to a parametric approach for the derivation of shift relations between Feynman integrals and a result on the number of master integrals. The shift relations are obtained from parametric annihilators of the Lee-Pomeransky
polynomial $mathcal{G}$. By identification of Feynman integrals as multi-dimensional Mellin transforms, we show that this approach generates every shift relation. Feynman integrals of a given family form a vector space, whose finite dimension is naturally interpreted as the number of master integrals. This number is an Euler characteristic of the polynomial $mathcal{G}$.
We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space IBP relations, which are well-known in the physics literature. Applying a result of Loeser and Sab
bah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.
Let f be a quasi-homogeneous polynomial with an isolated singularity. We compute the length of the D-modules $Df^c/Df^{c+1}$ generated by complex powers of f in terms of the Hodge filtration on the top cohomology of the Milnor fiber. For 1/f we obtai
n one more than the reduced genus of the singularity. We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the aforementioned quotient is nonzero when c is a root of the b-function of f (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these D-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.
For a smooth variety $Y$ over a perfect field of positive characteristic, the sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf of $PD$-differential operators) is known to be an Azumaya algebra over $T^*_{Y},$ the cotang
ent space of the Frobenius twist $Y$ of $Y.$ Thus to a sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of $T^*_{Y},$ called the $p$-support, namely the support of $M$ seen as a sheaf on $T^*_{Y}.$ We study here the family of $p$-supports assigned to the reductions modulo primes $p$ of a holonomic $mathcal{D}$-module. We prove that the Azumaya algebra of differential operators splits on the regular locus of the $p$-support and that the $p$-support is a Lagrangian subvariety of the cotangent space, for $p$ large enough. The latter was conjectured by Kontsevich. Our approach also provides a new proof of the involutivity of the singular support of a holonomic $mathcal{D}$-module, by reduction modulo $p.$
