Speaker
Description
Holonomic systems are a class of systems of linear (partial or ordinary) differential equations. One of the most fundamental properties of a holonomic system is that its solution space is finite-dimensional.
A Feynman amplitude is the integral of a rational function, or more generally, the product of complex powers of polynomials.
Hence we can, in principle, apply the following two facts in (computational) $D$-module theory:
1.
For (multivariate) polynomials $f_1,\dots,f_d$ and complex numbers $\lambda_1,\dots,\lambda_d$, the multi-valued analytic function $f_1^{\lambda_1}\cdots f_d^{\lambda_d}$ satisfies a holonomic system, which can be computed algorithmically.
2.
If a function satisfies a holonomic system, its integral with respect to some of its variables also satisfies a holonomic system, which can be computed algorithmically.
In the integration, it would also be natural to ragard the integrand as a local cohomology class associated with $f_1,\dots,f_d$, which roughly corresponds to the 'residue' of
$f_1^{\lambda_1}\cdots f_d^{\lambda_d}$ at $\lambda_1 = \cdots = \lambda_d = -1$ (at least in positive mass case for external diagrams).
There are also algorithms for computing a holonomic system for such a cohomology class.
However, actual computation, especially of integration, is hard in general because of the complexity.
I shall present some worked out examples together with an interpretation based on microlocal analysis.
