Intersection theory

Many integrals in physics share the same structure: a multivalued function, such as xa(1−x)b with non-integer exponents, integrated along a path. Euler's Beta function is the simplest example; Feynman integrals in dimensional regularization and string amplitudes are far richer ones. Because the integrand is multivalued, the "right" integration domains and the "right" integrands are not arbitrary: they live in two finite-dimensional vector spaces, called twisted homology and twisted cohomology.

Intersection numbers as scalar products

The key idea of intersection theory is that these vector spaces come with natural scalar products, the intersection numbers. Once you have a scalar product you can do linear algebra: project any integral onto a basis of master integrals directly, without solving huge linear systems, and derive the differential equations the integrals satisfy.

The same structure produces quadratic relations between integrals. A beautiful example from string theory: the Kawai–Lewellen–Tye relations, which write closed-string amplitudes as "squares" of open-string ones, turn out to be twisted versions of the classical Riemann period relations.

Beyond the standard framework

This machinery works smoothly when the geometry is "generic". Real physical problems often are not: singularities can be degenerate, and exponential factors introduce irregular behaviour. Part of my research is extending the framework to these cases, using tools from D-module theory, Hodge theory and exponential periods.

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