Models for the hidden dimensions of string theory

Ludvig Svensson will receive his doctoral degree in mathematics from Chalmers University of Technology and the University of Gothenburg in 2026. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor José Ignacio Burgos Gil at the Institute of Mathematical Sciences, Madrid, Spain. 

The project’s research area is complex analysis and its intersection with mathematical physics. Special geometric objects, Calabi–Yau manifolds, are of particular interest, and have been shown to provide solutions to the field equations in Einstein's general theory of relativity.

Calabi–Yau manifolds gained widespread attention through their role in modern physics’ string theory, which aims to unite quantum physics with general relativity. However, string theory predicts a ten-dimensional spacetime, while our world is essentially four-dimensional: three space dimensions and one time dimension. The remaining six dimensions of string theory are therefore thought to be so tiny that they are hidden to us. Calabi–Yau manifolds describe the geometry of these hidden extra dimensions. The manifolds also display mirror symmetry: each manifold has a mirror partner, and even if the two manifolds in the pair are geometrically different, they give rise to the same physics.

Of particular interest in mirror symmetry are situations in which the geometry of Calabi–Yau manifolds degenerates. This can be investigated using mathematical objects called period integrals. The period integrals associated with a Calabi–Yau manifold contain rich information about its geometry and arithmetic. 

When the Calabi–Yau geometry degenerates, the corresponding period integrals often become divergent, meaning they are infinite. Despite this, it is possible to extract a finite part from them. The project aims to investigate these finite parts of divergent Calabi–Yau period integrals and determine the extent to which they still have interesting arithmetic and/or geometric information, comparable to that of their convergent counterparts. 

Photo: Julia Romell