Dimensional & algorithmic reductions for Calogero-Ruijsenaars & Landau-Ginzburg models

In this thesis we look at two classes of models in which we explain complicated behaviour of a low-dimensional system by relating it to simple behaviour of a high-dimensional system. In both cases, the high-dimensional system provides insight that is hard to ob- tain directly in the low-dimensional system. The two classes to which we apply this pattern are Calogero–Ruijsenaars models and Landau–Ginzburg systems. The various Calogero–Ruijsenaars models describe n indistinguishable particles in one dimension subject to (in the simplest case) pairwise interactions. They are integrable systems: each has n mutually compatible conservation laws associated to its equations of motion. In the native description, however, these conservation laws are by no means obvious. From the point of view described above, the Calogero–Ruijsenaars models each arise from a higher-dimensional model by identifying orbits of a group action. The high-dimensional model is much simpler: in the simplest case it is free motion of a single particle. The higher-dimensional model therefore has “obvious” conservation laws. Because of the specifics of the group action and identification process (namely “Hamiltonian reduction”), these conservation laws carry over to the smaller system. This yields both an explanation for the conservation laws as well as an explicit way to compute them. In Part I of this thesis we give a detailed description of two instances of this process: the rational Calogero–Moser system and the trigonometric Ruijsenaars–Schneider system. Chapter 3 describes work previously published in [27], and largely follows the exposition there. As a new addition, we include a description of the search process that was used to find the non-generic counter-example from that article. Moreover, we describe improved optimizations and list more examples, including an example for a larger root system. This forms Section 3.4. In Part II of this thesis, we consider the Landau–Ginzburg model. It describes n scalar fields on a two-dimensional space-time with a polynomial in n variables as their interaction term. For compactifying such a model, we are led to consider an object called a matrix factorization of the polynomial. And from there it is a small step to generalize as follows: we consider several distinct domains of space-time in which different polynomials govern the interactions. This is possible as long as we find matrix factorizations connecting the polynomials wherever the domains share a boundary. The view we described at the start of this introduction now applies to the operation of fusing two boundaries. In computational terms, this fusion corresponds to the composition of the two associated matrix factorizations. This composition has a simple formula, but it results in an infinite-rank matrix factorization. We need to apply a reduction step to get a workable matrix factorization, and the result is very non-obvious. In this case, as in the previous, we see an interesting interplay between the high-dimensional description and the low-dimensional one. For example, one needs the high-dimensional version to establish basic properties such as associativity of this fusion process, but it is the low-dimensional version that gives computable results.