Abstract
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
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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.
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