Scientific goals

This CRC is a joint venture of mathematicians and physicists. It has equally strong motivations
from both fields. The interplay between mathematics and theoretical physics is highly beneficial
for both sides. Mathematics provides concepts for the formulation of physical theories, and in-
struments for deriving their predictions. Physics has suggested profound and surprising relations
between different parts of mathematics. This CRC brings both of these directions together.

Some of the most important questions about the origin and fate of our universe, and about the
nature of the basic constituents of matter are still wide open. While we do have very promising
candidates for fundamental theories that could answer such questions in principle, there exists a
large tension between the elegance and simplicity of the basic principles underlying fundamental
theories like quantum field theory and string theory, on the one hand, and the enormous difficul-
ties arising in applications to concrete physical questions, on the other hand. New mathematics
will be needed to overcome obstacles having prevented progress on fundamental problems of
physics for a long time. Three of the most central difficulties are the following ones.

First, it is often not clear a priori which quantity or mathematical object is best-suited to
exhibit the physical content of a given theory. Quantum field theory and string theory can be
formulated as theories of fields, quantities varying within space and time like the electric fields.
An important feature of all successful fundamental theories is the fact that two different field
configurations can describe the same state of a physical system. An operation mapping a field
configuration to another one describing the same state is called gauge symmetry. In order to
extract physical predictions one needs to describe the gauge symmetries precisely.

The values of fields at particular points are only auxiliary pieces of data, not per se revealing
physical information. There may, however, exist specific averages over the field values that do
have direct physical relevance. Such averages are observables measuring non-local correlations
called defects. Their definition depends on the choice of the region in space-time over which the
average is performed. Such regions can have varying dimensions, and may have the shapes of lines
or surfaces, for example. A lot of the physical information is contained in the relations among
defects associated to different regions. This leads to a rich interplay between the potentially
complicated geometry of the regions in space and time defining defects, and the problem to
exhibit the physical content of fundamental theories.

While the equations underlying fundamental theories may have a simple structure, finding
solutions usually turns out to be extremely hard. The known approximation schemes, in the
context of quantum field theory usually called perturbation theory, often do not provide enough
accuracy for applications. Effects that cannot be seen using perturbation theory are called non-
perturbative effects. The ubiquity of such effects is one of the main obstacles preventing many
applications of the known fundamental theories. Constructing and analysing solutions to the
basic equations defining fundamental theories is an enormous mathematical challenge.

New mathematics is needed to address these three issues.

Modern mathematics has recently begun to develop the relevant instruments, often driven
by purely mathematical motivations, and sometimes inspired by interactions with physics. The
terminology higher structures broadly refers to mathematical concepts suitable for describing
hierarchies of relations among mathematical objects associated to regions of varying dimensions,
and for the precise description of gauge symmetries. Moduli spaces are auxiliary geometric
spaces having points associated to the sets of field configurations whose elements can be related
to each other by gauge symmetries. Integrability is an additional feature that the equations of
mathematical physics can exhibit, allowing one to compute non-perturbative effects exactly. It
teaches us a lot about the mechanisms behind such effects, and about their consequences.

Our CRC will combine research on higher structures, moduli spaces and integrability in a
completely new way. Within mathematics our project will pave the way towards a mathematical
synthesis of results on these topics. The unifying theme is to build new mathematics at the
interface between algebra and geometry mediated by mathematical aspects of quantum field
theories and string theory, and to discover new relations between different parts of mathematics.
This type of interaction has already led to several mathematical breakthroughs like mirror
symmetry in the past, and much more mathematics of this depth remains to be discovered.

This new mathematics will furthermore allow us to develop powerful techniques to investigate
paradigmatic examples of quantum field theories and string theories in non-perturbative regimes.
In this way we will overcome obstacles which for a long time have hampered progress in these
directions. The expected benefits for mathematics and theoretical physics are equally high.