SFB 1624:  Higher structures, moduli spaces and integrability

• Subject Area: Mathematics ; Physics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 506632645

This CRC is a joint venture of mathematicians and physicists. It has equally strong motivations from both fields. The interaction is highly beneficial for both sides. Mathematics provides concepts for physical theories, and instruments for deriving the predictions. Physics suggests several profound and surprising relations between different parts of mathematics. Many important questions about the origin of our universe, and about the basic constituents of matter are still wide open. While we do have very promising candidates for fundamental theories, there are huge problems in deriving predictions for concrete physical questions from them. New mathematics is needed to overcome these obstacles. Three of the most central difficulties are: First, it is often not clear which quantity or mathematical object is best-suited to exhibit the physical content of a theory. Quantum field theory and string theory can be formulated as theories of fields, quantities varying within space and time. However, different field configurations can describe the same physics. Second, one needs to find averages over the fields called defects having direct physical relevance. Defects depend on the choice of the region in space-time over which the average is performed. One needs to describe the relations among defects associated to regions of varying geometric shapes. Third, solving the equations defining fundamental theories can be very hard. Constructing and analysing solutions is an enormous mathematical challenge. Modern mathematics develops instruments addressing these issues. Higher structures are mathematical concepts which can describe hierarchies of relations among mathematical objects associated to regions of varying dimensions. Moduli spaces are auxiliary geometric spaces having points associated to the sets of field configurations which can be considered equivalent. Integrability is an additional feature that the equations of mathematical physics can exhibit, allowing one to find exact solutions from which we can learn a lot. Our CRC will combine research on higher structures, moduli spaces and integrability in a completely new way. It will pave the way towards a mathematical synthesis of results on these topics and discover new relations between different parts of mathematics. This type of interaction has already led to several mathematical breakthroughs like mirror symmetry. This new mathematics will also allow us to develop powerful techniques to solve paradigmatic examples of quantum field theories and string theories exactly. In this way we will overcome obstacles which have hampered progress in these directions for a long time. An ideal blend of expertise in the relevant areas of pure mathematics and theoretical physics, combined with a very successful tradition of interactions between these disciplines, make Hamburg a perfect place for this line of research.

• DFG Programme: Collaborative Research Centres

Current projects

• A01 - Higher dimensions and categorification (Project Heads Reutter, David Jakob ;  Schweigert, Christoph ;  Wedrich, Paul )
• A02 - Stratified spaces and factorisation homology techniques (Project Heads Dyckerhoff, Tobias ;  Runkel, Ingo ;  Scheimbauer, Claudia ;  Schweigert, Christoph )
• A03 - Derived TFT and supersymmetric field theory (Project Heads Möller, Sven ;  Runkel, Ingo ;  Schweigert, Christoph ;  Teschner, Jörg )
• A04 - Quantized Chern-Simons observables as discrete integrable systems (Project Heads Runkel, Ingo ;  Schomerus, Volker )
• B01 - Generalised complex structures and homological algebra (Project Heads Cortés, Vicente ;  Holstein, Julian )
• B02 - Conformal manifolds and tt∗-geometry (Project Heads Pomoni, Ph.D., Elli ;  Weigand, Timo )
• B03 - Moduli Spaces and Quantum Gravity (Project Heads Weigand, Timo ;  Westphal, Alexander )
• C01 - Topological Fukaya categories, knots, and topological strings (Project Heads Dyckerhoff, Tobias ;  Teschner, Jörg ;  Wedrich, Paul )
• C02 - Quantum corrected moduli spaces and integrability (Project Heads Arutyunov, Gleb ;  Cortés, Vicente ;  Teschner, Jörg )
• C03 - Elliptic Integrability in 4D N = 2 gauge theories (Project Heads Arutyunov, Gleb ;  Pomoni, Ph.D., Elli )
• C04 - Elliptic Ruijsenaars-Schneider models and supersymmetric gauge theories (Project Heads Arutyunov, Gleb ;  Lawrie, Craig ;  Schomerus, Volker )
• MGK - Integrated research training group (Project Head Runkel, Ingo )
• Z - Central tasks of the CRC (Project Head Teschner, Jörg )

• Applicant Institution: Universität Hamburg
• Participating Institution: Deutsches Elektronen-Synchrotron (DESY)
• Participating University: Technische Universität München (TUM)
• Spokesperson: Professor Dr. Jörg Teschner