MACOLAB · Towards a mathematical conjecture for the Landau-Ginzburg/conformal field theory correspondence and beyond
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2017-08-01 → 2019-07-31
- EU contribution
- €165,599
- Participants
- 1
- Scheme
- MSCA-IF
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Results in brief
Towards a mathematical conjecture for the Landau-Ginzburg/conformal field theory correspondence and beyond
What is the problem/issue being addressed? The project aims to get a deeper understanding of the relation (suggested by physics, that we will call LG/CFT) between two apparently very different mathematical entities: matrix factorizations (MFs) and representations of vertex operator algebras (VOAs). Why is it important for society? This is a project within pure mathematics, and has little relevance for society. What are the overall objectives? 1) Obtain more equivalences (\mathbb{C}-linear and tensor) between categories of MFs and categories of representations of VOAs, 2) Study further properties shared between these two, 3) Attempt to construct a higher categorical framework where to embed all these equivalences.
Data: CORDIS, © European Union
Project objective
Initially a model to describe superconductivity, Landau-Ginzburg (LG) models were promoted in the late 80s to supersymmetric quantum field theories (QFTs) completely characterized by a polynomial W called potential. They gained importance in string theory and algebraic geometry as they play an interesting role in homological mirror symmetry. On the other hand, conformal field theories (CFTs) have been another kind of QFTs which display conformal symmetry. They have focused many efforts to understand the mathematical structures which encode them, e.g. inspiring the definition of vertex operator algebras (Borcherds, Fields medalist ’88) or pushing forward our knowledge of modular tensor categories. Despite seeming two very different topics, LG models and CFTs are intimately related via a result of theoretical physics — the LG/CFT correspondence— stating that the infrared fixed point of a LG model with potential W is a CFT of central charge c(W). Mathematically this implies equivalences of categories of matrix factorizations (which describe defects of LG models) and categories of representations of vertex operator algebras (which describe defects of CFT). Up to date, we lack a complete understanding of the LG/CFT correspondence and we only have a few examples. The main goal of this Marie Curie is to find a mathematical statement for it, via completing a list of examples, exploring their properties (e.g. tensoriality or even modularity of the categories) and then attacking the main goal. Utrecht University (host institution) is one of the few places in Europe hosting experts in representation, category and Galois theory and mathematical physics, providing exactly the necessary and complementary expertise required to achieve this goal. These results will build a surprising bridge between very different areas of mathematics, opening new research gates completely inspired by physics.
Original text from CORDIS.
Participants
- UNIVERSITEIT UTRECHT · UtrechtCoordinatorNetherlands
Links
Data: CORDIS, © European Union