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MACOLAB Report Summary

Project ID: 747555
Funded under: H2020-EU.1.3.2.

Periodic Reporting for period 1 - MACOLAB (Towards a mathematical conjecture for the Landau-Ginzburg/conformal field theory correspondence and beyond)

Reporting period: 2017-08-01 to 2019-07-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Research output in the period August 2017-May 2018: 3 preprints in preparation,
- 1 on spectral flows and conjugation morphisms in categories of MFs,
- 1 on algorithmic approaches to orbifold equivalence of potentials, and
- 1 on higher categorical structures within LG/CFT.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

"I have found interesting results implying the existence of structures (""spectral flows"") predicted by physics in categories of matrix factorizations. Jointly with my PhD student, I have developed an algorithmic approach that will help finding further equivalences in this setting. Last but not least, I also have made progress towards getting a clear understanding of the higher categorical structures lying behind this correspondence."

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