Geometric Models for Calabi-Yau Algebras and Homological Mirror Symmetry

Project Information

STABREP

Grant agreement ID: 838316

DOI

10.3030/838316

Project closed

EC signature date 10 April 2019

Start date 1 September 2020

End date 29 December 2022

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 212 933,76

EU contribution

€ 212 933,76

Coordinated by

UNIVERSITY OF LEICESTER
United Kingdom

Periodic Reporting for period 1 - STABREP (Geometric Models for Calabi-Yau Algebras and Homological Mirror Symmetry)

Reporting period: 2020-09-01 to 2022-08-31

Summary of the context and overall objectives of the project

In this project, I establish new connections between cluster algebras, mirror symmetry and representation theory through the introduction of geometric models. We also explored well-know geometric and combinatorial models of representation theory to explain some phenomena in cluster algebras and mirror symmetry.
Mirror symmetry is a natural phenomenon, first observed in superstring theory, consisting of two main approaches: the A- model, focused on the symplectic side of a Calabi-Yau manifold X, and the B-model, focused on the complex side of the manifold. Mirror symmetry is a duality between the two models. Based on this, Kontsevich formulated his famous homological mirror symmetry conjecture for categories. In this conjecture, the A-model is the Fukaya category of X, and the B-model is the derived category of coherent sheaves of X^. Cluster algebras were introduced in the early 2000s to provide a combinatorial framework for dual canonical bases. Many new ideas in representation theory have their origin in cluster algebras, bringing together category theory, particularly Calabi-Yau categories, combinatorics and the geometry of Riemann surfaces. In exciting recent developments cluster theory and homological mirror symmetry have been linked through scattering diagrams, opening up both theories.
In this project, I study the connections between cluster combinatorics and scattering diagrams through Calabi-Yau algebras, which appear in a natural way in cluster theory and mirror symmetry. My collaborators and I develop geometric models for the representation theory of skew-gentle algebras encoding, in particular, their (co)homology. This will lead to a complete understanding of these algebras and their role in the mirror symmetry program.
Dimer models are intrinsically linked to both cluster algebras and mirror symmetry. As part of my project, I started a generalization of the dimer models to the general setting of special multiserial algebras. Special multiserial algebras are of wild representation type and my geometric models will lead the way to an understanding of stability conditions for wild algebras

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

In this project, I establish new connections between cluster algebras, mirror symmetry and representation theory through the introduction of geometric models. We also explored well-know geometric and combinatorial models of representation theory to explain some phenomena in cluster algebras and mirror symmetry. We got four manuscript, two of them already published, the other two submitted, and we also have four ongoing projects, two of them almost ready. Please find below a list of the achievements during this project.

Papers
1. D. Labardini-Fragoso, S. Schroll, and Y. Valdivieso. Derived category of Skew-gentle algebras and orbifolds. To apper in Glasgow Mathematical Journal. Preprint arXiv:2006.05836.
2. Schroll, S., Treffinger, H., & Valdivieso, Y. (2021). On band modules and τ-tilting finiteness. Mathematische Zeitschrift, 299(3-4), 2405-2417.

Preprints
1. S. schroll, A. Tattar, H. Treffinger, Y. Valdivieso, and N. Williams. Stability spaces of string and band modules
2. E. Fernandez, S. Schroll, S. Trepode, H. Treffinger, and Y. Valdivieso. Split-by-nilpotent decompositions of algebras.

Works in progress:

1. A. Garcia-Elsener, V. Guazelli, and Y. Valdivieso. Skew-Brauer graph algebras and trivial extension of skew-gentle algebras. On preparation
2. A. Garcia-Elsener and Y. Valdivieso. m-cluster tilted algebras of geometric type. On preparation.
3. E. Banaian and Y. Valdivieso. Caldero-Chapoton functions for orbifolds and snake graphs. On preparation.
4. S. Schroll and Y. Valdivieso. Derived category of Skew-gentle algebras and orbifolds: part II. On preparation.

Dissemination talks:
1. Let’s talk about math 2021, Vasconcelos Library, Mexico.
Title: Robots also know math, July 2021.
2. 13ª Meeting of Business Mathematics, University Panameric, Mexico.
Title: Contextual bandits or what your clicks say about you, July 2021.

Invited talks:
1. Third Meeting of Mexican Women Mathematicians, Mexico, October 2021.
2. 54th National Congress of the Mexican Mathematical Society. Mexico, September 2021.
3. New developments in representation theory arising from cluster algebras, Isaac Newton Institute, Cambridge, UK, September 2021.
4. VIRTUAL ARTA 2021.Advances in Representation Theory of Algebras, Canada, May 2021.
5. Meeting on representation theory of algebras – Corona Edition, Canada.

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)

In this project, we established new connections between cluster algebras, mirror symmetry, and representation theory through the introduction of geometric models.
Mirror symmetry is a natural phenomenon. In particular, we obtained the following results.
[R1] We defined a geometric-combinatorial model for derived categories of a generalized version of the well-known gentle algebras.
[R2] We also give a combinatorial description of torsion classes containing band modules, a structure closely related to the space of the stability condition of the module.
[R3] We described the stability spaces of non-thin modules using the stability spaces of abstract string and band modules.
[R4] We gave a recognition theorem of trivial extension algebras in terms of quivers and relations.

Possible impact of our results.
[Im1] Having a geometric-combinatorial model for derived categories has proven to be helpful to compute some homological properties, like exceptional sequences, recollements, and silting theory. Using the geometric-combinatorial model for derived categories is also possible to compute trivial extensions of certain kinds of algebras, in an ongoing work with A. Garcia-Elsener and V. Guazzelli, we are exploiting the geometric model defined in [R1] to give a nice description of the trivial extension of skew-gentle algebras and the derived equivalence of repetitive algebras.
[Im2] Our geometric-combinatorial model for derived categories of skew-gentle algebras describes the indecomposable complexes in simple terms, in an ongoing project with S. Schroll, we are describing the homeomorphism spaces between two complexes in terms of respective geometric representation, which will help us to understand homological properties of the algebra itself.
[Im3] In general, stability conditions are hard to compute, using [R2] and [R3], we can compute in a simple way stability conditions and also the limits of stability spaces, which is helpful to understand a mathematical way to understand Π-stability for D-branes in string theory.
[Im4] Combining [R1], [R4], and a combinatorial description of trivial extensions of algebras given by E. Fernandez, in an ongoing work with A. Garcia-Elsener and V. Guazzelli, we describe, using orbifold dissections admissible cuts of trivial extension of skew-gentle algebras and their positive and negative reflections. This result might help to give a new technique to decide whether two algebras skew-gentle are derived equivalent.

Paper 1 Derived categories of skew gentle algebras

Papar 2 On band modules