Periodic Reporting for period 1 - NCMS (A new approach in curve counting theories and mirror symmetry)
Période du rapport: 2022-09-01 au 2024-08-31
[2204.00509] Relative quantum cohomology under birational transformations (arxiv.org). In my work, I studied how relative quantum cohomology, defined using orbifold Gromov—Witten theory, varies understand birational transformations. This property is known as the main difference between orbifold and logarithmic Gromov—Witten invariants. Instead of considering single invariants, I considered a generating function and studied the property on the structural level. I have been revising and improving this article, and now it is under review in Advances in Mathematics.
[2112.12891] Degenerations, fibrations and higher rank Landau-Ginzburg models (arxiv.org). This is joint work with C. Doran and J. Kostiuk. We generalize the Doran—Harder—Thompson conjecture beyond the Calabi—Yau setting and to more general and more complicated degenerations. We have been revising and improving this article, and now it is under review in Advances in Mathematics.
My paper (titled:” The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map”). Given a smooth log Calabi--Yau pair (X,D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau--Ginzburg potential and show that it is a generating function of two-point relative Gromov--Witten invariants of (X,D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem to compute two-point relative invariants. When D is nef, we compute the proper Landau--Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of Grafnitz—Ruddat--Zaslow that the proper Landau--Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.
I further generalize the result to consider the theta functions for more general pairs (simple normal crossing pairs): arXiv:2403.17077. We introduce a new type of orbifold invariants for snc pairs, called mid-age invariants, and use these invariants to define orbifold invariants associated with the broken line type. Then, we define the orbifold theta functions as generating functions of orbifold invariants with mid-ages. We show that these orbifold theta functions are well-defined and satisfy the multiplication rule.
In joint work with Yu Wang (arXiv:2403.17200) we are able to generalize the local-log-orbifold correspondence in an unexpectedly different direction. Given a smooth projective variety X and a smooth nef divisor D, we identify genus zero relative Gromov--Witten invariants of (X,D) with (n+1) relative markings with genus zero relative/orbifold Gromov--Witten invariants of a P1-bundle with n relative markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov--Witten invariants with genus zero absolute Gromov--Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov--Witten invariants.
[2112.12891] Degenerations, fibrations and higher rank Landau-Ginzburg models (arxiv.org). This is joint work with C. Doran and J. Kostiuk. We generalize the Doran—Harder—Thompson conjecture beyond the Calabi—Yau setting and to more general and more complicated degenerations. We have been revising and improving this article, and now it is under review in Advances in Mathematics.
My paper (titled:” The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map”). Given a smooth log Calabi--Yau pair (X,D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau--Ginzburg potential and show that it is a generating function of two-point relative Gromov--Witten invariants of (X,D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem to compute two-point relative invariants. When D is nef, we compute the proper Landau--Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of Grafnitz—Ruddat--Zaslow that the proper Landau--Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.
I further generalize the result to consider the theta functions for more general pairs (simple normal crossing pairs): arXiv:2403.17077. We introduce a new type of orbifold invariants for snc pairs, called mid-age invariants, and use these invariants to define orbifold invariants associated with the broken line type. Then, we define the orbifold theta functions as generating functions of orbifold invariants with mid-ages. We show that these orbifold theta functions are well-defined and satisfy the multiplication rule.
In joint work with Yu Wang (arXiv:2403.17200) we are able to generalize the local-log-orbifold correspondence in an unexpectedly different direction. Given a smooth projective variety X and a smooth nef divisor D, we identify genus zero relative Gromov--Witten invariants of (X,D) with (n+1) relative markings with genus zero relative/orbifold Gromov--Witten invariants of a P1-bundle with n relative markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov--Witten invariants with genus zero absolute Gromov--Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov--Witten invariants.
Project 1 studies a new approach to count curve with tangency conditions using orbifold Gromov—Witten theory of root stacks. The main question here is to understand the relation between the orbifold Gromov—Witten theory and the logarithmic Gromov—Witten theory. Along this direction, I have a preprint titled "Relative quantum cohomology under birational transformations".
Project 2 focuses on applications to mirror symmetry. The main question here is to study the so-called Doran—Harder—Thompson conjecture via various approaches. I have a paper this direction titled "Degenerations, fibrations and higher rank Landau-Ginzburg models". I also have a project with Charles Doran and David Favero on this conjecture via toric mirror symmetry.
Another important part of Project 2 is to study intrinsic mirror symmetry and use it to study the Doran—Harder—Thompson conjecture. I have several results along this direction and have developed necessary techniques to prove the conjecture via intrinsic mirror symmetry. This includes my paper titled:” The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map” and my paper titled: "Orbifold theta functions and mid-age invariants."
For Project 3 on the relation with other theories, I have a preprint with Yu Wang titled "Gromov--Witten theory beyond maximal contacts" on a generalization of the local-relative correspondence. For the relation with the SYZ mirror symmetry and open Gromov—Witten invariants, this is now an ongoing work with K Chan. S.-C. Lau, Y.S. Lin. This work is in progress.
Project 2 focuses on applications to mirror symmetry. The main question here is to study the so-called Doran—Harder—Thompson conjecture via various approaches. I have a paper this direction titled "Degenerations, fibrations and higher rank Landau-Ginzburg models". I also have a project with Charles Doran and David Favero on this conjecture via toric mirror symmetry.
Another important part of Project 2 is to study intrinsic mirror symmetry and use it to study the Doran—Harder—Thompson conjecture. I have several results along this direction and have developed necessary techniques to prove the conjecture via intrinsic mirror symmetry. This includes my paper titled:” The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map” and my paper titled: "Orbifold theta functions and mid-age invariants."
For Project 3 on the relation with other theories, I have a preprint with Yu Wang titled "Gromov--Witten theory beyond maximal contacts" on a generalization of the local-relative correspondence. For the relation with the SYZ mirror symmetry and open Gromov—Witten invariants, this is now an ongoing work with K Chan. S.-C. Lau, Y.S. Lin. This work is in progress.
I am building a novel direction with powerful tools linking various areas, become a leader of this new research direction. I learnt skills in the Gross–Siebert program that allow me to work on several projects in this program using my approach. These skills also strengthens my research in enumerative geometry and mirror symmetry. The projects have led to high quality publications, which strengthen my current research program and provide new career opportunities. Results that we obtained during the
fellowship are leading to further projects and some of the proposed projects are part of a long term program which will be continued after the fellowship. Upon the completion of the fellowship, I will start my position as an Assistant Professor at the University of Nottingham, where I had my secondment for the fellowship.
The research were disseminated to the mathematical and theoretical physics communities by posting preprints on arXiv.org publishing papers in prestigious journals and presenting results in seminars and workshops. Marie Sklodowska-Curie IF provided me more opportunities to present my work and build my network in Europe. The mobility in the European Research Area is beneficial for me.
fellowship are leading to further projects and some of the proposed projects are part of a long term program which will be continued after the fellowship. Upon the completion of the fellowship, I will start my position as an Assistant Professor at the University of Nottingham, where I had my secondment for the fellowship.
The research were disseminated to the mathematical and theoretical physics communities by posting preprints on arXiv.org publishing papers in prestigious journals and presenting results in seminars and workshops. Marie Sklodowska-Curie IF provided me more opportunities to present my work and build my network in Europe. The mobility in the European Research Area is beneficial for me.