Periodic Reporting for period 1 - FuSeGC (First Steps in Mirror Symmetry for Generalized Complex Geometry)
Período documentado: 2021-09-01 hasta 2023-08-31
FuSeGC is a research project in pure mathematics, specifically in geometry. Mathematics is an important tool in theoretical physics: It is the language in which physical theories are written, and large parts of modern fundamental physics are written in the language of geometry in particular. In order to formulate theories about the physics of the universe at all scales, physicists need a well-developed geometric language. Ideas from physics frequently inspire pure mathematical research, but also vice versa: Understanding in fundamental physics can often only advance once a suitable mathematical language to formulate theories has been developed. Developing the backbone of a part of this language is what this research project is about: Just as applied science and engineering need to rest on a solid foundation of fundamental and theoretical science to be successful, fundamental physics needs mathematical tools to open up new ways of thinking about and formulating theories.
This research project is inspired by (theoretical) discoveries in string theory in particular: String theory is one proposal for a unified theory encompassing both quantum physics and gravity, but it is less one theory than a whole toolbox for building many different theories. (It is an open problem to find the one theory that describes our particular universe, and it might of course well turn out that none does.)
At the core of string theory lies the idea that the macroscopic physics we observe could at very small scale arise from tiny vibrating strings (the fundamental objects of string theory) moving through a curved spacetime (the concept first introduced by Einstein in his general theory of relativity). The geometry of this spacetime heavily influences the physics observed at larger scales, but it turns out that two completely different spacetime geometries can result in the same physics: This is referred to as mirror symmetry, and the two spacetimes are said to be mirror partners.
The precise mathematical formulation of mirror symmetry still contains many open problems, and is one of the most active research areas in modern geometry – which spaces have mirror partners, and how to we find them? How general a phenomenon is mirror symmetry, and what insights can we gain about the geometry on either side of the duality by translating from one to the other? These questions are interesting to mathematicians independently of any physical applications because mirror symmetry relates two different types of geometry, which allows us to translate problems about one geometry to the other, where they might be much easier to solve. The answer can then be translated back into the original context.
With FuSeGC, my goal is to extend what is known about mirror symmetry so far to a new class of geometries. Generalized complex (GC) structures constitute a relatively new type of geometry that include both types of geometry involved in mirror symmetry as examples; more precisely as the two endpoints of the spectrum of GC structures. Thus there is a natural question: Is mirror symmetry fundamentally a generalized complex duality? Since modern mirror symmetry is a theory involving complex mathematical technology, so in order to begin to answer this question, much of this technology needs to be adapted and expanded to GC geometry. This is what FuSeGC does, for a selection of carefully chosen contexts.
This research project is inspired by (theoretical) discoveries in string theory in particular: String theory is one proposal for a unified theory encompassing both quantum physics and gravity, but it is less one theory than a whole toolbox for building many different theories. (It is an open problem to find the one theory that describes our particular universe, and it might of course well turn out that none does.)
At the core of string theory lies the idea that the macroscopic physics we observe could at very small scale arise from tiny vibrating strings (the fundamental objects of string theory) moving through a curved spacetime (the concept first introduced by Einstein in his general theory of relativity). The geometry of this spacetime heavily influences the physics observed at larger scales, but it turns out that two completely different spacetime geometries can result in the same physics: This is referred to as mirror symmetry, and the two spacetimes are said to be mirror partners.
The precise mathematical formulation of mirror symmetry still contains many open problems, and is one of the most active research areas in modern geometry – which spaces have mirror partners, and how to we find them? How general a phenomenon is mirror symmetry, and what insights can we gain about the geometry on either side of the duality by translating from one to the other? These questions are interesting to mathematicians independently of any physical applications because mirror symmetry relates two different types of geometry, which allows us to translate problems about one geometry to the other, where they might be much easier to solve. The answer can then be translated back into the original context.
With FuSeGC, my goal is to extend what is known about mirror symmetry so far to a new class of geometries. Generalized complex (GC) structures constitute a relatively new type of geometry that include both types of geometry involved in mirror symmetry as examples; more precisely as the two endpoints of the spectrum of GC structures. Thus there is a natural question: Is mirror symmetry fundamentally a generalized complex duality? Since modern mirror symmetry is a theory involving complex mathematical technology, so in order to begin to answer this question, much of this technology needs to be adapted and expanded to GC geometry. This is what FuSeGC does, for a selection of carefully chosen contexts.
The work performed for FuSeGC so far has chiefly focused on Objective 2 as outlined in the original proposal: Here, we study log symplectic surfaces as a low-dimensional analogue to the the GC structures which are to be studied for the other two objectives. While log symplectic structures are not GC, there are good reasons for studying them in the context of this project, as well as independently of questions about GC geometry:
- Log symplectic structures are relatively close to being symplectic, giving an indication of how to adapt and extend mirror symmetry techniques to them. At the same time, due to the degeneracies of their associated Poisson structures, they exhibit new behaviour that makes this generalization non-trivial and leads to the emergence of new phenomena.
- Log symplectic geometry is an important field within Poisson geometry, so the successful extension of techniques from symplectic geometry to log symplectic geometry is interesting irrespective of applications to GC geometry.
- In real dimension 2, many technical difficulties that already exist in the (well-studied) original symplectic context in higher dimensions disappear. By generalizing the theory first in this low dimension, we can focus on the challenges posed by the new type of geometry, rather than general technical difficulties. But on surfaces, no GC structures beyond the already known ones exist (new GC structures start appearing in 4 dimensions); however, log symplectic surfaces are very close in behaviour to so-called “stable” GC 4-manifolds, and can hence serve as a model for those.
The two main mathematical objects or techniques needed for the study of mirror symmetry on one side of the duality, the symplectic side, are called Lagrangian intersection Floer cohomology and Fukaya categories. The structures which we focus on in FuSeGC are all closer to the symplectic side of mirror symmetry, so it makes sense to start from the symplectic techniques and generalize these.
During this reporting period, both of these constructions were indeed successfully made for log symplectic surfaces; the completed results are presented in two research articles. The first of these is available as a preprint and is currently under peer review; the second is upcoming in Q4 of 2023. (The peer review in pure mathematics takes comparatively long and has been further slowed down by lingering effects of the Covid-19 pandemic.)
The remaining time of this reporting period has been spent on the detailed study of examples of Fukaya categories of specific log symplectic manifolds and their split generation, as well as initial work on establishing mirror symmetry and extending these results both to higher-dimensional log symplectic manifolds and genuine GC 4-manifolds. This work is in progress.
- Log symplectic structures are relatively close to being symplectic, giving an indication of how to adapt and extend mirror symmetry techniques to them. At the same time, due to the degeneracies of their associated Poisson structures, they exhibit new behaviour that makes this generalization non-trivial and leads to the emergence of new phenomena.
- Log symplectic geometry is an important field within Poisson geometry, so the successful extension of techniques from symplectic geometry to log symplectic geometry is interesting irrespective of applications to GC geometry.
- In real dimension 2, many technical difficulties that already exist in the (well-studied) original symplectic context in higher dimensions disappear. By generalizing the theory first in this low dimension, we can focus on the challenges posed by the new type of geometry, rather than general technical difficulties. But on surfaces, no GC structures beyond the already known ones exist (new GC structures start appearing in 4 dimensions); however, log symplectic surfaces are very close in behaviour to so-called “stable” GC 4-manifolds, and can hence serve as a model for those.
The two main mathematical objects or techniques needed for the study of mirror symmetry on one side of the duality, the symplectic side, are called Lagrangian intersection Floer cohomology and Fukaya categories. The structures which we focus on in FuSeGC are all closer to the symplectic side of mirror symmetry, so it makes sense to start from the symplectic techniques and generalize these.
During this reporting period, both of these constructions were indeed successfully made for log symplectic surfaces; the completed results are presented in two research articles. The first of these is available as a preprint and is currently under peer review; the second is upcoming in Q4 of 2023. (The peer review in pure mathematics takes comparatively long and has been further slowed down by lingering effects of the Covid-19 pandemic.)
The remaining time of this reporting period has been spent on the detailed study of examples of Fukaya categories of specific log symplectic manifolds and their split generation, as well as initial work on establishing mirror symmetry and extending these results both to higher-dimensional log symplectic manifolds and genuine GC 4-manifolds. This work is in progress.
As detailed in the previous paragraph, the results of FuSeGC up to this point have already advanced the state-of-the-art by providing the first extension of Lagrangian intersection Floer cohomology to and construction of a Fukaya category for a non-symplectic Poisson structure. These results serve as a proof of concept and open up a whole new field of research to explore. The success of this objective lays the basis for all further study into Fukaya categories of generically non-degenerate Poisson structures with localised degeneracies, as well as providing some indications for how to approach Fukaya categories of (some) regular Poisson manifolds (e.g. those underlying GC manifolds of constant type).
Until the end of the project, additional work will focus on extending these results to compact stable GC 4-manifolds, as well as establishing mirror symmetry for log symplectic manifolds. These aspects of FuSeGC are estimated to lead to 2 further publications beyond the 2 based on the completed results obtained so far.
Until the end of the project, additional work will focus on extending these results to compact stable GC 4-manifolds, as well as establishing mirror symmetry for log symplectic manifolds. These aspects of FuSeGC are estimated to lead to 2 further publications beyond the 2 based on the completed results obtained so far.