IN – Track A

We propose a radically new and rigorous construction of the Landau-Ginzburg open A-model via moduli spaces of solutions to Witten equations over a secondary polytope. The moduli degeneration exactly fits the “algebra of the infrared” as envisioned by Gaiotto-Moore-Witten and Kapranov-Kontsevich-Soibelman. We expect this to lead to proofs of their conjectures and open up several promising avenues for further research.

In many scenarios, one needs to consider a system of quantities which is allowed to vary under some constraints or hidden rules. This system can be regarded as a geometric object, and some additional basic properties (for example, achieved by approximation or simplification if not already intrinsically given) will give rise to various tools to study and understand the system. If the constraints are algebraic, this space is the common solution space of a set of polynomial equations, the study of which lies at the heart of algebraic geometry. If the system of quantities forms a regular space, it may instead belong to the realm of symplectic geometry. In each geometry, the ultimate goal is to classify these spaces, and for that we need finer and finer invariants to distinguish them.

In algebraic geometry, 1-dimensional curves are well-understood, and their algebraic behavior gives rise to degenerations that produce pinched/nodal curves. To understand higher-dimensional geometries, one fruitful approach is to probe them using 1-dimensional curves, like doing a CT scan. The resulting space is called a Gromov-Witten moduli space, and it can reveal much about the space being probed. In many important examples, the set of curves is a discrete set and one gets a numerical invariant by counting this discrete set; this curve-counting idea falls under the heading of enumerative geometry, and it is one of the oldest endeavors in pure mathematics. An interesting historical example is Steiner’s problem of finding the number of conics tangent to five plane conics in general position in the complex plane: the answer is 3264.

One can also leave the framework of algebraic geometry, and require only holomorphicity as a condition, namely by considering solutions to a nonlinear Cauchy-Riemann equation with a relaxed notion of complex multiplication. By doing this, one changes the scene from algebraic to symplectic geometry, a fast-developing and eclectic branch of differential geometry that makes liberal use of tools from PDE theory. In contrast to the algebraic setting, the symplectic version of this discussion allows for moduli spaces with interesting phenomena of real codimension one. An important example arises if one allows the probing complex curves to have boundaries with prescribed boundary conditions, giving rise to a much richer structure than a numerical invariant, called the Fukaya category. The boundary conditions in this setting are called Lagrangian submanifolds, and they do not fit into the framework of algebraic geometry—once one puts algebro-geometric examples into the more flexible symplectic framework and manipulates them in ways that are only admissible in the symplectic world, it is not clear how to return from there to the world of algebraic geometry.

Deep insights from physics have proposed a remarkable answer to the latter problem, called mirror symmetry. It conjectures in essence that the symplectic geometry (“A-model”) on one object and the algebraic geometry (“B-model”) on its “mirror” partner are really the same, essentially identifying and bridging two vastly different branches of mathematics. The justification from physics for this deep connection is that both settings should be regarded as two ways of describing the same physical reality. The attempt at a mathematical justification of these ideas forms a large area of active research.

The research in this project focuses on an open A-model theory. Mathematically, gradual progress has been made on confirming open/homological mirror symmetry by calculating each Fukaya category case-by-case and matching it with the algebraic counterpart of its expected mirror partner. The direction in this endeavor has always gone from A-models to B-models, so it is still quite far away from comprehensively using each geometry to harvest knowledge of the other. The spaces that form the background geometric objects in this project are called Landau-Ginzburg models; they have their origin in superconductivity and string theory. Recently D. Yang and collaborators introduced a rigorous construction of an LG Fukaya category for the open LG A-models. The formalism in this construction is expected to lead to a new moduli space construction for LG models, which should realize moduli spaces conjectured in the work of Gaiotto-Moore-Witten and Kapranov-Kontsevich-Soibelman with a certain real codimension 1 degeneration fitting into their proposed algebraic structure, called the algebra of secondary polytopes. Due to its context and interpretation in physics, this conjectured algebraic structure is also referred to as the “algebra of the infrared”.