2023 Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting

The first annual meeting was the true beginning of a new collaboration. A total of about hundred researchers from different areas of low-dimensional topology working on various aspects of the categorification program were brought together to establish new connections and learn about areas distant from one’s own. Excitement was definitely in the air.

The main goal of this first meeting was to forge a common language and to understand the key challenges, in order to facilitate an exchange of ideas and to explore new opportunities for collaboration. While we intend to continue to develop and evolve the collaboration, based on the lively discussions during the breaks and in the main program, the first step toward this goal was successful.

Aaron Lauda opened the scientific program with a survey of some of the main challenges that the Collaboration is going to tackle and then presented a new strategy based on reinterpreting Heegaard Floer homology as a categorified quantum invariant associated to the super Lie algebra \(gl(1|1)\). Since much is known about Heegaard Floer theory, its representation theoretic description can be used as a springboard for constructing more general homology theories of 3-manifolds, associated with other root systems.

The topic of algebraic structures relevant to Heegaard Floer homology and its many variants continued through most of the first day, in talks from Peter Ozsvath, Zoltan Szabo, and Raphael Rouquier. In particular, Peter Ozsvath presented an ongoing project with Robert Lipshitz and Dylan Thurston, generalizing their earlier work on bordered Heegaard Floer homology from the specialized (\(U=0\)) case to the unspecialized version. These two versions are related, respectively, to the simpler (“hat”) and more advanced (“minus”) versions of Heegaard Floer homology. For the unspecialized Heegaard Floer homology, Peter Ozsvath gave an explicit combinatorial description of a weighted \(A_{\infty}\) algebra associated to a 2-torus and described higher products and relations in this algebra. He then explained how one can construct modules associated to 3-manifolds with torus boundary and outlined various directions for future work.

Zoltan Szabo’s talk was devoted to similar versions of knot Floer homology and their bordered algebras: \(B\), \(C\), and \(P\). The latter, called the Pong algebra, is related to the most general case of knot Floer homology. Zoltan Szabo illustrated constructions of the algebras \(B(m,k)\), \(C(m,k)\), and \(P(m,k)\) with many explicit examples and explained the structure of modules relevant to construction of homological knot invariants.

Raphael Rouquier posed many questions and outlined a number of directions for studying algebraic structures relevant to 4-dimensional topology. In the first part of the talk, he explained that suitable analogues of representations of quantum groups should be 2-representations of Lie algebras, and pointed out difficulties with 2-tensor products. He then explained that such difficulties disappear in the case of positive part of \(gl(1|1)\) and, following his joint work with Andrew Manion, outlined how one can recover Heegaard Floer theory from higher representation theory in this special case. In the second part of his talk, Raphael Rouquier raised many interesting questions on what could become of vertex algebras, modular tensor categories, Vassiliev invariants, and the Drinfeld associator upon categorification.

Lisa Piccirillo presented two new constructions of exotic 4-manifolds, based on joint work in progress with Adam Levine and Tye Lidman. Both constructions are very explicit and use the same relatively simple cobordisms as building blocks. The first construction relies on a new invariant, derived from Heegaard Floer theory, and leads to exotic 4-manifolds with \(b_1 > 0\). One of the new interesting features of this construction is that it produces non-diffeomorphic 4-manifolds which are related by knot surgery in the case when the Alexander polynomial is trivial. The second part of the talk was devoted to the construction of simply-connected exotic 4-manifolds with \(b_2 = 10\). Using a symmetry of this construction, one can also obtain exotic examples of definite 4-manifolds with \(\pi_1 = \mathbb{Z} / 2\) and \(b_2 = 4\).

On the second day of the conference, Sergei Gukov presented several connections between gauge theory and quantum algebra in the context of multi-monopole 4-manifold invariants, Vafa-Witten invariants of 4-manifolds, and \(q\)-series invariants of 3-manifolds. Based on the joint work with Artan Sheshmani and Shing-Tung Yau, he gave an explicit description of the moduli spaces in Vafa-Witten theory and in Kapustin-Witten theory on certain 3-manifolds, and summarized several topological applications of the corresponding Floer homology groups. He also conjectured a 2-variable generalization of Vafa-Witten invariants of 4-manifolds, based on joint work in progress with Po-Shen Hsin and Du Pei.

Peter Kronheimer presented results of ongoing work with Tom Mrowka on singular instanton homology for knots and braids with local coefficients. After describing the general algebraic properties of this theory, he explained that, for simple braids (with \(n\) strands) in \(S^1 \times S^2\), the module structure of instanton homology admits an elegant geometric description in terms of a rank-1 sheaf on a curve \(C_n\). He presented an explicit form of \(C_n\) for small values of \(n\) and discussed several potential applications to topology and algebraic geometry, in particular to the question of Caporaso, Haris, and Mazur on a uniform upper bound for the arithmetic genus of the rational curves on quintic surfaces.

The program concluded with a talk by Mikhail Khovanov on universal construction of topological theories (a notion weaker than the standard definition of a TQFT) that he illustrated in three types of examples. The first class of examples, based in part on joint works with Radmila Sazdanovic and with Victor Ostrik and Yakov Kononov, illustrated the universal construction in the context of 2-dimensional topological theories. Curiously, already in this simple class one finds rich and interesting structures such as Deligne categories. For the second class of examples, Mikhail Khovanov described the combinatorial construction of \(sl(n)\) link homology from the perspective of the universal construction. Specifically, he reviewed the work of Louis-Hadrien Robert and Emmanuel Wagner on evaluation of closed \(sl(n)\) foams and made contact with the work of Peter Kronheimer and Tom Mrowka on homology of trivalent graphs. In the last part of his talk, Mikhail Khovanov presented yet another class of topological 1-dimensional theories associated to a group \(G\) and a finite-dimensional representation \(V\) over a field \(k\), that has connections to the theory of pseudocharacters in number theory. This part was based on very recent work with Mee Seong Im and Victor Ostrik.

Aaron Lauda
University of Southern California

Heegaard Floer Homology as a quantum invariant
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Crane and Frenkel’s dream was to categorify the theory of quantum groups and use them to construct new examples of 4-dimensional Topological Quantum Field Theories. The program to categorify quantum groups has made a great deal of progress, categorifying quantum groups, tensor powers of irreducibles, R-matrices, and associated quantum knot invariants. However, the program to construct new 4-manifold invariants from even the simples Lie algebra sl(2) has remained elusive.

In this talk I will explain a new strategy for categorifying quantum 4-manifold invariants by starting with the very successful theory of Heegaard Floer homology and working backwards to reinterpret its local formulation as categorifying quantum invariants associated to the super Lie algebra gl(1|1). Having a complete understanding of Heegaard Floer homology within this representation theoretic framework is likely to enhance our understanding of how to develop new 4-manifold invariants comings from other Lie algebras.
 

Peter Ozsvath
Princeton University

Constructions in bordered Floer homology

Bordered Floer homology, introduced by Lipshitz, D. Thurston, and me, allows one to express the U=0-specialized Heegaard Floer homology of a closed three-manifold Y in terms of a suitable decomposition of Y. This theory should have a generalization to the case of unspecialized Heegaard Floer homology. I will describe aspects of this generalization in the case of three-manifolds with torus boundary. This is joint work with Lipshitz and Thurston.
 

Zoltan Szabo
Princeton University

Knot Floer homology and Pong algebras

In this talk I will describe some recent joint work with Peter Ozsvath on differential graded algebras and knot Floer homology. The bordered Floer homology of Lipshitz, Ozsvath and Thurston is an invariant of three-manifolds with surface boundary, and can be used to study Heegaard Floer homology for closed three-manifolds. Adaptation of these methods, developed in a joint work with Peter Ozsvath, allows effective
computation for a version of knot Floer homology over the ring $$Z/2[U,V]/ (U\cdot V = 0)$$

In the talk I will describe some new constructions, some interesting problems and recent advances that are related to extending these methods.
 

Raphael Rouquier
UCLA

Algebra for 4d invariants
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A long-standing problem is to understand the algebraic structures underlying 4d topological field theories and to construct such structures. One issue is to clarify the precise types of field theories that exist, or are expected to exist. I will discuss the case of Heegaard-Floer theory. Other questions involve higher dimensional versions of structures arising in 3d.
 

Lisa Piccirillo
MIT

Building closed exotica by hand

Historically, closed exotic 4-manifolds are constructed using cut and paste operations, and their gauge theoretic invariants are computed using gluing formulas. We build closed exotic 4-manifolds out of simple 2-handle cobordisms, and we compute their Heegaard Floer mixed invariants explicitly using the surgery exact triangle. Exotic examples include definite 4-manifolds with \(b_2=4\) and fundamental group \(\mathbb{Z}/2\), and simply connected 4-manifolds with \(b_2=10\). We also develop new invariants for 4-manifolds with \(b_1\ge 1\), and we use these invariants to show that there are closed exotic manifolds with fundamental group \(\mathbb{Z}\) and homologically essential square zero spheres. We also show that one can change the smooth structure of a closed manifold by performing knot surgery in a fishtail neighborhood using a knot with Alexander polynomial 1. This is joint work in progress with Adam Levine and Tye Lidman.
 

Sergei Gukov
Caltech

Gauge theory and quantum algebra
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We will discuss gauge theory perspective on q-series invariants of 3-manifolds and 4-manifolds, as well as aspects that reveal interesting connections to quantum algebra.
 

Peter Kronheimer
Harvard University

Singular instanton homology for knots and braids: experiments and conjectures
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I will present some structural results (often experimental) about singular instanton homology, particularly for knots and links in S1 × S2 arising as the closures of spherical braids. There appears to be a potentially rich structure here, in that such a braid gives rise to a rank-1 sheaf on an interesting algebraic curve. Possible applications to braided surfaces in Σ × S2, where Σ is a surface with boundary, remain as future targets. This is a progress report on joint work with Tom Mrowka.
 

Mikhail Khovanov
Columbia University

Universal construction of topological theories and its applications

Foam evaluation gives an explicit algebraic approach to categorification of quantum GL(N) invariants of links. A necessary step in that approach is the universal construction for foam evaluations, giving state spaces that allow to build link homology groups. We review these notions and their applications to GL(N) link homology and to the Kronheimer-Mrowka 3-orbifold homology theory. The universal construction is starting to play a role beyond foam evaluation, naturally appearing in Deligne categories and their generalizations, noncommutative power series, automata and pseudo-representations, and this will be explained in the talk as well.