Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

Abstract: By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of $q$-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.

Overview of "Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants"

This paper by Sergei Gukov et al. explores the connections between Rozansky-Witten theory and various knot invariants, particularly the Akutsu-Deguchi-Ohtsuki (ADO) invariants and $Z$-series invariants. Rozansky-Witten theory, originally introduced as a three-dimensional topological quantum field theory (TQFT) associated with hyper-Kähler target spaces, provides a fertile ground for linking geometry with topology. The primary aim of the paper is to extend the framework of Rozansky-Witten theory to non-compact target spaces, thereby establishing novel ties with knot invariants with previously unexplored interpretations.

Rozansky-Witten Theory with Non-Compact Target Spaces

The authors begin by discussing the generalization of the Rozansky-Witten theory for target spaces that are non-compact. This generalization introduces subtlety in the theory's behavior, particularly concerning its non-compliance with the standard axioms of TQFTs when $X$ is non-compact. The space of states defined in terms of the cohomology on a genus-$g$ surface is infinite-dimensional. The authors develop a method to calculate Rozansky-Witten invariants using equivariant localization techniques, focusing on non-compact examples, such as $T^* C^1$ and ALE spaces.

Knot Invariants and Their Interrelations

In the field of knots and links, ADO invariants and $Z$-series invariants provide insights into the topology of 3-manifolds. ADO invariants, connected to the representation theory of quantum groups, are known to generalize the Alexander polynomial, while $Z$-series invariants offer a $q$-deformation of the inverse Turaev-Milnor torsion. The paper proposes that Rozansky-Witten theory can encompass these invariants by extending it to infinite-dimensional target spaces like affine Grassmannians.

Proposed Connections and Implications

The paper introduces several conjectures to formalize these connections. One such conjecture posits that the $Z$-series invariants are equivalent to Rozansky-Witten invariants with the target space identified as "T^*Gr". Another conjecture suggests that ADO invariants can be derived from $Z$-series invariants under specializations of the parameter $q$, effectively compactifying the parameter space and yielding finite-dimensional theories.

The implications of these connections are vast:

• Reformulation of Topological Invariants: The proposed equivalences provide novel formulations of traditional knot invariants in terms of Rozansky-Witten theory, potentially simplifying calculations and offering deeper geometric interpretations.
• Categorification: There is potential for categorifying ADO invariants, similar to knot Floer homology, leveraging infinite-dimensional homology theories.
• Algebraic Structure: This framework could lead to a better understanding of how TQFTs derived from non-semisimple modular tensor categories behave under operations like cutting and gluing.

Future Directions

The paper opens several avenues for future research. These include:

• Detailed analysis of the conjectured relation between affine Grassmannians and Rozansky-Witten invariants.
• Exploration of the geometric nature of the $q$ parameter and potential links to equivariant indices on moduli spaces.
• Development of explicit models for the proposed finite-dimensional target spaces $X_p$ corresponding to roots of unity.

The authors highlight how these explorations could influence both the mathematical understanding of knot invariants and the broader applications within theoretical physics, particularly in the context of 3-manifold topology and quantum field theory.

This paper stands as a significant contribution toward integrating higher-dimensional algebraic geometry and topological invariants, highlighting the rich interplay between geometry and topology within theoretical physics frameworks.