Supergroups, q-series and 3-manifolds
Francesca Ferrari, Pavel Putrov

TL;DR
This paper introduces supergroup analogues of 3-manifold invariants, focusing on superunitary groups like SU(2|1), and explores their properties, calculations, and conjectured relations to quantum invariants.
Contribution
It develops a framework for supergroup homological blocks, provides an explicit algorithm for their computation, and investigates their mathematical properties and connections to quantum supergroups.
Findings
Defined supergroup invariants for 3-manifolds
Provided an algorithm for computing q-series for plumbed 3-manifolds
Studied quantum modularity and resurgence properties
Abstract
We introduce supergroup analogues of 3-manifold invariants , also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups, and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Topological and Geometric Data Analysis · Algebraic structures and combinatorial models
