Related papers: Higher depth quantum modular forms and plumbed $3$…
The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsvath and Szabo for a family of plumbed rational homology 3-spheres. The main motivation is the study of the Seiberg-Witten type…
The physical 3d $\mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $\hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial…
We consider a connected negative definite plumbing graph, and we assume that the associated plumbed 3-manifold is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg-Witten invariant of this manifold. The…
Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical…
Gukov--Pei--Putrov--Vafa conjectured the existence of $ q $-series whose radial limits are Witten--Reshetikhin--Turaev invariants and called them homological blocks. For weakly negative definite plumbed 3-manifolds, Gukov--Pei--Putrov--Vafa…
Let M be a rational homology sphere plumbed 3-manifold associated with a connected negative definite plumbing graph. We show that its Seiberg-Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial.…
For a Seifert fibered homology sphere we show that the q-series Z-hat invariant introduced by Gukov, Pei, Putrov and Vafa is a resummation of the Ohtsuki serie. We show that for every even level k there exists a full asymptotic expansion of…
The three-manifold topological invariants $\hat Z$ capture the half-index of the three-dimensional theory with ${\mathcal{N}}=2$ supersymmetry obtained by compactifying the M5 brane theory on the closed three-manifold. In 2019, surprising…
$\widehat{Z}$ invariants, rigorously defined for negative definite plumbed 3-manifolds, are expected--on physical grounds--to exist for every closed, oriented 3-manifold. Several prescriptions have been proposed to extend their definition…
We introduce supergroup analogues of 3-manifold invariants $\hat{Z}$, 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…
In 2006 Habiro initiated a construction of generating functions for Witten-Reshetikhin-Turaev (WRT) invariants known as unified WRT invariants. In a series of papers together with Irmgard Buehler and Christian Blanchet we extended his…
For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more…
We study series invariants for plumbed 3-manifolds and knot complements twisted by a root lattice. Our series recover recent results of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri and apply more generally to 3-manifolds which are…
In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the…
In the spirit of arXiv:2501.12985, we propose an abelian categorification of $\hat{Z}$-invariants for negative definite plumbed 3-manifolds. It provides a blueprint for the expected dictionary between these $3$-manifolds and log VOAs; that…
We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple quantum…
The quantum modularity conjecture, first introduced by Don Zagier, is a general statement about a relation between $\mathfrak{sl}_2$ quantum invariants of links and 3-manifolds at roots of unity related by a modular transformation. In this…
We present a relation between the Witt invariants of 3-manifolds and the $\hat{Z}$-invariants. It provides an alternative approach to compute the Witt invariants of 3-manifolds, which were originally defined geometrically in four…
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in [arXiv:1404.7289]. In their construction the quantum parameter $q$ is a root of unity of order…
An example of a finite dimensional factorizable ribbon Hopf C-algebra is given by a quotient H=u_q(g) of the quantized universal enveloping algebra U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a surface of…