arXiv subjects

Marco De Renzi

Publications and source records attributed to Marco De Renzi.

17 recordsLinked to original sources

arXiv2025

Algebraic Presentation of $4$-Dimensional $2$-Handlebodies and $3$-Dimensional Cobordisms

In this paper, we give a new direct proof of a result by Bobtcheva and Piergallini that provides finite algebraic presentations of two categories, denoted $3\mathrm{Cob}$ and $4\mathrm{HB}$, whose morphisms are manifolds of dimension $3$ and $4$, respectively. More precisely, $3\mathrm{Cob}$ is the category of connected oriented $3$-dimensional cobordisms between connected surfaces with connected boundary, while $4\mathrm{HB}$ is the category of connected oriented $4$-dimensional $2$-handlebodies up to $2$-deformations. For this purpose, we explicitly construct the inverse of the functor $Φ: 4\mathrm{Alg} \to 4\mathrm{HB}$, where $4\mathrm{Alg}$ denotes the free monoidal category generated by a Bobtcheva--Piergallini Hopf algebra. As an application, we deduce an algebraic presentation of $3\mathrm{Cob}$ and show that it is equivalent to the one conjectured by Habiro.

math.GT↗

arXiv2025

Quantum Invariants of Ribbon Surfaces in $4$-Dimensional $2$-Handlebodies

We use unimodular ribbon categories to construct quantum invariants of ribbon surfaces in $4$-dimensional $2$-handlebodies up to $1$-isotopy. In the process, we recover invariants due to Bobtcheva-Messia, Broda-Petit, Gainutdinov-Geer-Patureau-Runkel (in collaboration with the second author), and Lee-Yetter. Our approach does not assume semisimplicity, and is based on a generalization of the Reshetikhin-Turaev functor to the category of labeled Kirby graphs which also yields invariants of framed links in the boundary of $4$-dimensional $2$-handlebodies up to $2$-deformations. The setup is very flexible, and allows for several different constructions, using central elements satisfying equations introduced by Hennings and Bobtcheva-Messia, modified traces, and modules over Frobenius algebras satisfying conditions dictated by the diagrammatic calculus for embedded surfaces developed by Hughes, Kim, and Miller.

math.GT↗

arXiv2024

On Integrality of Non-Semisimple Quantum Representations of Mapping Class Groups

For a root of unity $ζ$ of odd prime order, we restrict coefficients of non-semisimple quantum representations of mapping class groups associated with the small quantum group $\mathfrak{u}_ζ\mathfrak{sl}_2$ from $\mathbb{Q}(ζ)$ to $\mathbb{Z}[ζ]$. We do this by exhibiting explicit bases of states spaces that span $\mathbb{Z}[ζ]$-lattices that are invariant under projective actions of mapping class groups.

math.GT↗

arXiv2024

Hennings TQFTs for Cobordisms Decorated With Cohomology Classes

Starting from an abelian group $G$ and a factorizable ribbon Hopf $G$-bialgebra $H$, we construct a TQFT $J_H$ for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in $G$. When restricted to the subcategory of cobordisms with trivial decorations, our functor recovers a special case of Kerler-Lyubashenko TQFTs, namely those associated with factorizable ribbon Hopf algebras. Our result is inspired by the work of Blanchet-Costantino-Geer-Patureau, who constructed non-semisimple TQFTs for admissible decorated cobordisms using the unrolled quantum group of $\mathfrak{sl}_2$, and by that of Geer-Ha-Patureau, who reformulated the underlying invariants of admissible decorated $3$-manifolds using ribbon Hopf $G$-coalgebras. Our work represents the first step towards a homological model for non-semisimple TQFTs decorated with cohomology classes that appears in a conjecture by the first two authors.

math.GT↗

arXiv2023

Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies

In this paper we refine our recently constructed invariants of $4$-dimensional $2$-handlebodies up to $2$-deformations. More precisely, we define invariants of pairs of the form $(W,ω)$, where $W$ is a $4$-dimensional $2$-handlebody, $ω$ is a relative cohomology class in $H^2(W,\partial W;G)$, and $G$ is an abelian group. The algebraic input required for this construction is a unimodular ribbon Hopf $G$-coalgebra. We study these refined invariants for the restricted quantum group $U = U_q \mathfrak{sl}_2$ at a root of unity $q$ of even order, and for its braided extension $\tilde{U} = \tilde{U}_q \mathfrak{sl}_2$, which fits in this framework for $G=\mathbb{Z}/2\mathbb{Z}$, and we relate them to our original invariant. We deduce decomposition formulas for the original invariants in terms of the refined ones, generalizing splittings of the Witten-Reshetikhin-Turaev invariants with respect to spin structures and cohomology classes. Moreover, we identify our non-refined invariant associated with the small quantum group $\bar{U} = \bar{U}_q \mathfrak{sl}_2$ at a root of unity $q$ whose order is divisible by 4 with the refined one associated with the restricted quantum group $U$ for the trivial cohomology class $ω=0$.

math.GT↗

arXiv2023

Homological Construction of Quantum Representations of Mapping Class Groups

We provide a homological model for a family of quantum representations of mapping class groups arising from non-semisimple TQFTs (Topological Quantum Field Theories). Our approach gives a new geometric point of view on these representations, and it gathers into one theory two of the most promising constructions for investigating linearity of mapping class groups. More precisely, if $\varSigma_{g,1}$ is a surface of genus $g$ with $1$ boundary component, we consider a (crossed) action of its mapping class group $\mathrm{Mod}(\varSigma_{g,1})$ on the homology of its configuration space $\mathrm{Conf}_n(\varSigma_{g,1})$ with twisted coefficients in the Heisenberg quotient $\mathbb{H}_g$ of its surface braid group $π_1(\mathrm{Conf}_n(\varSigma_{g,1}))$. We show that this action intertwines an action of the quantum group of $\mathfrak{sl}_2$, that we define by purely homological means. For a finite-dimensional linear representation of $\mathbb{H}_g$ (depending on a root of unity $ζ$), we tweak the construction to obtain a projective representation of $\mathrm{Mod}(\varSigma_{g,1})$. Finally, we identify, by an explicit isomorphism, a subrepresentation of $\mathrm{Mod}(\varSigma_{g,1})$ that is equivalent to the quantum representation arising from the non-semisimple TQFT associated with quantum $\mathfrak{sl}_2$ at $ζ$. In the process, we provide concrete bases and explicit formulas for the actions of all the standard generators of $\mathrm{Mod}(\varSigma_{g,1})$ and of quantum $\mathfrak{sl}_2$ on both sides of the equivalence, and answer a question by Crivelli, Felder, and Wieczerkowski. We also make sure that the restriction of these representations to the Torelli group $\mathcal{I}(\varSigma_{g,1})$ are integral, in the sense that the actions have coefficients in the ring of cyclotomic integers $\mathbb{Z}[ζ]$, when expressed in these bases.

math.GT↗

arXiv2023

Kerler-Lyubashenko Functors on 4-Dimensional 2-Handlebodies

We construct a braided monoidal functor $J_4$ from Bobtcheva and Piergallini's category $4\mathrm{HB}$ of connected 4-dimensional 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category $\mathcal{C}$, which is not required to be semisimple. The main example of target category is provided by $H$-mod, the category of left modules over a unimodular ribbon Hopf algebra $H$. The source category $4\mathrm{HB}$ is freely generated, as a braided monoidal category, by a BPH algebra (short for Bobtcheva-Piergallini Hopf algebra), and this is sent by the Kerler-Lyubashenko functor $J_4$ to the end $\int_{X \in \mathcal{C}} X \otimes X^*$ in $\mathcal{C}$, which is given by the adjoint representation in the case of $H$-mod. When $\mathcal{C}$ is factorizable, we show that the construction only depends on the boundary and signature of handlebodies, and thus projects to a functor $J_3^σ$ defined on Kerler's category $3\mathrm{Cob}^σ$ of connected framed 3-dimensional cobordisms. When $H^*$ is not semisimple and $H$ is not factorizable, our functor $J_4$ has the potential of detecting diffeomorphisms that are not 2-deformations.

math.GT↗

arXiv2021

3-Dimensional TQFTs From Non-Semisimple Modular Categories

We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of previous non-semisimple TQFT constructions.

math.GT↗

arXiv2021

Diagrammatic Construction of Representations of Small Quantum $\mathfrak{sl}_2$

We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $\mathfrak{sl}_2$ at a root of unity $q$ of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley-Lieb category specialized at $δ= -q-q^{-1}$.

math.QA↗

arXiv2021

Modular Categories and TQFTs Beyond Semisimplicity

Vladimir Turaev discovered in the early years of quantum topology that the notion of modular category was an appropriate structure for building 3-dimensional Topological Quantum Field Theories (TQFTs for short) containing invariants of links in 3-manifolds such as Witten-Reshetikhin-Turaev ones. In recent years, generalized notions of modular categories, which relax the semisimplicity requirement, have been successfully used to extend Turaev's construction to various non-semisimple settings. We report on these recent developments in the domain, showing the richness of Vladimir's lineage.

math.GT↗

arXiv2021

Extended TQFTs From Non-Semisimple Modular Categories

We construct 3-dimensional once-Extended Topological Quantum Field Theories (ETQFTs for short) out of (possibly non-semisimple) modular categories, and we explicitly identify linear categories and functors in their image. The circle category of an ETQFT produced by our construction is equivalent to the full subcategory of projective objects of the underlying modular category. In particular, it need not be semisimple.

math.GT↗

arXiv2021

Mapping Class Group Representations From Non-Semisimple TQFTs

In [arXiv:1912.02063], we constructed 3-dimensional Topological Quantum Field Theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category $\mathcal{C}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of [arXiv:1912.02063] are equivalent to those obtained by Lyubashenko via generators and relations in [arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{C}$ is the category of finite-dimensional representations of the small quantum group of $\mathfrak{sl}_2$, the action of all Dehn twists for surfaces without marked points has infinite order.

math.GT↗

arXiv2021

Non-Semisimple 3-Manifold Invariants Derived From the Kauffman Bracket

We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $\mathfrak{sl}_2$ using purely combinatorial methods based on Temperley-Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten-Reshetikhin-Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.

math.GT↗

arXiv2020

Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Goups

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 invariants of closed 3-manifolds decorated with ribbon graphs and cohomology classes. When we consider the zero cohomology class, these quantum invariants are shown to coincide with the renormalized Hennings invariants coming from the corresponding small quantum groups.

math.GT↗

arXiv2019

Non-Semisimple Extended Topological Quantum Field Theories

We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by symmetric monoidal 2-functors which are defined over non-rigid 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes, and which take values in 2-categories of complete graded linear categories. In particular, our construction extends the family of graded 2+1-TQFTs defined for the unrolled version of quantum $\mathfrak{sl}_2$ by Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The non-semisimplicity of the theory is witnessed by the presence of non-semisimple graded linear categories associated with critical 1-manifolds.

math.GT↗

arXiv2018

Renormalized Hennings Invariants and 2+1-TQFTs

We construct non-semisimple $2+1$-TQFTs yielding mapping class group representations in Lyubashenko's spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum $\mathfrak{sl}_2$ to the setting of finite-dimensional non-degenerate unimodular ribbon Hopf algebras. The tools used for this construction are a Hennings-augmented Reshetikhin-Turaev functor and modified traces. When the Hopf algebra is factorizable, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a $2+1$-TQFT on a not completely rigid monoidal subcategory of cobordisms.

math.GT↗

arXiv2017

Quantum Invariants of 3-Manifolds Arising from Non-Semisimple Categories

This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, $\mathrm{N}_r$ and $\mathrm{N}^0_r$, using non-semisimple categories of representations of a quantum version of $\mathfrak{sl}_2$ at a $2r$-th root of unity with $r \geqslant 2$. The secondary invariants $\mathrm{N}^0_r$ conjecturally extend the original Reshetikhin-Turaev quantum $\mathfrak{sl}_2$ invariants. The authors also provide a machinery to produce invariants out of more general ribbon categories which can lack the semisimplicity condition. In the second paper a renormalized version of $\mathrm{N}_r$ for $r \not\equiv 0 \; (\mathrm{mod} \; 4)$ is extended to a TQFT, and connections with classical invariants such as the Alexander polynomial and the Reidemeister torsion are found. In particular, it is shown that the use of richer categories pays off, as these non-semisimple invariants are strictly finer than the original semisimple ones: indeed they can be used to recover the classification of lens spaces, which Reshetikhin-Turaev invariants could not always distinguish.

math.GT↗