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.Comment: 172 pages, 46 figures, entirely rewritten, several appendices adde

Milnor-Wood type inequalities

The Gauss-Bonnet Theorem, which was generalized by Shiing-Shen Chern in 1944 to all oriented closed even-dimensional smooth manifolds, correlates the curvature of the Levi-Civita connection of a Riemannian manifold with its Euler characteristic. This result provides a strong restriction on the kind of geometry such a manifold can support. For instance, let us consider Euclidean manifolds, i.e. manifolds which admit an atlas whose coordinate change functions are isometries of $\mathbb R^n$. Since the curvature of the Levi-Civita connection they inherit from $\mathbb R^n$ vanishes, the Euler characteristic represents an obstruction to the existence of Euclidean structures: indeed if $\chi(M) \neq 0$ then $M$ cannot support a flat metric. On the other hand, let us consider affine manifolds, i.e. manifolds which admit an atlas whose coordinate change functions are affine isomorphisms of $\mathbb R^n$. These can be characterized as those manifolds whose tangent bundle supports flat and symmetric connections. Although they may seem to be a mild generalization of Euclidean manifolds, the attempt to generalize the above result to affine manifolds resulted in the formulation of a long standing open conjecture: \textbf{Conjecture 1:} The Euler characteristic of a closed oriented affine manifold vanishes. The key point is that the Euler characteristic of a manifold cannot be computed from the curvature of an arbitrary linear connection $\nabla$, because it is essential for $\nabla$ to be compatible with a Riemannian metric. Now, although Conjecture 1 was shown to hold true for complete affine manifolds by Bertram Kostant and Dennis Sullivan, the non-complete case is much more difficult. There are known examples, due to John Smillie, of manifods with non-zero Euler characteristic and flat tangent bundle in every even dimension greater than 2. However, as William Goldman points out, since the torsion of these connections seems hard to control they do not disprove Conjecture 1. None of Smillie’s manifolds is aspherical, and indeed another open conjecture is: \textbf{Conjecture 1:} The Euler characteristic of a closed oriented aspherical manifold whose tangent bundle is flat vanishes. The first important breakthrough was made by John Milnor in 1958, when he proved both conjectures for closed oriented surfaces. He exploited the fact that the existence of a flat connection on a rank-$m$ vector bundle $\pi : E \rightarrow M$ is equivalent to the existence of a holonomy representation $\rho : \pi_1(M,x_0) \rightarrow \mathrm{GL}^+(m,\mathbb R)$ which induces the bundle. The study of all possible holonomy representations for closed oriented surfaces enabled him to establish a much more detailed result: he managed to characterize all flat oriented plane bundles over closed oriented surfaces by means of their Euler class, that is a cohomology class in the cohomology ring of the base space which generalizes the Euler characteristic. What happens is that the Euler class of flat bundles over a fixed surface $\Sigma$ is bounded, that is, just a finite number (up to isomorphism) of oriented plane bundles over $\Sigma$ can support flat connections. In particular, none of these is the tangent bundle if $\Sigma$ is not the torus. This remarkable result is now known as Milnor-Wood inequality (the name celebrates John Wood's generalization to $S^1$-bundles). While it has been proven that the boundedness of the Euler class of flat bundles generalizes to all dimensions, Conjectures 1 and 2 remain elusive. Indeed one needs explicit inequalities in order to determine whether the tangent bundle can be ruled out from the flat ones or not. Since Milnor's work very little progress has been made until very recently. In 2011 Michelle Bucher and Tsachik Gelander published a generalization of Milnor-Wood inequality to closed oriented manifolds whose universal cover is isometric to $(\mathbb H^2)^n$, thus confirming both conjectures for all manifolds which are locally isometric to a product of surfaces of constant curvature. Their work, which takes up the largest part of our exposition, uses the theory of bounded cohomology developed by Mikha\"{i}l Gromov in 1982 and some deep results about the super-rigidity of lattices in semisimple Lie groups due to Gregori Margulis

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

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 generated, as a braided monoidal category, by a 4-modular Hopf algebra object, 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 $J_3^\sigma$ defined on Kerler's category $3\mathrm{Cob}^\sigma$ 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.Comment: 55 pages. Version 2: Section 1.2 and Appendix C adde

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.Comment: 26 pages, to appear in Topology and Geometry: A Collection of Papers Dedicated to Vladimir G. Turaev, ed. A. Papadopoulos, European Mathematical Society Publishing House, Berlin, 202

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.Comment: 62 pages, Sections 3.2 and 5.1 were added in order to fix a sign proble

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

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.Comment: 31 page

Diagrammatic Construction of Representations of Small Quantum sl2

We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of sl2 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