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Christian Blanchet

Publications and source records attributed to Christian Blanchet.

At least 19 recordsLinked to original sources
arXiv2024

Abelian TQFTS and Schrödinger local systems

We construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a $q$-deformation of $U(1)$ for any root of unity $q$. We prove that for3-cobor\-disms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT. The full abelian TQFT provides a projective representation of the mapping class group $\mathrm{Mod}(Σ)$ on the Schrödinger representation,which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of $\mathrm{Mod}(Σ)$ on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.

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arXiv2024

Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds

We develop a complete obstruction theory for the $\mathbb{Z}_2$-index of a compact connected 4-dimensional manifold with free involution. This $\mathbb{Z}_2$-index, equal to the minimum integer $n$ for which there exists an equivariant map with target the $n$-sphere with antipodal involution, is computed in two steps using cohomology with twisted coefficients. The key ingredient is a spectral sequence computing twisted cohomology of the orbit space of a free involution on odd complex projective spaces. We illustrate the main results with various examples including computation of the secondary obstruction.

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arXiv2024

Heisenberg homology of ribbon graphs

We review Heisenberg homology of configurations in once bounded surfaces and extend the construction to the regular thickening of a finite graph with ribbon structure.

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arXiv2023

Action of subgroups of the mapping class group on Heisenberg homologies

In previous work we constructed twisted representations of mapping class groups of surfaces, depending on a choice of representation $V$ of the Heisenberg group $\mathcal{H}$. For certain $V$ we were able to untwist these mapping class group representations. Here, we study the restrictions of our twisted representations to different subgroups of the mapping class group. In particular, we prove that these representations may be untwisted on the Torelli group for any given representation $V$ of $\mathcal{H}$. When $V$ is the Schr\"odinger representation, we also construct untwisted representations of subgroups defined as kernels of crossed homomorphisms studied by Earle and Morita.

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arXiv2021

Heisenberg homology on surface configurations

Motivated by the Lawrence-Krammer-Bigelow representations of the classical braid groups, we study the homology of unordered configurations in an orientable genus-$g$ surface with one boundary component, over non-commutative local systems defined from representations of the discrete Heisenberg group. Mapping classes act on the local systems and for a general representation of the Heisenberg group we obtain a representation of the mapping class group that is twisted by this action. For the linearisation of the affine translation action of the Heisenberg group we obtain a genuine, untwisted representation of the mapping class group. In the case of the generic Schr\"odinger representation, by composing with a Stone-von Neumann isomorphism we obtain a projective representation by bounded operators on a Hilbert space, which lifts to a representation of the stably universal central extension of the mapping class group. We also discuss the finite dimensional Schr\"odinger representations, especially in the even case. Based on a natural intersection pairing, we show that our representations preserve a sesquilinear form.

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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 $δ= -q-q^{-1}$.

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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.

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arXiv2021

The Borsuk-Ulam theorem for 3-manifolds

We study the Borsuk-Ulam theorem for triple (M;τ; \R^n), where M is a compact, connected, 3-manifold equipped with a fixed-point-free involution τ. The largest value of n for which the Borsuk-Ulam theorem holds is called the Z_2-index and in our case it takes value 1, 2 or 3. We fully discuss this index according to cohomological operations applied on the characteristic class x \in H^1(N; Z_2), where N = M/τis the orbit space. In oriented case, we obtain an expression of the index from the linking matrix of a surgery presentation of the orbit space. We illustrate our results with examples, including a non orientable one.

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arXiv2020

Modified trace is a symmetrised integral

A modified trace for a finite k-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. We show that a non-degenerate modified trace defines a compatible with duality Calabi-Yau structure on the subcategory of projective objects. The modified trace provides a meaningful generalisation of the categorical trace to non-semisimple categories and allows to construct interesting topological invariants. We prove, that for any finite-dimensional unimodular pivotal Hopf algebra over a field k, a modified trace is determined by a symmetric linear form on the Hopf algebra constructed from an integral. More precisely, we prove that shifting with the pivotal element defines an isomorphism between the space of right integrals, which is known to be 1-dimensional, and the space of modified traces. This result allows us to compute modified traces for all simply laced restricted quantum groups at roots of unity.

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arXiv2020

Fukaya category of surfaces and mapping class group action

We construct the Fukaya category of a surface with genus greater than one and compute its Grothendieck group. We consider here a topological variant, in which we disregard the area form and use instead an admissibility condition borrowed from Heegaard-Floer theory which ensures invariance under isotopy. We also study action of the mapping class group of the surface on its Fukaya category and establish faithfulness.

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arXiv2018

Holonomy braidings, biquandles and quantum invariants of links with $SL_2(\mathbb C)$ flat connections

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group $U_qsl_2$ at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin-Turaev ribbon type functor for tangles with quandle representations. This theory applies to the unrestricted quantum group $U_qsl_2$ and produces an invariant of links with a gauge class of quandle representations.

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arXiv2017

Logarithmic Hennings invariants for restricted quantum sl(2)

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored by central elements and by trace classes, or elements in the $0^{\text{th}}$ Hochschild homology of $U$, respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of $U$, and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.

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arXiv2016

Non semi-simple TQFTs from unrolled quantum $sl(2)$

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:1202.3553 . They are invariants of $3$-manifolds together with a cohomology class which can be interpreted as a line bundle with flat connection. In arXiv:1404.7289 we have extended those invariants to graded TQFTs on suitable cobordism categories. Here we give an overview of constructions and results, and describe the TQFT vector spaces. Then we provide a new, algebraic, approach to the computation of these vector spaces.

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arXiv2014

Spin Modular Categories

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is played by objects which are invertible under tensor product. All known examples of cohomological or spin type refinements of the Witten-Reshetikhin-Turaev 3-manifold invariants are special cases of our construction. In addition, we establish a splitting formula for the refined invariants, generalizing the well-known product decomposition of quantum invariants into projective ones and those determined by the linking matrix.

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arXiv2014

An oriented model for Khovanov homology

We give an alternative presentation of Khovanov homology of links with strict functoriality result over integers. The construction uses an oriented $sl(2)$ state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces.

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arXiv2014

Non semi-simple sl(2) quantum invariants, spin case

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 $2r$ where $r>1$ is odd or congruent to $2$ modulo $4$. In this paper we consider the remaining cases where $r$ is congruent to zero modulo $4$ and produce invariants of $3$-manifolds with colored links, equipped with generalized spin structure. For a given $3$-manifold $M$, the relevant generalized spin structures are (non canonically) parametrized by $H^1(M;\mathbb C/2\mathbb Z)$.

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arXiv2014

Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants

We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in (arXiv:1202.3553) including the Kashaev invariant of links. Here the modular category framework does not apply and we use the ``universal construction''. Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a root of unity of order 2r. The functor is always symmetric monoidal but for even values of r the braiding on GrVect has to be the super-symmetric one, thus our TQFT may be considered as a super-TQFT. In the special case r=2 our construction yields a TQFT for a canonical normalization of Reidemeister torsion and we re-prove the classification of Lens spaces via the non-semi-simple quantum invariants defined in (arXiv:1202.3553). We prove that the representations of mapping class groups and Torelli groups resulting from our constructions are potentially more sensitive than those obtained from the standard Reshetikhin-Turaev functors; in particular we prove that the action of the bounding pairs generators of the Torelli group has always infinite order.

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