arXiv subjects

Bertrand Patureau-Mirand

Publications and source records attributed to Bertrand Patureau-Mirand.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2016

The trace on projective representations of quantum groups

For certain roots of unity, we consider the categories of weight modules over three quantum groups: small, un-restricted and unrolled. The first main theorem of this paper is to show that there is a modified trace on the projective modules of the first two categories. The second main theorem is to show that category over the unrolled quantum group is ribbon. Partial results related to these theorems were known previously.

math.QA↗

arXiv2016

Some remarks on the unrolled quantum group of sl(2)

In this paper we consider the representation theory of a non-standard quantization of sl(2). This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In the process of proving these results the paper acts as a survey of the known representation theory associated to this non-standard quantization of sl(2). The results of this paper are used extensively in [arXiv:1404.7289] to study Topological Quantum Field Theory (TQFT) and have connections with Conformal Field Theory (CFT).

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arXiv2016

Other quantum relatives of the Alexander polynomial through the Links-Gould invariants

Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show these Hopf algebras share properties with the $-1$ specialization of $U_q gl(n|1)$ leading to the proof of a conjecture of David De Wit, Atsushi Ishii and Jon Links on the Links-Gould invariants.

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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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arXiv2015

Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories

In this paper we construct invariants of 3-manifolds "à la Reshetikhin-Turaev" in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3-manifold invariants indexed by the integers. We prove this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Reshetikhin-Turaev-Witten invariants do not, and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links.

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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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arXiv2013

Relations between Witten-reshetikhin-turaev and non semi-simple sl(2) 3-manifold invariants

The Witten-Reshetikhin-Turaev invariants extend the Jones polynomials of links in S^3 to invariants of links in 3-manifolds. Similarly, in a preceding paper, the authors constructed two 3-manifold invariants N_r and N^0_r which extend the Akutsu-Deguchi-Ohtsuki invariant of links in S^3 colored by complex numbers to links in arbitrary manifolds. All these invariants are based on representation theory of the quantum group Uqsl2, where the definition of the invariants N_r and N^0_r uses a non-standard category of Uqsl2-modules which is not semi-simple. In this paper we study the second invariant N^0_r and consider its relationship with the WRT invariants. In particular, we show that the ADO invariant of a knot in S^3 is a meromorphic function of its color and we provide a strong relation between its residues and the colored Jones polynomials of the knot. Then we conjecture a similar relation between N^0_r and a WRT invariant. We prove this conjecture when the 3-manifold M is not a rational homology sphere and when M is a rational homology sphere obtained by surgery on a knot in S^3 or when M is a connected sum of such manifolds.

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arXiv2013

Modified quantum dimensions and re-normalized link invariants

In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of link invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots.

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arXiv2013

G-links invariants, Markov traces and the semi-cyclic Uqsl2-modules

Kashaev and Reshetikhin proposed a generalization of the Reshetikhin-Turaev link invariant construction to tangles with a flat connection in a principal G-bundle over the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to sl2, defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi, and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.

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arXiv2013

Polynomial 6j-Symbols and States Sums

For q a root of unity of order 2r, we give explicit formulas of a family of 3-variable Laurent polynomials J_{i,j,k} with coefficients in Z[q] that encode the 6j-symbols associated with nilpotent representations of U_qsl_2. For a given abelian group G, we use them to produce a state sum invariant tau^r(M,L,h_1,h_2) of a quadruplet (compact 3-manifold M, link L inside M, homology class h_1\in H_1(M,Z), homology class h_2\in H_2(M,G)) with values in a ring R related to G. The formulas are established by a "skein" calculus as an application of the theory of modified dimensions introduced in [arXiv:0711.4229]. For an oriented 3-manifold M, the invariants are related to TV(M,L,f\in H^1(M,C^*)) defined in [arXiv:0910.1624] from the category of nilpotent representations of U_qsl_2. They refine them as TV(M,L,f)= Sum_h tau^r(M,L,h,f') where f' correspond to f with the isomorphism H_2(M,C^*) ~ H^1(M,C^*).

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arXiv2013

Topological invariants from non-restricted quantum groups

We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, respectively. In this case we show that these invariants are equal and extend to what we call a relative Homotopy Quantum Field Theory which is a branch of the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our main examples of relative spherical categories are the categories of finite dimensional weight modules over non-restricted quantum groups considered by C. De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories are not semi-simple and have an infinite number of non-isomorphic irreducible modules all having vanishing quantum dimensions. We also show that these categories have associated ribbon categories which gives rise to re-normalized link invariants. In the case of sl(2) these link invariants are the Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.

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arXiv2012

Traces on ideals in pivotal categories

We extend the notion of an ambidextrous trace on an ideal (developed by the first two authors) to the setting of a pivotal category. We show that under some conditions, these traces lead to invariants of colored spherical graphs (and so to modified 6j-symbols).

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arXiv2011

Ambidextrous objects and trace functions for nonsemisimple categories

We provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role in low-dimensional topology as well as in studying the category itself. In particular, we prove they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the $(1,p)$ minimal model in conformal field theory, and quantum groups at a root of unity.

math.RT↗

arXiv2011

Non injectivity of the "hair" map

Kricker constructed a knot invariant Z^{rat} valued in a space of Feynman diagrams with beads. When composed with the so called "hair" map H, it gives the Kontsevich integral of the knot. We introduce a new grading on diagrams with beads and use it to show that a non trivial element constructed from Vogel's zero divisor in the algebra Λ is in the kernel of H. This shows that H is not injective.

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arXiv2010

Generalized trace and modified dimension functions on ribbon categories

In this paper we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are non-zero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.

math.RT↗

arXiv2009

Modified 6j-Symbols and 3-Manifold Invariants

We show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects.

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