By considering spaces of directed Jacobi diagrams, we construct a diagrammatic version of the Etingof-Kazhdan quantization of complex semisimple Lie algebras. This diagrammatic quantization is used to provide a construction of a directed version of the Kontsevich integral, denoted $Z_EK$, in a way which is analogous to the construction of the Reshetikhin-Turaev invariants from the R-matrices of the Drinfel'd-Jimbo quantum groups. Based on this analogy, we conjecture (and prove in a restricted sense) a formula for the value of the invariant $Z_EK$ on the unknot. This formula is simpler than the Wheels formula of [BGRT], but the precise relationship between the two is yet unknown.
We use the properties of the Melvin-Morton expansion of the colored Jones polynomial in order to prove that the trivial connection contribution converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant of rational homology spheres, as it was conjectured by R. Lawrence.
We prove that the SU(2) and SO(3) Witten-Reshetikhin-Turaev invariants of any 3-manifold with any colored link inside at any root of unity are algebraic integers.
The center Z(C) of a spherical fusion category C (over an arbitrary commutative ring) is modular. We give an algorithm for computing the Reshetikhin-Turaev invariant defined with Z(C). It is based on Hopf diagrams and an explicit description of the structure of the coend of Z(C).
This paper is a self-contained introduction to the theory of renormalized Reshetikhin-Turaev invariants of links defined by Geer, Patureau-Mirand and Turaev. Whereas the standard Reshetikhin-Turaev theory of a $\mathbb{C}$-linear ribbon category assigns the trivial invariant to any link with a component colored by a simple object of vanishing quantum dimension, the renormalized theory does not. We give a streamlined development of the renormalized Reshetikhin-Turaev theory of links for the category of weight modules over the restricted unrolled quantum group of $\mathfrak{sl}_2(\mathbb{C})$ at…
By methods similar to those used by Lisa Jeffrey, we compute the quantum $\mathrm{SU}(N)$-invariants for mapping tori of trace $2$ homeomorphisms of a genus $1$ surface when $N = 2,3$ and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the $\mathrm{SU}(2)$ case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace $-2$ homeomorphisms, obtaining -- in combination with Jeffrey's result…
On the asymptotic expansion of quantum invariants related to surgeries of Whitehead link I: Relative Reshetikhin-Turaev invariants and the Turaev-Viro invariants at $e^{\frac{2π\sqrt{-1}}{N+\frac{1}{2}}}$
In [AU2] we constructed the vacua modular functor based on the sheaf of vacua theory developed in [TUY] and the abelian analog in [AU1]. We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten-Reshetikhin-Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vogel to the vacua modular functor. This thus provides a geometric construction of the TQFT constructed first by Witten-Reshetikhin-Turaev from the quantum group associated to SL(N).
Costantino--Geer--Patureau-Mirand proved relations between the Reshetikhin--Turaev link invariants and the re-normalized link invariants for knots. Their theorem says that residues of the re-normalized link invariants are given by the Reshetikhin--Turaev link invariants. However, in the case of links, the residues are vanish, so we can not obtain any relations from the residues. In this paper, we prove that the Reshetikhin--Turaev link invariants appear in higher order terms of the re-normalized link invariants for links which are plumbed.
We prove the Volume Conjecture for the relative Reshetikhin-Turaev invariants proposed in [29] for all pairs (M,K) such that M\K is homeomorphic to the complement of the figure-8 knot in S^3 with almost all possible cone angles.
We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $\exp({2π\sqrt{-1}}/{r})$ instead of the standard $\exp({π\sqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $\infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten's Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard ro…
We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representation over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus and there are no colored points, the representations have finite image.
We present a mathematically clean review of our previous results on 1/K expansion of the colored Jones polynomial and on perturbative invariants of 3d rational homology spheres. We also prove that perturbative invariants defined through the stationary phase surgery formula are invariant under Kirby moves.
We provide a geometric construction of the boundary states for handlebodies which we in turn use to give a geometric formula for the Witten-Reshetikhin-Turaev quantum invariants. We then analyze the asymptotics of this invariant in the special case of a three manifold given by 1-surgery on a knot and we show that if the knot has an irreducible representation of its fundamental group into SU(2), then its quantum invariant cannot equal those of the three sphere. From this we conclude that if a knot has the same colored Jones polynomials as the unknot, it must be the unknot.
We formulate the Asymptotic Expansion Conjecture for the Witten-Reshetikhin-Turaev quantum invariants of closed oriented three manifolds. For finite order mapping tori, we study these quantum invariants via the geometric gauge theory approach to the corresponding quantum representations and prove, using a version of the Lefschetz-Riemann-Roch Theorem due to Baum, Fulton, MacPherson and Quart, that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus. Moreover, we show that the term corresponding to a component is a po…
We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754). We use these projectors to give a categorification of the sl_3 Reshetikhin-Turaev invariant of framed tangles.
We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity ζgives the the SU(2) Witten-Reshetikhin-Turaev invariant τ_ζ(M) of M at ζ. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of M. As a consequence, τ_ζ(M) is an algebraic integer. Moreover, it follows that τ_ζ(M) as a function on ζbehaves like an ``analytic function'' defined on the set of roots of unity. That is, the τ_ζ(M) for all roots of unity are determined by a "Taylor expansion" at any root of un…
We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.
We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of links in L(p,q) as a function of the level r-2. They are given by 1 over the square root of r times one of p Laurent polynomials evaluated at e to the 2 pi i divided by 4pr. The congruence class of r modulo p determines which polynomial is applicable. If p is zero modulo four, the meridian of L(p,q) is non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2) invariants. On the other hand, we show that one may recover the element in the Kauffman bracket skein module of L(p,q) represented by a link …
We study surface defects in three-dimensional topological quantum field theories which separate different theories of Reshetikhin-Turaev type. Based on the new notion of a Frobenius algebra over two commutative Frobenius algebras, we present an explicit and computable construction of such defects. It specialises to the construction in Carqueville et al., arXiv:1710.10214 if all 3-strata are labelled by the same topological field theory. We compare the results to the model-independent analysis in Fuchs et al., arXiv:1203.4568 and find agreement.
On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot
A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a p_1-structure, or an extended manifold structure. We perform the universal construction of Blanchet, Habegger, Masbaum and Vogel on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to t…
We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expressed in terms of the S- and T-matrices of the modular category. In another direction we derive a rational surgery formula, which states how the RT-invariants behave under rational surgery along framed links in arbitrary closed oriented 3-manifolds with embedded colored ribbon graphs…
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. This quantum trace satisfies conditions that render the link evaluation invariant under Kirby moves. If H is a suitable finite-dimensional Hopf algebra (not weak), our invariant reduces to the Kauffman-Radford invariant for the dual of H. If H is the Weak Hopf Algebra Tannaka-Krein …
A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $Δ$-separable symmetric Frobenius algebras and the line defects by "multi-modules" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where …
We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simply-connected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, …
We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category $\mathcal{C}$, using the language of defect TQFT from [arXiv:1705.06085]: (i) spherical fusion categories give orbifolds for the "trivial" defect TQFT associated to vect, (ii) $G$-crossed extensions of $\mathcal{C}$ give group orbifolds for any finite group $G$, and (iii) we construct orbifolds from commutative $Δ$-separable symmetric Frobenius algebras in $\mathcal{C}$. We also explain how the Turaev-Viro state sum construction fits into our framework by proving that it is isomorphic…
We study the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants proposed in \cite{WY4} for all pairs $(M,L)$ satisfying the property that $M\setminus L$ is homeomorphic to some fundamental shadow link complement. The hyperbolic cone structure of such $(M,L)$ can be described by using the logarithmic holonomies of the meridians of some fundamental shadow link. We show that when the logarithmic holonomies are sufficiently small and all cone angles are less than $π$, the asymptotic expansion conjecture of $(M,L)$ is true. Especially, we verify the asymptotic expansion c…
We propose the Volume Conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented $3$-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern-Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient $3$-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental …
We calculate the large quantum level asymptotic expansion of the RT-invariants associated to SU(2) of all oriented Seifert 3-manifolds X with orientable base or non-orientable base with even genus. Moreover, we identify the Chern-Simons invariants of flat SU(2)-connections on X in the asymptotic formula thereby proving the so-called asymptotic expansion conjecture (AEC) due to J. E. Andersen for these manifolds. For the case of Seifert manifolds with base $S^2$ we actually prove a little weaker result, namely that the asymptotic formula has a form as predicted by the AEC but contains some extr…
Gukov-Pei-Putrov-Vafa constructed $q$-series invariants called homological blocks in a physical way in order to categorify Witten-Reshetikhin-Turaev (WRT) invariants and conjectured that radial limits of homological blocks are WRT invariants. In this paper, we prove their conjecture for unimodular H-graphs. As a consequence, it turns out that the WRT invariants of H-graphs yield quantum modular forms of depth two and of weight one with the quantum set $\mathbb{Q}$. In the course of the proof of our main theorem, we first write the invariants as finite sums of rational functions. We second carr…
In the third paper in this series, we examine the Reshetikhin-Turaev and Turaev-Viro TQFTs at the level of surfaces. In particular, we show that for a closed surface $Σ$, $Z_{TV, \mathcal{C}}(Σ) \cong Z_{RT, Z(\C)}(Σ)$, thus extending the equality of 3-manifold invariants proved in an earlier paper to an equivalence of TQFTs. We also describe how to compute Turaev-Viro state sums for 3-manifolds with embedded ribbon graphs.
In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category $C$, to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center $Z(C)$. In the present paper we give a partial proof of this statement.
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.
Let C be a spherical fusion category. We prove that the Turaev-Viro-Barrett-Westbury state sum invariant of 3-manifolds derived from C is equal to the Reshetikhin-Turaev surgery invariant of 3-manifolds derived from Z(C), where Z(C) is the Drinfeld-Joyal-Street center of C.
Starting from the quantum group SL_q(2,C), we construct operator invariants of 3-cobordisms with spin structure, satisfying the requirements of a topological quantum field theory and refining the Reshetikhin--Turaev and Turaev--Viro models. We establish the relationship between these two refined models.
We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We then discuss the choice of the framing parameter. This leads, for any rank N and level K, to a modular category \tilde H^{N,K} and a reduced invariant \tildeτ_{N,K}. If N and K are coprime, then this invariant coincides with the known PSU(N) invariant at level K. If gcd(N,K)=d>1, then we show…
We prove in the context of quantum groups at even roots of unity that a Turaev-Viro type invariant of a 3-dimensional cobordism M equals the tensor product of the Reshetikhin-Turaev invariants of M and M*, where M* denotes M with orientation reversed.
We extend the definition of the U(1)-reducible connection contribution to the case of the Witten-Reshetikhin-Turaev invariant of a link in a rational homology sphere. We prove that, similarly ot the case of a link in S^3, this contribution is a formal power series in powers of q-1, whose coefficients are rational functions of q^{color}, their denominators being the powers of the Alexander-Conway polynomial. The coefficients of the polynomials in numerators are rational numbers, the bounds on their denominators are established with the help of the theorem proved by T. Ohtsuki in Appendix 2. Sim…
On the density and surjectivity of $\mathbf{SO(3)}$-Witten-Reshetikhin-Turaev quantum representations
Université Bourgogne Franche-Comté（勃艮第弗朗什-孔泰大学）；Institut Universitaire de France（法国高等研究院）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；Université Paris-Saclay（巴黎萨克雷大学）
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 infin…
In this paper we provide a general condition for the reducibility of the Reshetikhin-Turaev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are reducible. In particular for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin-Turaev theory we construct a decomposition for all even levels. We conjecture this deco…
We show that the Witten-Reshetikhin-Turaev SU(2) invariant and the Hennings invariant associated to the restricted quantum $sl_2$ are essentially the same for rational homology 3-spheres.
We show that when $r \geq 5$ is prime, the SO(3) Witten-Reshetikhin-Turaev quantum invariants for three-manifolds at the level $r$ form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.
The Reshetikhin - Turaev invariants arising from the quantum groups associated with the exceptional Lie algebras $G_2$, $F_4$ and $E_8$ at odd roots of unity are constructed and explicitly computed for all the lens spaces.
