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Francesco Costantino

Publications and source records attributed to Francesco Costantino.

33 records · Page 2Linked to original sources
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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arXiv2014

On the volume conjecture for polyhedra

We formulate a generalization of the volume conjecture for planar graphs. Denoting by the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain evaluations of grow exponentially as k goes to infinity and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is G (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar graphs, generalizing the Gordon-Schulten recursion for the quantum 6j-symbols. Assuming that does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schlafli equation); moreover, any small perturbation of the volume function that is still a solution to these equations, is a perturbation by an additive constant. In the appendix we also provide a proof outlined elsewhere of the conjecture for an infinite family of planar graphs including the tetrahedra.

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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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arXiv2012

An analytic family of representations for the mapping class group of punctured surfaces

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1,0] interpolate analytically between two natural geometric unitary representations, the SU(2)-character variety representation studied by Goldman and the multicurve representation induced by the action of the mapping class group on multicurves. The finite-dimensional representations converge analytically to the infinite-dimensional ones. We recover Marche and Narimannejad's convergence theorem, and Andersen, Freedman, Walker and Wang's asymptotic faithfulness, that states that the image of a non-central mapping class is always non-trivial after some level r. When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r in term of its dilatation.

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arXiv2011

On SL(2, C) quantum 6j-symbol and its relation to the hyperbolic volume

We generalize the colored Alexander invariant of knots to an invariant of graphs, and we construct a face model for this invariant by using the corresponding 6j-symbol, which comes from the non-integral representations of the quantum group U_q(sl_2). We call it the SL(2, C) quantum 6j-symbol, and show its relation to the hyperbolic volume of a truncated tetrahedron.

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arXiv2011

Generating series and asymptotics of classical spin networks

We study classical spin networks with group SU(2). In the first part, using gaussian integrals, we compute their generating series in the case where the networks are equipped with holonomies; this generalizes Westbury's formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.

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arXiv2009

Integrality of Kauffman brackets of trivalent graphs

We show that Kauffman brackets of colored framed graphs (also known as quantum spin networks) can be renormalized to a Laurent polynomial with integer coefficients by multiplying it by a coefficient which is a product of quantum factorials depending only on the abstract combinatorial structure of the graph. Then we compare the shadow-state sums and the state-sums based on $R$-matrices and Clebsch-Gordan symbols, reprove their equivalence and comment on the integrality of the weight of the states. We also provide short proofs of most of the standard identities satisfied by quantum $6j$-symbols of $U_q(sl_2)$.

math.QA↗
arXiv2007

6j-symbols, hyperbolic structures and the Volume Conjecture

We compute the asymptotical growth rate of a large family of $U_q(sl_2)$ $6j$-symbols and we interpret our results in geometric terms by relating them to volumes of hyperbolic truncated tetrahedra. We address a question which is strictly related with S.Gukov's generalized volume conjecture and deals with the case of hyperbolic links in connected sums of $S^2\times S^1$. We answer this question for the infinite family of fundamental shadow links.

math.GT↗
arXiv2007

3-manifolds efficiently bound 4-manifolds

It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the complexity of the 4-manifold produced. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the minimum number of n-simplices in a triangulation. It is an open question whether this quadratic bound can be replaced by a linear bound. The proof goes through the notion of "shadow complexity" of a 3-manifold M. A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M. We prove that, for a manifold M satisfying the Geometrization Conjecture with Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are the graph manifolds.

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arXiv2007

Stein domains and branched shadows of 4-manifolds

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact 4-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev's shadows suitably "smoothed"; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting "many" positive and negative Stein fillable contact structures, and prove a 4-dimensional analogue of Oertel's result on incompressibility of surfaces carried by branched polyhedra.

math.CV↗
arXiv2005

A short introduction to shadows of 4-manifolds

We give a self-contained introduction to the theory of Turaev's shadows as a tool to study 3 and 4-manifolds. The goal of the present paper twofold: on one side it is intended to be a shortcut to a basic use of the theory of shadows, on the other side it gives a sketchy overview of some of the recent results on shadows. No original result is proved here and most of the details of the proofs are not included.

math.GT↗
arXiv2005

Branched shadows and complex structures of 4-manifolds

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-handlebody is homotopic to a complex one.

math.GT↗
arXiv2005

A calculus for branched spine of 3-manifolds

We establish a calculus for branched spines of 3-manifolds by means of branched Matveev-Piergallini moves and branched bubble-moves. We briefly indicate some of its possible applications in the study and definition of State-Sum Quantum Invariants.

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