Analytic families of quantum hyperbolic invariants
We organize the quantum hyperbolic invariants (QHI) of -manifolds into sequences of rational functions indexed by the odd integers and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic -manifolds we generalize the QHI and get rational functions depending on a finite set of cohomological data called weights. These functions are regular on a determined Abelian covering of degree of a Zariski open subset, canonically associated to , of the geometric component of the variety of augmented -characters of . New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions depend on the weights as , and recover the volume for some specific choices of the weights.
Institut de Mathématiques et de Modélisation, Université Montpellier 2, Case Courrier 51, 34095 Montpellier Cedex 5, France (email@example.com)
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy (firstname.lastname@example.org)
- 1 Introduction
- 2 Structured triangulations and gluing varieties
- 3 Analytic configurations
- 4 Cusped manifolds
- 5 QHFT partition functions
- 6 -graph calculus
- 7 State sum invariance over weakly branched triangulations
- 8 Resolution of the sign anomaly
- 9.1 Notations
- 9.2 The figure-eight knot complement
- 9.3 The complement of the knot
- 9.4 The sister of the figure-eight knot complement
In the series of papers [3, 4, 5] we defined a family of complex valued quantum invariants of “patterns” of geometric nature, called quantum hyperbolic invariants (QHI) and indexed by the odd integers . Roughly, a pattern consists of a compact oriented -manifold equipped with a representation of the fundamental group into , plus some compatible cohomological datas (see the next subsections for a complete definition). The QHI generalize the Kashaev invariants of links in , subject to the celebrated Volume Conjecture. They define a -dimensional quantum field theory, and, when applied to mapping cylinders of surface diffeomorphisms, they coincide with the invariants derived from the local version of finite dimensional quantum Teichmüller theory (see ,  and ).
In [3, 4, 5], the invariance of the QHI was proved only up to sign and multiplication by -th roots of unity. The eventual existence and the meaning of such a phase anomaly, as well as the determination of the asymptotical behaviour of as , are two main open issues of the theory. In order to tackle them, it seemed necessary to develop a functional approach that would clarify the intrinsic nature of the various combinatorial and geometric ingredients involved in the definition of the QHI. To this aim, we achieve in this paper the following goals:
QHI of cusped manifolds: We extend the QHI of any one-cusped hyperbolic -manifold to invariants defined on the geometric component of the variety of augmented -characters of . In [4, 5] the QHI of cusped hyperbolic manifolds were defined only at the hyperbolic holonomy.
State sums over weakly branched triangulations: In order to achieve the previous goal it is necessary to introduce state sums over triangulations that do not support any branching, replaced by relaxed structures called weak branchings. These states sums give rise to QHI generalizing all the previously defined ones.
Analytic families of QHI: We recast all the QHI into sequences of families of complex analytic spaces and maps, indexed by the odd integers . Each family is associated to a topological support , and provides concrete models of geometric structures over called patterns. Patterns over cusped manifolds have an intrinsic meaning in terms of the -version of the -polynomial, and natural relationships with Chern-Simons theory.
Fixing the sign ambiguity: We include a sign correction in the state sum formulas of the QHI, which removes their sign ambiguity under a mild assumption on the “bulk -weight” (see below). The sign correction depends on the combinatorics of the weak branching and is a by-product of . It becomes trivial when dealing (when possible) with branched triangulations, so that the QHI of [3, 4, 5] are eventually defined up to multiplication by -th roots of unity.
In a sequel to this paper (in collaboration with C. Frohman), we will develop our approach concerning the asymptotical behaviour of the QHI. Basically, given a topological support and a sequence of patterns over , we study the limit of the family as , instead of for a single sequence . Then, assuming that is finite (this is the case for all natural sequences ), we consider the following problems:
Determine the nature (regularity) of as a function of the patterns over .
Describe the asymptotical behaviour of as in terms of classical geometric invariants of : Chern-Simons invariants, torsions, twisted cohomology, etc.
The Kashaev-Murakami-Murakami Volume Conjecture is a particular case of (2), for constant sequences of patterns associated to links in (see Theorem 1.2 below).
In the rest of this Introduction we describe with more details the content of the paper.
1.1. QHI of cusped manifold patterns
In this paper we call cusped manifold an oriented, connected, non–compact complete hyperbolic -manifold of finite volume with one cusp. Hence a cusped manifold is diffeomophic to the interior of a compact -manifold denoted by , with one torus boundary component.
A pattern over consists of a topological support together with additional geometric structures determined by the couple . The topological support takes the form , where is a so-called c-weight, defined by a “bulk -weight” and a “boundary -weight” satisfying
where is the reduction mod, and the map is induced by the inclusion map . The pattern is obtained by completing with a couple where is a -character of , i.e. a conjugacy class of representations of in , and is a so-called -weight (relative to ), defined by a “bulk -weight” and a “boundary -weight” satisfying the following constraint. Up to conjugacy the restriction of to the torus is valued in the group of complex affine transformations of the plane; the linear part of this restriction defines a class in . Let be the log of this class, with imaginary part in . One requires that for all ,
Collecting the bulk and boundary weights we will often write as , where
Notation. For every , we write “” to mean that and are equal up to multiplication by a power of . If is odd, then if and only if . We denote by the group of -th roots of unity, acting on by multiplication.
that is, for the pattern where is the hyperbolic holonomy of and all weights vanish. The following theorem summarizes our new results for patterns over (all terms are defined in Section 4).
Let be an arbitrary cusped manifold, the variety of augmented -characters of , and the irreducible component of . There is a canonical non empty Zariski open subset of containing such that:
(1) For every odd integer and every pattern such that , there is a quantum hyperbolic invariant satisfying .
(2) (Analytic Families) Fix a topological support and . For every odd integer , the invariants define a regular rational function on a determined -covering space of .
(3) (Resolution of the sign ambiguity) If mod, or mod and , then the statements (1) and (2) above hold true by replacing with .
a) is a complex algebraic variety and a regular point of . Hence there is a unique irreducible component of containing . As has only one cusp, is an algebraic curve and the complement of a finite set of points.
b) Theorem 1.1 (2) is a rough qualitative formulation of the -th analytic family associated to the topological support of .
1.2. QHI of other patterns
QHFT partition functions. The QHI of QHFT patterns have been defined in . The topological supports of QHFT patterns have the form where:
is a compact oriented connected -manifold with (possibly empty) boundary made by torus components; if we will use the notation instead of .
is a non–empty link in the interior of ;
the bulk boundary -weights satisfy (1).
The QHFT patterns are obtained by completing with a couple , where is any -character of and are bulk and boundary -weights satisfying (2) with respect to .
When is a closed -manifold, disappears so that . In [3, 4] we defined the QHI in that situation, for every character and weight . A specialization is , where is the trivial character of . In  we obtained the following result, which establishes a connection with Jones invariants.
For every link in and every odd integer we have
where is the link invariant defined by the enhanced Yang-Baxter operator extending the Kashaev R-matrix, and is the colored Jones polynomial, normalized so that on the unknot .
The second equality is due to . In [3, 4, 5] we quoted occasionally the first one as a motivating fact. Later we realized that we were unable to derive a complete proof from the existing literature (in particular [24, 25]), so we provided an independent one in , under the ambiguity . The above statement with follows from the state sum sign correction introduced in the present paper (see also Remarks 2.11 and 5.5).
When the QHFT partition functions are more sophisticated (see ). In particular the link contains an essential simple closed curve on each boundary component which actually encodes a Dehn filling instruction. Anyway, also in this case the invariants are defined for arbitrary characters and weights.
Relation with the QHI of cusped manifolds. The patterns over cusped manifolds and the QHFT patterns are complementary in the sense that the link is empty in the former. In [4, 5] the proof of invariance of differs to many extents from the one for the QHFT partition functions. It uses the “volume rigidity” for cusped manifolds (see eg. ), Thurston’s hyperbolic Dehn filling theorem, a construction of certain auxiliary invariants that depend a priori on an additional datum “”, and finally a surgery formula. Set , where is closed (see our notations above). By combining all these results we proved:
(, Section 6.2) Let be a sequence of closed hyperbolic Dehn fillings of whose holonomies (considered as -characters on ) converge to in . Denote by the geodesic core of the solid torus that fills to produce . For every odd and every additional datum “” we have
Hence “” is eventually immaterial, and the limit defines .
The normalization on the right side of (4) is a by-product of the proof. Under some additional assumptions on (for instance if is “very gentle” according to [4, 5]; then “”), we could avoid the delicate surgery argument and define the invariants for arbitrary weights relative to the hyperbolic holonomy.
1.3. State sums over weakly branched triangulations
For every topological support , denote by the compact space obtained by filling each boundary component of with the cone over it. If then ; has a finite set of non-manifold points, the vertices of the filling cones. For every pattern supported by , is computed by state sums over certain “decorated” triangulations of , depending on the choice of a point in the associated gluing variety such that represents the character (see Section 2), and on a suitable encoding of the weights . Moreover, is equipped with a branching; equivalently, carries a structure of -complex in the sense of .
In the case of QHFT partition functions this is not so demanding: can be a “quasi-regular” triangulation (every edge has distinct endpoints), a branching can be induced for example by a total ordering of the vertices, and can be realized by means of a so-called “idealization” of -valued 1-cocycles on .
On another hand, for a cusped manifold we use ideal triangulations of such that the gluing variety contains a point representing the hyperbolic holonomy , and having coordinates with non-negative imaginary part (we say that is non-negative). Such triangulations exist for every , for instance any maximal subdivision of the canonical Epstein-Penner cell decomposition has this property. However we do not know if every cusped manifold has such a triangulation admitting a branching . For instance, the canonical Epstein-Penner decomposition of the “figure-8-knot sister” (m003 in Snappea’s census) is made of two regular hyperbolic ideal tetrahedra and does not carry any branching. The “very gentle” manifolds mentioned after Theorem 1.4 admit by definition a branched triangulation with a non-negative point in .
In order to get Theorem 1.1 (1) we relax branchings to weak branchings which exist on every triangulation, and this leads us to include -face tensors in the state sum formulas. In this setup, as well as to cover arbitrary characters of in , the proof of the state sum invariance requires additional arguments with respect to [4, 5].
1.4. Plan of the paper
In Section 2 we recall a few general facts about triangulations endowed with pre-branchings, weak branchings, or branchings, and the associated gluing varieties of .
In Section 3 we construct the analytic configuration for every odd integer , weakly branched triangulation of , and rough charge on (suitably specialized “global charges” will eventually encode the -weights below). In particular contains an infinite Abelian covering of the gluing variety, , and an analytic function . In the case of QHFT patterns it is described qualitatively by:
For every topological support there is a weakly branched triangulation of and a global charge on such that:
encodes the -weight .
For every QHFT pattern with topological support , there is a point such that represents the character and for every odd the scalar does not depend on the choice of and .
In the case of patterns over a cusped manifold , we have:
For every topological support over , there is a determined Zariski open subset of containing the hyperbolic holonomy , and there is a weakly branched ideal triangulation of , and a global charge on such that:
encodes the -weight .
The gluing variety contains a non-negative point representing .
There is an irreducible component of , a Zariski open subset of containing , and a homeomorphism extending a regular rational isomorphism between Zariski open subsets, such that is the holonomy of represented by .
For every pattern over with topological support and for every , there is a point such that , and for every odd the scalar does not depend on the choice of and .
We will use concrete models of the finite coverings and regular rational maps in Theorem 1.1 (2) by considering a suitable factorization of .
In Section 5 we indicate briefly how to deal with QHFT partition functions.
In Section 6 we collect a few facts about a diagrammatic calculus for weakly branched triangulations. This calculus is used in Section 7, which contains the invariance proof of the state sums defined over such triangulations.
In Section 8 we prove Theorem 1.1 (3) and the analogous result for QHFT partition functions. As a by product we show that global compensations of the local sign ambiguities imply that the QHI defined by means of branched triangulations have no sign ambiguity. This applies to QHFT partition functions and to very gentle cusped manifolds.
In Section 9 we describe the quantum hyperbolic invariants of three cusped manifolds: the figure-eight knot complement, its “sister”, and the complement of the knot . We present the results of numerical computations showing that the functions depend on the weights as , and recover the volume for some specific choices of the weights.
Notations. In general we denote by the odd integers , including the case when specified, and we put and . The set is identified to the group and denotes the residue modulo of . We let (resp. ) if (resp. ).
Acknowledgments. The first author’s work was supported by the ANR projects Géométrie et Topologie Quantiques (ANR-08-JCJC-0114-01) and ”Extensions des théories de Teichmüller-Thurston” (ANR-09-BLAN-0116-01). Many thanks are due to Columbia University, Fukuoka University, the MSC at Tsinghua University and the University of Utah at Moab where parts of this work was presented. The second author’s work is supported by the italian FIRB project Geometry and topology of low-dimensional manifolds. The numerical computations presented in Section 9 were obtained by using the Maple software.
2. Structured triangulations and gluing varieties
Triangulations. Let be as in Section 1.2, that is, a compact oriented connected -manifold with (possibly empty) boundary made by toric components. Denote by the space obtained by taking the cone over each boundary component of . A triangulation of is a collection of oriented tetrahedra together with a complete system of pairings of their -faces via orientation reversing affine isomorphisms, such that the oriented quotient space
is homeomorphic to , preserving the orientations. We will distinguish between the -faces, edges and vertices of the disjoint union , and the ones of after the -face pairings. In particular we denote by and the set of edges of and respectively, and we write to mean that an edge is identified to under the -face pairings. The -faces of each tetrahedron have the boundary orientation defined by the rule: “first the outgoing normal”. When , the non-manifold points of are necessarily vertices of . A triangulation of is called ideal if the set of vertices of coincides with the set of non-manifold points of .
Gluing varieties. Let be as above. For every tetrahedron choose a vertex . Order the edges of the opposite -face so that the induced cyclic ordering is the opposite of the boundary orientation. Denote these edges by . Give a label to and the opposite edge, where and . The gluing variety of is the algebraic subset of with coordinates and defining equations ( is mod):
By the first set of “tetrahedral relations”, we see that the gluing variety is the graph of an explicit regular rational map defined on an algebraic subset of defined by the second set of “edge relations”. The auxiliary choices of ordered edges being immaterial, these algebraic varieties are canonically isomorphic. We denote them by .
Every point represents a -character of . Its components , , can be interpreted as the cross-ratio parameters of an isometry class of oriented hyperbolic ideal tetrahedra associated to , that we denote by . Their imaginary parts have a same sign (by convention if the imaginary parts are zero). By a classical result of Schläffli, the algebraic volume of is given by
where Vol is the geometric positive volume and D the Bloch-Wigner dilogarithm. When the components are real, is degenerate and both sides of (5) vanish. By summing the algebraic volumes of the s we get a volume function
If is non empty, for every point , coincides with the (intrinsically defined) volume of the character . In general might be empty, but has always triangulations such that is non trivial. A first general result concerns its dimension.
This result depends on the combinatorial properties of . If the interior of is a cusped manifold , then it has the canonical Epstein-Penner (EP) cell decomposition by embedded convex hyperbolic ideal polyhedra (see for instance ). Then we have:
The maximal subdivisions of the EP cell decomposition of define a finite set of ideal triangulations of , such that for every the gluing variety contains a non-negative point such that , and .
Every non-degenerate hyperbolic ideal tetrahedron of has strictly positive (geometric) volume, but in general one cannot avoid some degenerate tetrahedra.
For other manifolds the nature of is not so well-known. For instance consider the case of a closed -manifold . A triangulation of is called quasi-regular if every edge of has distinct vertices in . It is clear that every triangulation has a quasi-regular subdivision. Take one, and fix a total ordering of the vertices. For every edge with endpoints and , orient from to if . Every simplicial -valued -cocycle on , defined by using this edge orientation, represents a character of . Then, to any sufficiently generic cocycle one can associate a point such that , and we have ():
Let be a quasi-regular triangulation of . For every character of there is a point such that .
Clearly, the point is far to be unique. For instance, if every point of represents the trivial character. A similar, slightly more elaborated result holds for all other topological supports of QHFT patterns; it uses triangulations of obtained from quasi-regular relative triangulations of by adding a cone over each component of .
The method used to prove Proposition 2.3 is reminiscent of Thurston’s spinning construction, and is strictly related to it when is hyperbolic. In that case, the following result, which is proved by using the spinning construction, agrees with Proposition 2.3 in the case of quasi regular triangulations.
() Let be a triangulation of a closed oriented hyperbolic -manifold such that no edge is a null-homotopic loop in . Then there exists such that , and moreover its holonomy is the hyperbolic holonomy.
Variations on branched triangulations. Define a pre-branched tetrahedron as an oriented tetrahedron with a choice of co-orientations of the -faces, such that two co-orientations are ingoing and two are outgoing. As every -face has the boundary orientation, by duality can be interpreted as a system of -face orientations.
Figure 1 shows a pre-branched tetrahedron embedded in , with coordinates such that the plane of the picture is . We put on the orientation induced from , and assume that the two -faces above (resp. below) the plane are those with outgoing (resp. ingoing) co-orientations. This specifies two diagonal edges and four square edges. Every square edge is oriented as the common boundary edge of two -faces with opposite co-orientations. So the square edges form an oriented quadrilateral. Using the orientation of , one can also distinguish among the square edges two pairs of opposite edges, called -edges and -edges. The orientation of the diagonal edges is not determined.
An oriented tetrahedron becomes a -simplex by ordering its vertices. This is equivalent to a system of orientations of the edges, called (local) branching, such that the vertex has incoming edges (). The -faces of are ordered as the opposite vertices, and induces a branching on each -face . The branchings and define orientations on and respectively, the - and -orientations. The -orientation may coincide or not with the orientation of . We encode this by a sign, . The boundary orientation and the -orientation agree on two -faces. Hence defines a pre-branching . On another hand, given a pre-branching on there are exactly four branchings such that . They can be obtained by choosing an (resp. ) edge, reversing its orientation, and completing the resulting square edge orientations to define a branching (this can be done in a single way; see Figure 2). Note that (resp. ) if and only if we have chosen an (resp. ) square edge, and the square edge is . The diagonal edges are and .
One can extend these notions to triangulations of :
A pre-branched triangulation is formed by pre-branched tetrahedra such that the -face co-orientations match under the -face pairings.
A weakly-branched triangulation is formed by branched tetrahedra such that the induced pre-branched tetrahedra form a pre-branched triangulation .
A branched triangulation is formed by branched tetrahedra such that the branchings (ie. the edge orientations) match under the 2-face pairings.
Branched triangulations of and -complexes over  are equivalent notions. In particular the simplicial -chain represents the fundamental class in .
Let be a triangulation of . Denote by the compact -manifold with boundary obtained by removing a small open -ball around every vertex of which is a manifold point. Clearly, if and only if is an ideal triangulation. A pre-branched triangulation of can be described in a very concrete way in terms of the standard spine of dual to :
There is a -to- duality correspondence between the sets of pre-branchings of and those of , where a prebranching of is defined as an orientation of the singular locus such that every vertex has two outgoing and two ingoing edges.
The proof is evident, as every edge of is dual to a -face of . The notion of (weakly) branched triangulation has a natural dual counterpart as (weakly) branched spine .
The branched boundary of a pre-branched triangulation. The triangulation of yields a decomposition of into truncated tetrahedra. Their triangular -faces form a triangulation of . If is a pre-branched triangulation, then has a branching defined on the -faces of as in Figure 3.
The corners of every -simplex of have a label in , and a sign obtained by comparing the boundary and the -orientations. Such a -labelling may be defined more generally for any branched triangulation of a closed oriented surface.
Each vertex of a branched triangulation of a closed oriented surface has an even number of adjacent corners with the empty label . Hence every edge of a pre-branched triangulation of has an even number of diagonal edges such that .
Proof. For each -face the edges adjacent to the corner with label have opposite -orientations, while they agree elsewhere. The existence of an orientation on any transverse loop implies the first claim. The second follows by considering the triangulation .
Networks and -graphs. Consider an oriented graph as in Lemma 2.6. Put around every vertex the dual branched tetrahedron so that its -faces intersect transversely the edges of . Thus each edge connects a -face of the “initial” branched tetrahedron with a -face of the “final” one , identified in . The gluing map is determined by a color defined as follows. Set . Denote by the symmetric group on , by the subgroup of even permutations, and by and the vertices of and (). The map is determined by the permutation such that . Then we put
where is the isomorphism given by . We define as the oriented graph endowed with the -edge colors , and with the correspondence, for every vertex , between the -faces of and the germs of edges adjacent to . Clearly we have:
A weakly branched triangulation is branched if and only if all the -edge colors are equal to .
Any triangulation of admits pre-branchings and hence compatible weak-branchings.
We stress that this is no longer true for genuine branchings (see an example in Figure 7). If is quasi-regular, then every total ordering of the vertices induces a branching of .
It is useful to represent by a -graph , defined as follows. Basically is a planar immersion of with normal crossings. It has two kinds of vertices: essential crossings, represented by a solid dot, corresponding to the vertices of , and connected by the oriented edges of , and non essential (“immaterial”) crossings due to the immersion. An essential crossing encodes a branched tetrahedron as shown in Figure 4; the labels on the arc endpoints correspond to the -ordering of the dual -faces of . A full decoding is provided by Figure 5 in the case , showing a branched tetrahedron, the dual “butterfly” with co-oriented (hence oriented) wings, and its conversion into a portion of a oriented branched surface. An arc of corresponding to an edge of inherits the -color . If we omit it.
Given , an easy decoding procedure (extending the one of Figure 5) produces an embedding in of a closed regular neighbourhood of in the standard spine . We believe that it is enough to show this in the case of the simplest cusped manifolds: , the figure-eight-knot complement in , and its “sister” , which is the complement of a knot in the lens space (see ). In both cases the Epstein-Penner decomposition is an ideal triangulation made by two regular hyperbolic ideal tetrahedra. Denote by and the corresponding triangulations of and . Figure 6 (resp. Figure 7) shows an -graph and its decoding for a branching of (resp. weak branching of ). It is not hard to verify that does not carry any branching.
Edge decorations. We will use several labellings of the edges of a weakly branched triangulation, called decorations, such that for every tetrahedron opposite edges have the same label. An example is the decoration of the square edges with or according to the pre-branching, and of the diagonal edges with . Every decoration is determined on each tetrahedron by the triple . In terms of decoded -graphs, they are placed as in Figure 8, where we understand that the over/under crossing arcs are labelled by , and we show also , , .
In the case of genuine branchings, -graphs were used in  under the name of normal o-graphs, with the opposite convention for the sign (in accordance with the usual crossing signs of link diagrams). This choice gives the equality , where is the mirror image of the link . The present convention yields the statement of Theorem 1.2.
The model of the gluing variety. If is a weakly branched triangulation, we can use the weak branching to fix the auxiliary choices used in the definition of the gluing variety . On every branched tetrahedron let , and order the edges of as , , and . If , this ordering is compatible with the cyclic ordering induced by the opposite of the boundary orientation of . If this is no longer true, and since the coordinates , , of are the cross-ratio moduli of an isometry class of oriented hyperbolic tetrahedron associated to , should be replaced by in order to compensate the choice of opposite orientation. Hence we define a new system of coordinates of by labelling with , setting
This gives us the model of that we are going to use. We denote it by . The defining equations of are the edge relations, for all edges of , given by
where if is an edge of we have if and only if is or the opposite edge, and . The volume function on takes now the form . Clearly .
It is easy to recover the edge equations of from a decoded -graph representing it: at each essential crossing one places the cross-ratio variables like the decorations in Figure 8, and take the products of cross-ratio variables along the boundary lines of . For example, consider the cusped manifold and . Assign to the top crossing of Figure 7, and to the bottom one. Note that for both of them. Then we get the equations
Using the relation and the similar one for , they reduce to the unique quadratic equation
The parameter space of “positive solutions” is the half plane with the ray removed, where [FM]. The complete hyperbolic structure is realized at (two regular ideal tetrahedra). Similarly, using one recovers Thurston’s celebrated treatment of the figure-eight knot complement .
3. Analytic configurations
3.1. Local analytic configurations
Take an oriented -simplex . As in the previous section the -face is opposite to the vertex , and the edges of are ordered as
3.1.1. Quantum hyperbolic -simplices
Recall that the edge decorations of are equal on opposite edges, and hence specified by triples