The 3D index of an ideal triangulation and angle structures
Abstract.
The 3D index of DimofteGaiottoGukov a partially defined function on the set of ideal triangulations of 3manifolds with torii boundary components. For a fixed tuple of integers, the index takes values in the set of series with integer coefficients.
Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 32 moves, but not in general under 23 moves.
1991 Mathematics Classification. Primary 57N10. Secondary 57M25.
Key words and phrases: 3D index, tetrahedron index, quantum dilogarithm, gluing equations, NeumannZagier equations, hyperbolic geometry, ideal triangulations, 23 moves, pentagon, angle structures, index structures, qholonomic sequences.
Contents
1. Introduction
In a series of papers [DGG1, DGG2], DimofteGaiottoGukov studied topological gauge theories using as input an ideal triangulation of a 3manifold . These gauge theories play an important role in

ChernSimons perturbation theory (that fits well with the earlier work on quantum Riemann surfaces of [D1] and the later work on the perturbative invariants of [DG]),

categorification and Khovanov Homology, that fits with the earlier work [Wittenfivebranes].
Although the gauge theory depends on the ideal triangulation , and the 3D index in general may not converge, physics predicts that the gauge theory ought to be a topological invariant of the underlying 3manifold . When consists of torii, the low energy description of these gauge theories gives rise to a partially defined function
(1.1) 
for integers and , which is invariant under some partial 23 moves. The building block of the 3D index is the tetrahedron index defined by ^{1}^{1}1The variables are named after the magnetic and electric charges of [DGG2].
(1.2) 
where
and . If we wish, we can sum in the above equation over the integers, with the understanding that for .
Roughly, the 3D index of an ideal triangulation is a sum over tuples of integers of a finite product of tetrahedron indices evaluated at some linear forms in the summation variables. Convergence of such sums is not obvious, and not always expected on physics grounds. For instance, the following sum
converges in if and only if . This follows easily from the fact that the degree of the summand is given by
Our goal is to

prove that the 3D index exists if and only if admits an index structure (a generalization of a strict angle structure); see Theorem 2.12 below.

give a complete axiomatic definition of the tetrahedron index focusing on the combinatorial and holonomic aspects; see Section 3.

to show that the 3D index is invariant under moves, but not in general under moves, and give a necessary and sufficient criterion for invariance under moves; see Section 6.
2. Index structures, angle structures and the 3D index
2.1. Index structures
Consider two matrices and with integer entries and a column vector , and let .
Definition 2.1.
(a) We say that supports an index structure if the rank of is and for every there exists that satisfies
(2.1) 
and . The latter means that for every the following inequalities are satisfied:
(2.2) 
(b) We say that supports a strict index structure if the rank of is and there exists that satisfies (2.1), where is the set of positive rational numbers.
It is easy to see that if supports a strict index structure, then it supports an index structure, but not conversely. As we will see in Section 2.2, ideal triangulations give rise to matrices , and a strict index structure on is a strict angle structure on . On the other hand, index structures are new and motivated by Theorem 2.4 below.
The next definition discusses two actions on : an action of on the left which allows for row operations on , and a cyclic action of order three at the pair of the th columns of .
Definition 2.2.
(a) There is a left action of on , defined by
An index structure on is also an
index structure on .
(b) There is a left action of on acting
on the th columns of (and fixing
all other columns) given by
(2.3) 
where
(2.4) 
satisfies . We extend to act on an index structure of by
(2.5) 
and fixing all other coordinates of . It is easy to see that if is an index structure on and , then is an index structure of .
Definition 2.3.
Given , and consider the sum
(2.6) 
Theorem 2.4.
is convergent for all if and only if supports an index structure. In that case, is holonomic in the variables .
Remark 2.5.
holonomicity in Theorem 2.4 follows immediately from [WZ]. Convergence is the main difficulty.
Remark 2.6.
By definition, is a generalized Nahm sum in the sense of [GL2], where the summation is over a lattice.
Corollary 2.7.
Applying Theorem 2.4 to the case , , and the strict index structure
The next remark discusses the invariance of the index under the actions of Definition 2.2.
Remark 2.8.
Fix that supports an index structure. Then, for and , it follows that and also supports an index structure. In that case, Theorem 2.4 implies that , and are all convergent. We claim that
The first equality follows by changing variables in the definition of given by (2.6). The second equality follows from the fact that the tetrahedron index satisfies Equation (3.2); this is shown in part (a) of Theorem 3.7.
The next corollary follows easily from Theorem 2.4 and the definition of an index structure on .
Corollary 2.9.
Fix an matrix with integer entries and columns for , and let , and let . The following are equivalent:

converges.

and there exist for such that .
Question 2.10.
Compare the series with the vector partition functions of Sturmfels [Sturmfels] and BrionVergne [BV], and the hypergeometric systems of equations of [SST].
2.2. Angle structures
In this section we define the 3D index of an ideal triangulation. A generalized angle structure on a combinatorial ideal tetrahedron is an assignment of real numbers (called angles) at each edge of such that the sum of the three angles around each vertex is .^{2}^{2}2The sum of the 3 angles around each vertex is traditionally . It is easy to see that opposite edges are assigned the same angle, thus a generalized angle structure is determined by a triple that satisfies :
A generalized angle structure is strict if . Let denote an ideal triangulation of an oriented 3manifold with torus boundary. A generalized angle structure on is the assignment of angles at each tetrahedron of such that the sum of angles around every edge of is . A generalized angle structure on is strict if its restriction to each tetrahedron is strict. For a detailed discussion of angle structures and their duality with normal surfaces, see [HRS, LT, Ti]. Generalized angle structures are linearizations of the gluing equations, that may be used to construct complete hyperbolic structures, and intimately connected with the theory of normal surfaces on [Jaco].
The existence of a strict angle structure imposes restrictions on the topology of : it implies that is irreducible, atoroidal and each boundary component of is a torus; see for example [LT]. On the other hand, if is a hyperbolic link complement, then there exist triangulations which admit a strict angle structure, [HRS]. In fact, such triangulations can be constructed by a suitable refinement of the EpsteinPenner ideal cell decomposition of . Note that not all such triangulations are geometric [HRS].
2.3. The NeumannZagier matrices
Fix is an oriented ideal triangulation with tetrahedra of a 3manifold with torii boundary components. If we assign variables at the opposite edges of each tetrahedron respecting its orientation,
then we can read off matrices matrices , and whose rows are indexed by the edges of and whose columns are indexed by the variables. These are the socalled NeumannZagier matrices that encode the exponents of the gluing equations of , originally introduced by Thurston [NZ, Th]. In terms of these matrices, a generalized angle structure is a triple of vectors that satisfy the equations
(2.7) 
A quad for is a choice of pair of opposite edges at each tetrahedron for . can be used to eliminate one of the three variables at each tetrahedron using the relation . Doing so, Equations (2.7) take the form
The matrices have some key symplectic properties, discovered by NeumannZagier when is a hyperbolic 3manifold (and is welladapted to the hyperbolic stucture) [NZ], and later generalized to the case of arbitrary 3manifolds in [Neumanncombi]. NeumannZagier show that the rank of is , where is the number of boundary components of ; all assumed torii. If we choose linearly independent rows of , then we obtain matrices and a vector , which combine to . In addition, the exponents of meridian and longtitude loops at each boundary torus give additional matrices and of size .
Definition 2.11.
The 3D index of is defined by
(2.8) 
Implicit in the above definition is a choice of quad and a choice of rows to remove. However, the index is independent of these choices; see Remark 2.8. Keep in mind the action of given by acting on the th columns , and of , and by
(and fixing all other columns) and on the th coordinates of an angle structure by
(and fixing all other coordinates) and on the th columns and of and by
(and fixing all other columns). Since the rank of is and are matrices, it follows that admits a strict structure if and only admits a strict angle structure. In addition, admits an index structure if for every choice of quad there exist a solution of Equations (2.7) that satisfies the inequalities (2.2). Theorem 2.4 implies the following.
Theorem 2.12.
The index is welldefined if and only if admits an index structure. In particular, exists if admits a strict angle structure.
See Section 6.3 for an example of an ideal triangulation of the census manifold m136 [census] which admits a semistrict angle structure (i.e., angles are nonnegative real numbers), does not admit a strict angle structure, and which has a solution of the gluing equations that recover the complete hyperbolic structure. A casebycase analysis shows that this example admits an index structure, thus the index exists. This example appears in [HRS, Example 7.7]. We thank H. Segerman for a detailed analysis of this example.
2.4. On the topological invariance of the index
Physics predicts that when defined, the 3D index depends only on the underlying 3manifold . Recall that [HRS] prove that every hyperbolic 3manifold that satisfies
(2.9) 
(eg. a hyperbolic link complement) admits an ideal triangulation with a strict angle structure, and conversely if has an ideal triangulation with a strict angle structure, then is irreducible, atoroidal and every boundary component of is a torus [LT].
A simple way to construct a topological invariant using the index, would be a map
where is a cusped hyperbolic 3manifold with at least one cusp and is the set of ideal triangulations of that support an index structure. The latter is a nonempty (generally infinite) set by [HRS], assuming that satisfies (2.9). If we want a finite set, we can use the subset of ideal triangulations of which are a refinement of the EpsteinPenner celldecomposition of . Again, [HRS] implies that is nonempty assuming (2.9). But really, we would prefer a single 3D index for a cusped manifold , rather than a finite collection of 3D indices.
It is known that every two combinatorial ideal triangulations of a 3manifold are related by a sequence of 23 moves [Ma1, Ma2, Pi]. Thus, topological invariance of the 3D index follows from invariance under 23 moves.
Consider two ideal triangulations and with and tetrahedra related by a move shown in Figure 1.
Proposition 2.13.
If admits a strict angle structure structure, so does and .
For the next proposition, a special index structure on is given in Definition 6.2.
Proposition 2.14.
If admits a special strict angle structure, then admits a strict angle structure and .
Remark 2.15.
The asymmetry in Propositions 2.13 and 2.13 is curious, but also necessary. The origin of this asymmetry is the fact that 32 moves always preserve strict angle structures but 23 moves sometimes do not. If 23 moves always preserved strict angle structures, then all ideal triangulations of a fixed manifold would admit strict angle structures as long as one of them does. On the other hand, an ideal triangulation that contains an edge which belongs to exactly one (or two) ideal tetrahedra does not admit a strict angle structure since the angle equations around that edge should add to . Such triangulations are easy to construct, even for hyperbolic 3manifolds (eg. the knot).
3. Axioms for the tetrahedron index
In this section we discuss an axiomatic approach to the tetrahedron index. Let (resp., ) denote the ring of series of the form
where there exists such that for all (resp., ). For , its degree is the largest halfinteger (or infinity) such that . We will say that is positive if .
Definition 3.1.
A tetrahedron index is a function that satisfies the equations
(3.1a)  
(3.1b) 
for all integers , together with the parity condition for all and . Let denote the set of all tetrahedron indices, and denote the set of all positive tetrahedron indices.
Theorem 3.2.
(a) is a free holonomic module of rank .
(b) is a free holonomic module of rank .
(c) If , then it satisfies the equation
(3.2) 
for all integers and .
(d) If , then it satisfies the equations
(3.3a)  
(3.3b) 
for all integers .
Question 3.3.
What is a basis for ?
Remark 3.4.
The proof of part (a) of Theorem 3.2 implies that if is a tetrahedron index, then is a a unique linear combination of and where . For example, if , then where
Remark 3.5.
The proof of part (b) of Theorem 3.2 implies that if is a tetrahedron index, then is uniquely determined by . In particular, if , then are linear combinations of for . For example, we have:
In fact, it appears that is a linear combination of for , although we do not know how to show this, nor do we know of a geometric significance of this experimental fact.
The next lemma computes the degree of the tetrahedron index.
Lemma 3.6.
The degree of is given by:
(3.4) 
It follows that is a piecewise quadratic polynomial given by
The next theorem gives an axiomatic characterization of the tetrahedron index .
Theorem 3.7.
is uniquely characterized by the following equations:

,

satisfies the pentagon identity
(3.5) for all integers .
Remark 3.8.
The uniqueness part of Theorem 3.7 uses only the facts that , for all and satisfies the special pentagon
4. Properties of a tetrahedron index
4.1. Part (d) of Theorem 3.2
4.2. The rank of : part (a) of Theorem 3.2
An application of the HolonomicFunctions.m computer algebra package [Kou] implies that the linear difference operators corresponding to the recursions of Equations (3.1a) and (3.1b) is a Gröbner basis and the corresponding module has rank . Said differently, is a unique linear combination of and where and .
4.3. The rank of : part (b) of Theorem 3.2
Consider a function of two discrete integer variables which satisfies Equations (3.1a) and 3.1b. Section 4.1 implies that satisfies the 3term recursion
(4.1) 
for all integers . It follows that for every integer , is a linear combination of and where and . An induction on using the recursion relation (4.1) shows that for all we have
where are polynomials of maximum degree and constant term and respectively. For example, we have:
Let us write
If we assume that , this imposes a system of linear equations on the coefficients and of and . In fact, for fixed , the system of equations for is a triangular system of linear equations with unknowns for where all diagonal entries of the coefficient matrix are . For example, we have:
It follows that is a linear combination of for . This proves that the rank of the module is at most . Since (as follows from the proof of Theorem 3.7), it follows that the rank of the module is exactly . This proves part (b) of Theorem 3.2. ∎
Corollary 4.1.
The above proof implies that is uniquely determined by its initial condition . It follows that if , then
(4.2) 
for all integers and .
4.4. Proof of triality: part (c) of Theorem 3.2
In this section we prove part (c) of Theorem 3.2. Equation (3.2) concerns the following action on .
Definition 4.2.
Consider the action on a function given by:
(4.3) 
Proposition 4.3.
(a) We have: .
(b) If , then , and of course, also .
Part (c) of Theorem 3.2 follows from part (b) of the above proposition.
Proof.
(of Proposition 4.3) Part (a) is elementary. For part (b), assume that satisfies Equation (3.1a) for all . Replace by in (3.1a) and we obtain that
(4.4) 
Now, replace by in the left hand side of Equation (3.1a), and compute that the result is given by
The above vanishes from Equation (4.4).
Likewise, assume that satisfies Equation (3.1b) for all . Replace by in (3.1b) and we obtain that
(4.5) 
Now, replace by in the left hand side of Equation (3.1b), and compute that the result is given by
It follows that if , then the above vanishes from Equation (4.5). In other words, if then . To conclude that , it suffices to show (by part (a) of Theorem 3.2) that . If , , using Remark 3.4 we have:
This concludes the proof of Proposition 4.3. ∎
4.5. is a tetrahedron index
Observe that by its definition,
is given by a onedimensional sum of a proper hypergeometric term ([WZ, PWZ])
It follows by [WZ] that is holonomic in both variables and . Moreover, recursion relations for can be found by the creative telescoping method of [WZ]. For instance, satisfies the recursion
(4.6) 