The descendants of the 3d-index

Topological invariants of ideally triangulated 3-manifolds appeared in mathematical physics in relation to complex Chern–Simons theory [2] and its extension in the 3d-3d correspondence [5, 11]. Two of the best-known such invariants are the state-integrals of Andersen–Kashaev [2], which are analytic functions on $\mathbbm{C}\setminus(-\infty,0]$, and the 3D-index of Dimofte–Gaiotto–Gukov [8, 9], which is a collection of $q$-series with integer coefficients parametrized by the integer homology of the boundary of a 3-manifold. Although the state-integrals and the 3D-index are different looking functions, they are closely related on the mathematics side through the theory of holomorphic quantum modular forms developed by Zagier and the second author [22, 21], and on the physics side through the above mentioned 3d-3d correspondence.

The state-integrals and the 3D-index share many common features, stemming from the fact that on the physics side, under the 3d-3d correspondence [10, 26, 9, 7] (see [6] for a review) become the invariants of the dual 3d $N=2$ superconformal field theory on respectively $S^{3}$ and $S^{1}\times S^{2}$, both of which can be obtained by gluing two copies of $D^{2}\times S^{1}$ together.

On the mathematics side, both invariants are defined using combinatorial data of ideal triangulations of 3-manifolds whose local weights (namely the Faddeev quantum dilogarithm function, and the tetrahedron index, respectively) satisfy the same linear $q$-difference equations, whereas the invariants themselves are given by an integration/summation over variables associated to each tetrahedron.

A common feature to both invariants is their conjectured bilinear factorization in terms of the same holomorphic blocks $H(q)$, the latter being $q$-hypergeometric series defined for $|q|\neq 1$. This leads to bilinear expressions for the state-integral in terms of $H(q)$ times $H(\tilde{q})$ (where $q=e^{2\pi i\tau}$ and $\tilde{q}=e^{-2\pi i/\tau}$) and bilinear expressions for the 3D-index in terms of $H(q)$ times $H(q^{-1})$. This factorization is well-known in the physics literature [3] and interpreted as partition function of the dual 3d superconformal field theory on $D^{2}\times S^{1}$. They are also partially known for some examples of 3-manifolds in [17, 21]. We emphasize, however, that the bilinear factorization of state-integrals and of the 3D-index is conjectural, and so is the existence of the suitably normalized holomorphic blocks.

Another common feature to state-integrals and the 3D-index is that they are given by integrals/lattice sums where the integrand/summand has a common annihilating ideal. This implies that both state-integrals and the rotated 3D-index satisfy a pair of linear $q$-difference equations which are in fact conjectured to be identical, and equal to the homogeneous part of the linear $q$-difference equation for the colored Jones polynomial of a knot [19]. The conjectured common linear $q$-difference equations for state-integrals and for the 3D-index would also be a consequence of their common holomorphic block factorization. In physics these linear $q$-difference equations are interpreted as Ward identities of Wilson-’t Hooft line operators in the dual 3d superconformal field theory [8, 9].

Descendants appeared recently as computable, exponentially small corrections to the asymptotics of the Kashaev invariant of a knot, refining the Volume Conjecture to all orders in perturbation theory to a Quantum Modularity Conjecture [22]. One of the discoveries was that the Kashaev invariant of a knot is a distinguished $(\sigma_{0},\sigma_{1})$-entry in a square matrix of knot invariants at roots of unity. The rows and columns of the matrix are parametrized with boundary-parabolic $\mathrm{PSL}_{2}(\mathbbm{C})$-representations, with $\sigma_{0}$ denoting the trivial representation and $\sigma_{1}$ denoting the geometric representation of a hyperbolic knot complement. The above mentioned matrix has remarkable algebraic, analytic and arithmetic properties explained in detail in Section 5 of [22], and given explicitly for the $4_{1}$ and $5_{2}$ knots in Sections 7.1 and 7.2 of i.b.i.d. The rows of the matrix are supposed to be $\mathbbm{Q}(q^{1/2})$-linear combinations of fundamental solutions to a linear $q$-difference equation (homogeneous for all but the first row), thus the elements in each row are supposed to be descendants of each other. Although the existence of such a matrix is conjectured, its top row was defined in [16] for all knots in terms of the descendant Kashaev invariants of a knot.

The above mentioned matrix has three known realizations, one as functions at roots of unity mentioned above, a second as a matrix of Borel summable asymptotic series and a third as a matrix of $q^{1/2}$-series. The idea of descendants can be extended to the matrix of asymptotic series (whose first column are simply the vector of asymptotic series of the perturbative Chern–Simons theory at a $\mathrm{PSL}_{2}(\mathbbm{C})$-flat connection, and the remaining columns being descendants of the first column) as well as to a matrix of $q$-series. This extension was done for the case of the $4_{1}$ and $5_{2}$ knots by Mariño and two of the authors [13, Eqn.(13),App.A], with the later addition of the trivial $\mathrm{PSL}_{2}(\mathbbm{C})$-representation in [14, Sec.2.2,Sec.4.1].

To summarize, descendants are supposed to be the $\mathbbm{Q}(q^{1/2})$-span of a fundamental solution to a linear $q$-difference equation associated to the quantum invariants. It is becoming clear that this span is a fundamental quantum invariant of 3-manifolds, and we want to present further evidence for this using as an example an important quantum invariant, namely the 3D-index.

A detailed study of the 3D-index of a 3-manifold with torus boundary and its structural properties, namely holomorphic block factorization, linear $q$-difference equations, computations and asymptotics was recently done in [20].

The goal of the present paper is to extend the properties of the 3D-index by allowing observables, line operators, defects, descendants, all being synonymous names for the same object. On the topological side, an observable is a knot $L$ in a 3-manifold $M$, where in the case of interest to us, $M=S^{3}\setminus K$ is the complement of a knot in $S^{3}$. On the algebra side, the conjectural 3d-quantum trace map sends a knot $L\subset S^{3}\setminus K$ to an element $\mathcal{O}$ of a module over a $q$-Weyl algebra associated to an ideal triangulation $\mathcal{T}$ of $M$. We will postpone the description of the 3d-quantum trace map to a subsequent publication. Now $\mathcal{O}$ acts on the integrand/summand of the state-integral/3D-index, and by integrating/summing one obtains a state-integral/3D-index with insertion $\mathcal{O}$. On the physics side, $\mathcal{O}$ becomes a line-operator supported on a line $\gamma$ in the dual 3d $N=2$ superconformal field theory $T_{2}[M]$ under the 3d-3d correspondence [10, 26, 9, 7]. The 3d-3d correspondence can be understood as a consequence of compactifying 6d $N=2$ $A_{1}$ superconformal field theory on the three manifold $M$ and on $\mathbbm{R}^{3}$ with topological twist along $M$. The 6d theory has surface operators which can be supported on $L\times\gamma$, giving rise to the correspondence between the defect $L$ in $M$ and the line-operator on $\gamma\subset\mathbbm{R}^{3}$ in $T_{2}[M]$ [8, 9]. Our goal is to study the structural properties of the rotated, inserted, 3D-index $I^{\mathrm{rot}}_{\mathcal{T},\mathcal{O}}(q)$. Although this is a $\mathbbm{Z}\times\mathbbm{Z}$ matrix, we will see that it is determined from the uninserted rotated 3D-index $I^{\mathrm{rot}}_{\mathcal{T}}(q)$ in terms of a pair of linear $q$-difference equations and a finite size invertible matrix with coefficients in the field $\mathbbm{Q}(q^{1/2})$; see Conjectures 3.3 and 3.6 below, illustrated by examples in Section 4.

We emphasize that our paper concerns conjectural structural properties of topological invariants, such as the rotated inserted 3D-index, and not mathematical proofs. Nevertheless the structure of these invariants is rich, and leads to startling predictions and numerical conformations (see eg. Equation (36) below).

For simplicity, in the paper we will focus on the action of the quantum torus $\mathbbm{W}_{q}(\mathcal{T})$ on the 3D-index $I_{\mathcal{T}}$, and in fact in its rotated form $I^{\mathrm{rot}}_{\mathcal{T}}$ explained to us by Tudor Dimofte and studied extensively in [20]. To begin with, we fix an ideal triangulation $\mathcal{T}$ with $N$ tetrahedra of a 3-manifold $M$ whose torus boundary is marked by a pair of a meridian and longitude. The building block of the 3D-index is the tetrahedron index $I_{\Delta}(m,e)(q)\in\mathbbm{Z}[[q^{1/2}]]$ defined by

where $e_{+}=\max\{0,e\}$ and $(q;q)_{n}=\prod_{i=1}^{n}(1-q^{i})$. If we wish, we can sum in the above equation over the integers, with the understanding that $1/(q;q)_{n}=0$ for $n<0$.

The rotated 3D-index is given by

where

is assembled out of a product of tetrahedra indicies $I_{\Delta}$ evaluated to linear forms that depend on the Neumann–Zagier matrices $(A|B)$ of $\mathcal{T}$. The detailed definition of the Neumann–Zagier matrices is given in Appendix C.

Note that the degree $\delta(I_{\Delta}(m,e))$ of the tetrahedron index is a nonnegative piecewise quadratic function of $(m,e)$

It follows that for 1-efficient triangulations (see [15]) the degree of the summand in (5) is bounded below by a positive constant times $\max\{|k_{1}|,|k_{2}|,\dots,|k_{N}|\}$, thus the sum in (5) is a well-defined element of $\mathbbm{Z}((q^{1/2}))$.

The topological invariance of the 3D-index is a bit subtle, since the definition requires 1-efficient ideal triangulations, and the latter are not known to be connected under 2–3 Pachner moves. Nonetheless, in [15], it was shown that the 3D-index (and likewise, its rotated version) is a topological invariant of cusped hyperbolic 3-manifolds. An alternative proof of this fact was given in [18], where the rotated 3D-index was reformulated in terms of a meromorphic function of two variables.

From its very definition as a sum of proper $q$-hypergeometric series, it follows that $I^{\mathrm{rot}}_{\mathcal{T}}(n,n^{\prime})(q)$ is a $q$-holonomic function of $n$ and $n^{\prime}$ [27, 25]. But more is true. The rotated 3D-index factorizes into a sum of a product of pairs of colored holomorphic blocks. This holomorphic block factorization is a well-known phenomenon explained in [3], and most recently in [20] whose presentation we will follow. Let us recall how this works. We can assemble the collection $I^{\mathrm{rot}}_{\mathcal{T}}(n,n^{\prime})(q)$ of $q$-series indexed by pairs of integers into a $\mathbbm{Z}\times\mathbbm{Z}$ matrix $I^{\mathrm{rot}}_{\mathcal{T}}(q)$ whose $(n,n^{\prime})$ entry is $I^{\mathrm{rot}}_{\mathcal{T}}(n,n^{\prime})(q)$. Then, in [20] we explained the origin of the following conjecture for the rotated 3D-index.

When the triangulation is fixed and clear, we will drop it from the notation. If we denote the $(n,\alpha)$ entry of $H_{\mathcal{T}}(q)$ whose $(n,\alpha)$ entry by $h^{(\alpha)}_{n}(q)$, these functions are the so-called colored holomorphic blocks. It follows that the matrix $H(q)$ is a (properly normalized) fundamental solution to a pair of $q$-difference equations

where the operators act respectively by

Consequently the rotated 3D-index satisfies a pair of (left and right) linear $q$-difference equations

acting in a decoupled way on each of the rows and columns of $I^{\mathrm{rot}}_{\mathcal{T}}$.

The factorization (8) of the rotated 3D-index and the left and right linear $q$-difference equations (11) imply the following.

The holomorphic blocks satisfy the symmetry

for all $\alpha$ and all integers $n$, which together with Equation (8) implies the symmetries

and

for the rotated 3D-index.

Let us finally mention that the colored holomorphic blocks can be computed by the limit as $x\to 1$

of the $x$-deformed holomorphic blocks $B^{(\alpha)}_{\mathcal{T}}(x;q)$ and the latter can be determined from a factorization of an appropriate state-integral.

There is an important $\mathbbm{Q}(q)$-linear action of $\mathbbm{W}_{q}(\mathcal{T})$ on the set of functions $S_{\mathcal{T}}(k,n,n^{\prime})(q)$ giving rise to a map

which descends to a push-forward $\mathbbm{Q}(q^{1/2})$-linear map

Concretely, when $\mathcal{O}=\prod_{j=1}^{N}\widehat{z}_{j}^{\alpha_{j}}(\widehat{z}_{j}^{{}^{\prime\prime}})^{\beta_{j}}$, we have

where

Our conjecture relates the colored holomorphic blocks and the rotated 3D-index of $\mathcal{T}$ to those of $(\mathcal{T},\mathcal{O})$. Simply put, it asserts that inserting $\mathcal{O}$ simply changes the invariants ($\mathbbm{Z}((q^{1/2}))$-series) by multiplication of a matrix of rational functions, and changes the left $q$-difference equation whereas it preserves the right one. This implies that the $\mathbbm{Q}(q^{1/2})$-span of the collection $\{I^{\mathrm{rot}}_{\mathcal{T},\mathcal{O}}(q)\,|\,\mathcal{O}\in\mathcal{M}(\mathcal{T})\}$ is a finite dimensional $\mathbbm{Q}(q^{1/2})$-vector space.

Fix a 1-efficient ideal triangulation $\mathcal{T}$ of a 1-cusped 3-manifold $M$.

The above conjecture implies the following.

Another corollary of the above conjecture concerns the descendants of the rotated 3D-index, analogous to the descendants of the colored Jones polynomial of a knot defined in [16] and the descendants of the holomorphic blocks defined in [13, Eqn.(13), App.A]. To phrase it, let

denote the $\mathbbm{Q}(q^{1/2})$-span of the elements $I^{\mathrm{rot}}_{\mathcal{T}}(n,n^{\prime})$ of the ring $\mathbbm{Q}((q^{1/2}))$. Note that $DI^{\mathrm{rot}}_{\mathcal{T}}$ is a finite dimensional vector space over the field $\mathbbm{Q}(q^{1/2})$. Likewise, one defines $I^{\mathrm{rot}}_{\mathcal{T},\mathcal{O}}$. The next corollary justifies the title of the paper.

In other words, the descendants $DI^{\mathrm{rot}}_{\mathcal{T},\mathcal{O}}$ of the rotated 3D-index $DI^{\mathrm{rot}}_{\mathcal{T}}$ are expressed effectively by a finite-size matrix with entries in $\mathbbm{Q}(q^{1/2})$.

We now formulate a relative version of the AJ-Conjecture. Let $\widehat{A}(M,L)|_{q=1}=A(M,L)$ denote the classical limit of a linear $q$-difference equation. The AJ-Conjecture [12] relates the classical limit of the $\widehat{A}$-polynomial with the $A$-polynomial of a knot given in [4].

A consequence of Conjecture (3.3) (and Equation (22)) is that the all-order asymptotics of the colored holomorphic blocks $h^{(\alpha)}_{\mathcal{T},\mathcal{O},n}(q)$ and the $I^{\mathrm{rot}}_{\mathcal{T},\mathcal{O}}(n,n^{\prime})(q)$ are a $\mathbbm{Q}(q)$-linear combination of those of $h^{(\alpha)}_{\mathcal{T},n}(q)$ and $I^{\mathrm{rot}}_{\mathcal{T}}(n,n^{\prime})(q)$, respectively. The asymptotics of the latter were studied in detail in [20]. A corollary of this and Conjecture 3.6 is a resolution and an explanation from first principles, of the quantum length conjecture of [1].

The complement of the $4_{1}$ knot has an ideal triangulation with two tetrahedra. Using the gluing equation matrices

with the conventions explained in Appendix C, we obtain the matrices

in terms of which, the rotated 3D-index is given by

where $I_{\Delta}$ is the tetrahedron index given in (4). (The above formula agrees with [1, Eqn.(108)] after a shift $k_{1}\mapsto k_{1}-k_{2}$). Using Equation (7), it follows that the degree of the summand in (29) is bounded below by a positive constant times $\max\{|k_{1}|,|k_{2}|\}$, thus the sum in (29) is a well-defined element of $\mathbbm{Z}((q^{1/2}))$.

In this section we briefly summarize the properties of the rotated 3D-index of the $4_{1}$ knot following [20], namely its factorization in terms of colored holomorphic blocks, the linear $q$-difference equation, their symmetries and their asymptotics. All the functions in this section involve the knot $4_{1}$, which we suppress from the notation.

The rotated 3D-index is given by [20, Prop.9]

with the colored holomorphic blocks $h^{(0)}_{4_{1},n}(q)$ and $h^{(1)}_{4_{1}n}(q)$ given in the Appendix A.

The colored holomorphic blocks satisfy the symmetries

and

and the linear $q$-difference equation [20, Eqn.(63)]

where

We denote the corresponding operator of the $q$-difference equation (33) by $\widehat{A}_{4_{1}}(x,\sigma,q)=\sum_{j=0}^{2}P_{4_{1},j}(x,q)\sigma^{j}$.

We now consider two defects. The first one is the element

from [1, Eqn.(81)]. Computing the values of $I^{\mathrm{rot}}_{4_{1}}(n,n^{\prime})(q)$ and $I^{\mathrm{rot}}_{4_{1},\mathcal{O}}(n,n^{\prime})(q)$ for $0\leq n,n^{\prime}\leq 1$ up to $O(q^{121})$, we find out that the $2\times 2$ matrices

and

satisfy

illustrating the dramatic collapse of the $q$-series into short rational functions of $q^{1/2}$. This implies that the matrix $Q_{4_{1},\mathcal{O}}(q)$ is given by

with $\det(Q_{4_{1},\mathcal{O}})(q)=1+2q^{-1}$.

After computing the values of $I^{\mathrm{rot}}_{4_{1},\mathcal{O}}(n,0)(q)+O(q^{120})$ for $n=0,\dots,10$ and finding a short linear recursion among three consecutive values, and further interpolating for all $n$, we found out that the left $\widehat{A}$-polynomial of $I^{\mathrm{rot}}_{4_{1},\mathcal{O}}(q)$ is given by $\widehat{A}_{4_{1},\mathcal{O}}(x,\sigma,q)=\sum_{j=0}^{2}P_{4_{1},\mathcal{O},j}(x,q)\sigma^{j}$ where

The $\widehat{A}_{4_{1},\mathcal{O}}$ polynomial is palindromic, and together with the skew-symmetry of the $Q_{4_{1},\mathcal{O}}(q)$ matrix, it follows that the colored holomorphic blocks $h^{(0)}_{4_{1},\mathcal{O},n}(q)$ and $h^{(1)}_{4_{1},\mathcal{O},n}(q)$ satisfy the symmetries (31) and (32).

When we set $q=1$, we obtain

confirming Conjecture 3.6.

Equation (37) and the recursion (38) imply that for all integers $n$ and $n^{\prime}$, $I^{\mathrm{rot}}_{4_{1},\mathcal{O}}(n,n^{\prime})(q)$ is a $\mathbbm{Q}(q^{1/2})$-linear combination of the three $q$-series $I^{\mathrm{rot}}_{4_{1}}(0,0)(q)$, $I^{\mathrm{rot}}_{4_{1}}(0,1)(q)$ and $I^{\mathrm{rot}}_{4_{1}}(1,0)(q)$. For instance, Equation (23) implies that

and likewise for other values of $I^{\mathrm{rot}}_{4_{1},\mathcal{O}}(n,n^{\prime})(q)$. This reduces the problem of the asymptotic expansion of $I^{\mathrm{rot}}_{4_{1},\mathcal{O}}(n,n^{\prime})(q)$ for $q=e^{2\pi i\tau}$ to all orders in $\tau$ as $\tau$ tends to zero in a ray (nearly vertically, horizontally, or otherwise) to the problem of the asymptotics of colored holomorphic blocks and of the rotated 3D-index. This problem was studied in detail and solved in the work of Wheeler and the second author [20, Sec.5.7,5.8] for the $4_{1}$ knot.

As a second example, consider the element