$\hat{Z}$ invariants at rational $\tau$

$\hat{Z}$ invariants of 3-manifolds were introduced as series in $q=e^{2\pi i\tau}$ in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to $\tau=1/k$. However modularity properties suggest that all roots of unity are on the same footing. The main result of this paper is the expression connecting Reshetikhin-Turaev invariants with $\hat{Z}$ invariants for $\tau\in\mathbb{Q}$. We present the reasoning leading to this conjecture and test it on various 3-manifolds.


Introduction and summary
The main goal of this paper is to explore the behaviour ofẐ invariants of 3-manifolds at rational τ (in general τ ∈ H -the upper half-plane).Ẑ invariants were introduced in [1][2][3][4] as series in q = e 2πiτ with integer coefficients in order to enable the categorification of Witten-Reshetikhin-Turaev (WRT) invariants of 3-manifolds. It turns out that, apart from the topological applications,Ẑ invariants are very interesting from the point of view of physics and number theory.
PhysicallyẐ invariant is a 3d analogue of the elliptic genus introduced in [5]. More precisely it is a supersymmetrix index of 3d N = 2 theory with 2d N = (0, 2) boundary condition studied first in [6]. Detailed analysis of this interpretation can be found in [1,3] whereas [7] provides a lot of explicit results for various examples.Ẑ invariants are also related to 2d logarithmic conformal field theories [4] and newly proposed two-variable series for knot complements [8].
Due to their modular properties,Ẑ invariants are interesting from the point of view of number theory. A broad discussion of this subject can be found in [4]. For us the most important are two aspects. Firstly, for many 3-manifoldsẐ invariants can be expressed as a linear combination of false theta functions [2,4,9]. This fact plays an important role in explicit calculations in Sections 4.2 and 4.3. Analogous property for WRT invariants was studied earlier in [10][11][12][13][14][15][16].
In order to understand the second aspect, let us make a step back to the relation between WRT invariants andẐ invariants for plumbed 3-manifolds [1][2][3][4] WRT[M 3 (Γ); 1/k] = lim q→e 2πi k a∈CokerM e −2πik(a,M −1 a) b∈2CokerM +δ S abẐb 2 q 1/2 − q −1/2 , where M is the linking matrix of the plumbing graph Γ (for details see Section 2.1). Equation (1.1) corresponds to τ = 1/k. In this case there exists a well-known physical interpretation in the language of Chern-Simons theory, where k ∈ N is the quantum-corrected Chern-Simons level [17]. However from the point of view of number theory τ = 1/k is conceptually on the same footing as all other rational numbers [10]. Therefore there arises a natural question (which is the main motivation of this paper): What happens with (1.1) for τ = r/s?
Since for τ = r/s (r, s ∈ Z) there is no Chern-Simons theory interpretation, we will refer to the left hand side as the Reshetikhin-Turaev (RT) invariant -their combinatorial definition using quantum group representation theory [18] works for all τ ∈ Q. The main result of this paper is the following expression connecting the RT invariant with theẐ invariant We checked this formula in many examples and conjecture that it is true for all plumbed 3-manifolds. The form of (1.2), especially the summation over a ∈ Coker(rM ), is quite surprising. Is M → rM a purely computational phenomenon or does it have a topological interpretation? If the latter is true, should we view rM as the matrix defining a 3-manifold? What would be the relation to the initial one? We will come back to these questions in Sections 3 and 5. The plan of this paper is as follows. Section 2 contains the necessary preparations, focusing on plumbed 3-manifolds and an expression for RT invariant independent of (1.2). In Section 3 we derive and discuss our main result -the formula (1.2). Tests on various examples are presented in Section 4. Finally, Section 5 is devoted to the future directions.

Plumbed 3-manifolds
In this paper we focus on a very large class of 3-manifolds corresponding to decorated graphs which, for simplicity, are assumed to be connected. For a given graph Γ we can obtain the associated plumbed 3-manifold M 3 (Γ) by performing a Dehn surgery on L(Γ) -the corresponding link of framed unknots (see Figure 1). We are mainly interested in Seifert fibrations over S 2 which correspond to star-shaped graphs and are denoted by Among them there is a special class of Brieskorn homology spheres. They are defined as the inetrsection of the complex unit sphere with the hypersurface z p1 1 + z p2 2 + z p3 3 = 0 (p 1 , p 2 , p 3 are coprime integers) and denoted by Σ(p 1 , p 2 , p 3 ). Let us denote the set of vertices of Γ by V and the set of edges by E. L = |V | is equal to the number of components of L(Γ). A convenient way to encode the information given by the plumbing graph is the matrix M defined as v 1 and v 2 connected by the edge, From the link perspective M is the linking matrix of L(Γ). The cokernel of M is equal (setwise) to the first homology group of M 3 (Γ) The number of elements in each set is given by det M .

RT invariants from the Galois action on WRT invariants
We would like to find an expression for the RT invariant which is independent of (1.2) and could serve as a reference for cross-checks. We can do it using the following formula for the WRT invariant of where b + and b − are the number of positive and negative eigenvalues of the matrix M . The symbol ±1• denotes the plumbing graph with one vertex corresponding to the unknot with ±1 framing. In this paper we always assume q = e 2πiτ (2.4) and the WRT invariant corresponds to τ = 1/k. If we look at the WRT invariant from the point of view of the quantum group construnction, it transforms equivariantly under the Galois group Gal Q(e 2πi r s )/Q [10,19]. In consequence its generalisation to τ = r/s is given by substituting We will use this formula in many examples in Section 4, but it is interesting on its own. According to Turaev construction [20] we can associate a modular tensor category (MTC) to the 3d topological quantum field theory. The MTC comes equipped with modular S and T matrices which capture the structure of the topological partition function. For the plumbed 3-manifold this relation reads (see [21,22] for more details) Comparing (2.5) with (2.6) we can see that the form of F matches the structure of Z top for This is a projective representation of SL(2, Z), where the phase factor is an integer multiple of 1/8. In order to restore (ST ) 3 = ±1 we have to rescale T T mn → δ m,n q 2n 2 −s 8 . (2.8) The condition S 2 = ±1 is ensured by the normalisation factor 1 i √ 2s which cancels out in (2.5). Another important observation is the invariance of the formula (2.5) under r → r + ns symmetry (n ∈ Z). It is equivalent to the multiplication of every q by e 2πin = 1. The r → r + ns symmetry helps to solve the problem of choosing the branch of the complex root which often arises in Mathematica computations.

RT invariants fromẐ invariants
The reasoning leading to our main conjecture follows the structure of the Appendix A of [3], which presents the derivation of the following relation between the WRT invariant and theẐ invariant for The reasoning starts from expression (2.3) and, in the crucial step, uses the Gauss sum reciprocity formula where l ∈ Z L , (·, ·) is the standard pairing on Z L and σ = b + − b − is the signature of the linking matrix M . We would like to have an analogous derivation for τ = r/s, so we start from equation (2.5) and follow all the steps of the Appendix A. The crucial one is again the Gauss sum reciprocity formula. In order to deal with τ = r/s we have to rescale the formula, which is equivalent to considering (3.2) forM = rM andl = rl (we also write s instead of k). We obtain in denominator which has exactly r L terms and "compensates" this growth. For r = 1 equation (3.4) reduces to (3.1) which provides the first consistency check. Another interesting aspect of (3.4) is the fact that it preserves the r → r + ns symmetry but, in contrary to (2.5), it is difficult to see it directly.

Rational τ limit ofẐ invariants
For some simple 3-manifolds such as lens spaces L(p, 1) the τ → r/s limit of theẐ invariant is very easy to obtain (see Section 4.1), however these are exceptions rather than the rule. Fortunately for many 3-manifolds (e.g. Seifert manifolds with 3 singular fibers) theẐ invariant can be expressed as a linear combination of false theta functions defined as In this case the calculation of lim τ →r/sẐ is more difficult, but still possible. In [23] we find that Since this result is an essential tool in Section 4, it serves also as the guiding rule in choosing examples for testing our main conjecture.

Conventions
Before moving to examples let us discuss some conventional issues. In many papers, e.g. [1,2,4], the normalisation of the RT invariant (or the WRT invariant for τ = 1/k) is different. In our notation RT[S 3 ; r/s] = 1, whereas there RT CS [S 2 × S 1 ; r/s] = 1. (3.8) We write RT CS because this notation is based on the value of the Chern-Simons partition function for r/s = 1/k (many authors write Z CS instead of RT CS but we want to avoid the confusion withẐ). The relation between these two conventions is given by This convention is often called folded whereas ours -unfolded. The former is present in [1][2][3][4], we use the latter because it is inconvenient to divide Coker(rM ) by Z 2 for every considered r. We would like to stress that because of that ourẐ b differs from the folded one (denoted byẐ b ) by the factor of 2 if b is not a fixed point of Z 2 symmetry. Moreover, some papers use different numeration ofẐ b . Detailed discussion of this issue can be found in [3].

Examples
In this section we test our main conjecture (3.4) on various examples by comparing it to (2.5). All computations are done numerically using Mathematica.

Lens spaces L(p, 1)
For the lens space L(p, 1) the plumbing graph Γ is given by  Using Mathematica 1 we checked that (4.2) and (4.3) give the same result. We compared both formulas for p = 3, 5, 7, 9, 11 and r/s up to 16/17. For r = 1 this statement immediately follows from (3.1). HoweverẐ δ /2 q 1/2 − q −1/2 is defined for all τ ∈ H (q inside unit disk) with well-defined limits at all rational τ , so in this case there is no difference between r = 1 and other integers. Comparing which we numerically checked using Mathematica.

Σ(2, 3, 7)
The graph of the Σ(2, 3, 7) Brieskorn sphere is given by We number vertices in the following way (we do it for all 4-vertex graphs in the paper) In consequence the linking matrix reads Using Mathematica we checked -for all r/s up to 12/13 -that (4.8) and (4.10) give the same result.

Other Seifert manifolds
The Seifert manifold M −1; 1 2 , 1 3 , 1 9 can be described by the plumbing graph (4.19) We have In contrary to the Brieskorn spheres all terms are nontrivial. On the other hand (2.5) gives The Seifert manifold M −2; 1 2 , 1 3 , 1 2 has the following plumbing graph   Similarly to M −1; the necessity of calculating Coker(rM ) for each r made it easier to increase the parameter s (however in this case the cokernel is bigger) and we stopped at r/s = 5/21.

Open questions
The most interesting future direction seems to be the one towards the interpretation of our main conjecture. Do we really have another manifold associated to each r? The manifold corresponding to the matrix rM is not an r-fold cover of the one corresponding to M and it is difficult to find another topologically reasonable candidate. Or maybe the interpretation should not involve another manifold? But what would the summation over Coker(rM ) mean in this case? And how should we understand the denominator of (3.4)? The power L seems to refer to vertices of the plumbing graph, the number of terms r L = |Coker(rM )| suggests the correspondence between summands in the numerator and the denominator, but it is difficult to say something more concrete.
Another goal for future research is the the proof of our main conjecture. It would be also interesting to investigate non-Seifert manifolds and find the form of (3.4) valid for all 3-manifolds (not only plumbed).