$q$-nonabelianization for line defects

We consider the $q$-nonabelianization map, which maps links $L$ in a 3-manifold $M$ to links $\widetilde{L}$ in a branched $N$-fold cover $\widetilde{M}$. In quantum field theory terms, $q$-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional $(2,0)$ superconformal field theory of type $\mathfrak{gl}(N)$ on $M \times \mathbb{R}^{2,1}$, and we consider surface defects placed on $L \times \{x^4 = x^5 = 0\}$; in the IR we have the $(2,0)$ theory of type $\mathfrak{gl}(1)$ on $\widetilde{M} \times \mathbb{R}^{2,1}$, and put the defects on $\widetilde{L} \times \{x^4 = x^5 = 0\}$. In the case $M = \mathbb{R}^3$, $q$-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group $U(N)$. In the case $M = C \times \mathbb{R}$, when the projection of $L$ to $C$ is a simple non-contractible loop, $q$-nonabelianization computes the protected spin character for framed BPS states in 4d $\mathcal{N}=2$ theories of class $S$. In the case $N=2$ and $M = C \times \mathbb{R}$, we give a concrete construction of the $q$-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering $\widetilde{C} \to C$.


Introduction
This paper concerns a geometric construction which we call q-nonabelianization. In short, q-nonabelianization is an operation which maps links on a 3-manifold M to links on a branched N -fold cover M . A bit more precisely, q-nonabelianization is a map of linear combinations of links modulo certain skein relations, encoded in skein modules associated to M and M .
In this introduction we formulate the notion of q-nonabelianization rather generally: in particular, we discuss arbitrary N , and a general 3-manifold M . In the body of the paper, we describe a concrete way to construct q-nonabelianization in some detail, but only when N = 2 and M = C × R, with C a surface (which could be noncompact, e.g. C = R 2 ). The extensions to higher N and general 3-manifolds M involve similar ideas but various new difficulties, and will appear in upcoming work.
The q-nonabelianization map we describe is related to many previous constructions in the literature, as we discuss in the rest of this introduction, and particularly close to the works [1][2][3] which provided important inspirations for our approach; indeed this paper was motivated by the problem of understanding their constructions in a more covariant and local way. Our description of q-nonabelianization can also be viewed as an extension of an approach to the Jones polynomial described in [4], particularly Section 6.7 of that paper; from that point of view, what we are doing in this paper is explaining a way to replace R 3 by C × R.

Physical setup
Our starting point is the six-dimensional (2, 0) superconformal field theory of type gl(N ), which we call X[gl(N )]. We will consider the theory X[gl(N )] in six-dimensional spacetimes of the form where M is a Riemannian manifold. We adopt coordinates as follows: M is coordinatized by x 1,2,3 , R 2,1 by x 4,5,0 . The theory X[gl(N )] admits supersymmetric surface defects labeled by representations of gl(N ); in this paper we only consider the defect labeled by the fundamental representation. Given a 1-manifold (link) L ⊂ M , we insert this surface defect on a locus and call the resulting insertion S[L]. This kind of setup has been discussed frequently in the physics literature. As a tool for studying link invariants, it appears explicitly in [5], and its various dimensional reductions or M/string theory realizations have appeared in many other places, such as [6][7][8]. The case of M = C × R with L at a fixed x 3 -coordinate has also been studied in the context of line defects in class S theories, e.g. [9][10][11][12], while for more general M and L see e.g. [13].

The UV-IR map
Our approach to studying this setup is to pass to the "Coulomb branch" of the theory. The quickest way to understand what this means is to use the M-theory construction of the theory X[gl(N )] on M . This construction involves N fivebranes wrapped on the zero section M ⊂ T * M . To go to the Coulomb branch we consider instead one fivebrane wrapped on an N -fold cover M ⊂ T * M (possibly branched). This construction has been used frequently in the study of class S and class R theories, e.g. [4,12,[14][15][16].
Away from its branch locus the covering M is represented by N 1-forms λ i on M . In order to preserve supersymmetry, the λ i should be harmonic. In this paper, we will only consider two kinds of example: 1. M = R 3 , and the 1-forms λ i are constants, 2. M = C × R for a Riemann surface C, and the 1-forms λ i are the real parts of meromorphic 1-forms on C.
(Case 1 is actually a special case of case 2 where we take C = C, but it is interesting enough to merit mention on its own.) Now we ask the question: what does S[L] look like in the IR? In this limit the bulk theory is well approximated by the theory X[gl (1)] on M × R 2,1 [16]. In that theory we could consider surface defects S[ L] on the loci for links L ⊂ M . The picture we propose, similar to e.g. [11,17], is that S The meaning of the "coefficients" a( L) is a bit subtle since we are not talking about local operators but rather extended ones. Nevertheless, we want to compute something concrete, and we proceed as follows. What we will really compute is the Laurent polynomials α( L) = Tr V a( L) (−q) 2J 3 q 2I 3 ∈ Z[q, q −1 ] (see below).

Computing the UV-IR map
The construction of the UV-IR map (1.4) which we propose goes as follows. The links L ⊂ M are built out of two sorts of local pieces: • Lifts of segments of L to one of the sheets i of the covering M → M .  To each link L built from these pieces we assign a corresponding weight α( L) ∈ Z[q ±1 ], built as a product of elementary local factors. Most of the difficulty in constructing the UV-IR map is to get these factors correct.
In this paper we develop this scheme in detail only for N = 2. In the N = 2 case there are no trivalent string junctions, which enormously simplifies the situation: the only kinds of webs we have to deal with are the two shown below. There is an important difference between the detours and the exchanges. As we discuss in §4, each exchange comes with a factor of (q − q −1 ), so the contribution from exchanges vanishes at q = 1. (More generally, for N > 2, a web with k ends on L will come with a factor of (q − q −1 ) k−1 .) When we set q = 1, therefore, the only web contributions which survive are those from webs with exactly one end on L. Then our description of the UV-IR map reduces to a construction which already appeared in [12], for line defects in class S theories. In that context the webs attach to L only at places where L crosses the WKB spectral network. In contrast, when we allow q = 1, the webs can attach anywhere on L.
In [3] a version of the UV-IR map is described, in the language of spectral networks, which uses the detours but not the exchanges: in lieu of the exchanges one isotopes L to a specific profile relative to the spectral network, and then inserts by hand additional R-matrix factors. In our approach we do not make such an isotopy and we do not insert these R-matrix factors. The relation between the two approaches is roughly that, if we apply our approach to an L in this specific profile, then exchanges appear and produce automatically the off-diagonal parts of the R-matrix.

Skein relations
BPS quantities in the theory with surface defects S[L] inserted, such as supersymmetric indices, obey some relations. First, they depend only on the isotopy class of the link L, because deforming by an isotopy is a Q-exact deformation. Second, and more interestingly, there are also some Q-exact deformations which relate S[L] for non-isotopic links L.
We do not have a first-principles derivation of what these additional relations are, but to get a hint, we can use a picture from [5]: Euclideanize and compactify the x 0 direction, replace the x 4 -x 5 directions by a cigar and compactify on its circular direction. The resulting effective theory is N = 4 super Yang-Mills on M × R + , with a boundary condition at the finite end. The operators S[L] reduce to supersymmetric Wilson lines in the boundary M .
From this point of view one should expect S[L] to obey the skein relations of analytically continued Chern-Simons theory. We formulate this as the conjecture that up to Q-exact deformations, S[L] is determined by the class of L in the skein module Sk(M, gl(N )), described explicitly in §3.1 below. This sort of skein relation in supersymmetric field theory has been discussed in many different contexts, e.g. [3,10,11,[17][18][19]. Similarly we propose that the IR defect S[ L] depends only on the class of L in another skein module Sk( M , gl(1)), described in §3.2.
The UV-IR decomposition takes Q-exact deformations in the UV to Q-exact deformations in the IR. It follows that it should descend to a map of skein modules, F : Sk(M, gl(N )) → Sk( M , gl(1)).
(1.6) This is a strong constraint, which in particular is strong enough to determine all the weight factors α( L). In §7 we verify that our rules indeed satisfy this constraint in the N = 2 case. In case M = C × R, the skein modules are actually algebras, because of the operation of "stacking" links in the R direction (see §3.3); this algebra structure gets related to the OPEs of the operators S[L] or S[ L], and F is then a homomorphism of algebras. We should mention one subtlety we have been ignoring: the skein modules we consider involve framed links. The need to frame links, though familiar for Wilson lines in Chern-Simons theory, is not immediately obvious from the point of view of the theory X[gl(N )]. Nevertheless, it seems that they do need framing, and that shifting the framing in the gl(N ) theory by 1 unit is equivalent to changing the angular momentum of the defect in the x 4 -x 5 direction by N units. It would be desirable to understand this in a more fundamental way.

Framed BPS state counting
In this subsection we discuss one application of the UV-IR map: it can be used to compute the spectrum of ground states of the bulk-defect system. This point of view unifies the computation of link polynomials and that of framed BPS states in 4d class S theories. When we say "framed BPS state" we also include states associated to a slightly unfamiliar sort of line defect which breaks rotation invariance; we call these "fat line defects" and discuss them in §1.5.3 below.

Links in R 3
The simplest case arises when we take i.e. we consider X[gl(N )] in the spacetime R 5,1 . This theory has 16 supercharges. For generic L, the defect S[L] preserves 2 of these supercharges [20]. The setup also preserves a rotational U(1) P in the x 4 -x 5 plane and a U(1) R symmetry. Now we make a "Coulomb branch" perturbation M as mentioned above. The ground states of the system form a vector space depending on the link L and the perturbation M , which we call It was proposed in [4,5] following [7] that H L ( M ) is a homological invariant of the link L, likely closely related to the Khovanov-Rozansky homology. The action of U(1) P × U(1) R is responsible for the bigrading of the link homology. In particular, suppose we let J 3 denote the generator of U(1) P , and I 3 the generator of U(1) R . Then the proposal of [4,5] implies that the generating function is related to the (un-normalized) HOMFLY polynomial P HOMFLY (L). In our conventions the precise relation is where w(L) is the self-linking number of L. (In particular, this generating function is actually independent of the chosen perturbation M , although a priori it might have depended on it.) The generating function Ω(L, M ) can also be computed using the IR description of the theory. Indeed the IR skein module Sk( M , gl (1)) is easy to understand: every L is equivalent to the class of some multiple of the empty link [·]. It is not completely obvious what the contribution from the empty link to Ω(L, M ) should be (this amounts to counting the ground states of the system without a defect), but we propose that this contribution is just 1, and thus that (1.10) Thus, in this particular case, the UV-IR map described in §1.2 must reduce to a method of computing the knot polynomial (1.9). We discuss this more in §2 below.

Flat links in C × R
Now we consider the case where C is a Riemann surface (perhaps with marked points.) In this case we take a twisted version of X[gl(N )] in the spacetime C × R 3,1 . This kind of twist was used in [14,21]; it preserves 8 supercharges. If C is compact, then on flowing to the IR one arrives at an N = 2 theory of class S in R 3,1 ; we denote this theory X[C, gl(N )]. We are going to consider a link L ⊂ M and a surface defect S[L]. We fix a branched holomorphic N -fold covering C → C, where C ⊂ T * C. (Such a covering C is also a Seiberg-Witten curve, associated to a point of the Coulomb branch of the theory X[C, gl(N )], as described in [14].) Then we define (1.12) The symmetries preserved by the defect S[L] depend on how L is placed. In this section we consider the following: pick a simple closed curve ℘ ⊂ C and then take (1.13) We call such an L "flat" since it explores only 2 of the 3 dimensions of M . The remaining x 3 -direction combines with the x 4 -x 5 plane to give an R 3 , in which S[L] sits at the origin; thus in this setup we have 3-dimensional rotation invariance SU(2) P , and it turns out to also preserve R-symmetry SU(2) R , and 4 supercharges. For C compact and L flat, from the point of view of the reduced theory X[C, gl(N )], S[L] is a 1 2 -BPS line defect. These line defects have been studied extensively, beginning with [9][10][11]. 1 In particular, the vector space H L ( M ) of ground states has been studied, e.g. in [2,3,11,12]; it is called the space of framed BPS states of the line defect. 2 H L ( M ) admits a grading: Unlike the case of M = R 3 , here H L ( M ) does depend strongly on M ; as we vary M (moving in the Coulomb branch) the invariants Ω(L, M , γ) can change. This is the phenomenon of framed wall-crossing. (We discuss this phenomenon in more detail in §8, where we show that our map F obeys the expected framed wall-crossing formulas associated to BPS hypermultiplets and vector multiplets.) Once again, we can compute the invariants Ω(L, M , γ) using the IR description of the theory on its Coulomb branch. In this case the IR skein module Sk( M , gl(1)) is more interesting: it has one generator X γ for each class γ ∈ H 1 ( C, Z). Roughly speaking X γ is just represented by a loop on C in class γ; see §3.5 for the precise statement. A link in class X γ gives an IR line defect carrying charge γ, which contributes one state to H L,γ ( M ). This leads to the proposal i.e., q-nonabelianization gives the generating function of the protected spin characters.
1 More exactly, those works considered the case where the simple closed curve ℘ is not contactible. The case of contractible ℘ turns out to be a bit special, as we will see below.
2 The "framed" in "framed BPS states" is not directly related to the framing of links; we will discuss the role of framing of links below.
3 In comparing to [11] we have q here = −y there .

Non-flat links in C × R
Continuing with the case M = C × R, now we consider a more general link L ⊂ M . In this case we only have 2-dimensional rotation invariance U(1) P (in the x 4 -x 5 plane), and U(1) R , and 2 supercharges. From the point of view of symmetries, this is the same as the case of general links in M = R 3 which we considered in §1.5.1.
What does the defect S[L] look like from the point of view of the reduced theory X[C, gl(N )]? In the IR, the position variations of the defect in the x 3 direction are suppressed, so the effective support of the defect in the spatial R 3 is a point. Thus we obtain a 1 4 -BPS line defect in the theory X[C, gl(N )]. If L is isotopic to a flat link, then we expect that this defect is actually 1 2 -BPS in the IR, and all the IR physics should be the same as in §1.5.2. If L is not isotopic to a flat link, though, then the line defect we get is really only 1 4 -BPS. Moreover, in the UV the 1 4 -BPS defect breaks the rotational symmetry SU(2) P to the U(1) P rotation in the x 4 -x 5 plane, and the full SU(2) P need not be restored in the IR. Thus we expect that a general link L ⊂ C × R corresponds to an unconventional sort of line defect in theory X[C, gl(N )], which partially breaks rotation invariance in the spatial R 3 . We call these fat line defects.
As for a 1 2 -BPS line defect, we can consider a protected spin character Ω(L, M , γ) ∈ Z[q, q −1 ] for a fat line defect; it is defined by exactly the same equation (1.15) which we used before, with J 3 and I 3 the generators of U(1) P and U(1) R . We propose that this protected spin character is also computed by q-nonabelianization, just as in (1.16) above.

Positivity
It was conjectured in [11] that, when L is flat and not contractible, the action of SU(2) R on H L is trivial; this is the "framed no-exotics" conjecture 4 . If the framed no-exotics conjecture is true, then when L is flat, (1.15) reduces to the simpler This would imply that the coefficients in the (−q)-expansion of Ω(L, M , γ) are all positive, and moreover they have symmetry and monotonicity properties following from the fact that they give a character of the full SU(2) P , not only of U(1) P . For N = 2 the expected positivity has been established in [23,24], for the version of F given in [1]. The expected monotonicity property has not been proven as far as we know. 5 We will see in various examples below that the coefficients in Ω(L, M , γ) computed from our F do have the expected properties; it would be very interesting to give a proof directly from our construction of F .
In contrast, for non-flat links L there is no reason to expect any kind of positivity property (and indeed, for a link contained in a ball in M we get the polynomial (1.9), which is in general not sign-definite.) Likewise when L is contractible we do not necessarily expect positivity, and indeed for a small unknot in M (even a flat one) we will get F (L) = q + q −1 , which does not have a positive expansion in −q.

Connections and future problems
1. In the language of theory X[gl(N )], our UV-IR map can be interpreted roughly as follows. The direct lifts of L to M come from the usual symmetry-breaking phenomenon: moving to the Coulomb branch breaks the symmetry locally from gl(N ) to its Cartan subalgebra of diagonal matrices, and correspondingly decomposes the fundamental representation of gl(N ) into N one-dimensional weight spaces. The terms involving webs are contributions from massive BPS strings of theory X[gl(N )] on its Coulomb branch. In particular, a physical interpretation of the exchange factor for N = 2 has been given in [4]. Moreover its categorification in terms of knot homology was studied in [25]. We hope that similar interpretations also exist for generic web factors.
2. In the language of the M-theory construction of X[gl(N )], compactifying the time direction to S 1 to compute the BPS indices could optimistically be understood as computing a partition function in Type IIA string theory on T * M , with N D6-branes inserted on the zero section M ⊂ T * M , and D4-branes placed on the conormal bundle to L in T * M . Adding the parameter q would be implemented by making a rotation in the x 4 -x 5 plane as we go around the S 1 ; from the Type IIA point of view this corresponds to activating a graviphoton background. Such a partition function is computed by the A model topological string on T * M , with Lagrangian boundary conditions at the D-brane insertions, with string coupling g s where q = e −gs [6].
In this language the webs are interpreted as M2-branes in T * M whose boundary lies partly on M and partly on the conormal to L. For N = 2 this is indicated in Figure 1. 3. In the topological string approach to link invariants pioneered in [6], one takes the above setup with M = S 3 , but then passes through the conifold transition.
After the conifold transition we do not have the N D6-branes anymore, but we do have a compact holomorphic 2-cycle with volume N g s . In consequence, this sort of computation involves the variable N only through the combination a = e −N gs = q N ; so for example one gets the HOMFLY polynomial directly as a function of a, rather than its specialization (1.9) to a particular N . Recently this open topological string computation has been interpreted in the language of skein modules [26]. We want to emphasize that our computation is on the other side, "before" the conifold transition.
4. We discuss in this paper mainly the cases M = R 3 and M = C × R. Having come this far, it is natural to consider the case of a general 3-manifold M . There is a twist of X[gl(N )] on M × R 2,1 which preserves 4 supercharges. If M is compact, then on flowing to the IR one arrives at a theory X[M, gl(N )] in R 2,1 ; examples of these theories have been studied e.g. in [13,15,27]. The surface defect S[L] gives in the IR a 1 2 -BPS line defect in theory X[M, gl(N )]. As before we could imagine perturbing the theory to reach a "Coulomb branch" using an N -fold covering M → M . Unlike the cases we have discussed up to now, though, here we do not have a good understanding of the IR physics. In particular, it is not clear to us that we can define a meaningful ground state Hilbert space H L ( M ) in this case. Nevertheless, we could still think about the IR decomposition of line defects, and the corresponding map of skein modules. We believe that the same rules we use in this paper (in their covariant incarnation, § 6) will work for more general M , but there is a subtlety in formulating the skein module Sk( M , gl(1)) in this case: one needs to include corrections from boundaries of holomorphic discs in T * M . The q = 1 version of this problem will be treated in [28]; the case of general q is ongoing work.
5. One way of thinking of the skein modules is that they describe relations obeyed by Wilson lines in Chern-Simons theory. With that in mind, our q-nonabelianization map could be interpreted as part of a general relation between GL(N ) Chern-Simons theory on M and GL(1) Chern-Simons theory on M , with the latter corrected by holomorphic discs -or more succinctly, as a relation between the A model topological string on T * M with N D-branes on M and the A model topological string on T * M with 1 D-brane on M . The possibility of such a relation was proposed in [2,16]. Its avatar in classical Chern-Simons theory will appear in [28]. The existence of a relation between local systems on M and A branes in T * M is also known in the mathematics literature, e.g. [29][30][31].
6. Quantization of character varieties, skein modules and skein algebras have recently been very fruitfully investigated from the point of view of topological field theory [32][33][34][35]. Roughly speaking, the underlying idea is that an appropriate version of the gl(N ) skein module of M can be identified with the space of states of a topologically twisted version of N = 4 super Yang-Mills theory on M × R with gauge group U(N ). It seems natural to ask whether q-nonabelianization can be understood profitably in this language. 6 7. As we have mentioned, the map F of skein modules which we construct for N = 2 in this paper closely resembles the "quantum trace" map constructed first in [1] and revisited in [3]. Our map cannot be precisely the same as the map considered there, if only because the relevant skein modules are different: ultimately this is related to the fact that we consider the gl(2) theory while those references consider sl (2). We believe that after appropriately decoupling the central gl(1) factor, the maps are likely the same; we discuss this point a bit more in §9.
Our map is also closely related to the computation of framed BPS states for certain interfaces between surface defects given in [2]; in particular, the writhe used in [2] is essentially the same as the power of q which appears in relating a class [ L] ∈ Sk( M , gl(1)) to a quantum torus generator X γ , as described in §3.5.
Nevertheless, our way of producing F looks quite different from the constructions in [1][2][3], and has some advantages: it is more local, it can be covariantly formulated on a general M , it does not require us to make a large isotopy of the link L to put it in some special position, and it makes the connections to the strings of the theory X[gl(N )] or to holomorphic discs more manifest. We expect that these advantages will be useful for further developments. (In particular, for N > 2 our approach here is well suited for dealing with the case of theories X[C, gl(N )] at general points of their Coulomb branch, involving more general spectral networks than the "Fock-Goncharov" type considered in [3].) 8. In this paper we compute indexed dimensions of spaces H L ( M ) of framed BPS states. It would be very interesting to try to promote our construction to get the actual vector spaces H L ( M ), with their bigrading by U(1) P × U(1) R . When M = R 3 this is expected to give some relative of the Khovanov-Rozansky homology as discussed e.g. in [4,5,7,25,36].
9. At least for flat links in C × R, there is also an algebraic approach to studying the framed BPS states. In [37][38][39] it was proposed that for theories of quiver type as studied in [40][41][42][43][44][45][46], framed BPS spectra of line defects could be computed by methods of quiver quantum mechanics. More recently this idea has been further developed in [47][48][49][50]. It would be nice to understand better the connection between this algebraic method and our geometric approach. One possibility might be to compare the method of BPS graphs as in [19,51,52] to the framed BPS quivers.
10. As we have mentioned, when M = C × R our q-nonabelianization map is closely connected to the quantum trace of [1]. For other aspects of the quantum trace see e.g. [23,24,[53][54][55][56]. It is also closely related to the quantum cluster structure on moduli spaces of flat connections, discussed in [57][58][59]; for some choices of the covering C → C, we expect that the formula (1.16) gives the expansion of a distinguished element F ([L]) of the quantum cluster algebra, relative to the variables X γ of a particular cluster determined by C.
part by the Simons Collaboration on the Non-Perturbative Bootstrap and in part by the INFN.
2 The state sum model revisited We begin with the simplest case of our story. As we have discussed above, when M = R 3 , q-nonabelianization should boil down to a way of computing the specialization (1.9) of the HOMFLY polynomial. The precise way in which this works depends on the shape of the covering M → M . If M is given by N constant 1-forms λ i which are all nearly parallel, then qnonabelianization is essentially equivalent to a known method of computing (1.9), known as the state sum model. In this section we first review the state sum model and then describe the ways in which q-nonabelianization generalizes it.

The state sum model
Choose a distinguished axis in R 3 , say the x 3 -axis. This induces a projection R 3 → R 2 . We assume that L is in general position relative to this projection; if not, make a small isotopy so that it is. Then the projection induces an oriented knot diagram in R 2 .
A labeling of the knot diagram consists of an assignment of a label i ∈ {1, . . . , N } to each arc. The state sum is a sum over all labelings. Each labeling is assigned a weight α( ) ∈ Z[q, q −1 ] according to the following rules: • Each crossing gives a factor depending on the labels of the four involved arcs, as indicated in Figure 2. If the labels at any crossing are not of one of the types shown in the figure, then the factor for that crossing is 0 (and thus α( ) = 0.) • If the diagram includes crossings where the labels change, we "resolve" the crossings as shown in Figure 3. After so doing, the diagram consists of loops, each carrying a fixed label i. We assign each such loop a factor q w(N +1−2i) , where w is the winding number of the projection of the loop to R 2 (so e.g. for a small counterclockwise loop the winding number is +1.  Then, the state sum formula is where w(L) is the writhe of the knot diagram (ie the number of overcrossings minus the number of undercrossings.) Let us illustrate (2.1) with a few examples: • The most trivial example is the unknot, placed in R 3 so that its projection to R 2 is a circle. The HOMFLY polynomial for this knot is Since the diagram has only one arc, the state sum model just sums over the N possible labels for that arc. Since there are no crossings, the weight α( ) reduces to the winding factor, giving Our description of this factor is a bit different from what usually appears in the literature. One could equivalently define this factor by first making the stipulation that whenever there is a crossing, the orientations of both arcs in the crossing should have the same sign for their y-component (both up or both down), and then assigning factors q ± 1 2 (N −1−2i) to local maxima and minima of y along arcs; these locations are sometimes called "cups" and "caps" in the knot diagram. as desired.
• As a more interesting example, suppose we take L to be the left-handed trefoil, placed in R 3 so that its projection to R 2 is the diagram in Figure 4. The HOMFLY polynomial for this knot is Let us see how the state sum model reproduces this formula. The knot diagram in Figure 4 has 6 arcs; thus the state sum model involves a sum over N 6 different arc label assignments . The for which α( ) = 0 fall into five classes, as shown below: Applying the state sum model rules, we obtain This matches (2.1) as desired, since the writhe of the diagram in Figure 4 is w = −3.

Reinterpreting the state sum model
With an eye toward generalization, we now slightly reinterpret the state sum rules.
We think of the index i as labeling the i-th sheet of a trivial N -fold covering Each arc labeling with α( ) = 0 gets interpreted as representing some link L in M . If an arc in L is labeled i, it means L contains the lift of that arc to the i-th sheet in M . The simplest situation arises when all arcs in L carry the same label i: then L is just the lift of L to the i-th sheet. More generally, if the labels of the arcs change at the crossings, simply taking the lift of each arc would not give a closed link on M . At such a crossing involving labels i and j, we insert two segments, traveling along the z-direction on sheets i and j in opposite directions, to close up the link, as shown in Figure 3. We call this pair of segments an exchange. The resulting L is a disjoint union of closed links on various sheets of M . Now we reinterpret the weight factor α( ) in terms of the link L: • The weights q ±1 assigned to crossings where all arc labels are the same give altogether q N i=1 n i , where n i is the self-linking number of the part of L on sheet i. This factor can be understood as using the relations in Sk( M , gl(1)) to express L as a multiple of the empty link: • The weights ±(q − q −1 ) assigned to crossings where the arc labels change are interpreted as universal factors associated to exchanges in L.
• The winding factors q w(N +1−2i) can be interpreted as follows: for a loop of L on sheet i, we include a factor q w for each j > i, and a factor q −w for each j < i.

q-nonabelianization as a generalization
The q-nonabelianization map can be thought of as a generalization of the state sum model. Some of the key elements are: • The trivial covering of R 3 given in (2.5) above is replaced by the (in general nontrivial) branched covering M → M , with M ⊂ T * M . The labels i = 1, . . . , N which we used above are replaced by local choices of a sheet of this covering over a patch of M .
• The single projection R 3 → R 2 along the z-axis is replaced by many different projections, along the leaves of N 2 different locally defined foliations of M , labeled by pairs of sheets ij. The directions of the different projections are determined by the 1-forms λ i − λ j . To define the winding factor for an arc on sheet i, we sum the winding of N − 1 different projections of the arc to the leaf spaces of the N − 1 ij-foliations. (In the case of the state sum model all of the foliations coincide, and so all of the projections also coincide, but the leaf space R 2 gets a different orientation depending on whether i > j or i < j; this recovers the recipe above.) • Instead of simple exchanges built from segments traveling along the z-axis, we have to consider more general webs built out of segments of leaves of the N 2 foliations. Each end of each segment lies either on the link L, on the branch locus of M → M , or at a trivalent junction between three segments. Each such web comes with a weight factor generalizing the ±(q − q −1 ) we had above.
Even when M = R 3 , we can take M to be a general branched cover, and then qnonabelianization looks quite different from the state sum model, involving sums over various sorts of webs; nevertheless it still computes (1.9) in the end. We will show how this works in a few examples in §4.3 and §5.1 below.   8 The literature contains a number of variants of this skein module; the one we consider here was also considered in [26] where it is called the "U(N ) HOMFLYPT skein," except that q here = q 1 2 there .
In each skein relation, all the terms represent links which are the same outside a ball in M , and are as pictured inside that ball, with blackboard framing. Relation (II) can be thought of as a "change of framing" relation: the two links are isotopic as unframed links, but as framed links with blackboard framing, they differ by one unit of framing.

The gl(1) skein module with branch locus
Now consider an oriented 3-manifold M decorated by a codimension-2 locus F. (We sometimes call F the branch locus since in our application below, M will be a covering of M , branched along F.) The gl(1) skein module with branch locus, Sk( M , gl(1)), is the free Z[q ±1 ]-module generated by ambient isotopy classes of framed oriented links in M \ F, modulo the submodule generated by the following skein relations: In the skein relation (III) the orange cross represents the codimension-2 locus F ⊂ M . This skein relation says that we can isotope a link segment across F at the cost of a factor −1.
We remark that relations (I) and (II) imply a simple relation between links whose framings differ by one unit, parallel to what we had for gl(N ) above:

Skein algebras and their twists
Now we take M = C × R and M = C × R, 9 where C and C are oriented surfaces. We take the orientation of M (resp. M ) to be the one induced from the orientation of C (resp. C) and the standard orientation of R.
In this case Sk(M, gl(N )) and Sk( M , gl (1)) are algebras over Z[q ±1 ], where the multiplication is given by "stacking" links along the R direction: where LL is the link defined by superposing L with the translation L t of L in the positive x 3 -direction, so that all points of L t have larger R coordinate than all points of L . We emphasize that this algebra structure comes from the R factor: for a general 3-manifold M , there is no algebra structure on Sk(M, gl(N )).
We have to mention a little subtlety: the product structure that is most convenient for our purposes below is a twisted version of the usual one. Given two links L, L in C × R, let (L, L ) = ±1 be the mod 2 intersection number of their projections to C. Then we define the twisted product in Sk(M, gl(N )) by the rule In what follows we will only use the twisted products, not the untwisted ones. (The twisted and untwisted versions of the skein algebra are actually isomorphic, but not canonically so; to get an isomorphism between them one needs to choose a spin structure on C. For our purposes it will be more convenient not to make this choice.)

Standard framing
Another extra convenience in the case M = C × R is that we have a distinguished framing available: as long as the projection of a link L ⊂ M (resp. L ⊂ M ) to C (resp. C) is an immersion, we can equip the link with a framing vector pointing along the positive x 3 -direction. We call this standard framing and will use it frequently.

The gl(1) skein algebra is a quantum torus
When M = C × R we can describe the skein algebra Sk( M , gl (1)) explicitly as follows.
(Descriptions of the gl(1) skein algebra similar to what follows have been used before in connection with gl(1) Chern-Simons theory and BPS state counting, e.g. [2,43,60].) Given any lattice Γ with a skew bilinear pairing ·, · , the quantum torus Q Γ is a Z[q, q −1 ]-algebra with basis {X γ } γ∈Γ and the product law For later convenience we sometimes call the R-direction the "height" direction. Now let Γ be H 1 ( C, Z), with ·, · the intersection pairing. Then, there is a canonical isomorphism The construction of ι is as follows. Suppose given a link L on M with standard framing. Let n( L) be the writhe (number of overcrossings minus undercrossings) in the projection of L to C, and let n ( L) be the number of "non-local crossings" in the projection of L to C: these are places which are not crossings on C, but become crossings after further projecting from C to C. Then, we let To see that ι is really well defined, we must check that it respects the gl(1) skein relations. For this the key point is that when we perturb the link L across a branch point we shift n ( L) by 1, compatibly with relation (III) in Figure 7.
We also need to check that ι respects the algebra structures. For this consider two loops L, L . Let k( L, L ) be the signed number of crossings between L and L , and k ( L, L ) the number of non-local crossings. Then using the definition of ι we get directly so ι is indeed a homomorphism (recall the twisted algebra structure (3.2).) For an example of a quantum torus relation see Figure 9 below. The relation in Sk( M , gl (1)) shown there is or equivalently   Figure 9. A sample quantum torus relation. We show a patch of C, presented as a branched double cover of a patch of C. The covering is branched at the orange crosses, and branch cuts are shown as dashed orange segments. Labels 1 and 2 next to loop segments indicate which sheet of C they lie on. At a crossing between segments on the same sheet, the x 3 -direction is represented as the "height" (loops which are closer to the eye are ones with a larger x 3 coordinate.) At a "non-local crossing" where two segments on different sheets of C → C cross, we do not indicate the relative height.
4 q-nonabelianization for N = 2 For the rest of this paper, we will focus on the case N = 2 and M = C × R. In this section we spell out the concrete q-nonabelianization map in this case.

WKB foliations
As we have discussed, we fix a complex structure on C and a holomorphic branched double cover C → C. Locally on C we then have 2 holomorphic 1-forms λ i , corresponding to the two sheets of C. We define a foliation of C using these one-forms: the leaves are the paths along which λ i − λ j is real. The leaves are not naturally oriented, but if we choose one of the two sheets (say sheet i) then we get an orientation: the positive direction is the direction in which λ i − λ j is positive; see Figure 10. Thus the lift of a leaf to either sheet of C is naturally oriented.
i j Figure 10. A leaf with its two orientations labeled.
At branch points of C → C there is a three-pronged singularity as shown in Figure 11. 10 The three leaves ending on each branch point are called critical. 10 To understand this three-pronged structure note that around a branch point at z = 0 we have  Dividing C by the equivalence relation that identifies points lying on the same leaf, one obtains the leaf space of the foliation. This space is a trivalent tree, as indicated in Figure 12. It will be convenient later to equip it (arbitrarily) with a Euclidean structure.  The WKB foliation of C also induces a foliation of M = C × R: the leaves on M are of the form × {x 3 = c}, where ⊂ C is a leaf, and c is any constant; we illustrate this in Figure 13. Thus the leaf space of M is the product of a trivalent tree with R, i.e. it is a collection of 2-dimensional "pages" glued together at 1-dimensional "binders." The leaf space inherits a natural Euclidean structure, so locally each page looks like a patch of R 2 . The pages do not carry canonical orientations, but locally choosing a sheet i induces an orientation. This induced orientation is determined by the orientation of leaves on sheet i and the ambient orientation of M : our convention is that the induced orientation is the opposite of the quotient orientation. Note that switching the choice of sheet reverses the leaf space orientation. where L runs over all links in M built out of the following local constituents: • Direct lifts of segments of L to M : these are just the preimages of those strands under the covering map.
i Figure 14. The direct lift of a segment of L to sheet i of the covering M .
• When a segment of L intersects a critical leaf, L may include a detour along the critical leaf, as shown in Figure 15. Note that the orientation of the detour segments in L is constrained to match the orientations on the critical leaf; so e.g. in the situation of Figure 15 we can have a detour from sheet i to sheet j, but not from sheet j to sheet i. Again, the orientation of the lifted exchange in L is constrained to match the orientations on the ij-leaf.
We assume (by making a small perturbation if necessary) that L is sufficiently generic that detours and exchanges can only occur at finitely many places. (In particular, we always make a perturbation such that L is transverse to the "fixed height" slices C × {x 3 = c}, since otherwise exchanges could occur in 1-parameter families instead of discretely.) Once this is done, the sum over L is a finite sum.
For each L the corresponding weight α( L) ∈ Z[q ±1 ] is built as a product of elementary local factors, as follows: • At every place where the projection of L onto C is tangent to a leaf, we get a contribution q ± 1 2 to α( L), with the sign determined by the figure below:  (Note that this is an overall factor, depending only on L, not on L.) • Each detour in L contributes a factor of q ± 1 2 to α( L), with the sign determined by the tangent vector to L at the point where a strand of L meets the critical leaf. The factor is shown in Figure 18 below. up up down down Figure 18. Detour factors contributing to the overall weight α( L). Notation is as in Figure 17 above.
• Each exchange in L contributes a factor to α( L). This factor depends on two things: first, it depends whether the two legs of L cross the exchange in the same direction or in opposite directions when viewed in the standard projection; second, it depends whether the crossing in the leaf space projection of L is an overcrossing or an undercrossing. See Figure 19 for the factors in the four possible cases.  Figure 19. Exchange factors contributing to the overall weight α( L). The exchange factor depends on more data than we can represent in a single projection: we show the standard projection on top and the leaf space projection below. The picture represents L rather than L, so in the leaf space instead of a crossing we see its resolution. The words "up" and "down" here describe the behavior in the x 1 -direction, since that is the direction not visible in the leaf space projection; hence "up" means pointing out of the paper and "down" means pointing into the paper.
• Finally there is a contribution from the "winding" of L, or more precisely the winding of its projection to the leaf space of the foliation. This winding is defined as follows. Recall that the leaf space consists of "pages" each of which has a 2-dimensional Euclidean structure, glued together at 1-dimensional "binders." After perturbing L so that it meets each binder at a right angle, we can define a 1 2 Z-valued winding number for the restriction of L to each page. (Recall that L is lifted to one of the two sheets of M → M , say sheet i, and this picks out the i-orientation on the leaf space; we use this orientation to define the winding number.) Summing up the winding of L on all of the pages, we get a Z-valued total winding w( L). We include a factor q w( L) in the weight α( L).
For practical computations, it is convenient to have a way of computing the winding factors q w( L) without explicitly drawing the leaf space projections. Here is one scheme that works. We consider all of the places where the projection of L to C is tangent to a leaf of the foliation. 11 For each such place we assign a factor q ± 1 2 as indicated in Figure 20.  Figure 20. Factors contributing to the winding factor q w( L) . The notation is as in Figure 17, namely "up" and "down" refer to the tangency of L in the x 3 -direction. The difference from Figure 17 is that here we use the orientation of the ij-leaf and ignore that of the projection of L, while in Figure 17 we used the orientation of the projection of L and ignored that of the ij-leaf.
Although the elementary local factors can involve half-integer powers of q, the total weight α( L) is valued in Z[q ±1 ].

Simple unknot examples
In this section we illustrate concretely how q-nonabelianization works, in the simplest possible class of examples: we compute F ([K]) where K is the unknot in standard framing. In this case we have [K] = (q + q −1 )[·] in Sk(M, gl(2)), and so since F factors through Sk(M, gl (2)), the answer must be (4. 2) The details of how this works out depend on what the WKB foliation of C looks like and how K is positioned relative to that foliation. In this section we describe how it works in some simple cases. We give more interesting unknot examples in §5.1.1 below. First let us consider the case C = C, with a trivial double covering C → C, for which the WKB foliation is just given by straight lines in the x 1 -direction. We place the unknot K such that its projections to the x 1 -x 2 plane and the x 2 -x 3 plane are as shown in Figure 21. We use purple dots to mark the two points on K where the projection of K to C is tangent to a leaf. We will omit this information in the following figures. This is the simplest situation possible: there are no possible detours since the covering M → M is unbranched, and there are no exchanges since each leaf meets K at most once. Thus the only contributions to F ([K]) come from the direct lifts K 1 , K 2 to the two sheets. Their weights are given simply by the winding factors, which are q and q −1 respectively, all other contributions being trivial. Finally, each lift [ K i ] has self-linking number zero and thus is equivalent to the class [·] in Sk( M , gl(1)). We summarize the situation in the table: lift framing exchange detour winding [lift] total Thus we indeed get the expected answer (4.2). (In fact, the need to get this answer was our original motivation for including the winding factor in the q-nonabelianization map; see also [4] which includes a similar factor for a similar reason.) Next we consider a slightly more interesting case: again we take C = C with a foliation by straight lines in the x 1 -direction, but now take K as shown in Figure 22. In this case our path-lifting rules lead to three possible lifts: • We could lift the whole link K to sheet 1 or sheet 2; this gives two lifts K 1 and K 2 . Either of these lifts is contractible on M and has blackboard framing, so (1)). Each of these lifts has framing factor q, from the two places where the projection of K is tangent to the foliation of C. Each of these lifts has total winding zero, as we see from the leaf space projection on the right side of Figure 22; thus the winding factor is trivial. (Another convenient way to count the winding is to use the rules of Figure 20. The two tangencies contribute q 1 2 and q − 1 2 respectively, giving the total winding factor 1.) Thus these lifts have α( K 1 ) = α( K 2 ) = q, so they each contribute q[·] to F ([K]).
• There is also a more interesting possibility shown in Figure 23.
in Sk( M , gl (1)) and the total framing factor is q. This time, however, the total winding is −2 instead of zero; this arises because K 3 is divided into one loop on sheet 1 and one on sheet 2, and the 1-orientation and 2-orientation of the leaf space are opposite, so the windings from these two parts add instead of cancelling. Thus we get a winding factor q −2 . There is also an exchange factor of q(q −1 − q), as we read off from Figure 19; here we use the fact that the leaf space crossing is an undercrossing. Combining all these factors, this lift contributes (q −1 − q)[·] to F ([K]). lift framing exchange detour winding [lift] total Combining these three lifts we get as expected.
As we remarked earlier, in this case our computation is similar to the state sum model reviewed in §2, applied to the "figure-eight unknot" diagram we obtained by projecting to the leaf space (Figure 22, right.) There is a slight difference: the state sum model computes with the blackboard framing in leaf space, which in this case differs by one unit from our standard framing. Thus the state sum model gives q −2 (q −1 + q) instead of our result q −1 + q. Looking into the details of the computation one sees that the relative factor q 2 comes from two different places: our computation includes an extra q in the factors associated to the crossing, and also includes the framing factor q which has no direct analogue in the state sum model.

Knots in R 3
In § 4.3 we have shown how our q-nonabelianization map F correctly produces the Jones polynomial for the simplest unknots in R 3 . In this section we show how it works in a few more intricate examples, with more interesting knots placed in more interesting positions relative to the WKB foliations. In all cases we have to get the Jones polynomial: this follows from the fact that F is a well defined map of skein modules, which we prove in §7 below. Nevertheless it is interesting and reassuring to see how it works out explicitly in some concrete examples.

Unknots
We first look at an unknot K whose projection onto C is a small loop around a branch point of the covering C → C, as shown in Figure 24. This case is more interesting since we will meet detours as well as exchanges. In this case a direct lift of K is not allowed, since such a lift would not give a closed path on M : we need to include an odd number of detours to get back to the initial sheet. Indeed, according to our rules there are five lifts K n contributing to F ([K]): • There are three lifts which involve a single detour each, shown in Figure 25 below: Each of these lifts is contractible in M , and equipped with standard framing, so in Sk( M , gl(1)) we have Moreover, each of these lifts has a total framing factor q 3/2 and detour factor q −1/2 . Finally, each of these lifts has total winding w( K n ) = 0. Thus each of these lifts contributes q to F ([K]).
• There is one lift K 4 involving both a detour and an exchange, shown in Figure 26: In Sk( M , gl (1)) again we have [ K 4 ] = 1. The framing factor, detour factor, and winding factor for this lift are q 3/2 , q −1/2 , and q −2 respectively. The exchange carries a factor of q(q −1 − q). Combining all these factors, altogether this lift contributes q −1 − q.
• Finally there is one lift K 5 involving three detours, shown in Figure 27. Resolving crossings using the gl(1) skein relations, we find that [ K 5 ] is q times the class of the link shown at the right of Figure 27; in turn the class of that link is −[·] (the minus sign comes from deleting the loop in the middle, which winds once around the branch locus in M ); so altogether we get [ K 5 ] = −q. The framing factor is q 3/2 , and the detour factor is q −3/2 . Finally, the total winding of K 5 is zero, so there is no winding factor. Thus altogether this lift contributes −q.
lift framing exchange detour winding [lift] total Putting everything together, the image of [K] under q-nonabelianization is Again this matches the expected answer.
Next we look at an unknot K whose projection to C is a loop around two branch points of C → C, shown in Figure 28. This is the most detailed example which we will work out by hand. There are in total 17 lifts. Two of them, K 1 and K 2 , are the direct lifts of K to the two sheets of M . The next four lifts K 3 , K 4 , K 5 , K 6 each involve two detours at the same branch point: The next four lifts K 7 , K 8 , K 9 , K 10 each involve two detours at two different branch points: The lifts K 11 and K 12 have three detours at one branch point and one detour at the other branch point: The next two lifts K 13 and K 14 have two detours at each branch point: Finally there are three lifts K 15 , K 16 and K 17 which have an exchange path: This is our first example in which there is a nontrivial homology class on C, and thus the contributions to F ([K]) can be more interesting than just multiples of the unknot on C: they can involve the other quantum torus generators. Explicitly, consider the oriented loop p in Figure 34. According to the rules of §3.5 we define the quantum torus generator X γ = [p] ∈ Sk( M , gl(1)).  (1)).
The contributions from some of the lifts will involve the variable X γ . We will not describe in detail the computations for all 17 lifts; see the table below for the results. Summing the 17 terms together gives once again the expected answer, In particular, note that all the terms proportional to X γ cancel among themselves, as do those proportional to X −γ . This had to happen, since K is contained in a ball; in lift framing exchange detour winding such cases we always just get the polynomial (1.9) for K, just as for M = R 3 . In more interesting examples where K represents a nontrivial class in π 1 (M ), the X γ will not cancel out.

Trefoils
In §2 we obtained the Jones polynomial for a left-handed trefoil using the state-sum model. We could equally well apply our q-nonabelianization map to a left-handed trefoil K trefoil in a single domain, equipped with standard framing. The calculation goes through in a similar fashion to the state-sum model. Explicitly, there are 6 lifts, whose contributions sum to which matches (2.1). A more interesting example is a trefoil in the neighborhood of a branch point as shown in Figure 35. There are in total 18 lifts, whose contributions sum up to give the expected answer,

Figure-eight knot
In Figure 37 we show a figure-eight knot K figure-8 in standard projection and leaf space projection. There are in total 47 lifts, whose contributions sum up to matching P HOMFLY (K figure-8 , a = q 2 , z = q − q −1 ) as expected. 12   Figure 38. An unknot K whose standard projection is a small loop around a puncture (denoted as a blue dot) on C. For simplicity, in this and following examples, we do not draw the leaf space projection. Instead, we specify x 3 -coordinates (in red) at points where the unknot meets a critical leaf. Between these points the x 3 -coordinate varies as simply as possible: this means it varies monotonically, except for critical points which we place immediately before crossing a critical leaf. We also show cycles q 1 , q 2 representing homology classes µ 1 , µ 2 in H 1 ( C, Z) respectively.

Now we begin to consider examples of links which are homotopically nontrivial in M .
The simplest such example is a loop K whose standard projection encircles a puncture on C, as illustrated in Figure 38. 13 K corresponds to a pure flavor line defect in a theory of class S.
In this case the only lifts of K allowed are the direct lifts K 1 and K 2 on sheet 1 and sheet 2 respectively. Moreover, the framing factor and winding factor for each of these are trivial, so we simply have α( K 1 ) = α( K 2 ) = 1 and thus So far we have been considering class S theories of type gl (2), but in the following we will also discuss class S theories of type sl(2), in order to be able to compare directly to previous results in the literature. The projection from gl(2) to sl(2) is discussed in § 9; roughly it amounts to replacing X γ → X 1 2 (γ−σ(γ)) , where σ denotes the deck transformation exchanging the two sheets of C → C. In the following examples we first obtain the generating function F ([K]) in a class S theory of type gl (2), then apply this projection to get the result in the sl(2) theory.
For a first example, we revisit the pure flavor line defect, now in a theory of class S of type sl (2). The projection identifies µ 1 ∼ −µ 2 := µ, and the generating function is

SU(2) N = 2 * theory
Next we consider the SU(2) N = 2 * theory. This theory is obtained by giving a mass m to the adjoint hypermultiplet in the SU(2) N = 4 theory. Its class S construction is given by compactifying the 6d (2,0) A 1 theory on a once-punctured torus C. We choose m and the coupling τ such that the WKB foliation is as shown in Figure 39. We consider a line defect corresponding to a loop K whose projection to C wraps both A-cycle and B-cycle once, as shown in Figure 39.  Figure 39. WKB foliation on a torus with one puncture (blue dot). We also show the standard projection of a loop K, whose x 3 -profile is specified by the red numbers, following the convention we introduced in Figure 38. We will first compute the generating function in the gl (2) case. There are in total six lifts K i as illustrated in Figure 40. For each K i , the framing factor is trivial, while the winding factor and detour factors cancel out, so α( K i ) = 1. The question of finding F ([K]) thus reduces to expressing [ K i ] in terms of X γ i . Then according to the crossing-counting rules explained in §3.5, we have (recall that the factors of q come from the genuine crossings, and factors of −1 come from "non-local crossings.") Summing these up gives This result obeys the expected positivity and monotonicity properties discussed in §1. 5.4.
To obtain the spectrum in the sl(2) theory, we just perform the projection ρ, which has the effect of identifying γ 4 ∼ −γ 2 and γ 5 ∼ −γ 3 . The resulting generating function is This agrees (modulo some shifts in conventions) with [3], where the same line defect was considered. We also remark that using the traffic rules of [11] one could compute the vacuum expectation value of this line defect, which agrees with the classical limit of our generating function (5.5). (2) with N f = 4 flavors  Figure 41. Left: WKB foliation structure on CP 1 with four punctures (denoted as blue dots) with the parameters in (5.7). Right: a basis {γ 1 , γ 2 , µ 1 , µ 2 , µ 3 , µ 4 } for H odd 1 ( C, Z) with representative cycles s 1 , s 2 , q 1 , q 2 , q 3 , q 4 respectively.

SU
As our next example, we take C to be CP 1 with four punctures, corresponding to N = 2 SU(2) SYM with four fundamental hypermultiplets. Traditional 1 2 -BPS line defects in this theory have been systematically studied by many people, for example [9][10][11]61]. Here we will consider both traditional and fat line defects.
Let z be a coordinate on CP 1 . We choose a complex structure such that the four punctures are located at z = 1, i, −1, −i respectively. Moreover, we pick Coulomb branch and mass parameters such that the double cover C is The WKB foliation structure on C is shown in Figure 41.  Figure 42. Standard projection of two loops K 1 and K 2 in M = C × R. Height profiles for these loops are specified in the same way as in Figure 38.
As a more interesting example we consider the loop K 2 shown on the right of Figure 42. K 2 has in total 48 lifts whose contributions sum up to Once again the result has the expected properties: the coefficients of X γ form characters of SU(2) P representations, and framed BPS states that form even-(odd-) dimensional SU(2) P representations contribute to the protected spin character with a minus (plus) sign.
The most interesting charge sector is the charge 2γ 1 − γ 2 + µ 1 + µ 3 − 2µ 4 , where the framed BPS states form a direct sum of one-dimensional and three-dimensional SU(2) P representations. We show the four lifts realizing these framed BPS states in Figure 43. As our final example, we consider the link K 3 shown in Figure 44. K 3 has in total 52 lifts. Summing up their contributions, F ([K 3 ]) is given by a long expression, which can be conveniently written in terms of F ([K 2 ]) as follows: 14  Figure 44. Standard projection of an unknot K 3 in M = C ×R. Its height profile is specified in the same convention as in Figure 38.
In particular, the positivity is violated, and even if we ignore this, the coefficients of X γ in F ([K 3 ]) do not in general form SU(2) P characters (we have a q term but no corresponding 1/q). This is as expected since K 3 corresponds to a fat line defect, as the standard projection of K 3 contains a crossing.

(A 1 , A N ) Argyres-Douglas theories
In this section we briefly consider some theories whose class S construction involves irregular singularities: the (A 1 , A N ) Argyres-Douglas theories [43]. These theories are obtained by taking C = CP 1 with an irregular singularity at z = ∞, which prescribes that the coverings C that we consider have λ ∼ z N 2 +1 dz as z → ∞. 15 The local behavior of the WKB foliation in the neighborhood of an irregular singularity P is very different from that near a regular singularity. A generic leaf asymptotes tangentially to one of M 16 rays near P . If we draw an infinitesimal circle S 1 P around P bounding an infinitesimal disk D P , these M rays determine M marked points P i on S 1 P , evenly distributed.
In the examples that we will consider here C = CP 1 \ D P . Here the irregular singularity is placed at z = ∞. The green markers on the boundary circle correspond to P i , which are determined by the six rays going into the irregular singularity.
In the following we will use the (A 1 , A 3 ) Argyres-Douglas theory as an example. We choose a point in the parameter space of this theory such that the covering C is 15 The rules we discuss in this section also apply to the (A 1 , D N ) Argyres-Douglas theories, which have an irregular singularity at z = ∞ and also a regular singularity at z = 0. The spin content of framed BPS states for line defects in simple (A 1 , D N ) theories has also been studied in [62]. 16  given by where z is a coordinate on C such that D P is centered at z = ∞. Figure 45 shows the WKB foliation at this point in the moduli space.
In the presence of an irregular singularity, line defects do not correspond to ordinary links in C ×R anymore; we also need to include links which can have some endpoints on S 1 P ×R. Here we focus on flat line defects, i.e. we only consider L whose projection to C does not contain any crossings. The projection of such an L to C is an oriented version of what was called a lamination in [11] following [57]. A lamination is a collection of paths on C = CP 1 \ D P , which can be either closed or open with ends on the marked points P i . Each path carries an integer weight, subject to the constraint that paths that carry negative weights must be open and end at two adjacent marked points P i , P i+1 , and the sum of the weights of paths ending at each P i must be zero. For example, Figure 46 illustrates a lamination corresponding to a line defect in the (A 1 , A 3 ) Argyres-Douglas theory. Now we state the extra ingredients in our q-nonabelianization rules to accommodate the presence of irregular singularities. For simplicity we just consider the case of laminations where all weights on paths are ±1.
• Each open path carrying the weight −1 has to be lifted to a specific sheet of M : its orientation has to match with the orientation of the leaves near the boundary circle. In Figure 46 we show the lifts for the two open paths with −1 weight in that lamination.
• We enumerate all possible lifts L ⊂ M subject to these lifting constraints. Each L is a link in M , meeting S 1 P × R at a finite number of points. We then apply our q-nonabelianization rules as usual, except that for computing α( L ), we do not include winding factors or framing weights for the lifts of paths carrying weight −1.
Applying these rules to the line defect in Figure 46, we obtain the result: (5.10) This line defect was also studied in [11]; in particular the classical version of its generating function was given in (10.15) of that paper. Our q-nonabelianization computation refines the framed BPS state index from 2 to −(q + q −1 ) in the charge sector γ 1 − γ 3 .
Let us take a closer look at the term −(q + q −1 )X γ 1 −γ 3 , which turns out to be the sum of contributions from three lifts K 1 , K 2 and K 3 shown in Figure 47.  Figure 47. The standard projection of three lifts K 1 (left), K 2 (middle) and K 3 (right) whose contributions sum up to −(q + q −1 )X γ 1 −γ 3 . Here we have denoted the sheet numbers of lifted strands in blue.
For all three lifts K i , there are four places where the standard projection of a strand with +1 weight is tangent to a WKB leaf. The associated local framing weight factors cancel out, so in the end each K i has a trivial framing factor. The winding numbers of K 1 , K 2 and K 3 are −1, 0 and −1 respectively. Additionally K 1 has an exchange factor of q(q −1 − q), K 3 has a detour factor of q, and K 2 has trivial exchange and detour factors. In summary the total weights associated with the lifts are: Using the sign rules introduced in §3.5, we get Combining these gives

A covariant version of q-nonabelianization
In §4.2 above, we described the q-nonabelianization map F in a way that used the special structure M = C × R. For example, we always chose links with standard framing, and our explicit formulas for the weight factors α( L) involved the projection M → C. In this section we reformulate F in a more covariant way. This reformulation will be useful for the future generalization to other 3-manifolds M . It will also prove to be convenient for the proof that F is a map of skein modules in §7 below. Given a framed link L in M , we again write where L runs over links in M built from the same kinds of pieces as in §4.2 as above. Now, however, we need to explain what framings we use.
• Direct lifts of portions of L to M : these are equipped with a framing which just lifts the framing of L, using the projection to identify the tangent spaces to M and M .
• Detours: each detour is equipped with a distinguished framing, as follows. Let p denote the point where the link L crosses the critical leaf. At p we have two distinguished tangent directions: the tangent t w to the critical leaf (oriented toward the branch point) and the tangent t L to L. Define f p := t w × t L . Without loss of generality, we may assume that the framing of L at p is given by f p ; then the framing of L as we approach p will also be given by f p . (More generally if the framing of L is given by some f = f p , we simply insert a rotation of the framing of L from along some arc a from f to f p as we approach p along L, and insert the opposite rotation from f p back to f immediately after we leave p along L. The homotopy class of the resulting framing of L is then independent of the choice of arc a, since a change of a cancels out.) Then, the framing of the lifted detour must start out from the framing f p at the beginning of the lifted detour and end again at f p at the end of the lifted detour. To get this interpolating framing we just choose any trivialization of T M in a neighborhood of the detour and then use that trivialization to extend f p .
• Lifted exchanges: Let p denote either of the two points where the exchange attaches to the link L. At p we have two distinguished tangent directions: the tangent t e to the exchange (oriented away from p towards the exchange) and the tangent t L to L. Define f p := t e × t L ; without loss of generality, we may assume that the framing of L at p is given by f p (if not we add interpolating arcs as above.) The framing of the exchange needs to interpolate between the two framings f p at the two ends. Fortunately the normal bundle to the exchange has a canonical connection, coming from the fact that the exchange is a leaf of the foliation of M . We can use this connection to identify all the fibers with a single 2-dimensional vector space E. Then a framing of the exchange is a path in the circle E/R + , with its ends on the two points [f p ]. There are two distinguished such paths, one going the "short way" around the circle, the other going the "long way" around. We allow either of these framings; whichever one we choose, we use it on both sheets of the lift. (Thus when we consider a diagram involving n exchanges, the sum over L includes 2 n different terms associated to this diagram, differing only in their framing.) The weight factor α( L) is given as follows: • At every place where the framing of L is tangent to a leaf of the foliation of M , we include a contribution q ± 1 2 to α( L), with the sign determined as follows. We consider the normal planes to L near the point where the tangency occurs. These planes are naturally oriented (since L and M are), and each contains two vectors, one given by the framing, the other by the foliation. At the point of tangency these two vectors coincide; as we move along L in the positive direction, the framing vector rotates across the foliation vector, either in the positive or negative direction. The factor is q 1 2 in the positive case, q − 1 2 in the negative case.
To illustrate how this works, we revisit the first case in Figure 17. In that case the link L was taken to carry standard framing. In Figure 48 we show the framing vector and foliation vector in the normal plane to L near the tangency point. The framing vector rotates across the foliation vector in the positive direction, giving the framing factor q 1 2 .
up Figure 48. An example of the framing factor contribution to α( L) for L in standard framing. The framing vector is shown in purple while the foliation vector is shown in black.
• For each lifted exchange, as we have explained, there are two possible framings: one going the "short way" and one going the "long way." When the framing goes the "long way" we include an extra factor −1 in α( L).
• Finally, the winding contribution q w( L) to α( L) is defined just as in §4.2 (this definition was already covariant.) This completes our description of the covariant rules. There are two special cases worth discussing: • One can check directly that if we apply these covariant rules in the case where L has standard framing, we recover the rules of §4.2 above. (When we frame a lifted exchange, since we apply the same framing to the lifts on both sheets, the terms using the "short way" and "long way" differ by a total of 2 units of framing. Thus altogether the two choices of framing give lifts differing by a factor −q 2 . This is the origin of the exchange factors ±(q − q −1 ) in the rules of §4.2.) • Suppose M = R 3 with the foliation in the x 3 -direction. Then we can consider equipping L with "leaf space blackboard framing," i.e. choose the framing vector to point in the x 3 -direction. This framing is well defined provided that L is nowhere parallel to the x 3 -direction. There is a slight technicality: our covariant rules cannot be applied directly because the framing is not generic enough. We perturb by rotating the framing everywhere very slightly in (say) the clockwise direction. After so doing, the covariant rules reduce exactly to the N = 2 state sum rules of §2.1. In particular: -The framing factor in the covariant rules is trivial (there are no points where the framing is tangent to the foliation.) -The process of evaluating [ L] ∈ Sk( M , gl(1)) reduces to computing the selflinking number, which is given by a product over crossings, using the weights appearing in the first column of Figure 2.
-The sum over framings for an exchange produces the q − q −1 or q −1 − q appearing in the last column of Figure 2. For an example of how this works, see Figure 49 below. down up up down up down Figure 49. The sum over two framings for a lifted exchange, with the original link in blackboard framing. Middle: the blackboard framing far from the exchange, and the framings f p near the exchange. Left: framing interpolated the "short way." This framing differs from the blackboard framing by a factor of q. Right: framing interpolated the "long way." This framing differs from the blackboard framing by a factor of q −1 . Summing over the two framings gives the factor q−q −1 .
-The winding as defined in the covariant rules reduces to the winding of the link projection as used in the state sum rules.

Isotopy invariance and skein relations
In this section we give a sketch proof that F is well defined, i.e. that F (L) really only depends on the class [L] ∈ Sk(M, gl(N )). The basic strategy is as follows. First we need to prove that F (L) depends only on the framed isotopy class of L. For this purpose we use the covariant formulation of F , which we described in §6. We check first that F is covariant under changes of framing. Next we turn to the question of isotopy. From the covariant rules it follows immediately that F (L) is invariant under any isotopy which does not create or destroy exchanges and which leaves L untouched in some small neighborhood of the critical leaves. What remains is: • To deal with processes in which exchanges are created or destroyed; since exchanges correspond to crossings in the leaf space projection, this boils down to checking invariance under Reidemeister moves for that projection. We check this in §7.2 below.
• To deal with processes which change L in a neighborhood of a critical leaf; here again a small isotopy does not change F (L), but there are several Reidemeisterlike moves which have to be considered, which create or destroy detours, or move detours across one another. We check invariance under these in §7.3 below.
In each case we are free to choose any convenient framing; in practice the way we implement this is to choose a profile for the standard projection of L, and then use standard framing. Then for the actual computations we can use the concrete rules of §4.2.
After this is done, we check that F preserves the skein relations. This check is simplified by the fact that we have already verified isotopy invariance, so we are free to put the link L in a simple position relative to the WKB foliation.  (2)), and thus we must have F (L) = F (L ). This is relatively straightforward to check: as we vary the framing, the framing contributions q ± 1 2 to F (L) appear and disappear in cancelling pairs, while all other contributions vary continuously, so F (L) is constant.

Isotopy invariance away from critical leaves
With an eye towards future generalization to N > 2, we use the notion of ij-leaf space; in the case of N = 2 we simply have {i, j} = {1, 2}.
In the following we denote the open strand configurations before and after each isotopy as l A and l B respectively. Correspondingly we denote its lifts before and after the isotopy as l κ A and l κ B , where κ runs from 1 to the number of lifts.

The first Reidemeister move
In the following we consider the first Reidemeister move as shown in Figure 50. l A has two direct lifts to sheet i and sheet j, denoted as l 1 A and l 2 A respectively. l B has two direct lifts l 1 B and l 2 B , plus one lift l 3 B containing an exchange as shown in Figure 51. Lifts of l A carry no framing factor while lifts of l B have a framing factor q. Denoting the winding of l 1 A as w, we have Here  In Figure 52 we illustrate the second Reidemeister move. Here both l A and l B have four direct lifts. It is easy to see that contributions from these direct lifts match with each other. l B has two extra lifts l 5 B and l 6 B each containing an exchange, as shown in Figure 53. So we only need to prove that the contributions from these two lifts cancel with each other. This works out simply because [l

The third Reidemeister move
In this section we consider the third Reidemeister move as shown in Figure 54. First, using the relations in Sk( M , gl(1)) we observe that We denote the winding ofl 3 A andl 3 B as w 1 and w 2 . Then the weight factors associated to these six lifts are given as follows: Therefore contributions from these three lifts do match with each other:

Isotopy invariance near critical leaves
In this section we consider isotopies of open strands which involve critical leaves. All such isotopies can be perturbed into combinations of four basic moves, which are the analogues in this context of the Reidemeister moves. Each basic move happens in a neighborhood of a branch point with three critical leaves emanating from it. In such a neighborhood, the leaf space of M topologically looks like three pages glued together at a "binder" corresponding to the branch locus. For convenience, we label these three pages as I, II, III, and illustrate the leaf space projection within each page. We first consider an isotopy which moves a leaf space crossing from one page to another, as shown in Figure 56. There are in total eleven lifts on both sides. The comparison is reduced to matching contributions from the three lifts of l A and l B shown in Figure 57

We denote the weights of [l 1
A ] and [l 2 A ] as α 1 and α 2 . The weights of all the other lifts are then given by: We also have the following relations in Sk( M , gl(1)): Combining these we see that the contributions from these three lifts match on both sides.

Height exchange for detours
In Figure 58 we show another basic move in the neighborhood of a critical leaf. Here l A has nine lifts while l B has ten lifts in total. Eight lifts of l A and eight lifts of l B match on a term-by-term basis. The remaining two lifts of l B make the same total contribution as the remaining lift of l A . These terms are illustrated in Figure 59. Taking into account all the local factors, the weights associated to these lifts are related to each other as follows: We obtain the desired matching using the following relation in Sk( M , gl(1)):

Moving a strand across a critical leaf
In this section we consider a basic isotopy where a strand is moved across a critical leaf. This is illustrated in Figure 60. Here both l A and l B have two direct lifts with matching contributions. l B has two extra lifts involving detours, shown in Figure 61, whose contributions cancel each other.

Moving a strand across a branch point
In this section we consider the basic isotopy where a strand is moved across a branch point. This is illustrated in Figure 63. l A has three lifts which match three lifts of l B on a term-by-term basis. These matching lifts are shown in Figure 64. We take the pair l 1 A and l 1 B as an example. They clearly have the same winding. Moreover l 1 B has an extra framing factor q −1/2 and an extra detour factor q 1/2 . Therefore α([l 1 A ]) = α([l 1 B ]) and they make the same contribution to F ([l A,B ]). l B also has two more lifts whose contributions are supposed to cancel with each other. These two lifts are illustrated in Figure 65.

The skein relations
In this section we prove that the skein relations are preserved by the q-nonabelianization map.
We first look at the skein relation (I) in Sk(M, gl (2)). Without loss of generality, we pick a specific direction for the leaves, and monotonic height profiles for the open strands 18 , as shown in Figure 67.  (2)). On the right, we show the loci where exchanges can occur. 18 We can safely do so as we have already proved the q-nonabelianization map is isotopy invariant.  (2)).
We show lifts of these two terms in Figure 68. In the second term, in addition to the direct lifts, there are two lifts involving exchanges. Now subtracting these two sets of lifts, and using the skein relations of Sk( M , gl(1)) on the RHS, we get Figure 69. The RHS of that figure is indeed the lift of the right side of skein relation (I), as desired.  (2)).
Next we consider skein relation (II) in Sk(M, gl (2)) (the change-of-framing relation.) Again choose a generic direction for the foliation, and a simple monotonically decreasing height profile for the link strand. Due to the simple height profile the only lifts are direct lifts to sheet 1 and sheet 2. Each of these lifts comes with a framing fractor q 1 2 × q 1 2 = q from the two places where it is tangent to the foliation; combining this with the q from the skein relations in Sk( M , gl(1)) gives the desired factor q 2 . This is shown in Figure 70.  (1)).
Finally, skein relation (III) in Sk(M, gl (2)) follows from the fact that F assigns q + q −1 to an unknot, which we have already checked in §4.3 (and again in §5.1.1.)

Framed wall-crossing
As we have discussed, the q-nonabelianization map for M = C × R is defined using a WKB foliation associated to a covering C → C. As we vary the covering C → C continuously, there are discrete moments at which the topology of the WKB foliation can jump. These jumps were called K-walls in [12]. Following the topology change, the q-nonabelianization map is expected to jump according to the framed wall-crossing formula [11,12,14,58,[63][64][65]. In this section we verify that this indeed happens.

The framed wall-crossing formula
Let us first quickly recall the framed wall-crossing formula.
From the point of view of the 4d N = 2 theory X[C, g], the framed wall-crossing behavior is determined by the bulk BPS spectrum. At any point in the Coulomb branch, the one-particle BPS Hilbert space is graded by the IR charge lattice Γ, H = ⊕ γ∈Γ H γ . We factor out the center-of-mass degrees of freedom by writing and consider the protected spin character [11]: where J 3 , I 3 are Cartan generators of SU(2) P and SU(2) R respectively. The integers Ω n (γ) are conveniently packaged into the Kontsevich-Soibelman factor, K(q; X γ ; Ω n (γ)) : where E q (z) is the quantum dilogarithm defined as and σ(γ) = ±1 is a certain quadratic refinement defined in [14]. The K-walls are the (real codimension 1) loci in the Coulomb branch where, for some charge γ, the central charge Z γ ∈ R − and the Kontsevich-Soibelman factor K(q; X γ ; Ω n (γ)) = 1. ) before and after crossing a K-wall in the positive direction, i.e. the direction in which Im Z γ goes from negative to positive. The framed wall-crossing formula is [11,64]: In the following we will verify that our map F indeed satisfies (8.5), in the two simplest situations: • A BPS hypermultiplet K-wall: then K = E q (−X γ ) for some γ, which gives using (3.3) for γ, µ = 0. (8.6) • A BPS vector multiplet K-wall: then K = E q (−qX γ )E q (−q −1 X γ ) for some γ, which gives for γ, µ < 0. (8.7)

Relative skein modules
To compare F + to F − it will be convenient to observe that the definition of F is local, in the following sense. Let M • ⊂ M be a submanifold with boundary (and perhaps corners) such that no leaf meets ∂M transversely, i.e., the WKB foliation of M by 1-manifolds restricts to a foliation of ∂M by 1-manifolds. Also fix a 0-chain S in ∂M • , given by a finite subset of ∂M • with each point labeled by ±1. Then we can define a relative skein module Sk(M • , S, gl(2)), by applying exactly the rules of §3.1, except that instead of oriented links L, we use oriented tangles L, with boundary ∂L = S. Also let M • → M • be the branched double cover of M • obtained by restricting M → M . Then we define a relative skein module Sk(M • , S, gl(1)) by applying again the rules of §3.2 to oriented tangles L, where we allow ∂ L to be any 0-chain which projects to S. Our construction of F applies directly in this relative situation, to give a map F : Sk(M • , S, gl(2)) → Sk( M • , S, gl(1)).
where the · on the right is defined by concatenation of tangles. The relative skein module Sk( M • , S, gl (1)) is a two-sided module over the (absolute) skein algebra Sk( M • , gl(1)). For each relative homology class a we define an element X a ∈ Sk( M • , S, gl(1)), by the same rule we used to define X γ for ordinary homology classes γ in §3.5. Then we have the analog of (3.3), X γ X a = (−q) γ,a X γ+a . (8.10)

The hypermultiplet K-wall
We begin by considering a hypermultiplet K-wall. The transformation of the WKB foliation which occurs at such a K-wall is described in [12]. The only nontrivial change occurs in the neighborhood of a saddle connection on C, as shown in Figure 71.  Figure 71. A region C • of C bounded by generic leaves of the WKB foliation, and the critical leaves contained therein, before (left) and after (right) crossing a hypermultiplet K-wall. The charge γ of the hypermultiplet is also shown.
Let C • denote the region shown in Figure 71, and let l 1 , . . . , l m be the components of the intersection L ∩ (C • × R). By an isotopy of L, we may assume that each l n crosses C • in one of the simple ways shown in Figure 72. Moreover, we may assume that each l n has x 3 -coordinate monotonically increasing from a n to b n , with b n < a n+1 . Then we divide M into the disjoint regions M • n = C • × [a n , b n ] and the complementary region M • 0 . We will prove that, in each region M • n , we have Using the locality of F this then implies the desired formula (8.5). First we consider the region M • 0 . In this region the topology of the WKB foliation does not change as we cross the K-wall; (8.12) follows directly.
Next we consider a region M • n containing an open strand l n isotopic to the l in Figure 72. l has five liftsl 1 , . . . ,l 5 either before or after crossing the K-wall, as shown in Figure 73. Using p i for the standard projection ofl i , F ± ([l]) are: where w i is the winding of p i and we have used (8.10) to rewrite F ± ([l]). Now (8.14) is obtained from (8.13) by the substitutions X p 1 → X p 1 (1 − qX γ ), X p 2 → X p 2 (1 − q −1 X γ ) −1 , leaving all other X a alone; this matches the expected (8.11).

Finally we consider a region M •
n containing an open strand l n isotopic to the l in Figure 72. There are two direct lifts of l , which are the same on both sides of the K-wall; we denote their standard projections p 2 , p 3 . On the + side of the wall there is one lift containing a detour, with standard projection p 1 . On the − side there are two lifts containing detours. Altogether we find: This again agrees with the expected (8.11), using γ, p 1 = −1, γ, p 2 = 0, γ, p 3 = 0.

The vector multiplet K-wall
Now we turn to the vector multiplet K-wall, where the WKB foliation develops an annulus of closed trajectories. Here the transformation of the WKB foliation which occurs at the wall is subtler to describe. As described in [12,14], as we approach the K-wall from either direction, the WKB foliation in the annulus undergoes an infinite sequence of flips, which accumulate at the K-wall. The F ± we are after in this case therefore have to be understood as the limits of F as we approach the K-wall from the two sides. Moreover, the meaning of F ± is a bit subtle: they involve infinite sums of terms, so they really lie in some completion of the skein module; we will optimistically gloss over this point here, so what we are really giving here is a proof sketch rather than a complete proof.
Similarly to the classical case discussed in [12], we can compute F ± ([l]) by considering the limiting behavior of the critical leaves of the foliation as we approach the wall. A schematic picture of the limits is shown in Figure 74 (see also [12,14] for more such pictures). The degenerated critical leaves wind infinitely many times around the annulus. Concretely speaking, to compute F ± ([l]), we sum over direct lifts and detours along critical leaves in the usual way; the only tricky point is that even for a compact l, F ± ([l]) may involve an infinite sum over detours, since l may meet one of the degenerated critical leaves infinitely many times. Figure 74. A region C • of C bounded by generic leaves of the WKB foliation, and degenerated versions of the WKB foliation contained therein, immediately before (left) and after (right) crossing a vector multiplet K-wall. The picture is a bit schematic; the leaves which end in dotted lines should be understood as winding infinitely many times around the annulus, and l crosses each of these leaves infinitely many times. The charge γ of the vector multiplet is also shown on the left. Now we follow the same strategy we used for the hypermultiplet. The key computation is to compare the two limits F ± ([l]), where l is an open strand crossing C • as shown in Figure 74, with its x 3 -coordinate increasing monotonically.
9.2 Factorization of q-nonabelianization, for M = C × R Now suppose M = C × R. Let Γ = H 1 ( C, Z), and let σ be the involution on Γ induced by the deck transformation of C. Then we have the map which preserves the pairings and thus induces a map where the quantum tori Q 1 2 Γ odd , Q 1 2 Γ even are algebras over Z[(−q) ± 1 4 ], because the intersection pairings on these lattices take values in 1 4 Z. We conjecture that the q-nonabelianization map F can be factorized into odd and even parts, or more precisely, that there is a commuting diagram Sk(M, gl (2)) Sk( M , gl(1)) = Q Γ Sk(M, sl(2)) ⊗ Sk  ([L])), ρ(X γ ) = X 1 2 (γ−σ(γ)) , (9.6) and the map F odd coincides with the quantum trace map of [1]. One approach to verifying this factorization would be to interpret Q 1 2 Γ odd directly as a variant of the skein module which includes factors associated to non-local crossings. We will not undertake this verification here, but remark that a similar comparison (between the approach of [3] to q-nonabelianization and the quantum trace of [1]) was carried out in [55].