Quivers for 3-Manifolds: the Correspondence, BPS States, and 3D N =2 Theories

Prepared for submission to JHEP

Quivers for 3-manifolds: the correspondence, BPS states, and 3d N =2 theories

Piotr Kucharski Walter Burke Institute for Theoretical Physics, California Institute of Technology, 1200 E. California Blvd. Pasadena, CA 91125, USA Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland E-mail: [email protected]

Abstract: We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspon-

dence to Gukov-Manolescu invariants of knot complements (also known as FK or Zˆ). Apart from assigning quivers to complements of T (2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d N = 2 theories associated to both sides of the correspondence. We also make a step towards categoriﬁcation by proposing a t-deformation of all objects mentioned above. arXiv:2005.13394v2 [hep-th] 11 Aug 2020 Contents

1 Introduction 1

2 Prerequisites 3 2.1 Knot invariants 3 2.2 Quivers and their representations4 2.3 Knots-quivers correspondence4 2.4 GM invariants 6

3 Quivers for GM invariants7 3.1 Main conjecture 7 3.2 Corollary – BPS states and 3d N = 2 eﬀective theories8

4 Examples 10 4.1 Unknot complement 10 4.2 Trefoil knot complement 12 4.3 Cinquefoil knot complement 14 4.4 General T (2,2p+1) torus knot complements 17

5 t-deformation 18 5.1 Quivers 18 5.2 BPS states and 3d N = 2 eﬀective theories 20

6 Future directions 20

1 Introduction

In recent years many interesting relations between quantum field theory, string theory, low-dimensional topology, and representation theory have been studied. In this paper we focus on the knots-quivers correspondence [1,2] and the Gukov-Manolescu invariants of knot complements [3]. (We will abbreviate them to GM invariants and denote FK – often this symbol is also used as their name.) The core of the knots-quivers correspondence is an equality (after appropriate change of variables) between the generating series of symmetrically coloured HOMFLY-PT polynomials of a knot and a certain motivic generating series associated to the quiver. This implies that Labastida-Mariño- Ooguri-Vafa (LMOV) invariants of knots [4–6] can be expressed in terms of motivic Donaldson-Thomas (DT) invariants of associated quivers [7,8], which are known to be integer [9]. In consequence, by assigning a quiver to the knot, we automatically prove the integrality of LMOV invariants, which is the statement of the LMOV conjecture [4–6]. This was done for all knots up to 6 crossings and infinite families of T (2,2p+1) torus knots and twist knots in [2]. More systematic approach in terms of tangles provided a proof of the knots-quivers correspondence for all two-bridge knots [10] and recently for all arborescent knots [11], however finding a general proof remains to be an open problem.

– 1 – Geometric and physical interpretation of the knots-quivers correspondence given in [12, 13] con- nects it with Gromov-Witten invariants and counts of holomorphic curves, showing that the quiver encodes the construction of BPS spectrum from the basic states. This spectrum is shared by dual 3d N = 2 theories associated to the knot complement and the quiver, and respecitve counts of BPS states are given by LMOV and DT invariants. The moduli spaces of vacua of these theories are encoded in the graphs of A-polynomials [14–18] and their quiver versions [12, 13, 19, 20]. Another recent results on the knots-quivers correspondence include a relation to combinatorics of counting paths [19], a consideration of more general case of topological strings on various Calabi-Yau manifolds [20], and a connection with the topological recursion [21]. The origins of GM invariants lie in attempts of categorification of the Witten-Reshetikhin-Turaev (WRT) invariants of 3-manifolds. In order to solve the problem of non-integrality of WRT invariants Gukov, Pei, Putrov, and Vafa introduced new invariants of 3-manifolds [22, 23] (we will call them GPPV invariants, they are denoted by Zˆ and often this symbol is also used as their name). The GPPV invariant is a series in q with integer coefficients and the WRT invariant can be recovered 2πi from the q → e k limit (see [22–27] for details and generalisations). Physical origins of GPPV invariants lie in the 3d-3d correspondence [28–30]: Zˆ is a supersymmetrix index of 3d N = 2 theory with 2d N = (0, 2) boundary condition studied first in [31]. Detailed analysis of this interpretation, as well as the application of resurgence, can be found in [22–24], whereas [32, 33] contain some recent results on GPPV invariants coming from the study of 3d-3d corresponcendce. Another interpretation of GPPV invariants as characters of 2d logarithmic vertex operator algebras was proposed in [25]. On the other hand, this work – together with [34–36] – initiated an exploration of the intruguing modular properties of Zˆ. 3 GM invariants can be treated as knot complement versions of GPPV invariants, FK = Zˆ S \K [3] (because of that, sometimes GM invariants are denoted and called Zˆ, descending it from GPPV invariants). From the physical point of view, the GM invariant arises from the reduction of 6d N = (0, 2) theory describing M5-branes on the 3-manifold with the topology of the knot complement. Important properties analysed in [3] include the behaviour under surgeries, the agreement of the asymptotic expansion of FK with the Melvin-Morton-Rozansky expansion of the coloured Jones polynomials [37–40], relations to the Alexander polynomials, and the annihilation by the quantum A-polynomials introduced in [41, 42]. These relations suggest an interesting geometric interpretation of GM invariants in terms of the annuli counting presented in [43] and based on [44]. Another recent developments include a generalisation to arbitrary gauge group [45], an adaptation of the large colour R-matrix approach 2πi to GM invariants [46], and a relation to Akutsu-Deguchi-Ohtsuki invariant for q = e k [47]. From our perspective, the most important new results are closed form expressions for a-deformed 2 GM invariants provided in [43]. They reduce to the initial FK for a = q and turn out to be closely related to HOMFLY-PT polynomials. This allows us to propose a correspondence between knot complements and quivers via GM invariants and quiver motivic generating series. Moreover, since HOMFLY-PT polynomials admit t-deformation in terms of superpolynomials [48], we can follow [43] and conjecture results for a, t-deformed GM invariants. We also study the physical intepretation of the new correspondence (for clarity we will refer to the knots-quivers correspondence of [1,2] as the standard one), generalising the results of [12, 13]. It includes an exploration of the duality between 3d N = 2 theories associated to the knot complement and the quiver, as well as a study of their BPS states, which allows us to introduce knot complement analogs of LMOV invariants via DT invariants. The rest of the paper is organised as follows. Section2 provides a short introduction to the knots- quivers correspondence and GM invariants of knot complements. In section3 we state our main conjecture, assigning quivers to knot complements, and study consequences for 3d N = 2 theories

– 2 – and their BPS spectra. In section4 we show how these ideas can be applied and checked on concrete examples of unknot, trefoil, cinquefoil, and general T (2,2p+1) torus knots. Section5 contains a t- deformation of previous results inspired by the categoriﬁcation of HOMFLY-PT polynomials. In section6 we conclude with a discussion of interesting open problems.

2 Prerequisites

In this section we recall relevant aspects of the knots-quivers correspondence, its physical interpretation, and GM invariants of knot complements.

2.1 Knot invariants 3 If K ⊂ S is a knot, then its HOMFLY-PT polynomial PK (a, q) [49, 50] is a topological invariant which can be calculated via the skein relation. More generally, coloured HOMFLY-PT polynomials

PK,R(a, q) are similar polynomial knot invariants depending also on a representation R of the Lie algebra u(N). In this setting, the original HOMFLY-PT corresponds to the fundamental representation.

From the physical point of view, PK,R(a, q) is the expectation value of the knot viewed as a Wilson line in U(N) Chern-Simons gauge theory [51]. In the context of the knots-quivers correspondence, we are interested in the HOMFLY-PT generating series: ∞ X −r PK (λ, a, q) = PK,r(a, q)λ , (2.1) r=0 r where PK,r(a, q) are HOMFLY-PT polynomials coloured by the totally symmetric representations S (with r boxes in one row of the Young diagram), which for brevity we will call simply HOMFLY-PT polynomials. The unusual expansion variable with the negative power comes from the necessity of resolving the clash of four diﬀerent conventions present in the literature (and avoiding the confusion with the quiver variables):

• KRSS convention from [1,2, 52, 53], • EKL convention from [12, 13], • FGS convention from [3, 14, 16, 18, 45], • EGGKPS convention from [43].

The dictionary is given by

−1 −1 1/2 −1 λ = xKRSS = xEKL = yFGS = yEGGKPS, µ = yKRSS = yEKL = xFGSqFGS = xEGGKPS, (2.2) 2 2 2 2 a = aKRSS = aEKL = aFGS = aEGGKPS, q = qKRSS = qEKL = qFGS = qEGGKPS. HOMFLY-PT polynomials satisfy recurrence relations encoded in the quantum a-deformed A- polynomials [14–18]: ˆ Aˆ(ˆµ, λ, a, q)PK,r(a, q) = 0, (2.3) where r ˆ µPˆ K,r(a, q) = q PK,r(a, q), λPK,r(a, q) = PK,r+1(a, q). (2.4) For simplicity, in the remaining part of the paper we will drop “a-deformed” and call Aˆ(ˆµ, λ,ˆ a, q) the quantum A-polynomials. The LMOV invariants [4–6] are numbers assembled into the LMOV generating function N(λ, a, q) = P −r i j r,i,j Nr,i,jλ a q that gives the following expression for the HOMFLY-PT generating series: N(λ, a, q) P (λ, a, q) = Exp . (2.5) K 1 − q

– 3 – P n Exp is the plethystic exponential: if f(t) = n ant and a0 = 0, then " # X 1 k Y n an Exp [f(t)] = exp k f(t ) = (1 − t ) . (2.6) k n LMOV invariants can be extracted also from the A-polynomials, see [52, 53].

2.2 Quivers and their representations

A quiver Q is an oriented graph, i.e. a pair (Q0,Q1) where Q0 is a ﬁnite set of vertices and Q1 is a ﬁnite set of arrows between them. We number the vertices by 1, 2, ..., m = |Q0|. An adjacency matrix of Q is the m × m integer matrix with entries Cij equal to the number of arrows from i to j. If Cij = Cji, we call the quiver symmetric.

A quiver representation with dimension vector d = (d1, ..., dm) is the assignment of a vector space di dj of dimension di to the node i ∈ Q0 and of a linear map γij : C → C to each arrow from vertex i to vertex j. Quiver representation theory studies moduli spaces of stable quiver representations. While explicit expressions for invariants describing those spaces are hard to ﬁnd in general, they are quite well understood in the case of symmetric quivers [7–9, 54, 55]. Important information about the moduli space of representations of a symmetric quiver is encoded in the motivic generating series deﬁned as m d P i X 1/2 Cij didj Y xi PQ(x, q) = (−q ) i,j , (2.7) (q; q)di d1,...,dm≥0 i=1 where the denominator is the q-Pochhammer symbol:

n−1 Y k (z; q)n = (1 − zq ). (2.8) k=0 If we write Ω(x, q) P (x, q) = Exp , (2.9) Q 1 − q we obtain the generating series of motivic Donaldson-Thomas (DT) invariants Ωd,s [7,8]: ! X d s X Y di s Ω(x, q) = Ωd,sx q = Ω(d1,...,dm),s xi q . (2.10) d,s d,s i The DT invariants have two geometric interpretations, either as the intersection homology Betti numbers of the moduli space of all semi-simple representations of Q of dimension vector d, or as the Chow- Betti numbers of the moduli space of all simple representations of Q of dimension vector d, see [54, 55]. [9] provides a proof of integrality of DT invariants for the symmetric quivers.

2.3 Knots-quivers correspondence The knots-quivers correspondence [1,2] is a conjecture that for each knot K there exist a quiver Q and integers ni, ai, li, i ∈ Q0, such that

P (λ, a, q) = P (x, q)| n a l . (2.11) K Q xi=λ i a i q i If we substitute (2.5) and (2.9), we obtain the knots-quivers correspondence at the level of LMOV and DT invariants:

N(λ, a, q) = Ω(x, q)| n a l . (2.12) xi=λ i a i q i

– 4 – Since DT invariants are integer, this equation implies integrality of Nr,i,j, which is known as the LMOV conjecture. From the physical point of view, DT and LMOV invariants count BPS states in 3d N = 2 theories denoted by T [LK ] and T [QLK ] [12, 13]. T [LK ] is the eﬀective 3d N = 2 theory on the world-volume of M5-brane wrapped on the knot conormal inside the resolved conifold:

4 1 space-time : R × S × X ∪ ∪ 2 1 M5-brane : R × S × LK .

The structure of T [LK ] can be read from the semiclassical limit (q = e~ → 1) of the HOMFLY-PT generating series: ∞ r Z X −r q =µ dµ Y dzi 1 0 PK,r(a, q)λ −→ exp WfT [LK ](µ, λ, a, zi) + O(~ ) , (2.13) →0 µ z r=0 ~ i i ~

R dµ Q dzi where the integral corresponds to the gauge group U(1)M ×U(1)Z ×...×U(1)Z (we single µ i zi 1 k out U(1)M and its fugacity µ because for the knot complement theory it becomes a global symmetry).

WfT [LK ](µ, λ, a, zi) is the twisted superpotential with two typical kinds of contributions:

n nQ nM Zi Li2 a µ zi ←→ (chiral ﬁeld) , κ (2.14) ij log ζ · log ζ ←→ (Chern-Simons coupling) . 2 i j

Each dilogarithm is interpreted as the one-loop contribution of a chiral ﬁeld with charges (nQ, nM , nZi ) under the global symmetry U(1)Q (arising from the internal 2-cycle in the resolved conifold geometry) and the gauge group U(1)M × U(1)Z1 × ... × U(1)Zk . Quadratic-logarithmic terms are identiﬁed with Chern-Simons couplings among the various U(1) gauge and global symmetries, with ζi denoting the respective fugacities. For more details see [12, 16, 18, 28, 29, 56, 57].

Integrating over the gauge fugacities zi using a saddle-point approximation gives the eﬀective twisted superpotential of the theory:

eﬀ ∗ ∂WfT [LK ](µ, λ, a, zi) Wf (µ, λ, a) = WfT [L ](µ, λ, a, z ), where = 0. (2.15) T [LK ] K i ∂z i ∗ zi=zi

The moduli space of vacua of T [LK ], given by the extremal points of the eﬀective twisted superpotential, coincides with the graph of the classical A-polynomial:

∂Wfeﬀ (µ, λ, a) T [LK ] = 0 ⇔ A(µ, λ, a) = 0, ∂ log λ (2.16) A(µ, λ, a) = limAˆ(ˆµ, λ,ˆ a, q). q→1

In analogy to T [LK ], the structure of T [QLK ] is encoded in the semiclassical limit of the motivic generating series [12]:

d Z q i =yi Y dyi 1 0 PQ(x, q) −→ exp W (x, y) + O( ) , fT [QLK ] ~ ~→0 yi ~ i (2.17) C X Cii X ij WfT [Q ](x, y) = Li2(yi) + log (−1) xi log yi + log yi log yj. LK 2 i i,j

– 5 – Using the dictionary (2.13-2.14), we can interpret the elements of (2.17) in the following way:

• The integral R Q dyi corresponds to having the gauge group U(1)(1) × · · · × U(1)(m), i yi

(i) • Li2(yi) represents the chiral ﬁeld with charge 1 under U(1) ,

Cij eﬀ • 2 log yi log yj corresponds to the gauge Chern-Simons couplings, κij = Cij,

Cii • log (−1) xi log yi represents the Chern-Simons coupling between a gauge symmetry and its dual topological symmetry (the Fayet-Iliopoulos coupling).

The saddle point of the twisted superpotential encodes the moduli space of vacua of T [QLK ] and deﬁnes the quiver A-polynomials [12, 13, 19, 20]:

∂WfT [Q ](x, y) LK Cii Y Cij = 0 ⇔ Ai(x, y) = 1 − yi − xi(−yi) yj = 0. (2.18) ∂ log yi j6=i

Ai(x, y) is a classical limit of the quantum quiver A-polynomial, which annihilates the motivic generating series:

Aˆi(xˆ, yˆ, q)PQ(x, q) = 0,

xˆiPQ(x1, . . . , xi, . . . , xm, q) = xiPQ(x1, . . . , xi, . . . , xm, q), (2.19)

yˆiPQ(x1, . . . , xi, . . . , xm, q) = PQ(x1, . . . , qxi, . . . , xm, q).

The general formula for the quantum quiver A-polynomial corresponding to the quiver with adjacency matrix C is given by [13]

C ˆ 1/2 Cii Y ij Ai(xˆ, yˆ, q) = 1 − yî − xî(−q yî) yˆj (2.20) j6=i and we can see that

Ai(x, y) = limAˆi(xˆ, yˆ, q). (2.21) q→1

2.4 GM invariants 3 GM invariants FK = Zˆ S \K implicitly depend on the gauge group and results of [3] correspond to the simplest nontrivial case of SU(2). General case is analysed in [45], but it is often very involved from computational point of view. One of the goals of [43] was overcoming these diﬃculties and SU(N) studying the large-N behaviour of FK (µ, q) – the GM invariant corresponding to the symmetric representations of SU(N). It turns out that we can introduce a-deformed GM invariants FK (µ, a, q) which capture all N by the following relation:

N SU(N) FK (µ, a = q , q) = FK (µ, q). (2.22)

FK (µ, a, q) is annihilated by the quantum a-deformed A-polynomials: ˆ Aˆ(ˆµ, λ, a, q)FK (µ, a, q) = 0, (2.23) where ˆ µFˆ K (µ, a, q) = µFK (µ, a, q), λFK (µ, a, q) = FK (qµ, a, q). (2.24)

– 6 – Since we adapt the convention that the series expansion of FK starts from 1, as in [43], sometimes we have to rescale λˆ in order to compare to the literature. From our point of view, the crucial result of [43] is the connection between a-deformed GM invariants and HOMFLY-PT polynomials. In general, we expect it to be some version of the Fourier transform, but it is diﬃcult to deﬁne it properly. However, in some cases it reduces to a simple substitution:

FK (µ, a, q) = PK,r(a, q)|qr =µ . (2.25) It is not yet known what conditions are sufficient and which are necessary for this equation to hold. [43] contains an explicit check for the trefoil by comparing with F SU(N)(µ, q) from [3, 45]. Unfor- 31 tunately, already for the figure-eight knot the substitution qr = µ leads to ill-defined series in both µ and µ−1 and this situtation is ubiquitous. Nevertheless, solving equation (2.23) order by order in µ works well for 41, which suggests that there exists a well-defined F41 (µ, a, q), however not equal to P41,r(a, q)|qr =µ. On the other hand, there is a class of knots for which we know general expressions r (2,2p+1) for PK,r(a, q) and the substitution q = µ leads to well-defined series. They are T torus knots and for them (2.25) is conjectured to hold [43]. Closed form expressions for FK are essential for finding quivers, so T (2,2p+1) torus knots will be main focus of our interest. Since there exists a t-deformation of HOMFLY-PT polynomials provided by the superpolynomials [48], it is natural to consider the following t-deformation of GM invariants [43]:

FK (µ, a, q, t) = PK,r(a, q, t)|qr =µ . (2.26)

We will study this deformation for T (2,2p+1) torus knots in section5. SU(2) SU(N) For simplicity, in the remaining part of the paper we will refer to FK (µ, q), FK (µ, q), FK (µ, a, q), and FK (µ, a, q, t) broadly as GM invariants, specyfying the case explicitly only when it is not obvious from the context.

3 Quivers for GM invariants

In this section we introduce quivers for GM invariants and study their physical interpretation in terms of BPS state counts and 3d N = 2 eﬀective theories.

3.1 Main conjecture The connection between GM invariants and HOMFLY-PT polynomials, together with closed form expressions for FT (2,2p+1) (µ, a, q) found in [43], suggest that the idea of knots-quivers correspondence can be reformulated for GM invariants. In other words, we expect the following:

3 Conjecture 1 For a given knot complement MK = S \K, the GM invariant FK (µ, a, q) can be written in the form

m n d a d l d Pm i i i i i i X 1/2 Cij didj Y µ a q FK (µ, a, q) = (−q ) i,j=1 , (3.1) (q; q)di d1,...,dm≥0 i=1 where C is a symmetric m × m matrix and ni, ai, li are ﬁxed integers. In consequence, there exist a quiver Q which adjacency matrix is equal to C and the motivic generating series

m di Pm X 1/2 Cij didj Y xi FQ(x, q) = (−q ) i,j=1 (3.2) (q; q)di d1,...,dm≥0 i=1

– 7 – ni ai li reduces to the GM invariant after the change of variables xi = µ a q :

F (µ, a, q) = F (x, q)| n a l . (3.3) K Q xi=µ i a i q i

Let us stress that we may encounter negative entries of the quiver adjacency matrix: Cij < 0, so we can either accept having two types of arrows (ordinary arrows and “antiarrows” which annihilate each other) or use the change of framing to shift all entries (for details see [2]). Moreover, there is a possiblity of the sign diﬀerence in the knot complement and quiver variables which would lead to ji ni ai li more complicated change of variables xi = (−1) µ a q , but for all analysed examples it was not the case. We can also obtain quivers for SU(N) (and more speciﬁcally initial SU(2)) GM invariants by substituting a = qN in (3.3), which leads to

SU(N) F (µ, q) = F (x, q)| n l +Na . (3.4) K Q xi=µ i q i i 3.2 Corollary – BPS states and 3d N = 2 eﬀective theories

If there exist a quiver Q corresponding to the GM invariant of MK , we can compute the DT invariants using Ω(x, q) F (x, q) = Exp . (3.5) Q 1 − q Since Q is symmetric, we immediately know that DT invariants (which are coeﬃcients of Ω(x, q)) are integer numbers [9]. ni ai li We can also make a step back and apply the change of variables xi = µ a q to (3.5). This leads us to the deﬁnition of the knot complement analogs of LMOV invariants:

X r i j N(µ, a, q) = N µ a q = Ω(x, q)| n a l , (3.6) r,i,j xi=µ i a i q i r,i,j which implies   rn in jn N(µ, a, q) X Nr,i,jµ a q F (µ, a, q) = Exp = exp K 1 − q  1 − qn  n,r,i,j (3.7) −N Y r i j+l r,i,j Y r i j −Nr,i,j = 1 − µ a q = (µ a q ; q)∞ r,i,j,l r,i,j

Since DT invariants are integer numbers, so are the knot complement analogs of LMOV invariants.

This is consistent with the physical interpretation given in [12, 13]: we expect that Ωd,s and Nr,i,j invariants count the number of BPS states in a dual 3d N = 2 theories T [QMK ] and T [MK ]. 2 1 We identify T [MK ] with the eﬀective 3d N = 2 theory on R × S which can be engineered in two equivalent ways. One is the compactiﬁcation of N M5-branes on the knot complement:

4 1 ∗ space-time : R × S × T MK ∪ ∪ 2 1 N M5-branes : R × S × MK . When N → ∞, many protected quantities – such as the twisted superpotential – depend on the number of M5-branes only via the combination qN which can be treated as a separate variable a. The second

– 8 – way comes from the large-N transition from the deformed to the resolved conifold [4, 58]:

4 1 ∗ 3 space-time : R × S × T S 4 1 space-time : R × S × X ∪ ∪ ←→ 2 1 3 ∪ ∪ N M5-branes : R × S × S 2 1 2 1 M5-brane : R × S × LK . M5-brane : R × S × LK Then log a is a complexiﬁed Kähler parameter of X. More details, together with a description of a third way of engineering T [MK ], are available in [43]. The structure of T [MK ] can be read from the semiclassical limit of the GM invariant: Z Y dzi 1 0 FK (µ, a, q) → exp WfT [MK ](µ, a, zi) + O(~ ) . (3.8) →0 z ~ i i ~

Recalling the relation (2.25), we can see that WfT [MK ] is the same as the eﬀective twisted superpotential of the 3d N = 2 theory analysed in [16, 18]. On the other hand, the perspective of the large-N transition explains why WfT [MK ] is a Legendre transform of the superpotential of the theory T [LK ] discussed in section 2.3:

WfT [LK ](µ, λ, a, zi) = WfT [MK ](µ, a, zi) − log µ log λ. (3.9) We can also consider the eﬀective twisted superpotential

eﬀ ∗ ∂WfT [MK ](µ, a, zi) Wf (µ, a) = WfT [M ](µ, a, z ), where = 0, T [MK ] K i ∂z i ∗ zi=zi and introduce λ back as the variable dual to µ, which is equivalent to the saddle point equation for Weﬀ and the vanishing of the A-polynomial: fT [LK ]

∂Wfeﬀ (µ, a) ∂Wfeﬀ (µ, λ, a) T [MK ] = log λ ⇔ T [LK ] = 0 ⇔ A(µ, λ, a) = 0. (3.10) ∂ log µ ∂ log λ

In consenquence T [MK ] have the same moduli space of vacua as T [LK ] and both are described by the A-polynomial of K.

In analogy to (2.17), the structure of T [QMK ] is encoded in the semiclassical limit of FQ(x, q):

d Z q i =yi Y dyi 1 0 FQ(x, q) −→ exp W (x, y) + O( ) , fT [QMK ] ~ ~→0 yi ~ i (3.11) C X Cii X ij WfT [Q ](x, y) = Li2(yi) + log (−1) xi log yi + log yi log yj. MK 2 i i,j

W (x, y) is a twisted superpotential of the theory T [QM ] and each of its terms can be inter- fT [QMK ] K preted according to the dictionary described in section 2.3. The saddle point of the twisted superpotential encodes the moduli space of vacua of T [QMK ] and deﬁnes the quiver A-polynomials:

∂W (x, y) fT [QMK ] = 0 ⇔ Ai(x, y) = 0. (3.12) ∂ log yi

– 9 – Ai(x, y) is a classical limit of the quantum quiver A-polynomial, which annihilates the motivic generating series FQ(x, q):

1/2 Cii Y Cij A(x, y) = limAî(xˆ, yˆ, q), Aî(xˆ, yˆ, q) = 1 − yî − xî(−q yî) yˆ . (3.13) q→1 j j6=i

ni ai Q Applying the change of variables xi = µ a (q → 1) together with i yi = λ, we can transform Ai(x, y) into the classical A-polynomial A(µ, λ, a) given by (3.10).

4 Examples

This section is devoted to the explicit results: computations of quivers, BPS state counts, and 3d N = 2 theories for diﬀerent GM invariants of knot complements. The choice of examples on which we check ideas from section3 is determined by the fact that the closed form formulas for FK (µ, a, q) are completely new results, so far available only for the T (2,2p+1) torus knot complements [43]. We start from the unknot complement in the unreduced normalisation. For all other cases we use the reduced one, which corresponds to the division by the unknot factor. For more details see [43].1

4.1 Unknot complement The simplest example is the unknot complement, for which the GM invariant is given by [43]:

(µq; q)∞ F01 (µ, a, q) = . (4.1) (µa; q)∞ Using formulas for quantum dilogarithms we can write

F (µ, a, q) = F (x , x , q)| 1/2 , 01 Q 1 2 x1=µq , x2=µa

d1 d2 X 1/2 d2 x1 x2 (4.2) FQ(x1, x2, q) = (−q ) 1 . (q; q)d1 (q; q)d2 d1,d2≥0

We can see that the quiver for the unknot complement is given by

1 1 0 Q = ⇐⇒ C = , (4.3) 0 0 2 which is the same as in the standard knots-quivers correspondence for the unknot. However, now the variable a appears in the change of variables for the node without the loop. For the SU(N) GM invariants we obtain:

(µq; q) F SU(N)(µ, q) = ∞ = F (x , x , q)| . (4.4) 01 N Q 1 2 x =µq1/2, x =µqN (µq ; q)∞ 1 2

In case of original SU(2) GM invariants, we still have same quiver (4.3) and motivic generating se- 1/2 2 ries (4.2), but the change of variables reduces to x1 = µq , x2 = µq .

1Note that what we call unreduced normalisation is called fully unreduced in [43].

– 10 – Since

"P d s # 1/2 Ω(x, q) Ωd,sx q −q x + x F (x , x , q) = Exp = Exp d,s = Exp 1 2 , (4.5) Q 1 2 1 − q 1 − q 1 − q we have only two nonzero DT invariants:

Ω(1,0),1/2 = −1, Ω(0,1),0 = 1. (4.6) By deﬁnition, they lead to two nonzero knot complement analogs of LMOV invariants, exactly as in the case of the standard LMOV invariants for the unknot [4]:

N(µ, a, q) = Ω(x, q)| 1/2 x1=µq , x2=µa ( N1,0,1 = −1 X r i j =⇒ (4.7) Nr,i,jµ a q = − µq + µa N1,1,0 = 1. r,i,j Alternatively, we could have obtained this result by a direct comparison between (4.1) and (3.7). As discussed in section3, we can use the semiclassical limit of the motivic generating series and the GM invariant to obtain eﬀective twisted superpotentials of theories T [Q ] and T [M ], M01 01 whose BPS states are counted by DT invariants and knot complement analogs of LMOV invariants respectively. The limit (3.11) for the unknot complement quiver is given by

d Z q i =yi dy1 dy2 1 0 FQ(x, q) −→ exp WfT [QM ](x, y) + O(~ ) , →0 y y 01 ~ 1 2 ~ (4.8) 1 WfT [QM ](x, y) = Li2 (y1) + Li2 (y2) + log(−x1) log y1 + log x2 log y2 + log y1 log y1. 01 2 We can see that T [Q ] is a U(1)(1) × U(1)(2) gauge theory with one chiral ﬁeld for each group and M01 eﬀective Chern-Simons level one for U(1)(1), which is the same as T [Q ] [12]. According to (3.12), L01 the critical point of quiver twisted superpotential given by

∂WfT [QM ] 0 = 01 = log(−x ) + log y − log(1 − y ) , ∂ log y 1 1 1 1 (4.9)

∂WfT [QM ] 01 0 = = log x2 + log y2 − log(1 − y2) ∂ log y2 deﬁnes the quiver A-polynomials

A1(x, y) = 1 − y1 + x1y1,A2(x, y) = 1 − y2 − x2. (4.10)

A1 and A2 are decoupled, which originates from the lack of arrows between vertices 1 and 2 in Q. The quantum quiver A-polynomial, which annihilates FQ(x, q) and reduces to (4.10) for q → 1, is given by

1/2 Aˆ1(xˆ, yˆ, q) = 1 − yˆ1 +x ˆ1(−q yˆ1), Aˆ2(xˆ, yˆ, q) = 1 − yˆ2 − xˆ2. (4.11)

Applying the change of variables x1 = µ, x2 = µa, y1y2 = λ to (4.10), we recover the classical A-polynomial of the unknot2 [16]:

A01 (µ, λ, a) = 1 − aµ − λ + µλ, (4.12)

2 1/2 We have to take into account the rescaling of λ by a with respect to yFGS due to the fact the we start F01 (µ, a, q) from 1.

– 11 – which is in line with section3.

We also expect that the A-polynomial of the unknot encodes the moduli space of vacua of T [M01 ]

– the theory read from the semiclassical limit of F01 (µ, a, q): 1 F (µ, a, q) → exp W (µ, a) + O( 0) , 01 fT [M01 ] ~ ~→0 ~ (4.13) W (µ, a) = Li (µ) − Li (aµ). fT [M01 ] 2 2 Solving ∂WfT [M ](µ, a) log λ = 01 = log (1 − aµ) − log (1 − µ) (4.14) ∂ log µ we indeed ﬁnd an agreement with A01 (µ, λ, a) = 0.

4.2 Trefoil knot complement Let us move to the trefoil knot complement and start from the closed form expression found in [43]:

∞ (µ; q−1) (aq−1; q) X k1 k1 k1 F31 (µ, a, q) = (µq) . (4.15) (q; q)k1 k1=0 Using the formula k l2 X 1/2 l − l (q; q)k (ξ; q)k = −q ξ q 2 (4.16) (q; q)l(q; q)k−l l=0 −1 −1 for (µq ; q )k and the identity

−1 −1 l2+(k−l)2−k2 (q ; q )k 1/2 (q; q)k −1 −1 −1 −1 = −q , (4.17) (q ; q )l(q ; q )k−l (q; q)l(q; q)k−l we can write

∞ k1 2 2 −1 l (k1−l1) −k1 X X 1 (aq ; q)k1 k1+l1 k1+ 2 1/2 F31 (µ, a, q) = µ q −q . (4.18) (q; q)l1 (q; q)k1−l1 k1=0 l1=0 Then we use α2+γ2+2γl (ξ; q)k X α+γ − 1 (α+γ) 1/2 = ξ q 2 −q (4.19) (q; q)l(q; q)k−l α+β=l γ+δ=k−l to get

∞ k 2 2 2 2 l1 3 1 (k −l ) −k +α +γ +2γ l k1+l1 α1+γ1 k1+ − (α1+γ1) X X X 1/2 1 1 1 1 1 1 1 µ a q 2 2 F31 (µ, a, q) = −q . (q; q)α1 (q; q)β1 (q; q)γ1 (q; q)δ1 k1=0 l1=0 α1+β1=l1 γ1+δ1=k1−l1 (4.20) After the change of variables

α1 = d1, β1 = d2, γ1 = d3, δ1 = d4 (4.21) l1 = d1 + d2, k1 = d1 + d2 + d3 + d4

– 12 – we have

F (µ, a, q) = F (x, q)| n a l , 31 Q xi=µ i a i q i

4 di 2 2 x (4.22) X 1/2 −d +d −2d1d2−2d1d4−2d2d4 Y i FQ(x, q) = (−q ) 2 3 , (q; q)di d1,...,d4≥0 i=1

ni ai li with xi = µ a q given by

2 2 3/2 −1/2 x1 = µ a, x2 = µ q , x3 = µaq , x4 = µq. (4.23)

The quiver adjacency matrix reads:

 0 −1 0 −1   −1 −1 0 −1  C =   . (4.24)  0 0 1 0  −1 −1 0 0

This form will be useful for obtaining quivers for all T (2,2p+1) torus knots, but we can shift the framing by 1 to have only the non-negative entries:

 1 0 1 0   0 0 1 0  C =   ⇐⇒ Q = . (4.25)  1 1 2 1  0 0 1 1

For the trefoil the DT spectrum is inﬁnite, but we can ﬁnd the invariants order by order in xi. Let us focus on the linear one: Ω(x, q) F (x, q) = Exp Q 1 − q   4 di dn sn 2 2 x Ωd,sx q X 1/2 −d2+d3−2d1d2−2d1d4−2d2d4 Y i X (−q ) = exp  n  (4.26) (q; q)di 1 − q d1,...,d4≥0 i=1 n,d,s 1/2 −1 1/2 P d s x + (−q ) x + (−q )x + x Ωd,sx q 1 + 1 2 3 4 + O(x2) = 1 + d,s + O(x2). 1 − q i 1 − q i In consequence

Ω(1,0,0,0),0 = 1, Ω(0,1,0,0),−1/2 = −1, Ω(0,0,1,0),1/2 = −1, Ω(0,0,0,1),0 = 1. (4.27)

2 2 3/2 −1/2 Applying the change of variables x1 = µ a, x2 = µ q , x3 = µaq , x4 = µq, we obtain the (ﬁrst four) knot complement analogs of LMOV invariants for the trefoil knot complement:

N(µ, a, q) = Ω(x, q)|(4.23) ( N1,0,1 = 1,N2,0,1 = −1, X r i j 2 2 =⇒ (4.28) Nr,i,jµ a q = µ a − µ q − µa + µq N1,1,0 = −1,N2,1,0 = 1. r,i,j

– 13 – Ω and N count the BPS states in 3d N = 2 theories T [Q ] and T [M ] respectively. The d,s r,i,j M31 31 structure of the ﬁrst theory can be read oﬀ from the semiclassical limit of FQ(x, q):

4 d Z q i =yi Y dyi 1 0 FQ(x, q) −→ exp WfT [QM ](x, y) + O(~ ) , (4.29) →0 y 31 ~ i=1 i ~ 4 X WfT [QM ](x, y) = Li2 (yi) + log x1 log y1 + log(−x2) log y2 + log(−x3) log y3 + log x4 log y4 31 i=1 1 1 − log y log y − log y log y + log y log y − log y log y − log y log y . 1 2 2 2 2 2 3 3 1 4 2 4 Using the dictionary (2.13-2.14), we can see that the gauge group of T [Q ] is U(1)(1) × U(1)(2) × M31 (3) (4) (j) U(1) × U(1) and we have four chiral fields φi with charges δij under U(1) . We also have gauge eff Chern-Simons couplings determined by the quiver adjacency matrix: κij = Cij, and Fayet-Iliopoulos couplings between each gauge symmetry and its dual topological symmetry. Comparing with [12], we can see that the structure of T [Q ] is different than T [Q ]. The moduli space of vacua of T [Q ] M31 L31 M31 is given by the zero locus of the following quiver A-polynomials:

−1 −1 −1 −1 −1 A1(x, y) = 1 − y1 − x1y2 y4 ,A2(x, y) = 1 − y2 + x2y1 y2 y4 , (4.30) −1 −1 A3(x, y) = 1 − y3 + x3y3,A4(x, y) = 1 − y4 − x4y1 y2 , which are the classical limits of the annihilators of FQ(x, q):

ˆ −1 −1 ˆ −1/2 −1 −1 −1 A1(xˆ, yˆ, q) = 1 − yˆ1 − xˆ1yˆ2 yˆ4 , A2(xˆ, yˆ, q) = 1 − yˆ2 + q xˆ2yˆ1 yˆ2 yˆ4 , (4.31) ˆ 1/2 ˆ −1 −1 A3(xˆ, yˆ, q) = 1 − yˆ3 + q xˆ3yˆ3, A4(xˆ, yˆ, q) = 1 − yˆ4 − xˆ4yˆ1 yˆ2 .

The structure of the theory T [M31 ] is encoded in the semiclassical limit of F31 (µ, a, q): Z dz 1 F (µ, a, q) → exp W (µ, a, z) + O( 0) , 31 fT [M31 ] ~ ~→0 z ~ (4.32) W (µ, a, z) = log µ log z − Li (µ) + Li (µz−1) + Li (a) − Li (az) + Li (z). fT [M31 ] 2 2 2 2 2 Extremalisation with respect to z and the introduction of the variable λ dual to µ leads to

 ∂W (µ,a,z)  ∗ −1 ∗ fT [M3 ] µ(1−µ(z ) )(1−az )  1  0 = ∂z 1 = (1−z∗) =⇒ ∗ (4.33) ∂WfT [M3 ](µ,a,z) z (1−µ)  1 λ = ∗ −1 . log λ = ∂ log µ 1−µ(z ) Eliminating z∗ we obtain the zero locus of the A-polynomial

3 2 3 2 4 2 A31 (µ, λ, a) = (µ − 1)µ − 1 − µ + 2(1 − a)µ − aµ + a µ λ + (1 − aµ)λ , (4.34) which agrees with [16] after taking into account the rescaling of λ by a−1.

4.3 Cinquefoil knot complement For more complicated knot complements all expressions become more involved, so we focus on ﬁnding the corresponding quivers, having in mind that the analysis of BPS states and 3d N = 2 theories can be done analogously to the unknot and trefoil complement case. In this section we concentrate on (2,5) (2,2p+1) the 51 = T knot complement with a plan of generalisation to all T torus knot complements.

– 14 – We start from the formula for GM invariant given in [43]:

(aq−1; q) (µ; q−1) X k1+2k2 (k1+k2)−k1k2 k1 k1 k1 F51 (µ, a, q) = µ q , (4.35) (q; q)k1 k2 0≤k2≤k1 where we use the q-binomial: k (q; q) = k . (4.36) l (q; q)l(q; q)k−l Following the steps from section 4.2, we obtain

k 1 −2k k +(k −l )2−k2+α2+γ2+2γ l X X X 1/2 1 2 1 1 1 1 1 1 1 F51 (µ, a, q) = −q

0≤k2≤k1 l1=0 α1+β1=l1 γ1+δ1=k1−l1 k +2k +l α +γ k +k + 1 l − 3 (α +γ ) µ 1 2 1 a 1 1 q 1 2 2 1 2 1 1 k × 1 . (4.37) (q; q)α1 (q; q)β1 (q; q)γ1 (q; q)δ1 k2 Now we use the formula

n2 n1 X m1 n1 − m1 = q(m1−m2)(n2−m2) (4.38) n2 m2 n2 − m2 m2=0 in two iterations:

k2 k2 k1 X l1 k1 − l1 X X α1 β1 = q(l1−l2)(k2−l2) = q(l1−l2)(k2−l2) q(α1−α2)(l2−α2) k2 l2 k2 − l2 α2 β2 l2=0 l2=0 α2+β2=l2 X γ1 δ1 × q(γ1−γ2)(k2−l2−γ2) (4.39) γ2 δ2 γ2+δ2=k2−l2 to get

k 1 −2k k +(k −l )2−k2+α2+γ2+2γ l X X X 1/2 1 2 1 1 1 1 1 1 1 F51 (µ, a, q) = −q (4.40)

0≤k2≤k1 l1=0 α1+β1=l1 γ1+δ1=k1−l1

k2 X X 2(l1−l2)(k2−l2)+2(α1−α2)(l2−α2)+2(γ1−γ2)(k2−l2−γ2) −q1/2

l2=0 α2+β2=l2 γ2+δ2=k2−l2 k +2k +l α +γ k +k + 1 l − 3 (α +γ ) µ 1 2 1 a 1 1 q 1 2 2 1 2 1 1 × . (q; q)α1−α2 (q; q)β1−β2 (q; q)γ1−γ2 (q; q)δ1−δ2 (q; q)α2 (q; q)β2 (q; q)γ2 (q; q)δ2 The change of variables

α1 =d1 + d5, β1 =d2 + d6, γ1 =d3 + d7, δ1 =d4 + d8,

α2 =d5, β2 =d6, γ2 =d7, δ2 =d8 (4.41)

l1 =(d1 + d5) + (d2 + d6), k1 =(d1 + d5) + (d2 + d6) + (d3 + d7) + (d4 + d8)

l2 =d5 + d6, k2 =d5 + d6 + d7 + d8

– 15 – leads to

F (µ, a, q) = F (x, q)| n a l , 51 Q xi=µ i a i q i

8 di P8 (4.42) X 1/2 Cij didj Y xi FQ(x, q) = (−q ) i,j=1 , (q; q)di d1,...,d8≥0 i=1

ni ai li with xi = µ a q given by 2 2 3/2 −1/2 x1 =µ a, x2 =µ q , x3 =µaq , x4 =µq, (4.43) 4 4 5/2 3 1/2 3 2 x5 =µ aq, x6 =µ q , x7 =µ aq , x8 =µ q , and  0 −1 0 −1 −1 −1 0 −1   −1 −1 0 −1 −2 −2 0 −1     0 0 1 0 −1 −1 0 0     −1 −1 0 0 −2 −2 −1 −1    C =   . (4.44)  −1 −2 −1 −2 −2 −3 −2 −3     −1 −2 −1 −2 −3 −3 −2 −3     0 0 0 −1 −2 −2 −1 −2  −1 −1 0 −1 −3 −3 −2 −2 This is the adjacency matrix corresponding to the cinquefoil knot complement. Performed computations and the matrix itself may seem invloved, but they can be considered as a simple generalisation of the trefoil knot complement case. The most important is the fact that the change of variables (4.41) can be obtained from (4.21). For indices k1,l1,α1,β1,γ1,δ1, we just substitute di 7→ di + di+4. In case of k2,l2,α2,β2,γ2,δ2, we start from k1,l1,α1,β1,γ1,δ1 and shift di 7→ di+4. From the point of view of the quiver adjacency matrix, this corresponds to copying the 4 × 4 matrix C31 to ﬁll 8 × 8 entries: C C [C ] 7→ 31 31 . (4.45) 31 CT C 31 31 k1 We still need to include q−k1k2 and the q-binomial which, according to (4.39) and (4.41), con- k2 P C d d tribute to (−q1/2) i,j ij i j by

2(l1−l2)(k2−l2)+2(α1−α2)(l2−α2)+2(γ1−γ2)(k2−l2−γ2) 2(d1+d2)(d7+d8)+2d1d6+2d3d8 −q1/2 = −q1/2 (4.46)

−2k1k2 −2(d1+d2+...+d8)(d5+d6+d7+d8) −q1/2 = −q1/2 .

In consequence we have to modify (4.45) to D0 R0 [C31 ] 7→ T , (4.47) R0 D1 where  2 2 2 2   1 0 0 0   2 2 2 2   1 1 0 0  D0 = C31 ,D1 = D0 −   ,R0 = D0 −   . (4.48)  2 2 2 2   1 1 1 0  2 2 2 2 1 1 1 1 We can see that this is consistent with (4.44). The change of variables (4.43) is a direct consequence 2k2 k2 of (4.23) and (4.41), taking into account that in F51 (µ, a, q) we have an extra factor of µ q .

– 16 – 4.4 General T (2,2p+1) torus knot complements Now we are ready to move to the general case using recursion. We assume that we know the quiver for T (2,2p+1) torus knot complement and look for the quiver for T (2,2(p+1)+1) = T (2,2p+3). The general formula is given by [43]

X Pp 2(k1+...+kp)−k1 (k1+k2+...+kp)− i=2 ki−1ki FT (2,2p+1) (µ, a, q) = µ q

0≤kp≤...≤k1 (4.49) (aq−1; q) (µ; q−1) k k × k1 k1 1 ··· p−1 , (q; q)k1 k2 kp so when we go from p to p + 1, the summand in FT (2,2p+1) is multiplied by

k µ2kp+1 qkp+1 (−q1/2)2kpkp+1 p . (4.50) kp+1

This means that we copy the rightmost (also downmost and diagonal) 4 × 4 matrix blocks right

(respectively down and further on the diagonal), exactly like we did for C31 7→ C51 , with the same modiﬁcation as in (4.47-4.48). Therefore the quiver for T (2,2p+1) torus knot is given by   D0 R0 R0 ...R0 R0  RT D R ...R R   0 1 1 1 1   T T   R0 R1 D2 ...R2 R2  C =  . . . . .  , (4.51)  ......   ......   T T T   R0 R1 R2 ...Dp−2 Rp−2  T T T T R0 R1 R2 ...Rp−2 Dp−1 where

 0 −1 0 −1   2 2 2 2   −2n −2n − 1 −2n −2n − 1   −1 −1 0 −1   2 2 2 2   −2n − 1 −2n − 1 −2n −2n − 1  Dn =   − n   =   ,  0 0 1 0   2 2 2 2   −2n −2n −2n + 1 −2n  −1 −1 0 0 2 2 2 2 −2n − 1 −2n − 1 −2n −2n (4.52)  1 0 0 0   −2n − 1 −2n − 1 −2n −2n − 1   1 1 0 0   −2n − 2 −2n − 2 −2n −2n − 1  Rn = Dn −   =   .  1 1 1 0   −2n − 1 −2n − 1 −2n −2n  1 1 1 1 −2n − 2 −2n − 2 −2n − 1 −2n − 1

We can see that, similarly to the standard knots-quivers correspondence [1,2], increasing p does not change any previously determined entries of the matrix, so it makes sense to consider the limit p → ∞ leading to an inﬁnite quiver. ni ai li (2,2p+1) The change of variables xi = µ a q for T torus knot complement comes directly from

– 17 – the generalisation of (4.23, 4.43) according to (4.50) and is given by X nidi =2(d1 + d2) + 1(d3 + d4) i

+4(d5 + d6) + 3(d7 + d8) . .

+2p(d4p−3 + d4p−2) + (2p − 1)(d4p−1 + d4p), (4.53) X aidi =d1 + d3 i

+d5 + d7 . .

+d4p−3 + d4p−1,

X 3 1 l d =0d + d − d + 1d i i 1 2 2 2 3 4 i 5 1 +1d + d + d + 2d 5 2 6 2 7 8 . . 2p + 1 2p − 3 +(p − 1)d + d + d + pd . 4p−3 2 4p−2 2 4p−1 4p We can summarise (4.51-4.53) in the concise form of the correspondence between T (2,2p+1) torus knot complements and quivers:

F (2,2p+1) (µ, a, q) = F (x, q)| n a l T Q xi=µ i a i q i

4p d 4p i (4.54) X 1/2 P C d d Y xi = (−q ) i,j=1 ij i j (q; q)di d1,...,d4p≥0 i=1 n a l xi=µ i a i q i .

5 t-deformation

In this section we propose a t-deformation of the results from sections3-4 using FK (µ, a, q, t) given in [43]. These results are based on the t-deformation of GM invariants following the generalisation of HOMFLY-PT polynomials to superpolynomials, which is reﬂected in equations (2.25) and (2.26).

5.1 Quivers The connection between GM invariants and superpolynomials [43] suggest the following t-deformed correspondence:

3 Conjecture 2 For a given knot complement MK = S \K, the GM invariant FK (µ, a, q, t) can be written in the form

m n d a d l d t d Pm i i i i i i i i X 1/2 Cij didj Y µ a q (−t) FK (µ, a, q, t) = (−q ) i,j=1 , (5.1) (q; q)di d1,...,dm≥0 i=1

– 18 – where C is a symmetric m × m matrix and ni, ai, li, ti are ﬁxed integers. In consequence, there exist a quiver Q which adjacency matrix is equal to C and motivic generating series

m di Pm X 1/2 Cij didj Y xi FQ(x, q) = (−q ) i,j=1 (5.2) (q; q)di d1,...,dm≥0 i=1

ni ai li ti reduces to the GM invariant after the change of variables xi = µ a q (−t) :

F (µ, a, q, t) = F (x, q)| n a l t . (5.3) K Q xi=µ i a i q i (−t) i Let us look at examples, starting from the unknot complement (in unreduced normalisation). Basing on [43], we have

(µq; q)∞ F01 (µ, a, q, t) = 3 . (5.4) (−µat ; q)∞ We immediately see that the quiver is the same as in (4.3), but the change of variables is given by

1/2 3 x1 = µq , x2 = −µat . (5.5)

For all T (2,2p+1) torus knots (including trefoil and cinquefoil) the situation is similar. The general formula (in reduced normalisation) [43]

X Pp 2(k1+...+kp)−k1 (k1+k2+...+kp)− i=2 ki−1ki 2(k1+...+kp) FT (2,2p+1) (µ, a, q, t) = µ q t

0≤kp≤...≤k1 (5.6) (−aq−1t; q) (µ; q−1) k k × k1 k1 1 ··· p−1 (q; q)k1 k2 kp leads to the quiver (4.51-4.52), the only diﬀerence with respect to section 4.4 lies in the change of ni ai li ti variables. Now xi = µ a q (−t) , where X tidi =3(d1 + d3) + 2(d2 + d4) i +5(d + d ) + 4(d + d ) 5 7 6 8 (5.7) . .

+(2p + 1)(d4p−3 + d4p−1) + 2p(d4p−2 + d4p).

For the trefoil it leads to the change of variables

2 3 2 3/2 2 −1/2 3 2 x1 = −µ at , x2 = µ q t , x3 = −µaq t , x4 = µqt . (5.8)

Comparing with the quiver adjacency matrix (4.24), we can see that

ti 6= Cii, (5.9) in contrary to the standard knots-quivers correspondence [1,2].

– 19 – 5.2 BPS states and 3d N = 2 eﬀective theories In section 3.2 we constructed knot complement analogs of LMOV invariants basing on the DT in- ni ai li ti variants. Knowing the form of the t-deformed change of variables xi = µ a q (−t) , we can deﬁne the t-deformed knot complement analogs of LMOV invariants:

X r i j k N(µ, a, q, t) = N µ a q (−t) = Ω(x, q)| n a l t , (5.10) r,i,j,k xi=µ i a i q i (−t) i r,i,j,k which implies   rn in jn kn N(µ, a, q, t) X Nr,i,j,kµ a q (−t) F (µ, a, q, t) =Exp = exp K 1 − q  1 − q  n,r,i,j (5.11) Y r i j+l k−Nr,i,j,k Y r i j k −Nr,i,j,k = 1 − µ a q (−t) = (µ a q (−t) ; q)∞ . r,i,j,k,l r,i,j,k

Since DT invariants are integer numbers, so are Nr,i,j,k. In case of the unknot complement, the GM invariant given in terms of inﬁnite q-Pochhammers

(µq; q)∞ F01 (µ, a, q, t) = 3 (5.12) (µa(−t) ; q)∞ immediately leads to

N1,0,1,0 = −1,N1,1,0,3 = 1. (5.13)

From the point of view of 3d N = 2 eﬀective theory T [MK ], the t-deformation of the semiclassical limit (3.8) is given by Z Y dzi 1 0 FK (µ, a, q, t) → exp WfT [MK ](µ, a, t, zi) + O(~ ) (5.14) →0 z ~ i i ~ and can be interpreted as the introduction of the global R-symmetry U(1)F with fugacity (−t), associated to rotations in the directions normal to M5-brane inside R4 (for more details see [12, 14, 16, 18]).

Then the moduli space of vacua of T [MK ], obtained by the extremalisation of WfT [MK ] with respect to zi and the introduction of variable λ dual to µ, is described by the graph of the super-A-polynomial [16]:

∂Wfeﬀ (µ, a, t) T [MK ] = log λ ⇔ A(µ, λ, a, t) = 0. (5.15) ∂ log µ

A(µ, λ, a, q, t) is the classical limit of the quantum super-A-polynomial, which annihilates superpolynomials (see [14, 16, 18, 48]).

6 Future directions

We conclude with a brief description of interesting directions for future research:

• It would be desirable to understand the relation between quivers found in this paper and in [1,2]. We could see that for the unknot and the unknot complement quivers are the same, but the quiver corresponding to the trefoil has 3 nodes, whereas the one associated to the trefoil complement has 4 nodes. The core of this issue probably lies in the relation between HOMFLY-PT

– 20 – polynomials and GM invariants. It seems that in general it should be some transformation anal- ogous to the Fourier transform, which for some cases (including T (2,2p+1) torus knots) reduces to the substitution µ = qr. Investigating the relation between the standard and new correspondence could help in ﬁnding quivers for other knot complements. Probably it would also provide a new insight to the duality web of associated 3d N = 2 theories studied in [12, 13].

• Since GPPV invariants exhibit peculiar modularity properties and are related to logarithmic con- formal ﬁeld theories, it would be interesting to perform a study of these aspects for GM invariants in the context of the correspondence with quivers.

• GM invariants still lack a proper mathematical deﬁnition and a proof that they are ineed topological invariants of knot complements. These goals seem ambitious, but achieving them would potentially allow to properly state and prove the conjectures proposed in this work.

• One can look for suggestions and constraints on GM invariants coming from the existence of a corresponding quiver. Such approach applied to the case of the standard knots-quivers corre-

spondence enabled ﬁnding the formulas for HOMFLY-PT polynomials PK,r(a, q) for 62 and 63 knots [2].

• The form and meaning of the t-deformation proposed in [43] and applied here are still uncertain. Better understanding of this aspect is crucial for the categoriﬁcation of 3-manifold invariants, which was the main motivation of introducing GPPV and GM invariants.

Acknowledgements

I would like to thank Tobias Ekholm, Angus Gruen, Sergei Gukov, Pietro Longhi, Sunghuk Park, and Piotr Sułkowski for insightful discussions. My work is supported by the Polish Ministry of Science and Higher Education through its programme Mobility Plus (decision no. 1667/MOB/V/2017/0).

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– 24 –