3-Manifolds and VOA Characters

By studying the properties of $q$-series $\widehat Z$-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for $\widehat{Z}$-invariants leads to many infinite families of new fermionic formulae for VOA characters.


Introduction and Summary of Results
The main goal of this paper is to build a new bridge between different areas of mathematics and mathematical physics. Namely, we explore the relation between characters of vertex operator algebras (VOAs), on the one hand, and q-series invariants of manifolds in low-dimensional topology, on the other hand.
A prototypical example of such a duality (or, correspondence) that goes back to the mid-90s involves Vafa-Witten invariants of 4-manifolds [1]. Starting with the seminal work of Nakajima [2], these q-series invariants of 4-manifolds can be interpreted as VOA characters, see e.g. [3] for a recent account and identification of cutting-and-gluing operations on both sides of the correspondence. The Vafa-Witten invariants have the following general form: Namely, they depend on the choice of the 4-manifold X, a compact Lie group G, the variable τ with values in the upper-half plane, τ ∈ H, and the extra "decoration" data on the 4-manifold b ∈ H 2 (X; π 1 (G)). The physical definition of the invariants exists, at least in principle, for general 4-manifolds. However, the corresponding moduli spaces turn out to be non-compact and, as a result, rigorous mathematical definitions are currently limited to Kähler surfaces, see e.g. [4][5][6] for some recent work in this direction. The correspondence explored in this paper can be considered as a 3-dimensional analogue of the long-studied relation between Vafa-Witten invariants and VOA characters [7]. With its roots in the 3d-3d correspondence, the 3-manifold analogue of the Vafa-Witten q-series invariant, called Z G b (X; τ ) 1 , also depends on the choice of gauge group G, 3-manifold X, variable q = e 2πiτ in the unit disk, |q| < 1, and extra data b given by the generalised Spin c -structure. In part due to the fact that topology of 3-manifolds is much simpler than the topology of 4-manifolds, the qseries invariants in dimension three are easier to define and compute. For example, Z G b (X, τ ) can be defined [8] via Rozansky-Witten theory [9] based on affine Grassmannians, and Rozansky-Witten theory has rigorous mathematical definitions due to Kapranov [10] and Kontsevich [11].

(1.2)
These techniques provide much more data for the explicit form of Z G b (X; τ ) than we can hope to match with VOA characters. In fact, starting with rather simple classes of surgeries, such as Seifert 3-manifolds or plumbed 3-manifolds, our main goal will be to identify the Z with characters of vertex algebras, when overall powers of q and η(τ ) are ignored. Compared to the four-dimensional version mentioned earlier, one interesting feature of this relation for 3-manifolds is that in most examples the vertex algebra is logarithmic. We will refer to them as logarithmic vertex operator algebras (log VOAs). In particular, the algebras we discuss in the paper all have irreducible but indecomposable modules. For a review, see for example [16][17][18].
Specifically, the results of this paper mainly focus on Seifert manifolds with three or four exceptional fibers. Apart from studying the relation (1.3) between Z G b (X) and VOA characters, we also explore a more refined relation among the integrand of the contour integral leading to Z G b (X), and the "triplet" type VOA, containing the "singlet" type VOA as a subalgebra. Namely, we have and χ b (τ, ξ ) is given by characters of certain triplet vertex algebras. The results are summarised in Table 1. We say a Seifert manifold with N exceptional fibers, with Seifert data X Γ = M (b; {q i /p i } i=1,...,N ), is pseudo-spherical if 1 e p i ∈ Z for all i = 1, . . . , N, (1.5) where e = b + k q k p k is the orbifold Euler characteristic, which is specifically always the case when X Γ is an integral homological sphere. In the present work we focus on the cases of negative Seifert manifolds, namely those with e < 0.
To summarize, in the present paper we show the following.
• Let X be any pseudo-spherical negative Seifert manifold with three exceptional fibers, G be any choice of simply-laced Lie group, and b be any choice of the generalised Spin c structure. Then the integrandχ b of the three-manifold invariant Z G b (X, τ ), up to overall powers of η(τ ) and q, is given by a virtual generalised character of the so-called triplet vertex algebra corresponding to the Lie algebra g, reviewed in §3.1.
As a consequence, again up to overall powers of η(τ ) and q, the three-manifold invariant Z G b (X; τ ) is given by a virtual generalised character 3 of the so-called singlet vertex algebras corresponding to the Lie algebra g.
The above corresponds to the entries with in Table 1 and precised in Theorem 4.4 and Corollary 4.5.
• Let X be any negative Seifert manifold with three exceptional fibers, G be any choice of Lie group, and b be any choice of the generalised Spin c structure. Then there is a sum over the generalised Spin c structures including b, such that the corresponding sum of the integrand χ b (τ, ξ ), up to an overall power of η(τ ) and q as well as a rescaling of τ , is given by a virtual generalised character of the so-called triplet vertex algebras corresponding to the Lie algebra g.
As a consequence, again up to an overall power of η(τ ) and q as well as a rescaling of τ , the corresponding sum of three-manifold invariants Z G b (X; τ ) is given by a virtual generalised character of the so-called singlet vertex algebras corresponding to the Lie algebra g.
The above corresponds to the entries with in Table 1 and precised in Theorem 4.3.
• Let X be any negative Seifert manifold with four exceptional fibers that is an integral homological sphere, G = SU (2), and b be any choice of the Spin c structure. Then the integrand χ b (τ, ξ ) of the three-manifold invariant Z G b (X; τ ), up to overall multiplicative constants and powers of η(τ ) and q, is given by an integral linear combination of generalised characters of the so-called (p, p ) triplet vertex algebras corresponding to the Lie algebra g, reviewed in §3.2.
As a consequence, again up to overall powers of η(τ ) and q, three-manifold invariant Z G b (X; τ ) is given by a virtual generalised character of the so-called (p, p ) singlet vertex algebras.
The precise version of the above is the content of Theorem 4.7 and Corollary 4.8.
• Following the consideration of §4 of [19], we also investigate the effect of including Wilson operators in the theory on the relation between the BPS partition function Z and the characters of log VOAs. We found that the relation continues to exist in the presence of Wilson operators, but gets modified in ways that depend on to which node the Wilson operator is associated to.
Upon the including of a Wilson operator associated with an end node, Theorem 4.3 and as a result Theorem 4.4 and Corollary 4.6 continue to hold, as well as Theorem 4.7, with a modification of parameters that is given by (4.71). When the Wilson operator is associated with the central node, Corollary 4.5 gets modified into (4.77), and similarly for Corollary 4.6. This "shifting" phenomenon has been observed for the special Lens space example in [19].
The relation undergoes a more drastic modification when the added Wilson operator is associated to an intermediate node (a vertex in the plumbing graph with two other vertices connected to it). Namely, the statements in Corollaries 4.5-4.6 are modified into (4.85), and homological blocks are no longer given by a virtual VOA character up to an overall multiplicative constant and powers of η(τ ) and q. Instead, they are given by a virtual generalised character of a log VOA, each modified by an individual rational q-power, up to an overall multiplicative constant and powers of η(τ ).
• Until recently, various ways 4 of computing Z-invariants would typically produce an explicit form of the q-series up to any desired order in q, but not a closed form expression. This makes the study of modular properties and other related questions quite challenging in general. Recent insights from enumerative geometry and the knot-quiver correspondence provide a new and surprising solution to this problem, which we discuss in §5 and expect to be a powerful tool in the future work on Z-invariants. In particular, the closed form expressions produced by a version of the knot-quiver correspondence are perfectly suited for identifying the spectrum of quasiparticles in integrable massive deformations of the 2d logarithmic CFTs. . See equation (4.6). S w,w 1 ,w 2 ,...,w N ; b The set given by { 0 |( 0, −w1( ρ), · · · , −wN ( ρ), 0, · · · , 0) ∈ ΓM,G + w( b)}. See (4.18).

Various Log VOAs and Their Characters
In this section we will briefly review the logarithmic vertex operator algebras relevant for our study of homological blocks and in particular their characters.
We take G to be a simply-laced Lie group, use g to denote the associated Lie algebra and let Λ = Λ G be the corresponding root lattice. We will denote by Φ s = { α i } a set of simple roots and { ω i } the corresponding fundamental weights, Φ ± the set of positive resp. negative roots, and by the set of dominant integral weights, where ·, · is a quadratic form given by the Cartan matrix of G. For x ∈ C ⊗ Z Λ, we define the norm | x| 2 := x, x as usual.
In §3.1 we review the VOAs log-VΛ(m) and log-V 0 Λ (m), associated with g. They are also often referred to as the triplet and singlet (1, m) log VOAs, respectively. In §3.2 we review the VOAs log-VΛ(p, p ) and log-V 0 Λ (p, p ), associated with g = A 1 . They are often referred to as the triplet and singlet (p, p ) log VOAs, respectively.

log-VΛ(m)
The logarithmic vertex operator algebra log-VΛ A 1 (m), also known as the triplet model and sometimes denoted as W(m) in the literature, was first constructed in [20,21]. The analogous algebras are later defined for arbitrary simple-laced semisimple Lie algebra g in [22]. We mainly follow [22,23].
Let ϕ α i (z) be the chiral scalar field associated to the root α i ∈ Φ s , satisfying the following operator product expansion In terms of the mode expansion we modify the commutation rule of the zero modes such that The vertex operators are defined as The lattice VOA VΛ can be constructed directly from these fields. The irreducible modules of this VOA are specified by an element λ ∈Λ ∨ /Λ, whereΛ := √ mΛ. It is convenient to decompose λ into the following two parts [22]: where λ ∈ Λ ∨ /Λ and The VΛ irreducible modules can be written in terms of the Fock module F λ , corresponding to the vertex operator V λ (z), as

Now we choose an energy momentum tensor
ρ is the Weyl vector, and C ij denotes the ij-entry of the inverse of the Cartan matrix. The Virasoro algebra has thus central charge and the vertex operator V λ (z) with λ ∈Λ ∨ has conformal dimension The log-VΛ(m) algebra can be described as a subalgebra of VΛ by considering the intersection of the kernels of the screening charges in the original lattice VOA. Let be the screening operators. They commute with the energy momentum tensor and in addition e i commutes with f j for i, j = 1, . . . , rankG. The log-VΛ(m) algebra is the vertex operator subalgebra of VΛ defined by [22,23] log-VΛ(m) : Instead of taking the kernel over the whole lattice algebra VΛ, in order to define the log-V 0 Λ (m) "singlet" algebra, we restrict to the charge zero subalgebra F 0

Characters
As described in the previous section, a module X λ for log-VΛ(m) is specified by an element λ ∈Λ ∨ /Λ, parametrized as in (3.7). Let be the usual Weyl denominator of the Lie algebra g, where l(w) is the length of w in the Weyl group W . With an abuse of notation of using λ below to denote any arbitrary representative of the λ ∈ Λ ∨ /Λ, the character of X λ is given as follows [22]: Note that, to go from the first line to the second line in (3.15), we have used w( λ) ≡ λ mod Λ if λ ∈ Λ ∨ , and the fact that: if ρ + λ lies on the boundary of a Weyl chamber. We have also used the Weyl character formula to write the multiplicity function dim(V λ ( β)) in a highest weight module of the Lie algebra g: For later use, we will also introduce the generalised characters, given by (3.15) but with µ ∈Λ ∨ lying outside the range indicated in (3.8).
Taking ξ = 0, we have (3.18) Another way to view the log-VΛ(m) characters in (3.15) is as a generating function for the singlet characters. Assuming that the log-VΛ(m) module X λ is completely reducible as the module for the corresponding singlet model (cf §5 of [24]), we can write the log-VΛ(m) character in terms of the singlet characters χ 0 λ , labelled by λ ∈Λ ∨ , as where we have written λ = √ m λ + µ analogously to (3.7). In particular, the corresponding atypical singlet characters then read In the above, we have used "CT x (f (x))" to denote the "x 0 (constant) terms of the polynomial f (x) in x". In what follows we will often use the notation ξ = i ξ i α i , z i := e ξ, ω i = e ξ i , and hence e ξ, α = α i ∈Φs z α i , α i , and denote the corresponding vector by z. In (3.20), we say the left hand side is a generalised character of log-V 0 Λ (m) when χ λ (τ, ξ) is a generalised character of log-VΛ(m).
where ξ = ξ α and z = e ξ , and the log-VΛ (m) modules are labelled by where we use α to denote the simple root. Their characters are given by (3.23) The corresponding singlet characters are (3.24) Taking the z-constant term in the first equation in (3.23) we obtain is the false theta function [25]. Namely, the above singlet characters are given by the false theta functions when multiplied by the eta function, and differ from the false theta function by a finite polynomial in q in the case of generalised characters, for which s does not lie in the range s = 1, 2, . . . , m.
Interestingly, note that terms of the form q c η(τ ) also have the interpretation as a character of the typical module character of the singlet (1, m) model log-V 0 Λ (m). As a result, one can interpret the above identity (3.25) as the fact that a generalised SU (2) singlet character can always be expressed as an integral linear combination of the typical and atypical module characters. We will see in (3.33) a somewhat similar phenomenon for the case G = SU (3).

log-VΛ(p, p )
Apart from the log-VΛ(m) algebra, often referred to as the (1, m) log VOA, one can consider a more general family of algebras. They are often referred to as the (p, p ) log VOA, labelled by two coprime integers p and p , and their definition reduces to the one for log-VΛ(m) when setting p = 1, p = m. In the following we will focus on the case when Λ = Λ A 1 is given by A 1 root lattice. In this section, we mostly follow [26]. 5 See also [27][28][29][30].
2pp Z, and consider the lattice VOA VΛ. It will turn out that the algebra log-VΛ(p, p ) is a subalgebra of VΛ. For the rest of this section we restrict to the case of Λ = A 1 . In this case we define a general vertex operator of VΛ(p, p ) as, 5 Note: our definition of ϕ differs by √ 2 from that of [26].
The lattice VOA VΛ has 2pp irreducible modules Y ± r,s for 1 ≤ r ≤ p and 1 ≤ s ≤ p . They can be decomposed into Fock modules F r,s;n corresponding to the vertex operator V r,s;n (z) as follows: where Y + p−1,p −1 is the vacuum module, and Let F 0 := F 1,1;0 . Now we define and choose an energy momentum tensor such that its modes span a Virasoro algebra with The general vertex operators V r,s;n have conformal dimension ∆ r,s;n := (ps − p r + pp n) 2 − (p − p ) 2 4pp with respect to this choice of T (z). Finally, let J r,s;n denote the irreducible Virasoro module of highest weight ∆ r,s;n . In order to define the log-VΛ(p, p ) VOA, we can start with the screening operators which commute with the energy-momentum tensor, [e + , T (z)] = [f − , T (z)] = 0. Let where 1 ≤ r ≤ p − 1 and 1 ≤ s ≤ p − 1. Then the log-VΛ(p, p ) VOA is defined to be the subalgebra of VΛ with underlying vector space K + 1,1 . It is strongly generated by the energy momentum tensor T (z) and two Virasoro primaries W ± (z) of conformal dimension (2p − 1)(2p − 1). In terms of screening charges and vertex operators, these primaries are given by (3.36) There are 2pp + 1 2 (p − 1)(p − 1) irreducible modules of the log-VΛ(p, p ) VOA. These come in two categories: These modules for the (p, p ) Virasoro minimal model are also modules of the log-VΛ(p, p ) VOA and are annihilated by the maximal VOA ideal.
• 2pp number of irreducible modules X ± r,s for 1 ≤ r ≤ p and 1 ≤ s ≤ p . These can be described as The log-VΛ(p, p ) VOA admits an sl(2, C) action which commutes with the Virasoro algebra generated by T (z) and the currents W ± (z) are highest and lowestweight components of an sl(2, C) triplet. This is where the name "triplet algebra" derives from. As Virasoro and sl(2, C) bimodules, the X ± r,s decompose as and where n is the n-dimensional irreducible representation of sl(2, C). The J r,s are sl(2, C) singlets. One can also define the closely related log-V 0 Λ (p, p ) VOA (also called the (p, p ) singlet model) as a subalgebra of F 0 via (see, e.g., [29])

Characters
From the discussion in the previous section, one can compute the characters ch ± r,s (τ, ξ) := Tr χ ± r,s q L 0 −c/24 z J 0 = n≥0 chJ r,p −s;2n+1 (τ )ch 2n+1 (ξ) n≥1 chJ r,p −s;2n (τ )ch 2n (ξ) Beginning with the case of ch + r,s (τ, ξ), we have Now we use the identity for f (n) = 2n+1 (z) and g(k, n) = q ∆(n,k) for each of the four terms in (3.43) to rewrite this as where we have used the fact that Taking k → −k − 2 in the second and fourth terms, we can rewrite this as which, after plugging in the explicit forms of ∆ r,s;n and c(p, p ) and shifting k → k−1, finally yields whereμ r,s,1 := ps + p r ,μ r,s,2 := ps − p r. In the next section, we will also consider the generalised characters, namely functions defined as in (3.46) with r, s ∈ Z that are not necessarily in the range (3.40).
When considered as such generalised characters, we see that they have the symmetry property ch r,s = r s ch r r, ss for r , s ∈ {1, −1}. (3.48) A similar computation leads to a formula for the character ch − r,s (τ, ξ), which is given by where we have used that Note that the above characters can be expressed in terms of sums of theta functions and their derivatives [26]. Taking the z-constant term of (3.46) gives the corresponding singlet character . (3.52) Notice the relation between the above characters and Eichler integrals of theta functions. In [7] we propose that for four exceptional fibers the following building blocks Ξ m,r play a role analogous to that of the false theta functions in the case of Seifert manifolds with 3 exceptional fibers: where r ∈ Z/2m, satisfying Ξ m,r = Ξ m,−r .  which moreover coincides with Ξ pp ,ps+p r − Ξ pp ,ps−p r (τ ) for r and s in the range (3.40).

Spectral Curves for Vertex Algebras
When a log VOA has affine Kac-Moody symmetryĝ, its module V is naturally graded by this symmetry and the character χ V (τ, ξ) := Trq L 0 − c 24 z J is a function of z that takes values in the maximal torus of G, such that g = Lie(G). We wish to explore the z-dependence of characters in log VOAs, in particular q-difference operators that annihilate χ V (τ, ξ):Â χ V (τ, ξ) = 0 (3.59)

Example: the Triplet Algebra log-VΛ(m)
Recall that the character of the triplet (1, m) model log-VΛ(m) is (3.23): where we made explicit the dependence on the parameters m and s ∈ {1, . . . , m}.
We claim that χ 1−m 2 √ m α (τ, ξ) is annihilated by the following q-difference operator: whereẑ andŷ form the algebraŷẑ = qẑŷ, usually called the quantum torus. On a function f (τ, ξ) these operators act aŝ It is easy to see that they indeed satisfy the desired q-commutation relation. Let us sketch the derivation of (3.62). First, it is convenient to remove the denominator and introduce an auxiliary function: We then observe that it has a structure similar to the unknot (and, more generally, torus knots), and so as in [31] we make the following ansatz for the operator that annihilates F m,m (τ, ξ):Â with some rational function R(ξ, τ ) that needs to be determined. Then, it is easy to show thatÂ The final step in getting to (3.62) requires passing from F m,m to χ 1−m 2 √ m α (τ, ξ), which at the level of q-difference is achieved by conjugating withẑ −ẑ −1 : Multiplying byẑ −ẑ −1 q mẑ −q −mẑ−1 from the left, we get (3.62). In the classical limit q → 1, the quantum curve (3.62) becomes a hyperelliptic curve y m + y −m = z −2 + z 2 (3.68) It would be interesting to extend this calculation to other values of s and to more general logarithmic vertex algebras.

Fermionic forms of log VOA characters
Later, in section 5 we explain how connections between knot theory and physics (or, knot theory and enumerative geometry) can teach us useful lessons about the structure of the Z-invariants for many closed hyperbolic 3-manifolds. Until recently, this was the major obstacle in understanding the modular properties of Z b (X) for hyperbolic X and identifying vertex algebras dual to 3-manifolds in the sense of (1.3). By connecting knot theory to physics and enumerative geometry, this obstacle can be removed and one finds new avenues for exploring the modular properties of the BPS q-series invariants and connections to vertex algebras. Relegating a more complete account of these developments to section 5, here we briefly recall the relevant structure in the triplet vertex algebra [32].
Let us consider the logarithmic vertex operator algebra log-VΛ A 1 (m) discussed in §3.1. We are interested in the 2m irreducible representations whose characters are given in (3.23). In particular, we are interested in the linear combination of modules (in a notation consistent with (3.38)-(3.39)): (3.69) One may write characters of these modules in the "bosonic form," as in (3.23). Another way to write characters of these modules is via embedding the local chiral algebra of the (1, m) model into a larger algebra A(m): To obtain the algebra A(m), we first consider the sl(2, C) doublet of fields: where the operators a ± have the same conformal dimension (3p − 2)/4. The OPE of a ± has the form where each H n (w) has conformal dimension n. The algebra A(m) generated by these operators is graded by the weight lattice of sl(2, C): which can be viewed as the origin of the sl(2, C) symmetry in the triplet log VOA. In particular, this leads to the decomposition of the highest-weight irreducible modules 6 X 1,s generated from the vector |s, m , a ± − 3m−2s 4 +n |s, m = 0, s ∈ {1, · · · , m}, n ∈ N. (3.74) into Vir ⊕ sl(2, C) modules: (m). 6 These modules have conformal dimension s 2 −1 4m + 1−s 2 .
The fermionic form of characters then comes from the filtration on the graded algebra A(m). Relegating the details to [32], we reproduce here the resultant character formula for the irreducible module X 1,s , with the fugacity z set to z = 1: 6 · · · 6 · · · · · · · · · · · · · · · · · · · · · m − 1 m − 1 2 4 6 · · · 2(m − 1) Here, c denotes the central charge of the logarithmic (1, m) model and 0 in v s occurs s − 1 times. Next, we explain how many elements of this paper find their natural home in the framework of 3d supersymmetric quantum field theory, related to the study of 3-manifold invariants via 3d-3d correspondence.

Log VOAs and 3d N = 2 Theories
Two-dimensional logarithmic VOAs and CFTs are relevant to many physical phenomena, including quantum Hall effect (QHE) plateau phase transition, percolation, and self-avoiding walks [33]. Curiously, their characters arise from supersymmetric theories in one extra dimension, in a way akin to holography, namely as half-indices of 3d N = 2 theories with 2d (0, 2) boundary conditions [34].
The half-indices are basically 3d analogues of elliptic genera [35] in two-dimensional systems that count local operators inQ + -cohomology on the boundary of the 2d-3d combined system. Such combined systems naturally appear in the study of a 6d fivebrane theory partially twisted along a 4-manifold, especially in operations involving cutting and gluing [36]. The resulting 3d N = 2 theory is then topologically twisted along one of its directions and holomorphically twisted along two other directions. 7 The study of such partial and holomorphic twists in supersymmetric QFTs goes back to [37][38][39][40] and has been an area of active research in recent years.
There are several ways to formulate the half-index of the 2d-3d combined system. In radial quantization, counting local operators in QFT d requires surrounding such operators by a sphere S d−1 and studying the Hilbert space H(S d−1 ). In the case of 3d theory with 2d boundary, a local operator on the boundary is surrounded by a disk D 2 , and so the analogue of radial quantization involves taking the trace over H(D 2 ) or, equivalently, computing the partition function on where this way of writing S 1 × q D 2 reminds us that the result depends on complex structure τ = 1 2πi log q of the boundary torus T 2 = ∂ S 1 × D 2 . Sometimes, the S 1 × q D 2 partition function is also called K-theoretic vortex partition function (with Omega-background along D 2 q ). The basic ingredients of 3d N = 2 Lagrangian theories include two types of supermultiplets: chiral and vector. Similarly, there are two types of matter supermultiplets in 2d (0, 2) theories, Fermi and chiral, so that basic elements of 2d (0, 2) gauge theories are Fermi, chiral, and vector multiplets. Below we summarize their contribution to the index (3.77): • The contribution of a 2d (0, 2) Fermi multiplet to the elliptic genus and, hence, also to the index (3.77) is basically a theta-function, where x is the fugacity for the global U (1) symmetry and we also indicate which modes of the Fermi multiplet contribute to various terms. Half of this contribution, shown in red, is the contribution to (3.77) of a 3d N = 2 chiral multiplet with Dirichlet boundary conditions.
• The contribution of a 2d (0, 2) chiral multiplet to the elliptic genus and, hence, to the index (3.77) is the inverse theta-function, Shown in red is the contribution to (3.77) of a 3d N = 2 chiral multiplet with Neumann boundary conditions.
• Finally, gauging a U (1) symmetry with fugacity z in the index (3.77) has the effect of integrating over z. This operation has a clear physical meaning as it picks out gauge-invariant operators, i.e. the "constant term" in the xdependent part of the integrand. To summarize, a 2d (0, 2) vector multiplet or, equivalently, a 3d N = 2 vector multiplet with Neumann boundary conditions corresponds to the simple rule: We can use these ingredients to (re)produce characters of older and more familiar logarithmic CFTs/VOAs. The best known examples of log VOAs include the following three infinite families: • Symplectic fermions are basically βγ-systems, labelled by an integer d > 0 (the number of symplectic fermions) and with central charge Note, that negative values of the central charge signal non-unitarity, which is a general feature of logarithmic theories.
• Triplet (1, m) models, denoted as log-VΛ The last two families correspond to sl(2) Lie algebra, which is not obvious in a short summary given here. They admit generalizations to other Lie algebras g and to (p, p ) models labelled by two integers p and p , all of which are less studied. Note that, for d = 1 and m = 2 the central charges (3.81) and (3.83) take equal value c = −2. This is a manifestation of the relation between the simplest symplectic fermions with d = 1 and the singlet (1, 2) model, which are equivalent. 8 The corresponding character is easily obtained by writing the invariant combinations of ψ n andψ m , the modes of two fermions [21,[41][42][43]: = 1 + q 2 + 2q 3 + 3q 4 + 4q 5 + 6q 6 + 8q 7 + 12q 8 + 16q 9 + . . . 8 To be more precise, they are related by gauging a Z2 symmetry.
Naturally, this is called the fermionic form of the character, which we already encountered in section 3.4 and that will be discussed in more detail in section 5. It also has a bosonic form 9 which will be useful in what follows. In particular, we will demonstrate how this character arises from a 3d N = 2 theory.
Since the character of this c = −2 logarithmic model is constructed as the space of neutral (charge-0) states of two fermions ψ andψ with charges −1 and +1, it is already in the form that can be easily converted to the supersymmetric index (3.77) of a 2d-3d combined system. Namely, the modes of the 2d chiral fermions ψ and ψ each comprise the field content of a 2d (0, 2) "half-Fermi" multiplet. Since they carry charges +1 and −1, respectively, they contribute to the half-index factors where e 2πiξ = z is the fugacity for the global symmetry (that we are about to gauge). Therefore, the elliptic genus of two such multiplets (complex fermions) with charges −1 and +1 is (z −1 q; q) ∞ (zq; q) ∞ . Introducing a 2d (0, 2) vector multiplet and gauging this symmetry of the fermions means taking the constant term in this infinite product or, equivalently, integrating over z, cf. (3.80): This clearly agrees with (3.84)-(3.85). So, we have our first result: we managed to find a 2d (0, 2) physical system whose elliptic genus equals the character of the c = −2 log VOA. More precisely, our realization of symplectic fermions in supersymmetric QFT involves a 2d (0, 2) theory on a boundary of 3d N = 2 theory. Indeed, a pure twodimensional gauge theory with half-Fermi multiplets carrying charges +1 and −1 has gauge anomaly and, by itself, would be inconsistent. It has − 1 2 − 1 2 = −1 units of gauge anomaly, which can be compensated by anomaly inflow from 3d N = 2 gauge theory with G = U (1) and supersymmetric Chern-Simons term at level k = +1. In fact, this model is just a special case of a more general class of 2d-3d coupled systems in [36].
In particular, half-Fermi multiplets naturally arise from 3d N = 2 chiral multiplets with Dirichlet boundary conditions [36]. So, we conclude that the character (3.84)-(3.85) of the symplectic fermions is equal to 2d-3d half-index of the following system: This theory is a special instance of Theory A: 3d N = 2 gauge theory with gauge group U (N c ), Chern-Simons level k > 0, and N f pairs of charged chirals with R-charge R which by the famous Giveon-Kutasov duality [44] is dual to Namely, our original Theory A has N c = 1, k = 1, N f = 1, R = 0. Therefore, its dual Theory B is a U (1) −1 gauge theory with the following field content 3d N = 2 multiplet boundary condition U (1) vector with k = −1 super-CS Neumann chiral with charge +1 and R = 1 Neumann chiral with charge −1 and R = 1 Neumann chiral with charge 0 and R = 0 Dirichlet and the cubic superpotential (3.88). Here we also wrote the dual boundary conditions for all the fields. 10 Using the rules (3.78)-(3.80), it is easy to see that the combined 2d-3d half-index (3.77) with these boundary conditions produces another integral expression for our character (3.84)-(3.85), similar to (3.87): Here, we intentionally focused on the simplest non-trivial example of a log VOA character realized as the combined 2d-3d half-index, to illustrate how the failure of classical modular properties and the logarithmic nature of the VOA originate from three dimensions. If our system was entirely two-dimensional as a consistent QFT, its elliptic genus would be well-defined and exhibit familiar modular properties. However, as we saw in this simple example, the two-dimensional part of our system by itself is anomalous and requires three-dimensional "bulk" which, in turn, spoils modular properties. This simple example can be easily extended to more general systems related to characters of other logarithmic VOAs, old and new. In particular, via 3d-3d correspondence, many Z-invariants of 3-manifolds provide such examples.

Z G -invariants for Seifert Manifolds
In this section we will define our main object Z G b , labelled by a simply-laced Lie group G, a weakly negative plumbed three-manifold X Γ , a choice of generalized Spin c structure b, and potential a Wilson line operator W νv * . Subsequently, we will study its relation to log VOAs reviewed in the previous section.
Denote the plumbing graph by Γ and the resulting plumbed manifold by X Γ . We write its adjacency matrix as M and denote by V its vertex set. For a given simply-laced Lie group G, we introduce the following notation: Also, we define the norms: We also define the lattice with norm given as in (4.2). In what follows, we choose the set B of b to be isomorphic Z |V | ⊗ Z Λ/Γ M,G . In particular, we let Then, following [48], we define homological blocks for the three manifold X Γ .
Definition 4.1. Given a simply-laced Lie group G, the homological blocks for a weakly negative plumbed three-manifold X Γ are defined as: In the above equation, W is the Weyl group and w( b) denotes the diagonal action The integration measure is given by and the contour C is given by the Cauchy principal value integral around the unique circle in the z i,v -plane. Recall that weakly negative means that M −1 defines a negative-definite subspace in C |V | spanned by the so-called high-valency vertices with deg(v) > 2 [7].
Letting π M be the number of positive eigenvalues of M and σ M the signature of M , according to [48], where Φ + is a set of positive roots for G and ρ is a Weyl vector for G. In the case of G = SU (r + 1), we have and the factor becomes In what follows, we will specialise the above definition to our main cases of interest in this paper. First, we specialise to the "N -leg star graphs" which contain only a single node with degree N larger than two, which we will refer to as the central node v = v 0 . The resulting plumbed manifolds are Seifert three-manifolds. We will restrict to weakly negative star graphs, and the weak negativity simply means M −1 v 0 ,v 0 < 0 in this case. We say that the corresponding manifold is negative Seifert. From (A.24), we see that these are precisely the manifolds Proposition 4.2. Fix a simply-laced Lie group G. Consider an N -leg star graph that corresponds to a negative Seifert manifold The homological blocks, defined in Definition 4.1, are given by where the contour C is as described in Definition 4.1, the integrand is eitherχŵ ; b = 0, or there exists a unique κŵ ; b ∈ Λ/DΛ such that where δ and Aŵ are given as and Proof. For a star graph, we can separate the vertices into the central node v 0 , the end nodes with degree one and the intermediate nods with degree two: (4.14) Integrating over the intermediate vertices, we obtain while integrating over the end vertices gives where we have made use of the denominator identity (3.14). We are then left with an integral over the central node. Writing ξ = ξ v 0 and where the three groups of vectors correspond to the three subsets of the vertices (4.14), we get where we define the set so we can rewrite the integral as where we have introduced the notationŵ to denote (w 1 , . . . , w N ), and define Explicitly, we have Combined with Lemma A.1, this leads to with the integrand given bỹ where Moreover, as shown in Lemma A.1, we can write the above in terms of the Seifert data M (b; {q i /p i } i ) for the plumbed manifold as in (4.12). (See also Lemma A.1 for an alternative expression for δ.) Next, we observe from the form of the vectors b (4.4) that the set (4.18) can be expressed as More precisely, we have for the central node, end nodes and the other nodes respectively. In terms of the root basis λ = the above set is given by and the condition D 1 e p i ∈ Z, when combined with (A.27), ensures that is an integer. For a given choice of α k and v ∈ V , the condition is the greatest common divisor of each pair in the triplet (d v , c from the definition of D, we conclude that (4.27) for all v ∈ V either has no solution or has a unique solution in λ (k) ∈ Z/DZ. As a result, for given b andŵ either there exists a unique κŵ ; b ∈ Λ/DΛ such that 28) or S 1,ŵ; b = ∅. Put into (4.23), we obtain (4.11) in the first case and zero in the second case.

Specialisation: Integral Homology Spheres
Now, in Proposition 4.2, we restrict to the graphs Γ with one central node and N legs with a unimodular plumbing matrix. The only choice (up to a trivial shift with elements in Γ M,G ) for b in (4.4) is given by It then follows that κŵ ; b 0 = 0 and the set (4.18) is given by As a result, choosing any representative W of the coset W ⊗N /W , one can express the only non-trivial homological block as Since A wŵ = w( Aŵ), a convenient choice of W is given by thoseŵ with the corresponding Aŵ ∈ P + .

Seifert Manifolds with Three Exceptional Fibers
Given a simply-laced Lie group G, we study the integrandsχŵ ; b and the resulting invariants Z G b (X Γ ) for arbitrary negative three-leg graphs, leading to Seifert manifolds with three exceptional fibres. As seen in Proposition 4.2, the integrand is a finite sum of specific q-series labelled by the set W ⊗3 . We find that these q-series closely resemble the characters of the (1, m) triplet algebra log-VΛ, for a given positive integer m which we will specify shortly. Subsequently, it follows from the relation (3.19) between the singlet and triplet characters that the Z G b (X Γ ) closely resembles a particular linear combination of the characters of the (1, m) singlet algebra log-V 0 Λ . Taking this observation as a starting point, we will establish the various relations betweenχŵ ; b , Z G b (X Γ ) and log VOA characters summarised in Table 1.
To start, we first establish the following. For N = 3, we can rewrite the integrand (4.11) in Proposition 4.2 in terms of the Lie algebra characters (3.17) as In terms of the notation (3.7) analogous to that of the log VOA modules, we can write (4.36) Note that the expression in (4.34) has the same structure as the triplet character (3.15) multiplied by η(τ ) rankG , with the only difference being that in the Z-integrand χŵ ; b (4.34) we restrict the sum to DΛ instead of Λ. Performing the contour integral in order to obtain the invariant Z G b (4.22), we obtain an answer which again shares the same structure of the singlet characters (3.20). The structure of the restricted lattice sum inχŵ ; b for a given b leads to the following interesting result: fixingŵ, a sum ofχŵ ; b over a specific class of b coincides with, up to an overall factor and a rescaling τ → Dτ , a generalised character of the triplet algebra log-VΛ.

From (4.25) it is clear that the same holds for
It then follows that which leads to the theorem when comparing to the generalised character (3.15).
Restricting to the case with D = 1, we obtain the following result for the pseudospherical Seifert manifolds.
Theorem 4.4. Fix a simply-laced Lie group G. Consider a three-leg graph corresponding to a negative Seifert manifold with three exceptional fibers which has integral inverse Euler number, and let If m p i ∈ Z for p = 1, 2, 3, then the integrands of the homological invariants (4.22) are equal, up to an overall rational q-power and the factor η rankG , to a virtual generalised character (3.15) of the (1, m) triplet algebra log-VΛ(m) with given G. More precisely, we have either χ b,ŵ = 0, or q −δ η rankGχŵ; b = χ µŵ (4.41) where δ is given in (4.12), χ λ is given as in (3.15) with Proof. Note that D = 1 in the notation of Proposition 4.2. As a result, from the expression (4.26) we have for all v ∈ V 11 , and S 1,ŵ; b = ∅ otherwise. In the latter case, we have simplyχŵ ; b = 0.
In the former case we havẽ where µŵ is given as in (4.42). Comparing with (3.15) establishes the result.
From the result of Theorem 4.4 and performing the contour integral (4.22), we obtain the following corollary.
From the relation between the generalised characters and the actual characters when G = SU (2) or G = SU (3), and from the special properties of the G = SU (2) characters, we can further conclude Corollary 4.6.
1. Consider G = SU (2). For any three-leg graph corresponding to a negative Seifert manifold X Γ , and for any b, there exists a function Z G b (τ ; X Γ ) on the upper half plane such that . 11 Note that this is always true when XΓ is a intgeral homological sphere.
It would be interesting to investigate the relation between the generalised characters and the actual characters for general G.
In particular, for G = SU (2) we can write Moreover, note that the Cauchy principal value integral gives and we can therefore adjust the integrand in the following way: Clearly, we haveχ where −ŵ denotes multiplyingŵ ∈ W ⊗4 by the non-trivial element of W ∼ = Z 2 diagonally, in accordance with (4.31). Comparing with the characters (3.46) for the (p, p ) triplet model, we see that the form of the integrand χ ŵ; b (τ, ξ) is tantalisingly close to that of the (p, p ) characters.
In the following we will give a proof that this relation to (p, p ) triplet model always holds when where ch + r,s is the characters (3.46) of the (p, p ) triplet model log-VΛ(p, p ) with p = p 4 , p = p 1 p 2 p 3 , and and similarly for all permutations of (p 1 , p 2 , p 3 , p 4 ). Beyond these four possible pairs of (p, p ) and their images under p ↔ p , there are no other choices of (p, p ) algebras for which the relation (4.54) between the homological block integrands and triplet characters holds.
Proof. Since D = 1, we have b = b 0 and κŵ ; b = 0 for allŵ ∈ W ⊗4 in (4.53), and we haveχ So, if p, p ∈ Z + with m = pp and given a pair (ŵ,ŵ ) such that (−1) (ŵ)+ (ŵ )+1 = 1, we can find r, s such that µŵ = ±(ps + p r), µŵ = ±(ps − p r) (4.57) leading to: For p = p 4 , we have p = p 1 p 2 p 3 and p| m p i since for i = 1, 2, 3, (p 4 , p i ) = 1. As a result, we can choose w i = w i for i = 1, 2, 3 and w 4 = 1 = −w 4 , for which case we have µŵ = −ε w 1 ,w 2 ,w 3 sgn(q 4 ) ps w 1 ,w 2 ,w 3 + ε w 1 ,w 2 ,w 3 (−1) (w 4 ) p r (4.60) with Apart from a symmetry in the exchange of p with p one can show that no other splitting of m into p and p will offer a similar result. To see this, note that a pairing ofŵ andŵ satisfying (−1) (ŵ)+ (ŵ )+1 = 1 must have w i = w i for three or for one i ∈ {1, 2, 3, 4}, and the two cases are in fact equivalent sinceχ . As a result, we must have either p|p 4 , m/p|p 1 p 2 p 3 or a permutation of (p 1 , p 2 , p 3 , p 4 ) or swapping p and p , from which we conclude that the solutions given in the theorem are the only possibilities.
Finally, performing the contour integral of (4.54) and using the relation (3.57) between the generalised and the true singlet characters, we obtain the following Corollary.
Corollary 4.8. Consider G = SU (2). For any X Γ as in Theorem 4.7, let p, p , s w 1 ,w 2 ,w 3 as in Theorem 4.7. There exists Z for some a r ∈ C.

Z-invariants with Line Operators
As discussed in §4 of [19], one can also consider homological blocks when Wilson operators, corresponding to half-BPS line operators in the 3d N = 2 SCFT, are incorporated. Here we consider the homological blocks, modified by Wilson operators W νv * associated to a node v * ∈ V in the plumbing graph, corresponding to a highest weight representation with highest weight ν ∈ Λ ∨ : is the character of the representation of G with highest weight ν, where m ( ν) σ is the multiplicity of the weight σ in the highest weight module with highest weight ν. They are building blocks of the half index of the three-dimensional theory with line operators included, and reduce to the homological blocks without Wilson operators when one sets the highest weight ν = 0.
In what follows, as in the rest of the paper, we mainly focus on Seifert manifolds, and consider Wilson operators associated with the end nodes, the central node, and the intermediate nodes of the plumbing graph. We will see that in each of the three cases, the Wilson operator leads to a different modification of the relation to log VOA characters discussed in the earlier part of the section.

Wilson Operator at an End Node
First consider including a Wilson operator associated with an end node, say v 1 ∈ V 1 . While integrating over the end vertices with v = v 1 gives integrating over ξ v 1 gives As a result, it is easy to check that the following statement, which is completely analogous to Proposition 4.2, holds for Z G b (W νv 1 ). Namely, and the integrand is eitherχŵ ; b = 0, or there exists a unique κ w,ŵ; b ∈ Λ ∨ /DΛ such that with δ and Aŵ as given in (4.12), but now with ρ replaced by ρ v , namely and κŵ ; b ∈ Λ ∨ /DΛ such that Consequently, we have the following proposition.
where the highest weight is given by ν = ν ω.

Wilson Operator at the Central Node
Next we consider a Wilson operator associated with the central node v 0 . The integral over ξ v 0 in (4.64) reads where we have dropped the subscript in ξ v 0 . As a result, we see that the statement in Proposition 4.2 is modified in a very simple way by the inclusion of a Wilson operator associated to the central node: whereχŵ ; b is given as in (4.11). The two changes in the integrand on right-hand side upon including Wilson operators are 1) a sum over the weights σ that appear in the corresponding highest weight module, and 2) a multiplication by a factor e ξ, σ . These changes alter but do not destroy the form of the relation between the homological blocks and the generalised singlet and triplet characters, in the case of negative Seifert manifolds with three singular fibers, and we have the following proposition.
where a particular sum of the integrands is given by a polynomial in z i times a generalised character of the log VOA algebra log-VΛ : and analogously for Theorem 4.4. As a result, given that the original homological block satisfies for some a µ ∈ Z as in Corollary 4.5, the homological block with Wilson operator is given by Note that this is precisely the "shifting" phenomenon that has been observed for the special case of a Lens space example in [19].

Wilson Operator at an Intermediate Node
Finally we will consider the case when a Wilson operator associated to an intermediate node in a star graph, say v int ∈ V 2 , is added. In this case we have (4.78) As a result, the relevant sets are now where −w ( σ) is the vector corresponding to the vertex v int , satisfying S w,w ,ŵ; b ∼ = S w * w,w * w ,w * ŵ; b , (4. 79) with the isomorphism given by 0 → w * ( 0 ). Putting everything together, we get whereχŵ ,w ; b vanishes when S 1,w ,ŵ; b = ∅, and is given bỹ Moreover, compared to the case without Wilson operators (4.24), the data for the homological blocks are modified as in the notation of Proposition 4.2. As a result, we see that Theorem 4.3 and Theorem 4.4 hold analogously forχŵ ,w ; b (τ, ξ) in this case, with the modification of the data as given above. Note that the q-power δŵ ,w is no longer independent ofŵ and w due to the last term. Consequently, the homological block is no longer given by a sum of log VOA characters up to an overall factor. For instance, the statement of Corollary 4.5 gets modified into and similarly for Corollary 4.6.

Quivers, Nahm Sums, and Fermionic Characters
The main theme of this paper is the identification of q-series invariants of 3-manifolds with VOA characters. In this section, we show how this identification can be used to produce new fermionic forms of characters for logarithmic VOAs: We will focus on Z G with G = SU (2), and often drop the superscript for notational convenience.
The main idea is to use the enumerative interpretation of Z-invariants and their connection with quiver (COHA) generating series. This perspective on Z-invariants allows one to write the invariants of knot and link complements in the fermionic form (also known as the quiver form or Nahm sum form). Then, it is easy to see that surgery formulae preserve this form, so that Z G b (X, τ ) for closed 3-manifolds can be expressed as a linear combination of fermionic characters (5.1). In fact, this is the same linear combination of characters of log VOAs we saw earlier, so that individual terms can be matched and provide (new) fermionic expressions for (combinations of) log VOA characters.
The fermionic form of VOA characters has a long history and goes as far back as the original work of Hans Bethe [49]. Its modern form is rooted in the relation between 2d CFTs and vertex algebras, on the one hand, and their massive integrable deformations, on the other hand. 12 Underlying this integrable structure are quantum groups, Bethe ansatz equations, Yangian symmetry, and various other symmetries discovered and studied throughout the 1980s, by the Zamolodchikov brothers [50][51][52], by the Leningrad school [53,54] where the explicit form (5.1) appeared in connection with the Kostka polynomials [55,56], and by many other groups.
The name for the structure of the q-series (5.1) was coined by Barry McCoy and the Stony Brook group in the early 1990s [57][58][59][60], where various properties of (5.1) were studied, including the Rogers-Ramanujan type identities as manifestations of the bosonization/fermionization [61,62]. In all these developments, the matrix C in the q-series (5.1) is one of the main ingredients; in particular, the size of C is equal to the number of quasi-particles in the integrable massive deformation of a CFT.
On the other hand, the study of finite-size effects and the Bethe ansatz equations mentioned earlier involve dilogarithms and dilogarithm identities [63,64], which is one of the main conceptual reasons why one should expect these developments from the early 1990s to have direct relation to the 3d-3d correspondence where dilogarithms and dilogarithm identities also play a key role. Building on these developments, in 1995 E. Frenkel and A. Szenes [65] proposed a relation to algebraic K-theory, and in 2004 W. Nahm [66] proposed a relation between the fermionic characters (5.1) and the Bloch group. Sometimes the fermionic expressions (5.1) are called Nahm sums, though it is not clear whether this term was intended to be the same or different from the original notion introduced by B. McCoy and others.
In a completely different line of developments -that has origins in the relation between quantum topology and enumerative geometry -a very interesting correspondence between knots and (equivalence classes of) quivers was proposed about four years ago [71]. In this correspondence, studied e.g. in [72][73][74][75][76][77], the combinatorial and algebraic data associated to the quiver encodes wealth of information about the knot. For example, the structure of the quiver matrix C has a close relation to the structure of the (uncolored) superpolynomials and triply-graded knot homology [78]: the size of C is equal to the number of generators in the reduced superpolynomial / HOMFLY-PT homology, so that the diagonal values of C are given by the homological t-degrees of the generators, etc. One of the main statements in this correspondence is that all HOMFLY-PT polynomials of a knot K coloured by Young tableaux that consists of a single row (column) can be combined in a generating series ∞ n=0 P n (a, q)x n = d 1 ,...,dm≥0 where the right-hand side is the motivic DT generating series of the corresponding quiver. Again, the key ingredient is the quiver matrix C, accompanied by the vectors t, a, and l. One of the conceptual underpinnings of the knot-quiver correspondence, from which it derives its strength and leads to expressions like (5.2), has to do with the HOMFLY-PT variable a. Namely, instead of working with quantum group invariants of a fixed rank, passing to HOMFLY-PT homology or polynomial invariants allows one to see a much richer structure, associated with enumerative and BPS invariants. In fact, an even richer structure can be uncovered 13 by incorporating another variable t, which keeps track of the homological grading, but we will not need it here.
What will be important to us is that many lessons from the physical interpretation of the HOMFLY-PT homology in terms of BPS states and enumerative invariants extend to Z-invariants. In the language of enumerative geometry, this means that the structure of differentials d N and spectral sequences can be transferred from knot conormal Lagrangian submanifolds in Calabi-Yau geometry to knot complements. The first hints for this were seen [79] already for very simple closed 3-manifolds, such as X = S 3 . Very recently, it was realized that the knot-quiver correspondence can be extended to Z-invariants of knot complements as well [76,80].
In particular, this means that Z-invariants of knot complement also can be written in the fermionic form a la (5.1) or (5.2). Then, it quickly follows that the same is true for closed 3-manifolds obtained via surgery operations. This last step is very simple, so let us start by explaining why the surgery operation preserves the fermionic form. Suppose (in the conventions of [80]): where c ∈ Z, and |d| = i d i . Then, the surgery formula [12] gives: 14 where L (0) −1/r : x u → q ru 2 . As a result, we obtain the following formula for Z 0 : Here, E is the matrix where every entry is 1. More generally, if The reason for this is that, with both a-and t-gradings manifest, one can see a rich structure of differentials acting on the space of (refined) BPS states, and this plays an important role in many aspects of the knot-quiver correspondence.
14 Note that in (5.4), the convention of FK follows [80]. In the convention of [12], FK is normalized in a way that its Laplace transformation reads: Z0(S 3 −1/r (K)) = L we can still use (5.5), with the more general modified norm, |d| = i n i d i , and matrix E whose ij entry is Various methods for producing the quiver/fermionic form of F K (x, q) can be found in [80]. They include non-trivial dualities, e.g. to enumerative geometry or to 3d N = 2 theories T [X], as well as more direct diagrammatic techniques, e.g. based on the R-matrix approach [13,14]. In the rest of this section, we illustrate how, starting with such expressions for knot complements, one can obtain analogous fermionic/quiver forms for Z-invariants of closed 3-manifolds, namely the so-called "small" surgeries on various knots:

New Fermionic Forms Related to log-V 0 Λ (m)
A simple infinite family of examples can be obtained by considering surgeries on a torus knot. For concreteness, and to avoid dealing with Spin c structures, we can consider the so-called "small surgeries" on (s, t) torus knots, which all give Brieskorn spheres: In fact, as a warm up, we can start with the simplest knot of all, the right-handed trefoil knot 3 r 1 = T 2,3 : For the right-handed trefoil knot, F 3 r 1 (x, q) can be written in the quiver form (5.3) with [80]: Therefore, after the surgery we get a linear combination of fermionic forms (5.5) with Comparing this expression with (5.5) and rearranging the sum, we obtain a new fermionic form for the log-V 0 (m) (virtual) character (3.25) with 4 × 4 matrix (5.10).

Fermionic characters from 4 1 , 5 2 , and 6 2 knots
Next, we turn to other classes of knots, twist knots 4 1 , 5 2 , and 6 2 . Small surgeries on these knots produce infinitely many distinct hyperbolic manifolds. Another advantage of this family of examples is that all manifolds that result from small surgeries have H 1 (X) = 0, so that there is a unique Spin c structure, and we do not need to worry about the labels of Z-invariants.
For the 4 1 knot, its F K invariant is given by (with a = q 2 specialization): To be explicit, we will make the following choice: With (5.4), we obtain the following data (C, t, l) to compute the Z-invariants.
Similarly, for the 5 2 knot, the matrix C K can be written as follows: (5.20) In this notation, we find: t = (0, 1, 1, 0, 0, 1, 1, 0), As a result, we obtain the following data (C, t, l): For the 6 2 knot, its F K invariant is given by: where the matrix C K and the auxiliary vector l K are given by: After a (−1/r)-surgery along the 6 2 knot, the relevant data (C, t, l) to compute Z can be explicitly given as: t = (1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0) As we already mentioned earlier, it would interesting to understand which logarithmic VOAs can have (linear combinations of) characters that match these fermionic forms.
In all of the above examples, the VOAs have the effective central charge which can be obtained directly from the growth of the integer coefficients in the qexpansion, cf. [80]. This result is somewhat surprising since generic vertex algebras, logarithmic or non-logarithmic, can have (and do have!) other values of c eff . When the fermionic forms are available, c eff can be obtained from the Thermodynamic Bethe Ansatz controlled by matrix C, and comes out to be a sum of special values of dilogarithms, evaluated at the roots of the Bethe equations [52,63,64]. Therefore, a generic C-matrix would produce values of c eff very different from 1, and it is not completely clear at present why VOA charactes that come from Zinvariants (1.3) have this property. There must be something special about fermionic forms that come from Z-invariants. It seems that in our examples (5.24) has to do with the rank of G = SU (2). However, as far as we know, the analogue of (5.24) has not been tested for groups of higher rank, nor do we know if it holds for all closed 3-manifolds, even when G = SU (2). All these questions are excellent subject for future work.

Examples
In this section we explicitly demonstrate the relations found in this paper between the Z-invariant, its integrandχ, and log VOA characters through concrete examples. This section is divided in three subsections, reflecting the three cases analyzed in §4.
In §6.1, we focus on Seifert manifolds with three exceptional fibers. We provide a general analysis for integral homology spheres (which we sometimes refer to as "spherical manifolds" or as "Brieskorn spheres") and In §6.2 we focus on Seifert manifolds with four exceptional fibers. We provide two examples to demonstrate Theorem 4.7 and thus Corollary 4.8. The first example presented in §6.2 is that of a spherical manifold. With the second example, we also demonstrate that results similar to Theorem 4.7 and Corollary 4.8 apply to select cases of pseudo-spherical manifolds.
Lastly, in §6. 3 we provide examples of Seifert manifolds with three exceptional fibers with Wilson line operator insertions. As mentioned in §4.3, for Seifert manifolds the line operators can be associated to the central, intermediate and end nodes of the plumbing graph. Examples with operator insertions at end nodes for spherical, pseudo-spherical and non spherical Seifert manifolds will serve to demonstrate Proposition 4.9. Proposition 4.10 is then demonstrated on a spherical and a pseudospherical example, by insertion of a Wilson operator at the central node. We will conclude this section by providing examples of insertions of Wilson line operators at intermediate nodes of a spherical and a non-spherical Seifert manifold and relating the Z-invariant to linear combination of log VOA characters, as in (4.85).
In this section we will make explicit reference to the Weyl groups of SU (2) and SU (3). The Weyl group of SU (2) is isomorphic to Z 2 . We will write the elements of Weyl length zero and one respectively as 1 and −1. Elements of the Weyl group of SU (3), isomorphic to D 3 , will be written in terms of group elements a, b, corresponding to reflections with respect to planes orthogonal to the two simple roots α 1 , α 2 . The SU (3) Weyl group is then given by {1, a, b, ab, ba, aba = bab}.
Furthermore, we will always use the weight basis when we explicitly write the vectors as tuples. For instance, we write s = i s i ω i =: (s 1 , s 2 ).
All q-series and topological quantities were computed using "pySeifert": a computational toolkit written using Sage [83,84].

Seifert Manifolds with Three Exceptional Fibers
In this section we give examples of Seifert manifolds with three exceptional fibers. After presenting general results for spherical manifolds, we compute Z-invariant integrands,χŵ ; b , for spherical, pseudo-spherical and non-spherical manifolds. We then verify their relation to the log VOA characters as described in Theorems 4.3, 4.4 and Corollaries 4.5 and 4.6.

General Spherical Examples
The simplest class of examples we can consider is that of Brieskorn spheres In these cases, we have As explained in §4, spherical examples have a single b 0 , whose expression can be found in equation (4.29). From (4.42) and (4.36), we obtain for a given choice ofŵ ∈ W ⊗3 . Integrands of the Z-invariant for Brieskorn spheres enjoy a symmetry property (invariance up to a sign, see (4.31)) under the diagonal action of the Weyl groupŵ → w ŵ. Under diagonal action of the Weyl group, equation (4.36) shows that s is mapped to w( s). Hence one can always choose a representative s that lies in the fundamental Weyl chamber. Equivalence under diagonal action allows for the reduction of the number of inequivalent s from |W ⊗N | to |W ⊗N −1 |. In the SU (2) case we thus only have |W | N −1 = 2 2 inequivalent s = s ω, which can be obtained by fixing one of the components ofŵ to any w ∈ W . One possible choice is to fix w 1 to the identity 1. The s values we obtain with this choice are   s 2 ). Using the diagonal Weyl group action to fixŵ 1 to be the identity element, the pairs s = (s 1 , s 2 ) corresponding to even resp. odd Weyl length can be found in Table  2. Alternatively, symmetry under the diagonal action of the Weyl group (4.31) can be used in most cases to fix s so that the its weight components fall into the range s i ∈ {1, 2, . . . , m} corresponding to the expected range for characters of log VOA representations (as opposed to generalized characters).
This restriction on the magnitude of s i depends on the relative magnitude of the p i coefficients. For G = SU (2) the only manifold not to have this simplification is the Poincaré sphere, (p 1 , p 2 , p 3 ) = (2, 3, 5); for G = SU (3) the property fails on all spheres with at least one p i < 4 15 .

One spherical example
We explicitly work out the Brieskorn sphere example X Γ = M −1; 3 5 , 2 7 , 1 9 = Σ (5,7,9). By expanding the continued fractions we can compute the plumbing matrix for this Seifert manifold so the plumbing matrix is The plumbing graph of the Brieskorn sphere can also be found in Figure 2. Of the three legs two are of length two (corresponding to |q i | > 1) and one is of length one. In total this graph has |V | = 6 nodes which coincide with the dimensions of the plumbing matrix.
Other important topological data needed to compute the integrand of Symmetry with respect to the components of s is a general property of Brieskorn spheres and a direct consequence of equations (4.24) and (4.12).
For eachŵ, and thus for each pair (s 1 , s 2 ), we can computeχŵ ; b explicitly. If we setŵ = (1, 1, 1), using equation (4.34): where: In Table 4 the reader can find a representative of the set of equivalent s's for the case G = SU (3) in terms of the p i 's, such that all s 1 , s 2 are within the range between 1 and m for X Γ = Σ (5,7,9). The explicit values of s's for G = SU (2) and G = SU (3) can be found in Table 3. Finally, we can demonstrate Corollary 4.5 by summing over allχŵ ; b (τ, ξ) and integrating: A pseudo-spherical example Pseudo-spherical Seifert manifolds share similar features to spherical Seifert manifolds, with the crucial difference that not allŵ contribute to the Z−invariant. Such manifolds have non-unimodular plumbing matrix, but because M −1 v 0 ,v ∈ Z ∀v ∈ V the lattice dilation factor D is one. One such example is given by X Γ = M −1; 1 2 , 1 3 , 1 9 .
From the inverse plumbing matrix we can compute m = −DM −1 v 0 ,v 0 = 42. As for the other non-spherical case and the pseudo-spherical case, the sets (4.18) are empty for some choices ofŵ. For eachŵ with non-empty set Sŵ ; b we compute the vector κŵ ; b 0 as in (4.28). For b 0 ,ŵ = (1, 1, 1), we compute κŵ ; b 0 = (22,22) and find with δ = − 11 42 and s 1 = s 2 = −29. In (6.23) we have put in dots the q-powers between q −2 and q 298 to highlight the fact thatχŵ , b 0 is included in the log VOA character To recover the full character we sum over all possible λ ∈ Λ/DΛ,

Seifert Manifolds with Four Exceptional Fibers
In this subsection we will provide two examples Seifert manifolds with four exceptional fibers to demonstrate the results of §4.2. The first example, which will be of a spherical manifold, will give a numerical confirmation of Theorem 4.7 and its Corollary 4.8. The second example in this subsection will be of a pseudo-spherical Seifert manifold, even though this case is not covered by Theorem 4.7. This demonstrates that the relation between log-V 0 Λ (p, p ) and three-manifolds holds more generally than what is proven in Theorem 4.7.

A Spherical Example
Our first example is the spherical manifold X Γ = M −2, 1 2 , 2 3 , 2 5 , 3 7 . We analyse this manifold to demonstrate the relation between Z SU (2) b and log-V 0 Λ (p, p ) characters described in Theorem 4.7. The plumbing matrix of X Γ is   Using (4.61) we can produce all possible s w 1 ,w 2 ,w 3 for each pair. Independent s w 1 ,w 2 ,w 3 are collected in Table 7, while the remaining ones are obtained with an extra a minus sign.
Using the data above one may verify the main claim in Theorem 4.7: (τ, ξ).

(6.27)
A Pseudo-Spherical Example In pseudo-spherical and non-spherical cases, non-integer entries in the inverse plumbing matrix cause the Sŵ ; b set to be empty for someŵ, and the corresponding χ ŵ; b to vanish. In most cases, this results in the absence of pairings between Z integrands withŵ andŵ such that w i = −w i and w j = w j , i = j which are a key assumption necessary for Theorem 4.7.

Z-invariants with Line Operators
As discussed in §4.3, the insertion of Wilson operators allows us to access different log VOA characters through the Z-invariants. In this subsection we provide examples of Propositions 4.9, 4.10 and (4.85). Without further remark, in this subsection we exemplify the new phenomena when incorporating line operators using the same manifolds as those in §6.1, and simply referring them as the "spherical manifold", "pseudo-spherical manifold" etc.
Wilson Operators at an End Node Theorems 4.3 and 4.4 can also be applied to spherical, non-spherical and pseudospherical Seifert manifolds when Wilson operators are inserted at end nodes of the plumbing graphs legs. Such generalizations merely require a substitution of δ and Aŵ with the definition in (4.71). Theorem 4.4 will apply to the spherical and pseudo-spherical cases. Let ν = (1, 4) be the highest weight of the A 2 representation. For the spherical case we find which provides a numerical confirmation of Theorem 4.4 for spherical manifolds. A similar result can also be obtained for the pseudo-spherical case. With the same highest weight ν we get Aŵ = 13 9 ω 1 + 53 18 ω 2 , δ = 125/18, √ mµ = 39ω 1 + 42ω 2 and, therefore, forŵ = (1, 1, 1) = q δ η 2 (τ ) χ λ 0,0,s 1 ,s 2 (τ, ξ).

Wilson Operators at the Central Node
When a Wilson operator is inserted at the central node of the plumbing graph, integrands of Z G b obtain an extra polynomial in z i (cf. (4.73)) as a multiplicative factor. As argued in Proposition 4.10, for spherical and pseudo-spherical Seifert manifolds with three exceptional fibers, the Z invariant can still be expressed as a linear combination of singlet characters (3.20).

Discussions and Future Directions
We close this work with a list of questions and suggestions for future research: • Quantum spectral curves like (3.59) are ubiquitous in SL(2, C) Chern-Simons theory and in 3d-3d correspondence. However, their role in vertex algebra is less clear, aside from the obvious fact that such q-difference equations encode dependence of VOA characters on the fugacity associated with a symmetry of the VOA. In particular, even the classical limit of the quantum curve that we worked out for the triplet algebra requires a better understanding and interpretation, from the VOA perspective.
• Following [34], in the earlier work [7] and in §3.5 of this paper we offered a physical explanation for the departure from the classical modular properties of the BPS half-indices in 3d N = 2 theories with 2d (0, 2) boundary conditions. Since characters of logarithmic vertex algebras exhibit similar deviations from traditional modularity, it is perhaps not surprising to find many relations of the form (1.3), which is indeed one of the main results of this work. While we were able to build a large dictionary between q-series invariants of families of 3-manifolds and characters of log VOAs, it would be interesting to develop new tools that allow us to access other aspects of a log VOA, e.g. other conformal blocks, directly from the data of a 3-manifold. We hope this can be achieved by a further study of the 3d-3d correspondence and better understanding of the relation between the category of log VOA modules and the algebraic structure MTC[X] of line operators in 3d theory T [X], along the lines of [7,8,15,79,85,86].
• The relations between non-traditional ("exotic") forms of modularity, logarithmic vertex algebras, and q-series invariants of 3-manifolds also involve quantum groups at generic values of q and their various specializations: log VOAs

Z-invariants
quantum groups quantum modular forms (7.1) In this work we focused on the upper and the left, and the lower to a lesser extent, nodes in this diagram. It is however important to stress that the relation to quantum groups also plays an important role in these connections, see e.g. [13,14] and the upcoming work [87].
• The relation discussed in the present work between three-manifold invariants and log VOAs is far from being one-to-one. In particular, in Proposition 4.2 and Theorem 4.4, the log VOA in question only depends on m = −DM −1 v 0 ,v 0 . A natural question is hence whether there is an extension of the log-VΛ algebra such that Z G (X Γ ) is related to the algebra in an even closer way? In [7] we discussed the Weil representation attached to Z SU (2) (X Γ ) when X Γ is a negative Seifert manifold with three exceptional fibers. We believe that this is a crucial property that provides important hints for the search of such an extended algebra V G X Γ .
• In Theorem 4.3, we see that the integrands of a specific combination of Z G b , with the summand labelled by Λ/DΛ, are given by log VOA (generalised) characters. While Z G b with different generalised Spin c structures are independent topological invariants, it has been noticed that sometimes one has to combine different b to recover various known topological invariants. See [88] for interesting examples. In this sense, what we have found in this work is an analogous phenomenon. In this regard, natural questions include the following. What is the topological meaning of the parameter D, and in particular, what is the meaning for the manifold to be "pseudo-spherical", namely to have D = 1? For the case of D > 1, does an individual Z G b with a given b have an interpretation in terms of the log VOA?
• The results in §4.1 relate the Z G invariants and log VOA characters for all simply-laced gauge groups G. The relation in particular holds for G = SU (N ) for all positive integers N . It is therefore natural to consider the large-N behaviour of the log-VΛ SU (N ) (m) models. More specifically, an effective variable a = q N is expected to play a role in the homological blocks Z SU (N ) [75,79,80]. In the large-N limit, then, could there be a triply graded version of the log-VΛ model, with an additional a-grading? We expect the answer to be in affirmative and the corresponding log-VOA to be an analogue of the triplet algebra where a finite-dimensional symmetry is replaced by an infinite-dimensional symmetry a la Yangian.
• It has been conjectured that the homological invariants Z are closely related to quantum modular forms (defined by Zagier [89]) in some way [7]. See [7,90,91] for earlier results for the G = SU (2) case. At the same time, the quantum modular properties of the log VOA characters have been an active area of research [24,28,92,93], and this immediately leads to some results on the quantum modular properties of Z G for G = SU (2). An in-depth analysis of the modular properties of Z G for G = SU (3) will appear in an upcoming paper [94]. It would be interesting to further develop the triangular relation between quantum modular forms, log VOAs, and homological blocks, as depicted in (7.1).
• In this work, we mainly focus on negative Seifert manifolds for the sake of concreteness. It would be very interesting to explore the VOAs corresponding to other weakly negative plumbed three-manifolds. In particular, it would be very interesting if one could construct the VOAs with a procedure reflecting the operations on plumbing graphs, such as connecting weighted graphs into a larger one.
• In the present work we focus on weakly negative plumbed three-manifolds, and in particular negative Seifert manifolds. In [7], it was proposed that the role of (higher rank) false theta functions in these cases will be replaced by (higher depth) mock modular forms in the case of plumbed manifolds that are not weakly negative, and in particular positive Seifert manifolds. A natural question is thus to find the VOAs connected to the homological blocks of such three-manifolds, in a systematic manner analogous to the results presented in the present paper.
• Understanding of the relations in (7.1) can be greatly facilitated by the fermionic form of log VOA characters and Z-invariants. In this paper, we only made some initial steps in this direction, leaving many interesting questions to future work. For example, the appearance of classical and quantum dilogarithms on both sides of the 3d-3d correspondence suggests many connections to cluster algebras which, while natural, so far did not appear in the study of Z-invariants of log VOA. The relation between cluster algebras and Z TQFT is also expected because the latter provides a non-perturbative definition of SL(2, C) Chern-Simons theory, whereas the relation between cluster algebras and log VOAs is natural in view knot-quiver correspondence and recent work [77,95].
• The quiver/fermionic form of q-series invariants also offers new ways of addressing long-standing questions in logarithmic vertex algebras. For example, it offers a fresh new perspective on the "semi-classical" limit (m → ∞) of the log-V 0 Λ (m) model and going from the "positive zone" in Kazhdan-Lusztig correspondence to the "negative zone," on which we plan to report elsewhere. The counterpart of this question in quantum topology is the reversal of orientation on X and the relation between Z(X) and Z(−X).
• The quiver/fermionic forms discussed in this paper are virtual characters of the familiar vertex algebras like triplet and singlet log VOAs. In other words, these fermionic formulas are linear combinations of the characters of irreducible modules. As was mentioned in the fourth bullet point of this section, this seems to suggest that logarithmic VOAs associated to 3-manifolds are extensions of these familiar algebras, at least in simple examples. This point should be important for upgrading the dictionary between 3-manifolds and characters to actual VOAs.  Given the explicit form of the matrices B and X, it is straightforward to compute that the (N + 1) × (N + 1) matrix BX −1 B T is diagonal and takes the form and (A.18) The above expressions and (A.14) immediately leads to the identity (A.1). It will be illuminating to express the above quantities in terms of the Seifert data (as opposed to plumbing data) directly. To do so, let us compute the entries of X −1 i . To avoid an overload of indices, we will momentarily suppress the index i labelling the different legs of the graph when the context is clear.
Note that X is a tridiagonal matrix. Using the recursion formula for its inverse, we obtain similarly θ k = θ  To prove (A.3), we choose the continued fraction expression (A.22) for −p i /q i that is effective, namely the {a (i) k } satisfying a (i) k ≤ −2 for all k > 0. We will see that the final expression we have derived using this particular choice is invariant under the 3d Kirby moves 16 , which clarifies that making this choice does not lead to a loss of generality. With this choice, it is easy to see that θ k = (−1) k |θ k | and similarly for θ k , and in particular θ = θ = (−1) |q|.

B The A 2 Character Identity
Let χ 0 a,b;s 1 ,s 2 be generalized A 2 characters defined as in (3.32). In this appendix we prove the identity: Lemma B.1. where s 1 , s 2 are defined as in (3.28).
Note that the identity in Lemma B.1 holds for all s 1 , s 2 ∈ Z and is not subjected to the condition s 1 , s 2 ∈ {1, 2, · · · , m}, which is natural from the point of view of representation theory of the singlet algebras. Does this imply the existence of some interesting isomorphism among modules of log VOA? We will leave the question for future work and simply prove the algebraic identity here.