An $\mathcal{N}=1$ 3d-3d Correspondence

M5-branes on an associative three-cycle $M_3$ in a $G_2$-holonomy manifold give rise to a 3d $\mathcal{N}=1$ supersymmetric gauge theory, $T_{\mathcal{N}=1} [M_3]$. We propose an $\mathcal{N}=1$ 3d-3d correspondence, based on two observables of these theories: the Witten index and the $S^3$-partition function. The Witten index of a 3d $\mathcal{N}=1$ theory $T_{\mathcal{N}=1} [M_3]$ is shown to be computed in terms of the partition function of a topological field theory, a super-BF-model coupled to a spinorial hypermultiplet (BFH), on $M_3$. The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on $M_3$. Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d $(2,0)$ theory. We also consider a correspondence for the $S^3$-partition function of the $T_{\mathcal{N}=1} [M_3]$ theories, by determining the dimensional reduction of the M5-brane theory on $S^3$. The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on $M_3$, whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic $G_2$-manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the $S^3$-partition function of $T_{\mathcal{N}=1}[M_3]$ is given by the Witten-Reshetikhin-Turaev invariant of $M_3$.

. The so-called 3d-3d correspondence [4,5] similarly relates the S 3 -partition function of 3d N = 2 theories T N =2 [M 3 ] to the partition function of complex Chern-Simons theory on M 3 . Finally, for d = 2 there is a correspondence for 2d (0, 2) theories labeled by four-manifolds M 4 . The half-topologically twisted sphere partition function is conjectured to be computed by a topological sigma-model into the monopole moduli space [6], whereas the Witten index is identified with the Vafa-Witten (VW) [7] partition function of 4d N = 4 Super-Yang Mills theory on M 4 [8].
Much of the progress in establishing these conjectures relies on the computational advances that were made for sphere partition functions of N = 2 supersymmetric theories thanks to localization techniques [9,10] -for a recent review see [11]. For less supersymmetry, many of these tools are not quite as well developed thus far. One may hope that formulating similar correspondences for N = 1 theories could give further insight into their structure. An initial step towards developing an N = 1 version of the AGT correspondence has been made in [12], with the goal to relate the sphere partition function of the class S k theories [13] to 2d conformal blocks.
The goal of this paper is to develop an N = 1 version of the 3d-3d correspondence, motivating it from first principles by starting with the 6d (2, 0) theory. As is well-known the T N =2 [M 3 ] are obtained by wrapping M5-branes on special Lagrangian (sLag) three-cycles in Calabi-Yau three-folds. To retain N = 1 in 3d, we will show that the natural setup for 3 ] is to realize M 3 as an associative three-cycle in a G 2 -holonomy manifold 1 .
A priori we do not know what the topological theories are, which would complement the 3d N = 1 theories in such a 3d-3d correspondence. To determine these, it is useful to recall the approach applied in the N = 2 setting: the topological theory, whose partition function computes the sphere partition function of the T N =2 [M 6−d ] theories can be determined from the sphere reduction of the 6d (2,0) in an N = 2 preserving conformal supergravity background [14,6,15]. We will employ this approach in the following to determine the topological theories, which compute the following two observables of T N =1 [M 3 ]: the T 3 -partition function, i.e.
Witten index, and the S 3 -partition function.
For 3d N = 1 the Witten index [16] is a well explored observable, much more so than the S 3 -partition function. For this reason we will focus much of our attention on this observable and provide non-trivial checks of the proposed correspondence. We will derive the 'dual' topological theory by considering the 6d (2, 0) theory first on T 3 , which gives 3d N = 8 SYM, which we then topologically twist along M 3 , while preserving two topological supercharges.
This twist corresponds to the embedding of M 3 as an associative cycle in a G 2 -manifold. In summary, the theory, whose partition function computes the Witten index of the T N =1 [M 3 ] will be shown to be a supersymmetric BF-theory coupled to a spinorial hypermultiplet (BFH), which is a section of the normal bundle N M 3 of M 3 inside the G 2 -manifold.
We should at this point elaborate briefly on the geometry of associative three-cycles in G 2 -holonomy manifolds [17,18], which will play an important role in the behavior of the Witten index. The normal bundle of an associative three-cycle is N M 3 = S ⊗ V , where S is the spin bundle and V an SU (2)-bundle, in particular sections of the normal bundle are twisted harmonic spinors on M 3 , satisfying a twisted Dirac equation. On an odd-dimensional manifold the Dirac operator has vanishing index, which implies that the dimension of the kernel (infinitesimal deformations) equals that of its co-kernel (obstructions to these deformations), however the index does not reveal any information about the non-triviality of each of these spaces. For non-generic choices of G 2 -structure, there can be twisted harmonic spinors, which are accompanied with non-trivial obstructions of the deformations, which they parametrize.
This fact will reflect itself in the discontinuity/wall-crossing of the Witten index  We find that already in the abelian case the index is discontinuous under metric deformations, and jumps depending on the existence of twisted harmonic spinors. The fact that the partition function of the BFH-model is only topological up to wall-crossing can, as noted earlier, be traced back to the deformation theory of associative three-cycles within G 2 -manifolds.
At the location of the walls the normal deformations, appearing in the gSW equations, are obstructed. This means that M gSW can become singular and its Euler characteristic can jump.
A second less-explored observable for 3d N = 1 theories is the S 3 -partition function (for a discussion of this observable for SCFTs see [19]). Whereas the N = 2 3d-3d correspondence is studied for the S 3 -partition function, and many computational results are available thanks to localization methods [20] (and see [11] for a recent review), the situation for N = 1 is much less explored. In particular localization will not be applicable for computing the sphere partition functions with 3d N = 1 supersymmetry. Here we will nevertheless determine what the 'dual' topological field theory is, whose partition function on M 3 would provide a conjecture for the S 3 -partition function of T N =1 [M 3 ]. To do so, we determine the conformal supergravity background similar to [14,6,15] and perform the reduction of the 6d (2, 0) theory on a threesphere, first to 5d SYM and then on an S 2 to 3d, whilst preserving N = 1 supersymmetry.
The resulting topological theory is shown to be real Chern-Simons gauge theory on M 3 coupled to a twisted harmonic spinor φ, i.e. a Chern-Simons-Dirac theory whose equations of motion are the generalized Seiberg-Witten equations (for a review see [21]). For generic associatives 2 The analog for N = 2 is the moduli space of complex flat connections.
in G 2 -manifolds there will be no twisted harmonic spinors, and the theory reduces to real Chern-Simons theory. In this case the topological partition function is given in terms of the Witten-Reshetikhin-Turaev invariant [22,23], which we conclude must compute the S 3 - The most interesting physical application arises when viewing the M5-branes as domain walls in the 4d N = 1 theory obtained by M-theory on the G 2 -holonomy manifold. For Lens spaces this case has been studied in [24] and we will connect these results, when discussing concrete examples. This may in particular be of interest in recent constructions of new G 2holonomy manifolds in [25][26][27] and singular limits thereof [28] that realize non-abelian gauge groups.
Finally, we should remark that in the case that there is a non-trivial IR fixed point, the M5-branes on associatives have a holographic dual description in terms of AdS 4 -solutions, where M 3 is a hyperbolic three-manifold [29]. This means the metric has constant sectional curvature −1, and by Schur's lemma the metric is Einstein. Examples of such associatives exist in the Bryant-Salamon G 2 -manifolds [30], which are the total space of the spin bundle over M 3 , with M 3 of constant sectional curvature ±1. For metrics on M 3 with negative scalar curvature the associatives can indeed have obstructions, which are determined by zero modes of the Dirac operator. It would be interesting to explore this from a holographic point of view.
The plan of the paper is as follows: section 2 serves as an overview and summary of background material, starting with a concise statement of the proposed N = 1 3d-3d correspondence in section 2.1. In section 2.2 we provide some background on G 2 -holonomy manifolds, calibrated cycles and their deformation theory. We conclude this section with a discussion of 3d Seiberg-Witten equations and the non-abelian generalization that we encounter. In section 3 we derive the abelian theory T N =1 [M 3 , U (1)] and its Witten index. The generalization to non-abelian gauge groups is discussed in section 4 by either considering a specialization of the three-manifold to Lens spaces, or by considering a reduction to 5d SYM and subsequently a derivation of the circle-reduction of T N =1 [M 3 ] to a 2d sigma-model into the moduli space of generalized 3d Seiberg-Witten equations. In section 5 we derive the topological field theory side of the 3d-3d correspondence in the case of the Witten index, and provide several checks. In section 6 we determine the topological theory whose partition function computes the S 3 -partition function of the 3d N = 1 theory. We conclude in section 7 with a discussion and outlook. Various appendices summarize our notation and provide further computational details.
2 Overview and Background 2.1 An N = 1 3d-3d Correspondence Consider 6d (2, 0) theory with gauge group G on a three-manifold M 3 . Depending on the topological twist, the resulting 3d theory after compactification along M 3 , can preserve either N = 2 or N = 1 supersymmetry. We denote the Lorentz and R-symmetry of the 6d theory by SO(1, 5) L × Sp(4) R . With the space-time decomposition R 1,2 × M 3 , the two twists are realized as follows: (2.1) Twisting the underlined R-symmetry groups with the local Lorentz group SO(3) M of M 3 , results in two types of supersymmetric 3d theories: The N = 2 case has been studied extensively in the standard 3d-3d correspondence [4,5] and the twist is realized geometrically in terms of embedding M 3 as a special Lagrangian (sLag) cycle in a Calabi-Yau three-fold. The correspondence states the equivalence between the S 3 -partition function Z S 3 of the N = 2 , and the complex Chern-Simons partition function on M 3 for general gauge group G. This correspondence is by now not only supported by computational evidence in terms of examples of S 3 -partition functions, but in [14] the complex Chern-Simons theory was derived by a dimensional reduction of the 6d (2, 0) theory on the three-sphere, coupled to a suitable conformal supergravity background.
The twist in (2.1) that preserves N = 1 in 3d will be the subject of this paper and we denote the corresponding 3d theories by T N =1 [M 3 ] 3 . Geometrically this twist is realized by embedding M 3 as an associative three-cycle (i.e. calibrated and thus supersymmetric cycle) in a G 2 -holonomy manifold. The observable we consider first is the T 3 -partition function, or Witten index [16]. We determine the 3d topological field theory, whose partition function on M 3 will be conjectured to compute the Witten index of the T N =1 [M 3 ] theory: first we dimensionally reduce the 6d (2, 0) theory on T 3 to 3d N = 8 SYM, which is then topologically twisted along M 3 to preserve two scalar supercharges. This results in a super-BF-model coupled to a Hypermultiplet, denoted by BFH-model in the following, and we propose the following identification of observables Here φ αα is a section of the normal bundle of the associative cycle, i.e. a section of the spin bundle twisted with an SU (2)-bundle V , and A is a gauge connection for the gauge group  Note that for 3d N = 2 theories on compact three-manifolds 4 , obtained from the 6d (2, 0) theory on special Lagrangian three-manifolds inside a Calabi-Yau three-fold, it was argued The complex flat connections arise from the gauge field coupling to sections of the normal bundle of the special Lagrangian three-cycle given by the cotangent bundle. In the present case, we obtain the generalized Seiberg-Witten equations (2.4), which couple sections φ αα of the normal bundle of the associative cycle to real connections on M 3 . If M 3 does not admit any twisted harmonic spinors, i.e. d / D (M 3 , g) = 0, the solutions to gSW M 3 ,G are real flat G-connections, and the N = 1 correspondence equates vacua of In the next subsections we will first give a lightning summary of the geometry of associative  Before turning to this, we also summarize our findings with regards to another, less ex- Very little is known about this observable in 3d N = 1 theories. We will nevertheless propose a variant of the 3d-3d correspondence for this observable, by determining the topological field theory, whose partition function computes the S 3 -partition function of the N = 1 theory. The strategy is again to start with the 6d (2, 0) theory and dimensionally reduce on S 3 in a suitable conformal supergravity background -the setup is shown in figure 2. We find that the topological theory is a Chern-Simons-Dirac (CS-Dirac) theory, i.e. real Chern-Simons gauge theory at level 1 coupled to a twisted harmonic spinor φ, whose equations of motion are precisely the generalized Seiberg-Witten equations (2.4). This leads us to conjecture that the sphere partition function is computed by the M 3 -partition function of this CS-Dirac theory As we will discuss, generically in G 2 -manifolds associatives will not have twisted harmonic spinors in which case the right hand side simply reduces to the partition function of real CS-theory on M 3 , which computes the Witten-Reshetikhin-Turaev (WRT) invariant This variant of the 3d-3d correspondence will be derived in section 6.

G 2 -holonomy, Associatives and Deformation Theory
This section gives a brief review of manifolds with holonomy group G 2 and calibrated, socalled associative, three-cycles, that play a key role in this paper. For more in depth reviews on these geometries in mathematics and string theory see [18,34]. A G 2 -holonomy manifold is a pair (X 7 , Φ), where X 7 is an oriented manifold of real dimension seven and Φ is a harmonic three-form, the G 2 -form. This manifold admits a metric g Φ , constructed from Φ, with reduced holonomy group G 2 ⊂ SO (7). Key to the present discussion are calibrated or supersymmetric cycles. Associative three-cycles M 3 are calibrated by Φ, 5 i.e.
and their properties including deformation theory have been discussed in [17,35]. Let us fix a G 2 -holonomy manifold (X 7 , Φ) and an associative submanifold M 3 . The cotangent space of X 7 can be identified with Im H ⊕ H containing elements (u, v). On M 3 , we now identify u as a one-form and v as a section of the conormal bundle. In this language the G 2 -form is for some initial element (u 0 , v 0 ). The space of deformations, which leave the structure invariant can be described by an SO(4)-action on Im H ⊕ H. Note that SO(4) appears naturally as it is the largest group that does not mix u and v, which is necessary to preserve the associative.
To recover the usual transformation of one-forms we expect u 0 → gu 0ḡ = u under the group action, whereḡ = g −1 . In order for Φ to transform as a three-form (i.e. in the same way as u) the group action on v 0 is v 0 → hv 0ḡ = v. From these transformations one can deduce that the structure of the normal bundle consists of two parts. The group action v → vḡ identifies v as transforming in the 2 of SU (2) M , and similarly also in the 2 of SU (2) V . From these two transformations of v one finds that the normal bundle of M 3 is where S is the spinor bundle of M 3 and V is the principle bundle associated to SU (2) V .
The space of linear deformations is given by sections of the normal bundle, i.e. the kernel of Note that d / D (M 3 , g) depends on the metric g [36]. An example of an associative that admits twisted harmonic spinors, given by an associative T 3 inside T 7 , was discussed in [37]. Although over a three-manifold every SU (2)-bundle is trivial, i.e. it admits a trivial connection, this does not imply that we can simply set the connection A on V to zero, as the connection relevant here is the one determined by the embedding of the associative cycle into the G 2manifold. The connection is therefore fixed, and not necessarily equal to the trivial connection.
In the examples studied by Bryant-Salamon [30], the G 2 -holonomy manifolds are constructed as the total space of the spin bundle over a constant sectional curvature three-manifold, so that A = 0. In fact, whenever A = 0 there are three distinct cases depending on the scalar curvature R: 1. R > 0: For positive curvature there are no harmonic spinors, i.e. d / D (M 3 , g) = 0 [38], which implies that the associative three-cycle is rigid. The index of the Dirac operator vanishes on a closed three-manifold, so that its obstruction space is of the same dimension as the deformation space [17,37]. This implies that the virtual dimension of the moduli space of associative deformations is always zero. It is believed [18,37] that for generic M 3 there is no obstruction space, so there are also no linear deformations and d / D (M 3 , g) = 0. This structure is generically preserved under small deformations of the metric, or equivalently the G 2 -form Φ. However for non-generic G 2 -manifolds, the obstruction and deformation spaces can be non-trivial.
An example for how deformations can lead to obstructed associative cycles was given in [40].
Consider families of G 2 -manifolds (X 7 , Φ(a)) ≡ X(a) and embeddings where Y is some three-manifold, that vary smoothly with independent parameters a, b ∈ R.
It was argued in [40] that one can choose the embeddings such that the has a non-zero linear deformation space, d / D (M 3 , g) = 1. Clearly, this logic breaks down at higher order in ε so the deformations are obstructed. Throughout this paper we will discuss variations of associative three-cycles M 3 (b) in a fixed homology class and retain a description after wall-crossing, which occurs at the points where d / D (M 3 , g) = 0.

Generalized Seiberg-Witten Equations
The topological field theories that will play a role in the 3d-3d correspondence for both T 3and S 3 -partition functions, are closely related to a set of generalized Seiberg-Witten equations on the three-manifold M 3 (2.14) Here φ αα is a section of the normal bundle of the three-cycle M 3 in the G 2 -manifold N M 3 = S⊗V and thus a 'bispinor' with respect to the SU (2) M and SU (2) V . Both the gauge connection A and φ are in the adjoint of the gauge group G, which we take to be U (N ), with N > 1. 6 The Dirac operator acting on the spinor φ is covariantized to the twisted Dirac operator, including a connection on V where ∇ is the covariant derivative with respect to the spin connection on M 3 .
The above equations are non-abelian generalizations of the well-known 3d SW equations.
Let us briefly recapitulate their origin and then contrast other generalizations to the one we encounter here (2.14). The 4d Seiberg-Witten or monopole equations [41][42][43] consist of a U (1) gauge field A coupled to a positive chirality spinor Φ and its complex conjugate where the Dirac operator is twisted by the connection B of the SU (n)-bundle. The virtual dimension of the associated moduli space, given by the difference of the dimension of the space of deformations and obstructions, is zero [37], however the count of solutions is not necessarily independent of the metric on the three-manifold. This is an intrinsic property of the moduli space of twisted harmonic spinors, which are the zero modes of / D A⊗B . Consequently, the solutions to (2.14), which correspond to a non-abelian generalization of (2.17) in the case of n = 2, are also expected to be metric-dependent.
The abelian 3d Seiberg-Witten equations with n = 2 have appeared in relation to deformations of associative three-cycles in manifolds with (not necessarily torsion-free) G 2 -structure in [47][48][49], where the twisted harmonic spinor condition of [17] is supplemented with an additional Seiberg-Witten-like equation, which couples a U (1) gauge field to sections of the normal bundle (2.12), in order to make the space of deformations compact and zero-dimensional. In the context of G 2 -strings, a non-abelian version was shown to arise as the equations of motion of the world-volume theory of topological 3-branes wrapped on an associative three-cycle in a G 2 -manifold [50]. In this context the spinor φ αα arises exactly from considering the normal modes of the associative, and the equations derived there, have some resemblance to (2.14).
It would be interesting to understand the precise relation between these two setups.
We now turn to deriving the 3d N = 1 theory

Topological Twist for Associative Three-Cycles
We consider the 6d N = (2, 0) theory on where M 3 is an associative three-cycle in a G 2 -holonomy manifold. The local model for this configuration is the total space of the normal bundle (2.12). The relevant topological twist was first discussed in [51]. We will here determine the dimensional reduction to the theory and its non-abelian generalization. The Lorentz and R-symmetry group of the 6d theory is given by SO(1, 5) L × Sp(4) R ⊂ OSp(2, 6|4), the superconformal group in 6d. We consider where SO(1, 2) L and SO(3) M are the local Lorentz groups acting on R 1,2 and M 3 , respectively.
The supersymmetry parameter transforms in the (4, 4) which decomposes as As explained in (2.1) we twist the local Lorentz group of M 3 with the R-symmetry SU (2) r under which the supersymmetry parameters transform as There are two supercharges that transform trivially under the twisted Lorentz symmetry on M 3 , and thus N = 1 supersymmetry is preserved in the transverse 3d space-time.
To substantiate this, let us now consider the reduction of the 6d abelian tensor multiplet, consisting of five scalars Φmn, a self-dual three-form H = dB, and fermions αm satisfying the symplectic-Majorana-Weyl condition (A.7). A summary of our conventions can be found in appendix A, wherem,n = 1, · · · , 4 denotes the fundamental of Sp(4) R . 7 Under (3.5) these become Finally, x, y and a, b are flat space-time indices on R 1,2 , and M 3 , respectively. The key point to observe here is that from the 6d scalars we obtain a field φ αα transforming as a spinor under both the twisted Lorentz SU (2) twist and the remnant flavour symmetry SU (2) . We therefore identify the SU (2) with the SU (2)-bundle V in (2.12), which makes φ αα a section For a single M5-brane, the dimensional reduction follows from the abelian tensor multiplet [52], and we can determine T N =1 [M 3 , U (1)] explicitly. In appendix B we carry out the reduction and show T N =1 [M 3 , U (1)] to be a supersymmetric Chern-Simons theory with free scalar multiplets.
as well as the number of twisted harmonic spinors d / D (M 3 , g) on M 3 . These fields organize into 3d N = 1 multiplets (some basic properties of such theories are summarized in appendix C). The supersymmetry transformations of the abelian 6d N = (2, 0) theory are The topological twist is implemented by requiring that the invariant supersymmetry parameter where Σ a and Σâ are the generators of SU (2) M and SU (2) r , respectively, as defined in appendix A.2. The solution to (3.9) is given by where ε αm is the anti-symmetric two-tensor. The 6d symplectic-Majorana-Weyl condition implies the 3d reality condition with 1 and 2 real. The dimensional reduction of the supersymmetry variations yields Then, the variation of δH + xyz reduces to δh in (3.12). We can now compare this to the 3d N = 1 multiplet structure, which is summarized in appendix C. 9 Note that for the scalar multiplet with leading component ϕ we obtain the full off-shell supersymmetry transformations. The field content of T N =1 [M 3 , U (1)] can be interpreted as consisting of the following free 3d N = 1 multiplets: 9 By an appropriate rescaling of the fermions → √ 2 and {λ, ξ, ρ} → − 1 √ 8 {λ, ξ, ρ}, the supersymmetry variations can be brought into the standard form.

Witten Index of
The observable for the proposed 3d-3d correspondence is the T 3 -partition function, or Witten index, of T N =1 [M 3 ], which is defined by I = Tr(−1) F [53], and for N = 1 theories is the most natural and well-explored observable to consider. We will discuss the much less explored S 3 -partition function in section 6. We now compute the Witten index for the abelian Chern- The key geometric input from M 3 are its first Betti number b 1 (M 3 ), the torsion numbers p m in (3.7) and the number d / D (M 3 , g) of zero-modes of the twisted Dirac operator, with g being the metric arising from the restriction of the G 2 -metric onto M 3 . Since the abelian theory is non-interacting the multiplets decouple and the vacua can be written as tensor products.
Thus, the Witten index is a product with the following contributions: 1. Independent of the details of the compact M 3 the theory includes a free scalar multiplet consisting of a bosonic and a fermionic state. Thus, the ground states always come in pairs with opposite fermion number and the full Witten index vanishes. Since this does not yield any information about the associative three-cycle we will exclude this center of mass multiplet from the computation of I = Tr (−1) F in the same fashion as in [24].
3. The final piece is the set of r vector multiplets with Chern-Simons self-interactions at levels p m given by the torsion numbers (3.7). For a single U (1) gauge field with Chern-Simons level k the Witten index is k [22]. Since the r gauge fields are independent of each other the total contribution of this sector is I CS = r m=1 p m .
The total Witten index I , excluding the center of mass contribution, is thus given by Under smooth deformations of the metric of the G 2 -holonomy manifold, or equivalently the G 2 -form, this index is discontinuous, whenever the kernel of / D is non-vanishing. There is no guarantee that the index is invariant after passing through these loci, as the deformation space of an obstructed associative three-cycle can be singular. This phenomenon is known as wall-crossing.
We should comment on how the wall-crossing that we observe compares with the one proposed in [40] for associatives in G 2 -manifolds. We are interested only in variations of the G 2 -form, which result in metric deformations on the associative three-cycle, while keeping its class in the third homology fixed. In particular we do not consider deformations, where the topology changes or the associative ceases to exists, or splits. The type of wall-crossing considered by Joyce in [40] is more closely connected to the M2-instanton partition function, as recently discussed in [54].

Non-abelian Generalization
Ideally at this point we would provide a generalization to non-abelian theories for general, compact three-manifolds, however since the non-abelian 6d theory is unknown this is hard to come by. Thus, a precise dictionary between the associatives and the non-abelian theories in general is beyond the scope of the current paper. However, we provide two alternate ways to obtain some information about the non-abelian generalizations. One approach is to first reduce from 6d to 5d on a circle and study the 5d SYM theory on M 3 . This results in a 2d sigma-model with (1, 1) supersymmetry, whose target space is the moduli space of the gSW equations. The alternative is to use a specialization of M 3 , when the reduction is known, such as for M 3 a Lens space. More generally one can consider circle-fibrations and use methods such as [14] to compute the dimensional reduction in those cases. We leave this for future work and focus here on the Lens spaces.

Circle Reduction to 2d N = (1, 1) Sigma-Model
To generalize the correspondence to non-abelian theories, without specializing the threemanifold, we first dimensionally reduce the 6d theory to 5d, and consider the non-abelian 5d SYM theory on M 3 . This will not immediately reveal the T N =1 [M 3 ] theory, however we will be able to generalize our results to the non-abelian version of this theory dimensionally where F a b is the field strength of the 5d gauge field A b andm = 1, · · · , 4 is the R-symmetry index. Consider the topological twist discussed in section 3.1, which is unaltered by the reduction to 5d. The resulting field content is given by The twist of the 5d N = 2 vector yields gauge fields A b on M 3 and A ± on R 1,1 respectively.
The decomposition of the spinor and scalar sectors is similar to (3.6). This preserves two supercharges of opposite chirality corresponding to 2d N = (1, 1). In order to put this theory on a general associative M 3 the Dirac operator has to be covariantized with respect to the spin connection on M 3 as well as the SU (2) V connection A as discussed in appendix B. 11 To dimensionally reduce on M 3 we introduce a length scale s of the three-manifold and consider the theory in the limit s → 0. To this end we rescale the fields and coordinates on M 3 to make their s-dependence explicit. The reduction proceeds by solving the BPS equations, which are the most divergent terms in the s-expansion of the supersymmetry variations

Specializing
Another approach to obtaining further insight into the non-abelian generalizations is to consider special cases of three-cycles M 3 . In fact one of the most common associative cycles in compact G 2 -manifolds that are known are three-spheres or simple modifications thereof -see e.g. the twisted connected sum constructions in [25][26][27], where associatives are either S 3 or diffeomorphic to S 2 × S 1 , or more recently the conjecture for an infinite family of associative three-cycles with topology S 3 in these geometries [54].
It is thus useful to consider specializations of the three-cycles M 3 and determine the nonabelian theories using special properties of the geometries. This strategy has already been successfully applied for the N = 2 version of the 3d-3d correspondence. In particular we focus on Lens spaces L(p, q), defined as Z p quotients of S 3 , where the action of the quotient on S 3 , embedded as a unit sphere in C 2 with complex coordinates (z 1 , z 2 ), is given by For (p, q) coprime this action has no fixed points, and is therefore free. Lens spaces appear as associative three-cycles in the G 2 -manifolds considered in [24]. In these cases the embedding of the associative three-cycle is trivial and there are no twisted harmonic spinors. For q = 1, the Lens space is a Hopf fibration S 1 → L(p, 1) → S 2 . The reduction of the 6d N = (2, 0) theory along the fiber direction yields 5d N = 2 SYM in the presence of p units of graviphoton flux. This theory possesses a Lagrangian description for general gauge group and the subsequent reduction on S 2 can be performed explicitly. This yields a 3d SYM theory with a Chern-Simons term at level p coupled to a scalar multiplet in the adjoint. The bosonic part of the action is given by where the scale of the gauge coupling is given by the radius of the Lens space g −2 ∼ r. In the abelian case G = U (1) this matches the description in section 3.2 as H 1 (L(p, q), Z) = Z p .
Since the theory only depends on the topological data of the associative this can be generalized to all q.
For these 3d N = 1 Chern-Simons theories T N =1 [L(p, q)] the Witten index was computed in [16,24] to be To compute this one uses the fact that the vacua of the Chern-Simons theory (4.

BFH-Model on M 3
We now turn to the topological field theory side of the correspondence, i.e. the right hand side of figure 1, whose partition function is conjectured to compute the Witten index of the 3d Here, the hatted and dotted indices denote the representations under the internal symmetries SU (2) and SU (2) I , respectively. This 3d topological action is given by the 3d super-BFmodel [58,59] coupled to a spinorial hypermultiplet, which first appeared in [60]. In addition, we will gauge the SU (2) flavour symmetry, which is identified with the bundle V in (2.12), to ensure that M 3 is embedded as an associative three-cycle.
This theory can also be obtained as the T 2 -reduction of the twisted 5d N = 2 theory introduced in section 4.1. After the reduction to three dimensions, the U (1) symmetry on the torus gets enhanced to SU (2) I , and the transformations of the fields can be repackaged into the content in (5.1). Inspired by the description in [60] we find that the reduction of the supersymmetry variations can be written in an off-shell form with the introduction of two auxiliary fields, B a and W αα , which can be written in a Q-exact form where We identify the gauge coupling constant e as where v T 2 is the T 2 -volume and r is the radius of the M-theory circle. Note that the radii of the three circles enter the gauge coupling differently, as we first reduce to 5d, where the fields are rescaled by the corresponding radius r to ensure canonical scaling dimension. 12 The auxiliary fields can be integrated out to give such that the bosonic action can be written as (5.9) Note that the above derivation holds for curved M 3 by covariantizing the derivatives with respect to the spin connection on M 3 and the connection A of the SU (2) bundle V . Indeed, the energy momentum tensor T ab , the variation of S BFH with respect to the metric on M 3 , can be written in a Q-exact form 13 For the Euler characteristic to be well-defined the moduli space needs to be compact. This was shown to be the case for the 3d abelian Seiberg-Witten equations coupled to n spinors transforming under an SU (n)bundle in [46]. The moduli space of the non-abelian generalisation in 4d can be non-compact, but has a natural compactification analogous to the Uhlenbeck compactification of anti-self-dual instantons. This has been studied for the case of P U (N )-monopoles in [62,63], however to our knowledge the compactification of the moduli space of 3d non-abelian Seiberg-Witten equations has not been studied.

The Abelian BFH-Model
Let us look at the abelian case more closely, where we are able to perform concrete compu- This has the consequence that the partition function of the abelian theory is not completely invariant under metric deformations, but can jump, whenever the metric deforms to allow for harmonic spinors. Although this is not explicitly verified in the non-abelian case we expect this type of wall-crossing to persist in the more general case as well.
In the case of T 7 expressed as the product T 3 × T 4 , it is noted in [37] that there exist a family of flat G 2 -structures for which T 3 is an associative three-cycle with a deformation space of dimension four. In fact this deformation space is exactly the transverse T 4 , and one finds a confirmation of the conjecture (5.16). In this case, the twisted harmonic spinors simply correspond to the covariantly constant spinors on T 3 .
We note that in the well-studied case of 3d N = 2 supersymmetry the reduction of the 6d abelian theory differs from section 3.2 only in the scalar and spinor sectors. From the N = 2 topological twist and reduction on M 3 we obtain instead b 1 (M 3 ) chiral multiplets in addition to the centre of mass multiplet. The Witten index in this case is given by On the topological field theory side the BFH-model is replaced by a complex super-BF-model that localizes on complex flat G-connections. In the abelian case these coincide with real U (1)- 14 A similar situation arises in Donaldson theory where the moduli space is described by the pair (A, φ) satisfying F + = 0 , Dµφ = 0 .
The moduli space becomes non-compact for solutions with non-zero scalar field φ [65], which correspond to reducible connections. From the results of Uhlenbeck, there exists a natural compactification of the anti-selfdual instanton moduli space from which one computes the Donaldson invariants.
connections and the moduli space is T b 1 (M 3 ) × M T . Thus, its Euler characteristic coincides with the index (5.17).

S 3 -Partition Function and Chern-Simons-Dirac Theory
In this last section we will discuss a much less explored observable for 3d N = 1 theories, the S 3 -partition function. Much progress in computing this observable has been made for N = 2 thanks to localization results, however these seem to be not applicable in the minimal

4). We thus propose that the
The derivation of this proceeds by considering M5-branes on S 3 in a conformal supergravity background that preserves N = 1 supersymmetry. The circle-reduction along the fiber of the Hopf fibration S 1 → S 3 → S 2 gives 5d SYM on S 2 with radius where R S 2 is the radius of the two-sphere and r is the radius of the Hopf circle. For the N = 2 preserving background this reduction from 5d to 3d after non-abelianization was carried out in [14]. Due to the non-trivial geometry of the S 2 it is necessary to couple to conformal supergravity [66,67]. We determine the values of the background fields by solving the Killing spinor equations. The coupling to supergravity then leads to additional mass terms and interactions.
After determining the supergravity background we perform the topological twist on the equations of motion coupled to supergravity, which preserves two scalar supercharges on M 3 , and dimensionally reduce the theory on S 2 , to determine the 3d topological theory. Most of the technical details can be found in appendix D. In the following, we will summarize the salient features and the results.

Supergravity Background for 5d SYM
The 6d (2, 0) theory on M 3 × S 3 can be described in terms of a supergravity background of 5d SYM on M 3 × S 2 , that is obtained after dimensional reduction along the Hopf-fiber. Here M 3 will be as before, an associative three-cycle in a G 2 -manifold. The metric of the background for 5d SYM is where (θ, φ) are the spherical polar coordinates. As the Hopf fibration is non-trivial, there is a non-vanishing graviphoton C = cos 2 θ 2 dφ, in 5d with field strength where x, y are flat indices on S 2 , and ε xy is the rank 2 antisymmetric tensor such that G is proportional to the volume form of the unit two-sphere. In the following we determine the supergravity background fields, that ensure N = 1 supersymmetry for the theory along S 3 (or equivalently, two scalar supercharges along M 3 ).
We begin with the 5d N = 2 SYM coupled to background supergravity, which was derived in [68,67] from a dimensional reduction of the 6d N = (2, 0) tensor multiplet coupled to conformal supergravity [66]. such that the ansatz for the singlet can be written as where we have included a factor of r for later convenience.
The Killing spinor equations are solved in appendix D.1, and we find a one-parameter family of solutions, parametrized by v, of the form (6.14) The supersymmetry parameter is solved to be constant along M 3 and satisfies

5d SYM on M 3 × S 2
The action for 5d SYM on R 3 × S 2 in the supergravity background of section 6.1 is derived in appendix D.2 using the decomposition of the 5d fields as in (4.2). We find a one-parameter family of theories, where the masses of the fields, with the exception of the 3d gauge field and its superpartner, are dependent on the background parameter v. We note that the final action with v = 0 matches the action derived in [69] for M5-branes on R 1,2 × S 3 from an alternative, deconstruction point of view starting with the BLG-theory. Whereas in [69] the action was argued to be unique, here we find, by coupling to off-shell supergravity, that there is a one parameter family of solutions for this background. The key difference lies in the presence of the spinorial kinetic term for the field φ αα of the form This term is absent in [69], but can be included while preserving the same amount of supersymmetry by adding mass terms for the bosonic fields ϕ and φ αα and their superpartners. For the spinor φ αα on M 3 we obtain where D is the covariant derivative on the three-manifold with respect to the metric, with Riemann tensor R ab cd , the gauge connection A and the SU (2) V connection A, with field strength F. To cancel the contribution of these terms in the supersymmetry variation of the action it is necessary to introduce the terms No additional corrections are required for the other fields. In appendix D.3 the additional curvature terms are determined by turning on an R-symmetry gauge field, to cancel the spin-connection on M 3 , in the background supergravity. We find that the terms required for preserving supersymmetry on a curved three-manifold agree, however in order to solve the supergravity Killing spinor equations M 3 is required to be Einstein.
In the action the additional terms (6.18) can be combined with the covariantized kinetic term for φ αα using the Lichnerowicz-Weitzenböck formula for the twisted Dirac operator, derived in [70] Making use of this identity we generalize the action derived in section D.2 to curved M 3 (6.21) after imposing (6.15) on the supersymmetry parameter.

Reduction to Chern-Simons-Dirac Theory
The reduction of the action and supersymmetry variations proceeds by expanding the fields in terms of harmonics on S 2 , which are detailed in appendix D.4.1. For the gauge fields we first note that S 2 does not admit any non-trivial one-forms and therefore there are no zero modes for the gauge field A x . However, we need to integrate out F xy which sets it to (6.22) leading to an additional mass term for ϕ. Taking the zero mode of A b on the sphere leads to the 3d action where all higher modes become massive and decouple. In the limit r → 0 the kinetic term is correspondence is to hold, the topological theory has to be sensitive to the twisted harmonic spinors φ on M 3 . Even for the abelian theory it is clear that non-trivial d / D (M 3 , g) will result in additional scalar multiplets that contribute to the sphere partition function. This motivates us to consider the case v = 2, which has a massless twisted harmonic spinor. It would indeed be very nice to have another first principle way to fix v from the reduction in the supergravity background.
Recall that the 3d gauge field is massless for all values of v and its action is given by (6.23). The case v = 2 is the simplest one for which there is also a massless field coming from φ αα , namely the one that is constant along the S 2 . The reduction of its kinetic term is Crucially, the spinorial kinetic term is leading in the limit r → 0. The massless field content is completed by spinors λ (2,j) and scalars ϕ (1,m) as can be seen from the conditions (D.41) and (D.42). Since the kinetic term of λ couples to ξ in (6.20) the massive modes ξ a (2,j) have to be integrated out correctly. This same procedure was also used in [14] and will be written out in detail in appendix D.4.3 for the (slightly simpler) case v = 1. Crucially, the kinetic terms of the λ (2,j) become bosonic and scale with r. The same naturally happens for the kinetic terms of the scalars ϕ (1,m) . Here, we will only consider terms at leading order in r, so these fields do not receive a kinetic term. 15 Nevertheless, they still appear in the action to order O(r 0 ) in the form of non-vanishing Yukawa couplings. However, these fields do not couple to the gauge field or the bispinor and are thus non-dynamical. This interaction term thus vanishes on-shell. At leading order in r we are left with the action This theory is non-abelian Chern-Simons theory coupled to twisted harmonic spinors i.e. nonabelian Chern-Simons-Dirac theory. The equations of motion are given by the generalized that have already appeared at various points throughout this paper.

Chern-Simons-Dirac Partition Function and WRT-Invariants
As we have repeatedly stated, not much is known about the S 3 -partition function for 3d N = 1 theories. In fact this motivated studying the topological theory, which would under a 3d-3d correspondence compute this quantity in terms of the partition function on M 3 . Taking stock we should assess how concrete this proposal can be made. We have seen that the topological theory is a Chern-Simon-Dirac theory, whose equations of motion are the generalized Seiberg- e.g. [21], however the generalization that we find here seems to not have appeared so far in this literature. However recall that in a generic G 2 -manifold, the associatives will not have any deformations/obstructions (see the discussion in section 2.2). It is thus also of interest to consider the case when φ = 0 and the topological theory reduces to real Chern-Simons. In this case much more is known about three-manifold partition functions, which we will briefly summarize now.
The partition function of real Chern-Simons theory is a topological invariant of threemanifolds, the Witten-Reshetikhin-Turaev invariant [22,23]. In [23] an oriented three-manifold invariant was constructed, which was proposed to be equal to Witten's invariant, the partition function of real Chern-Simons theory. Analytic expressions of these have been computed for M 3 = S 3 and L(p, q = ±1) Lens spaces in [22,71]. For level k and gauge group G = SU (2) these are For general Lens spaces (and gauge groups) the partition function was determined in [72].
Futhermore for Seifert manifolds exact expressions were determined for any simply-laced gauge groups in [73][74][75]. For k = 1 our result suggests that these are the values of the S 3 -partition functions of the T N =1 [M 3 ] theories for M 3 without any deformations. Needless to say, this would be indeed very interesting to check. Perhaps a more accessible framework similar to the one applied for N = 1 AGT in [12] is to start with quotients of N = 2 theories for which the S 3 -partition function has been computed.

Discussions and Outlook
We proposed an N = 1 3d-3d correspondence relating two observables of 3d N = 1 theories Unfortunately not much is known about localization results for 3d theories preserving N = 1 supersymmetry, unlike their higher-supersymmetric cousins, which limits the scope of checks in the case of the S 3 -partition function. It would indeed be interesting to use the results in section 6 and either gain insight into the computation of S 3 -partition functions of 3d N = 1 through the correspondence with CS-theory, or to compare with direct computations by other means, e.g. from orbifolds of N = 2 theories. Thanks to some resurgence in interest in dualities in 3d theories without [76][77][78][79] and with minimal [80][81][82][83] supersymmetry, as well as new geometric constructions [84], further progress on 3d N = 1 theories may be on the horizon.
Another interesting direction to pursue is the relation of the N = 1 3d-3d correspondence to an N = 1 AGT type correspondence, much along the lines of [4], where the 3d theories are defects in the 4d N = 1 theories.
We have seen that there is a wall-crossing phenomenon for the Witten index and it would be interesting to further investigate this, in tandem with a better understanding of the moduli space of generalized Seiberg-Witten equations for metrics on M 3 , which admit twisted harmonic spinors. In the abelian theories T N =1 [M 3 , U (1)] these special metrics give rise to additional zero modes, which result in the vanishing of the Witten index, and one natural question is the non-abelian generalization of this.
One possible method of accessing this information is the following. It was conjectured in [85] that the Euler characteristic of the moduli space of solutions to the non-abelian 3d Seiberg-Witten equations, coupled to additional matter multiplets, is proportional to the Rozansky-Witten invariant [86] of M 3 . The set of equations in [85] take the form where T A I are the generators of G in representation R I , and N f is the number of flavors, and the generalized Seiberg-Witten equations (2.4) are obtained from (7.1) for N f = 2 and R I =Adj.
It is therefore natural to conjecture, when the bundle V is trivial, that the partition function of the BFH-model can be computed by the Rozansky-Witten invariant for the sigma-model on M 3 with target the Coulomb branch of a 3d N = 4 theory. The relevant 3d N = 4 theory is singled out, in that after the topological twist, its BPS equations should be given by the gSW equations. To make use of the proposal in [85] knowledge of the full non-perturbative corrections to the Coulomb branch of 3d N = 4 SU (N ) SYM in the presence of matter is required, which are not known in general. However, it would be interesting to explore this further for the case of two additional adjoint hypermultiplets in the light of the correspondence proposed in this paper.

A Conventions
In this appendix we summarize our conventions for indices, gamma matrices and spinors in this paper.

A.2 Gamma Matrix and Spinor Conventions
There are two types of gamma matrices involved in the calculations, describing either the space-time or the R-symmetry.
The 6d chirality matrix Γ = − a Γ a , the charge conjugation matrix C and the reality matrix B are given by and satisfy the relations The antisymmetrized product of gamma matrices is defined as where S n is the symmetric group of order n. The natural index structure of the gamma matrices is Γ α β acting on the spinors Ψ β . The spinor indices are raised and lowered by the charge conjugation matrix The spinors of SO(1, 5) L obey a symplectic-Majorana-Weyl condition which can be written in terms of the Dirac conjugate spinorΨ αm = Ψ * βm Γ 0 Our conventions for the 5d gamma matrices Γ a , where a = 0, · · · , 4, on R 1,1 × M 3 are In the Euclidean case on S 2 × M 3 we take the same gamma matrices (A.10) but with a Wick rotation on Γ 0 such that Γ Eucl 0 = σ 1 ⊗ 1 2 . For both the Minkowski and Euclidean gamma matrices the charge conjugation matrix is given by raising and lowering the spinor indices as

Gamma matrices of Sp(4) R
The R-symmetry of the full 6d theory is Sp(4) R ∼ = SO(5) R . The gamma matrices in the The index structure is γâ mn whereâ =1, · · · ,5 is the vector index of SO(5) R . The

A.3 Spinor Decomposition
In the following we will give the spinor decompositions used throughout this paper.
The symplectic-Majorana-Weyl condition can then be imposed by setting Ψ σ = Ψ 1 iΨ 2 , where the fields Ψ 1 and Ψ 2 are real. The spinor indices are raised and lowered by The gamma matrices reduce in the obvious way such that on the two three-dimensional spaces they are given by We will also use that the 6d Dirac operator acting on an (anti-)chiral spinor Ψ α ± decomposes as

R-symmetry Conventions
Under the decomposition of the R-symmetry Sp(4) R → SU (2) r × SU (2)  where H = dB, since H closes trivially. In terms of the scalar H xyz = −ε xyz h dual to the three-form this action gives rise to the equation of motion [87] h = 0, and we obtain a scalar field which is set to zero by its equation of motion. We will see from the supersymmetry variations that this scalar can be identified with the auxiliary field in the center of mass scalar multiplet.
By this direct reduction it is not possible to detect gauge fields with non-zero Chern-Simons term. The action of a single such field takes the form where g is the gauge coupling and k is the integer Chern-Simons level. In the presence of such a Chern-Simons term the gauge field acquires a mass [88] m = g 2 k 4π . (B.12) Despite the massive nature of the gauge field the theory is not completely trivial at low energies [16]. It was argued in [89] that the integral first homology group of the three-manifold is presented by the matrix of Chern-Simons levels in a U (1) R theory, where R = b 1 (M 3 ) + r.
In other words, the Chern-Simons levels induce the relation 13) where γ N are the generators of the full integral homology H 1 (M 3 , Z) as in (3.7). 16 Thus, given and we obtain a single massless scalar in 3d for ϕ constant on M 3 . As discussed in section 3.1, the fields φ αα are identified with sections of the normal bundle of M 3 . For general threemanifolds the additional SU (2) V -bundle is twisted with the spin bundle and can act nontrivially. Thus, the derivative acting on φ αα needs to be covariantized not only with respect to the spin connection on M 3 , but also an SU (2) V connection. The equation of motion for φ αα therefore takes the form where / D is the twisted Dirac operator on M 3 explicitly given by where ω a bc is the spin connection on M 3 and A aαβ is the SU (2) V connection. Since the kernels of / D and / D 2 are equal φ αα is expanded in a basis of twisted harmonic spinors ζ αα The number of massless modes is given by the dimension, 4. An auxiliary real scalar field h.
5. d / D (M 3 , g) real scalars φ i and spinors ρ σi , where i = 1, · · · , d / D (M 3 , g) is the number of twisted harmonic spinors on M 3 , which depends on the metric on M 3 .

C 3d N = 1 Supersymmetry
In this appendix we review the basic multiplet structure of 3d N = 1 in signature (− + +), following [90]. The gamma matrices are given by γ x = {iσ 1 , σ 2 , σ 3 }. We begin with defining coordinates θ σ , where σ = 1, 2, on the superspace. Throughout this discussion all spinors are of the form Ψ σ = Ψ 1 iΨ 2 with real components Ψ 1,2 , where the indices are raised and lowered as in (A.18). Next, define derivatives ∂ σ on the superspace with the convention Note that written as a spinors ∂ σ and θ σ fulfill the above reality condition on the components.
To define supersymmetry transformations the usual Poincaré algebra is extended by supercharges Q σ which obey where P x = −∂ x is the momentum operator generating translations. The supercharges are given by Since the partial derivative acting on the superspace coordinates is not invariant under Q, one defines a covariant derivative The fields invariant under supersymmetry are functions of space-time coordinates and the θ σ . Since the latter anticommute it is possible to expand the fields in powers of θ σ terminating at θ 2 . There are two types of superfields that we will discuss in more detail.

Scalar Multiplet
The scalar multiplet A ϕ contains a real scalar ϕ, a spinor λ σ and a real auxiliary scalar h Its transformation under the supersymmetry is generated by the supercharges Q σ where σ is an infinitesimal supersymmetry parameter. Direct calculation yields From these scalar multiplets one can build supersymmetric actions that are invariant under (C.7). Since D σ is invariant, every Lorentz invariant function of the superfields and its derivatives can be inserted into If we choose f (A, DA, · · · ) = − 1 2 (D σ A ϕ ) 2 then we obtain the kinetic term Interaction terms can be added by picking more general functions.

Vector Multiplet
Let us now couple such a scalar multiplet to a real gauge field A x . In order to preserve supersymmetry we can again make an ansatz for a vector multiplet V σ A consisting of the gauge field A x and a spinor ξ σ . Then the covariant derivative changes as to ensure gauge invariance using the same logic as for the non-supersymmetric case. In a particular gauge, the Wess-Zumino gauge, the vector multiplet and its associated gauge invariant field strength can be written as (C.11) The two simplest gauge invariant actions are (C.12) The first action describes the canonical kinetic term for a gauge field with gauge coupling g. The Chern-Simons term is unusual as it involves the vector multiplet, which is not gauge invariant on its own. However, for integer k the quantum theory is gauge invariant. This construction is unique to three dimensions and gives the field strength an effective mass.

D M5-branes on S 3
This appendix contains details of the reduction of M5-branes on S 3 in section 6.

D.1 The Killing Spinor Equations on S 2 × R 3
In this section we solve the Killing spinor equations and determine the resulting conditions on the supersymmetry parameter and the supergravity background.

D.1.1 Gravitini-variation
In our background, the non-vanishing components of the supersymmetry variation of the gravitini ψm a , are [67] δψm a = D a m + ir 2 We impose that the supersymmetry parameter solves the equation for the topological twist on M 3 (3.9), and therefore, using the spinor decompositions detailed in appendix A.3, the non-vanishing components can be written as where µ = 1, 2, α = 1, 2 andm = 1, 2 denote the spinor representations of SO(2) L , SO(3) M and SU (2) r , respectively. 17

Flat directions
Let us first consider the components of (D.1) along R 3 . Inserting the ansätze for the background fields and taking the supersymmetry parameter to be constant along R 3 , the Killing spinor equation reduces to which is solved for 5) and no further restrictions on µ arise.

S 2 directions
To solve the equations along the S 2 we have to include an explicit dependence of the supersymmetry parameter on the coordinates ξ = (θ, φ) . The non-vanishing contributions are Taking the two angles individually and inserting the singlet ansatz (D.3) for the supersymmetry parameter one obtains (D.7) 17 Here we do not decompose the spinors on S 2 into one component positive and negative chirality spinors labelled by ± as in section 4.1. Instead we will leave them as two component spinors for compactness. Similarly, we keep x = 1, 2 as the flat vector index on the S 2 .
which is equivalent to is the Dirac operator on the unit two-sphere. Since the operator (σ 3 / D S 2 ) has two eigenvectors with eigenvalue −1 we find that the background preserves two supercharges, which transform as scalars on M 3 .

D.1.2 Dilatino-variation
The non-vanishing components of the second Killing spinor equation, given by the supersymmetry variation of the dilatino χmnr, are where the curvature is given by The trace terms, determined in [66], ensure that the variation fulfills the symmetries of the dilatino Ωmnχmnr = χmnm = 0.
Using the ansätze for the background fields derived in section 6.1 the second Killing spinor equation is solved for where we used (D.5) for the second equality.
In this section we determine the 5d action coupled to off-shell supergravity fields using the ansätze determined in sections 6.1 and D.1. The field content of the SYM after the topological twist is given in (4.2). We find a one parameter family of actions where the masses of the twisted scalars (ϕ, φ αα ), and their superpartners, depend on the free parameter v.

Gauge field
The 5d action of the gauge field is given by The graviphoton C is only non-vanishing in the φ-direction on S 2 and therefore from the second term in the 5d action we obtain a Chern-Simons term on R 3 . The action thus reduces to S A = r 32π 2 d 3 x sin θdθdφ Tr where we have taken out the factor of r 2 4 in the metric on the S 2 in (6.3). From here on curved indices on S 2 are therefore raised and lowered using the metric on a unit radius two-sphere.

Scalars
The 5d action of the scalars is given by In the 5d theory the covariant derivative and mass term for the scalars are given by where D a = ∂ a + [A a , ·]. Inserting in the form of the background fields and the field decomposition under the topological twist the mass of ϕ takes the form For the spinor φ αα on M 3 there is an additional contribution to its mass besides the explicit 5d mass term in (D. 16), originating from the R-symmetry gauge field in the covariant derivative.
This term is proportional to V 2 and the combination of these two mass contributions gives The linear coupling with the R-symmetry gauge field, arising from the covariant derivative, in the scalar equations of motion gives rise to a spinorial kinetic term for φ αα . The full scalar action becomes (D. 19) From this action we observe that only for certain values of v are either ϕ or φ αα massless. We will see the same feature arising for the superpartners of these fields in the next section.

Fermions
The theory includes a set of fermions ρm whose 5d action is given by (D.21) Using the spinor decomposition ρ α m → λ µ ε αm + iξ µ a (σ a ) α β ε βm + ρ µαα , (D. 22) the kinetic terms and mass terms for the fermions take the form where D S 2 is the covariant derivative on S 2 with respect to the metric and the gauge field A x . The mass terms for the fermions include contributions from the explicit mass term in 5d and the coupling to the R-symmetry gauge field in the 5d covariant derivative. Note that the supersymmetry partners for the gauge fields on M 3 , corresponding to ξ σ a , are always massless, while the other fermions have v dependent mass terms.

Interactions
The final piece of the 5d action are the interaction terms given by

D.3 Generalization to Curved M 3
In defining the theory on general three-manifold the derivatives need to be covariantized with respect to the curvature on M 3 and the connection A of SU (2) as discussed in section 2.2.
In section 6.2 we added the correction terms (6.18) necessary to preserve supersymmetry.
Alternatively, in this section we use the supergravity background to derive the required curvature terms. Let us perform the twist using the R-symmetry gauge field in the supergravity multiplet. In this we keep the ansatz for the other two fields Tmn a b and Dmnrŝ the same and introduce the additional prefactors d twist and t twist where the curvature of the R-symmetry gauge field is defined in (D.11). Using the solution for V twist we can write the curvature as

D.4 Reduction on S 2
The next step is to reduce the action (6.20) on the S 2 , keeping only the massless modes on M 3 . To this end we expand the 5d fields in terms of eigenvectors of the relevant differential operators.

D.4.1 Spherical Harmonics and Eigenspinors on S 2
For the Laplacian ∆ S 2 = D x D x the eigenvectors are given by the spherical harmonics Y m k , where k is a non-negative integer and |m| ≤ k counts the multiplicity. They fulfill 35) and are normalized as The modified Dirac operator (σ 3 / D S 2 ) µ ν , with / D S 2 as in (D.9), has eigenvalues ±n, for n a positive integer. This can be seen by noticing that (σ 3 / D S 2 ) 2 = −( / D S 2 ) 2 and acknowledging that the usual Dirac operator on the unit sphere has eigenvalues ±in, see [91]. Let us call the corresponding eigenspinors Θ µ n,j , where the subscript j is the eigenvalue under i∂ φ leading to a degeneracy of 2n. They thus fulfill (σ 3 / D S 2 ) µ ν Θ ν n,j = nΘ µ n,j . whereas ξ always receives a mass. Since there is no closed form for the 3d theory for general v, we have to fix v first and then carry out the reduction. In section 6.3 it was argued that for the value v = 2 we obtain real Chern-Simons-Dirac theory. In the following we will carry out the reduction for the values v = 0, 1 in more detail and obtain real Chern-Simons theory.

D.4.3 Real Chern-Simons Theory
: v = 0, 1 In the case v = 1 the massless field content consists of the gauge field A b as well as a scalar field ϕ, which is constant on the S 2 , and two fermions λ ± . To obtain the correct action we also have to integrate out the massive fields ξ ± a andλ ± . In terms of the eigenspinors Θ µ n,j of the Dirac operator as in (D.38) these fields are defined by (D.43) All the modes coming from the fields φ αα and ρ µαα are massive and can safely be ignored.
The 3d action is then given by A very similar action has appeared in the case for N = 2 supersymmetry in [14]. There it was shown that the fields with kinetic terms scaling with r can be interpreted as ghosts. These can be integrated out to gauge fix the complex Chern-Simons action. In the case at hand we obtain exactly half the field content of [14]. We can thus again interpret ϕ and λ ± as gauge parameters. Consequently, we can safely take the limit r → 0, without fixing a gauge, and obtain real Chern-Simons theory