M5-branes on S^2 x M_4: Nahm's Equations and 4d Topological Sigma-models

We study the 6d N=(0,2) superconformal field theory, which describes multiple M5-branes, on the product space S^2 x M_4, and suggest a correspondence between a 2d N=(0,2) half-twisted gauge theory on S^2 and a topological sigma-model on the four-manifold M_4. To set up this correspondence, we determine in this paper the dimensional reduction of the 6d N=(0,2) theory on a two-sphere and derive that the four-dimensional theory is a sigma-model into the moduli space of solutions to Nahm's equations, or equivalently the moduli space of k-centered SU(2) monopoles, where k is the number of M5-branes. We proceed in three steps: we reduce the 6d abelian theory to a 5d Super-Yang-Mills theory on I x M_4, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M_4 is a Hyper-Kahler manifold, we show that the dimensional reduction gives rise to a topological sigma-model based on tri-holomorphic maps. Deriving the theory on a general M_4 requires knowledge of the metric of the target space. For k=2 the target space is the Atiyah-Hitchin manifold and we twist the theory to obtain a topological sigma-model, which has both scalar fields and self-dual two-forms.


Introduction
The six-dimensional N = (0, 2) superconformal theory (SCFT) with an ADE type gauge group is believed to describe the theory on multiple M5-branes. The equations of motion in six dimensions are known only for the abelian theory [1,2], and a Lagrangian formulation of this theory is believed to not exist. However, in the last few years, much progress has been made in uncovering properties of this elusive theory by considering compactifications to lower dimensions. Compactification of the 6d theory on a product S d × M 6−d has resulted in correspondences between supersymmetric gauge theories on d-dimensional spheres S d and conformal/topological field theories on a 6 − d dimensional manifold M 6−d . The goal of this paper is to consider the compactification of the 6d theory on a four-manifold M 4 times a twosphere S 2 and to determine the topological theory on M 4 . The particular background that we consider is a half-topological twist along the S 2 , together with a Vafa-Witten-like twist on M 4 , and we will find that the theory on M 4 is a twisted version of a sigma-model into the moduli space of SU (2) monopoles with k centers, where k is the number of M5-branes, or equivalently, the moduli space of Nahm's equations [3] with certain singular boundary conditions. This suggests the existence of a correspondence between this topological sigma-model on M 4 and a two-dimensional (0, 2) theory, with a half-twist. This fits into the correspondences studied in the last years, which we shall now briefly summarize.
For d = 4, the Alday-Gaiotto-Tachikawa (AGT) correspondence [4] connects 4d N = 2 supersymmetric gauge theories on S 4 with Liouville or Toda theories on Riemann surfaces M 2 . Correlation functions in Toda theories are equal to the partition function of an N = 2 supersymmetric gauge theory, which depends on the Riemann surface M 2 . Such 4d N = 2 gauge theories obtained by dimensional reduction of the 6d N = (0, 2) theories were first studied by Gaiotto in [5], generalizing the Seiberg-Witten construction [6]. For d = 3, a correspondence between 3d supersymmetric gauge theories, labeled by three-manifolds M 3 , and complex Chern-Simons theory on M 3 was proposed in [7,8], also refered to as the 3d-3d correspondence. This correspondence has a direct connection to the AGT correspondence by considering three-manifolds, which are a Riemann surface M 2 times an interval I, M 3 = M 2 × ϕ I, whose endpoints are identified modulo the action of an element ϕ of the mapping class group of M 2 . On the dual gauge theory side, the mapping class group action translates into a generalized S-duality, and the three-dimensional gauge theories, dual to complex Chern-Simons theory are obtained on duality defects in the 4d N = 2 Gaiotto theory. The 3d-3d correspondence was ultimately derived from a direct dimensional reduction of the 6d (0, 2) theory on a three-sphere via 5d by Cordova and Jafferis [9,10].
Other dimensional reductions concern the case of T d ×M 6−d . The circle-reduction is known to give rise to N = 2 5d Super-Yang-Mills (SYM) [11]. The case of d = 2 gives rise to N = 4 SYM with the Vafa-Witten twist [12] along M 4 [13], which yields a duality between a 2d N = (0, 2) gauge theory on T 2 and the Vafa-Witten theory on M 4 . Some results on twisted M5-branes have appeared in [14].
Both the AGT and 3d-3d correspondences uncovered very deep and surprising relations between supersymmetric gauge theories and two/three-manifolds, their geometry and moduli spaces. In view of this a very natural question is to ask, whether we can obtain insights into four-manifolds, as well as the dual two-dimensional gauge theories obtained by dimensional reduction of the 6d (0, 2) theory. Here, unlike the AGT case, the theory on the four-manifold is a topological theory, and the gauge theory lives in the remaining two dimensions and has (half-twisted) N = (0, 2) supersymmetry. A schematic depiction of this is given in figure 1.
More precisely, we propose a correspondence between a 4d topological sigma-model and a 2d half-twisted N = (0, 2) gauge theory. In particular we expect that topological observables in 6d (0,2) on S 2 x M 4 (0,2) SCFT on S 2 Topological -model on M 4 Vol(S 2 ) 0 Vol(M 4 ) 0 Figure 1: 4d-2d correspondence between the reduction of the 6d (0, 2) theory on M 4 to a 2d (0, 2) SCFT on S 2 , and the 'dual' 4d topological sigma-model from M 4 into the Nahm or monopole moduli space, which is obtained in this paper by reducing the 6d theory on a two-sphere.
the 4d theory can be mapped to the partition function and other supersymmetric observables of the 2d theory. Note that the S 2 partition function defined with the topological halftwist [15] is ambiguous as explained in [16]. However the analysis of counter-terms (and therefore ambiguities) must be revisited in the context of the embedding in 6d conformal supergravity, which is our set-up. In particular, the 2d counterterms should originate from 6d counter-terms. Recent results on localization in 2d (0, 2) theories have appeared in [17], albeit only for theories that have (2,2) loci. The theories obtained from the reduction in this paper do not necessarily have such a (2, 2) locus.
From a brane picture, the theory we consider can be obtained by compactifying k M5branes on a co-associated cycle in G 2 [18,19]. The two-dimensional theory that is transverse to the co-associative cycle has (0, 2) supersymmetry, and we consider this on a two-sphere, with an additional topological half-twist.
The first question in view of this proposal is to determine what the topological theory on M 4 is. There are various ways to approach this question. The simplest case is the abelian theory, which on S 2 × R 1,3 gives rise to a 4d free N = 2 hyper-multiplet [20], which we shall view as a sigma-model into the one-monopole moduli space. On a general four-manifold M 4 , we will show that in the topologically twisted reduction, the abelian theory integrates indeed to a "twisted version" of a hyper-multiplet, where the fields are a compact scalar and self-dual two-form on M 4 .
For the general, non-abelian case, this 4d-2d correspondence can in principle be connected to the 3d-3d correspondence by considering the special case of M 4 = M 3 × ϕ I, where I is an interval, similar to the derivation of the 3d-3d correspondence from AGT. In this paper we will refrain from considering this approach, and study instead the reduction via 5d SYM, in the same spirit as [9,10].  Figure 2: The dimensional reduction of the 6d N = (0, 2) theory on an S 2 , viewed as a circle-fibration along an interval I, is determined by dimensional reduction via 5d SYM. The scalars of the 5d theory satisfy the Nahm equations, with Nahm pole boundary conditions at the endpoints of the interval. The 4d theory is a topological sigma-model into the moduli space of solutions to these Nahm equations, or equivalently the moduli space of monopoles.
We first consider the dimensional reduction on flat M 4 , and then topologically twist the resulting 4d N = 2 theory. We restrict to the U (k) gauge groups, but in principle the analysis holds also for the D and E type. To determine the flat space reduction, we view the S 2 in terms of a circle-fibration over an interval, where the circle-fiber shrinks to zero-radius at the two endpoints. We determine the 5d supergravity background, which corresponds to the dimensional reduction of the 6d theory on S 2 . The resulting theory is 5d SYM on an interval, where the scalars satisfy Nahm pole boundary conditions [21,22]. Further dimensional reduction to 4d requires to consider scalars, that satisfy Nahm's equations. The resulting theory is a 4d sigma-model into the moduli space of solutions of Nahm's equations, which is isomorphic to the moduli space of k-centered monopoles [23] and has a natural Hyper-Kähler structure. Much of the geometry of the moduli space is known, in particular for oneor two-monopoles [24], and a more algebraic formulation in terms of Slodowy-slices exists following [25][26][27]. The latter description is particularly amenable for the characterization of N = 2 Gaiotto theories with finite area for the Riemann surface as studied in [22]. Figure 2 summarizes our dimensional reduction procedure.
The 4d N = 2 supersymmetric sigma-model for flat M 4 falls into the class of models obtained in [28,29]. We find that the coupling constant of the 4d sigma-model is given in terms of the area of the two-sphere. To define this sigma-model on a general four-manifold requires topologically twisting the theory with the R-symmetry of the 4d theory. One of the complications is that the SU (2) R-symmetry of the 4d theory gets identified with an SU (2) isometry of the Hyper-Kähler target. The twisting requires thus a precise knowledge of how the coordinates of the monopole moduli space transform under the SU (2) symmetry. This is known only in the case of one-and two-monopoles, where a metric has been determined explicitly [24]. In these cases, we shall describe in section 6 the topological sigma-models, which have both scalars and self-dual two-form fields on M 4 . The sigma-model into the onemonopole moduli space S 1 ×R 3 , corresponding to the reduction of the abelian theory to a free 4d hypermultiplet, gives rise upon twisting to a (free) theory on M 4 with a compact scalar and a self-dual two-form, and belongs to the class of 4d A-model of [30]. The sigma-model into the two-monopole moduli space, which is closely related to the Atiyah-Hitchin manifold, gives rise to an exotic sigma-model of scalars and self-dual two-forms obeying constraints. Sigma-models in 4d are non-renormalizable and infrared free, however, the observables of the topologically twisted theory are independent of the RG flow and can in principle be computed in the weak coupling regime.
In the case of M 4 a Hyper-Kähler manifold, the holonomy is reduced and the twisting does not require knowledge of the R-symmetry transformations of the coordinate fields. This is discussed in section 5.1, and the topological sigma-model that we find upon twisting is the one studied in [31] by Anselmi and Frè for almost quaternionic target spaces.
In this paper we focus on the reduction of the 6d (0, 2) theory on a two-sphere, however, as we emphasize in section 3, the reduction would proceed in the same way with the addition of two arbitrary 'punctures' on the two-sphere, characterizing BPS defects of the 6d non-abelian theory. In the intermediate 5d theory, it would result in different Nahm-pole boundary conditions for scalar fields at the two ends of the interval and the final flat space four-dimensional theory would be a sigma-model into the moduli space of solutions of Nahm's equations with these modified Nahm-pole boundary conditions. We should also remark upon the connection of our results to the paper by Gadde, Gukov and Putrov [13], who consider the torus-reduction of the M5-brane theory. The topological twist along M 4 is the same in their setup as in our construction. Thus, the dictionary to the data of the 2d theory as developed in [13], such as its dependence on the topological/geometric data of M 4 , should hold in our case as well. For instance, the rank of the 2d gauge group is determined by b 2 (M 4 ). The key difference is however, that we consider this 2d theory on S 2 , and topologically twist the chiral supersymmetry. Interestingly, the reduction of the 6d theory on either T 2 or S 2 with half-twist gives rather distinct 4d topological theories: in the former, the 4d N = 4 SYM theory with Vafa-Witten twist, in the latter, we find a four-dimensional topological sigma-model into the monopole moduli space, which for general M 4 has both scalars as well as self-dual two-forms. The appearance of self-dual two-forms is indeed not surprising in this context, as the topological twist along M 4 is precisely realized in terms of M5-branes wrapping a co-associative cycle in G 2 , which locally is given in terms of the bundle of self-dual two-forms Ω 2+ (M 4 ) [32].
The plan of the paper is as follows. We begin in section 2 by setting up the various topological twists of the 6d N = (0, 2) theory on S 2 × M 4 , and provide the supergravity background and Killing spinors, for the S 2 reduction with the half-twist. In section 3 we dimensionally reduce the 6d theory to 5d SYM on an interval times R 4 , with Nahm pole boundary conditions for the scalar fields. In particular we study this with a generic squashed metric on S 2 and in a special 'cylinder' limit. The reduction to 4d is then performed in section 4, where we show that the fields have to take values in the moduli space of Nahm's equations, and determine the N = 2 supersymmetric sigma-model on R 4 . The action can be found in (4.30), as well as in the form of the models of [28,29] in (4.35). In sections 5 and 6 we study the associated topological sigma-models: in section 5 we consider the case of M 4 a Hyper-Kähler manifold, and show that this gives rise to the topological sigma-model in [31]. The action can be found in (5.12). We furthermore connect this to the dimensional reduction of the topologically twisted 5d SYM theory and show that both approaches yield the same 4d sigma-model in appendix F. In section 6, we let M 4 be a general four-manifold, but specialize to the case of one-or two-monopole moduli spaces, and use the explicit metrics to determine the topological field theory. In this case, the bosonic fields are scalars and self-dual two-forms on M 4 . The action for k = 1 is (6.12) and for k = 2 we obtain (6.35). We close with some open questions in section 7, and provide details on our conventions and computational intricacies in the appendices.

Topological Twists and Supergravity Backgrounds
This section serves two purposes: firstly, to explain the possible twists of the 6d N = (0, 2) theory on a two-sphere S 2 , and secondly, to determine the supergravity background associated to the topological half-twist on S 2 .

Twists of the M5-brane on M 4
We consider the compactification of the M5-brane theory, i.e. the six-dimensional N = (0, 2) theory, on M 4 ×S 2 , where M 4 is a four-dimensional manifold. More generally, we can consider the twists for reductions on general Riemann surfaces Σ instead of S 2 . We will determine the 4d theory that is obtained upon dimensional reduction on the S 2 , and consider this theory on a general four-manifold M 4 . Supersymmetry of this theory requires that certain background fields are switched on, which correspond to twisting the theory -both along M 4 as well as along S 2 . The twisting procedure requires to identify part of the Lorentz algebra of the flat space theory with a subalgebra of the R-symmetry. The R-symmetry and Lorentz algebra of the M5-brane theory on R 6 are 1 sp(4) R ⊕ so(6) L . (2.1) The supercharges transform in the (4,4) spinor representation (the same representation as the fermions in the theory, see appendix A). The product structure of the space-time implies that we decompose the Lorentz algebra as We can consider the following twists of the theory along M 4 . Either we identify an su (2) subalgebra of both Lorentz and R-symmetry, or we twist with the full so(4).
On M 4 there are two su(2) twists that we can consider. In the first instance consider the decomposition of the R-symmetry as and the su (2) is twisted by su(2) R . That is we replace su(2) by the diagonal su(2) twist ⊂ su(2) ⊕ su(2) R and define the twisted su(2) generators by so that the twisted theory has the following symmetries This twist is reminiscent of the Vafa-Witten twist of 4d N = 4 SYM [12]. The supercharges decompose under (2.2) and (2.3) as which after the twist becomes This yields two scalar supercharges on M 4 , which are of the same negative 2d chirality under so(2) L Upon reduction on M 4 , this twist leads to a 2d theory with N = (0, 2) supersymmetry. In this paper we are not concerned with the reduction on M 4 , but focus on the reverse, namely the theory on M 4 . This twist is compatible with a further twist along S 2 or more generally an arbitrary Riemann surface Σ, which identifies so(2) L with the remaining R-symmetry so(2) R .
This is the setup that we will study in this paper. In the following we will first perform the reduction (and topological twisting) along the S 2 , and then further twist the resulting four-dimensional theory on M 4 .
Finally, let us briefly discuss alternative twists. We can use a different su(2) R-symmetry factor to twist the theory along M 4 , namely we can use su (2) This twist leads upon reduction on M 4 to a 2d theory with N = (0, 1) supersymmetry.
We can in fact further twist the su(2) 2 with the remaining su(2) r Lorentz symmetry on M 4 . This corresponds to a total twist of the full so(4) R with so(4) L and is analogous to the geometric Langlands (or Marcus) twist of 4d N = 4 SYM theory on M 4 [33,34] Twist 3 : which has two scalar supercharges of opposite 2d chiralities so that this twist leads upon reduction on M 4 to a 2d theory with N = (1, 1) supersymmetry.
It is not compatible with a further topological twist on S 2 . Interestingly it was found in [35] that supersymmetry can be preserved by turning on suitable background supergravity fields on M 4 . We will not study this background in this paper, but will return to this in the future.
We will now consider the setup of twist 1 and carry out the reduction of the 6d N = (0, 2) theory on S 2 ×M 4 . As explained in the introduction our strategy is to find the 6d supergravity background corresponding to the twisted theory along S 2 , taking M 4 = R 4 to begin with, and carry out the reduction to 4d, where we will finally twist the theory along an arbitrary M 4 .

Twisting on S 2
For our analysis we first consider the theory on S 2 × R 4 and the twist along S 2 . The Lorentz and R-symmetry groups reduce again as in (2.2) and (2.3). The twist is implemented by identifying so(2) R with so(2) L and we denote it so(2) twist u(1) twist , whose generators are given by As we have seen this is compatible with the twist 1, discussed in the last subsection. 14) The residual symmetry group and decomposition of the supercharges and fermions is then There are eight supercharges transforming as singlets on S 2 and transforming as Weyl spinors of opposite chirality on M 4 and doublets under the remaining R-symmetry. The fields of the 6d (0, 2) theory decompose as follows (2.16) Note from the point of view of the 4d N = 2 superalgebra, some of these fields transform in hyper-multiplets, however with a non-standard transformation under the R-symmetry, under which some of the scalars form a triplet. The standard transformation of the hyper-multiplet can be obtained using an additional SU (2) symmetry [36]. However, in the present situation, we have to use the R-symmetry as given in the above decomposition. Twisting with the su (2) Lorentz with the remaining su(2) R , i.e.
the resulting topological theory has the following matter content (2. 18) In the following it will be clear that the 6d scalars Φ give rise to scalars and a self-dual twoform on M 4 . The fermions give rise to either vectors, or scalars and self-dual two-forms as well. The fields appearing in the decomposition of the two-form B are not all independent due to the constraint of self-duality of H = dB. They will give rise to a vector field and a scalar on M 4 . This matter content will be visible in the intermediate 5d description that we reach later in section 3, however, after reducing the theory to 4d and integrating out massive fields, the matter content of the final 4d theories will be different.

Supergravity Background Fields
Before describing the details of the reduction, we should summarize our strategy. Our goal is to determine the dimensional reduction of the 6d (0, 2) theory with non-abelian u(k) gauge algebra. For the abelian theory, the dimensional reduction is possible, using the equations of motions in 6d [1,2]. However, for the non-abelian case, due to absence of a 6d formulation of the theory, we have to follow an alternative strategy. Our strategy is much alike to the derivation of complex Chern-Simons theory as the dimensional reduction on an S 3 in [10]. First note, that the 6d theory on S 1 gives rise to 5d N = 2 SYM theory. More generally, the dimensional reduction of the 6d theory on a circle-fibration gives rise to a 5d SYM theory in a supergravity background [9] (for earlier references see [37,38]). This theory has a non-abelian extension, consistent with gauge invariance and supersymmetry, which is then conjectured to be the dimensional reduction of the non-abelian 6d theory.
More precisely, this approach requires first to determine the background of the 6d abelian theory as described in terms of the N = (0, 2) conformal supergravity theory [39,40]. The 5d background is determined by reduction on the circle fiber, and is then non-abelianized. We can further reduce the theory along the remaining compact directions to determine the theory in 4d. For S 3 , there is the Hopf-fibration, used in [10] to derive the Chern-Simons theory in this two-step reduction process. In the present case of the two-sphere, we will fiber the S 1 over an interval I, and necessarily, the fibers will have to become singular at the end-points.
In the following we will prepare the analysis of the supergravity background. By requiring invariance under the residual group of symmetries g res preserved by the topological twist on S 2 , we derive ansätze for the background fields in 6d N = (0, 2) off-shell conformal supergravity fields. In the next section we will consider the Killing spinor equations and fix the background fields completely.
To begin with, the 6d metric on S 2 × R 4 is given by Auxiliary 3-form 5 with (θ) = r sin(θ) for the round two-sphere and θ ∈ I = [0, π]. More generally, (θ) can be a function, which is smooth and interpolates between We choose the frame The corresponding non-vanishing components of the spin connection are and also for the R-symmetry The bosonic supergravity fields of 6d off-shell conformal maximal supergravity were determined in [9,37,[39][40][41]. They are the frame e A µ and where dV and db denote the field strength of the R-symmetry and dilatation gauge fields, We shall now decompose these in turn under the residual symmetry group g res ∼ = su(2) ⊕ su(2) r ⊕ su(2) R ⊕ u(1) twist and determine the components that transform trivially, and thus can take non-trivial background values.

T [BCD]
A : The decomposition under g res is given by

(2.28)
We see that we have a singlet that corresponds to turning on a flux on the S 2 and an ansatz for V is given by where x, y run over the components B, C = 4, 5, and the other components of V vanish.
3. b A : The field strength (db) [AB] decomposes under g res as There is a singlet, which corresponds to turning on a field strength on the S 2 . In the following we will not consider this possibility. Note that any other choice can always be obtained by a conformal transformation with K, which shifts b A [40]. In the following we thus set The decomposition under g res is given by There is one singlet corresponding to the ansatz with other components vanishing. The relative coefficients are fixed by the tracelessness condition on D ( A B) .

Killing spinors
where for the round two-sphere (θ) = r sin(θ), and the Killing spinor is constant and satisfies the following constraint The value of the R-symmetry gauge field V 56 = − (θ) r dφ = ω 56 and the fact that the preserved supersymmetries are generated by constant spinors indicates that this supergravity background realizes the topological twist on S 2 , as expected.
Finally, recall that we chose a gauge for which b A = 0. Note that the background field b A can be fixed to an arbitrary other value by a special conformal transformation (see [40]). The special conformal transformation does not act on the other background fields (they transform as scalars under these transformations), nor on the spinor m , however it changes the spinor In this way one can recover the gauge choice b µ = α −1 ∂ µ α (with α = 1/ in our conventions) of [9], although we will keep our more convenient choice b µ = 0. For our gauge choice, the dimensional reduction to 5d is rederived in appendix C.
3 From 6d (0, 2) on S 2 to 5d SYM We now proceed with the dimensional reduction of the six dimensional N = (0, 2) theory on S 1 to obtain 5d maximally supersymmetric Yang-Mills theory, as in [9,37]. The main distinction in our case arises in subtle boundary conditions, which will have to be imposed on the fields along the 5d interval. All our conventions are summarized in appendix A. We should remark on an important point in the signature conventions: the reduction to the 5d SYM theory is accomplished in Lorentzian signature, R 4 → R 1,3 , where fields admit 6d reality conditions, however it would go through in Euclidean signature upon complexifying the fields in 6d and then imposing reality conditions in 5d. This amounts to Wick-rotating the Lorentzian 5d theory. In later sections, when we study the 5d theory on a generic M 4 , we adopt the Euclidean signature, which is compatible with the twist on M 4 .

The 6d (0, 2) Theory
The abelian 6d N = (0, 2) theory contains a tensor multiplet, which is comprised of a twoform B with field strength H = dB, five scalars Φ m n , and four Weyl spinors ρ m m of negative chirality, which are symplectic Majorana. The scalars satisfy Φ m n = −Φ n m and Ω m n Φ m n = 0. The equations of motion are (we will use the conventions of [40]) T m n µνσ Γ µνσ ρ n = 0 . (3.1) Here H ± = 1/2(H ± H) and the R-symmetry indices of the background fields have been transformed from A → m n using the Gamma-matrices as in (B.3). The covariant derivatives are defined as follows Here R 6d is the 6d Ricci scalar. These equations are invariant under the following supersymmetry transformations The dimensional reduction of these equations yields abelian 5d SYM in a general supergravity background. We will perform this reduction in a gauge choice where b A = 0, which is for instance different from the choice in [9]. The details of this general reduction are given in appendix C. The 6d supergravity fields decompose as follows where we used again the index conventions in appendix A. The action of abelian 5d SYM theory in a general background is where with all mass matrices defined in appendix C andF is defined as

5d SYM in the Supergravity Background
We can now specialize to the 6d background R 4 × S 2 , including the background supergravity fields of section 2 and determine the 5d SYM theory in the background, which corresponds to the 6d (0, 2) theory on S 2 , by performing the dimensional reduction along the circle fiber. As shown in section 2.3, the only background fields for the 5d SYM theory, which are compatible with the residual symmetry group, are D m n r s and V m n φ ≡ S m n . With these background fields, and the action of the 5d SYM theory in a general background, that we derived in appendix C in the gauge b A = 0, we can now determine the non-abelian 5d action in our background.
For our background the metric, graviphoton, C A , and the dilaton, α, are given by which means that G = dC = 0. Imposing these conditions and turning on only the background fields D m n rs and S m n the full action is given by (3.10) Here, we non-abelianized the theory, and the covariant derivatives and mass matrices where the five dimensional Ricci scalar vanishes, because we have a flat metric on the interval.
In the non-abelian case we can add the following interaction terms where the non-vanishing background fields are where and denote first and second derivatives of with respect to θ. The action is invariant under the following supersymmetry transformations 4 (3.14) Note that the Killing spinor m m satisfies the relation (2.35) which now reads So far we have kept the sp(4) R R-symmetry indices explicit. However the background breaks the R-symmetry to su(2) R ⊕ so(2) R . To make the symmetry of the theory manifest, we decompose the scalar fields Φ m n into a triplet of scalars ϕ a , transforming in the 3 0 of su(2) R ⊕ so(2) R , and the complex field ϕ, which is a singlet 1 1 . This can be achieved as follows The spinors ρ m decompose into the two doublets ρ p , transforming in (2) 1 ⊕ (2) −1 , as detailed in appendix A.3. We also split the gauge field (singlet of the R-symmetry) into the components A µ along R 4 and the component A θ along the interval.
The spinor n parametrizing supersymmetry transformations decomposes under the R- p (see appendix A.3). The projection condition (3.15) becomes For any 5d spinor χ we define as the four-dimensional chirality. The action for the gauge field is and for the scalars we find with the mass term which for the round sphere is m 2 ϕ = cot(θ) 2 /r 2 and diverges at the endpoints of the interval. We will return to this matter when discussing the boundary conditions. The action for the fermions is Finally, the interaction terms in this decomposition read as follows (3.23) The complete 5d action is 24) and the supersymmetry variations for this action, decomposed with regards to the R-symmetry, are summarized in appendix D. The action above should be supplemented with appropriate boundary terms, which ensure that supersymmetry is preserved and that the action is finite.
This will be addressed subsequently.
We need to determine the boundary conditions of the 5d fields at the endpoints of the θ interval. To proceed we first notice that the complex scalar ϕ has a mass term m(θ) 2 which diverges at the boundaries θ = 0, π 5 (3.25) Finiteness of the action requires that ϕ behaves as (3.26) The boundary conditions on the other fields are most easily determined by the requirement of preserving supersymmetry under the transformations generated by (1) p and (2) p presented in appendix D. We obtain at θ = 0: 27) and the counterpart at θ = π. The fields ϕ a , A θ are constrained by supersymmetry to obey modified Nahm's equations as they approach the boundaries, given by These equations are compatible with a singular boundary behaviour of the fields at the endpoints of the θ-interval. For simplicity let us assume the gauge A θ = 0 in a neighborhood of θ = 0, then the above modified Nahm's equations are compatible with the polar behavior at θ = 0 denotes a Lie algebra homomorphism from su(2) to u(k), see e.g. in [21,22] and τ a are related to the Pauli matrices σ a as follows Moreover the O(1) term is constrained to be in the commutant of in u(k). The reduction that we study, from a smooth two-sphere to the interval, corresponds to being an irreducible embedding [22].
More generally the Nahm pole boundary condition (3.28) is compatible with any homomorphism and is associated with the presence of 'punctures' -or field singularities -at the poles of the two-sphere in the 6d non-abelian theory [5]. An embedding can be associated to a decomposition of the fundamental representation k under su(2) and can be recast into a partition [n 1 , n 2 , · · · ] of k. The irreducible embedding is associated to the partition = [k] and corresponds to the absence of punctures in 6d, and is therefore the sphere reduction that we consider here. The boundary conditions at θ = π are symmetric to the ones at θ = 0 and are also characterized by Nahm pole behaviour with irreducible embedding = [k].
The remaining fermions ρ + appear in the supersymmetry variations of ϕ a and hence 32) and similarly at θ = π.
The boundary condition (3.29) for the scalars ϕ a introduces two difficulties: the supersymmetry variation of the action results in a non-vanishing boundary term and the poles of the scalar fields make the action diverge. These two problems are cured by the addition of the following boundary term The second line gives S bdry as a total θ-derivative and we shall take this as the definition of the boundary term. This additional term ensures supersymmetry and makes the 5d action finite. In particular, taking the derivative along θ we find, where the first piece cancels the cubic scalar interaction in the 5d action and the second term combines to give which is the square of modified Nahm's equations. The 5d action is finite since the scalar fields ϕ a obey modified Nahm's equations at the boundaries.
We notice that the modified Nahm's equations (3.28) can be recast into the form of standard Nahm's equations by a change of coordinate to and a similar Nahm pole behavior at the other end of the θ interval. We conclude then that the moduli space of solutions of the modified Nahm's equations is the same as the moduli space of solution of the standard Nahm's equations.

Cylinder Limit
For general hyperbolic Riemann surfaces, with a half-topological twist, the dimensional reduction depends only on the complex structure moduli [5]. The two-sphere has no complex structure moduli, however, there will be a metric-dependence in terms of the area of the sphere, which enters as the coupling constant of the 4d sigma-model [22]. We do not expect the reduction to depend on the function (θ), except through the area of the sphere. This can be checked explicitly by performing the reduction keeping (θ) arbitrary. However, for simplicity we consider here the special singular limiting case, when the two-sphere is deformed to a thin cylinder. This is achieved by taking the metric factor (θ) as follows (θ) → smooth caps for θ < , π − < θ , and then taking the limit → 0. The limit is singular at the endpoints of the θ-interval, since at finite , the two-sphere has smooth caps, (θ) ∼ rθ, while at = 0, (θ) = is constant on the whole θ interval and describes the metric on a cylinder, or a sphere with two punctures.
One may worry that such a singular limit is too strong and would change the theory itself. We will argue below in section 3.4 that the reduction of the theory with constant leads to the same four dimensional sigma model as for arbitrary (θ). The reason for choosing constant is only to simplify the derivation.
We rescale the fields as follows (3.38) The action in this limit simplifies to (3.39) The supersymmetry variations of the 5d action summarized in appendix D simplify in the cylinder limit and for the bosonic fields are and for the fermions (3.41) The theory we obtain is nothing else than the maximally supersymmetric N = 2 SYM in 5d. A similar reduction of the 6d (0,2) theory on a cigar geometry was considered in [21].
This five-dimensional SYM theory is defined on a manifold with boundaries, which are at the end-points of the θ-interval and half of the supersymmetries are broken by the boundary conditions. It is key to study the boundary terms and boundary conditions in detail, which will be done in the next subsection.

Nahm's Equations and Boundary Considerations
The boundary conditions at the two ends of the θ interval are affected by the singular cylinder limit. They can be worked out in the same way as in section 3.2 by enforcing supersymmetry at the boundaries. In the cylinder limit of the two-sphere (θ) → the mass term m(θ) 2 goes to zero everywhere along the θ-interval except at the endpoints θ = 0, π where it diverges, forcing the scalar ϕ to vanish at the boundary, as before. The other boundary conditions are found by requiring supersymmetry under the eight supercharges. This requires that the scalars ϕ a obey the standard Nahm's equations close to the boundaries Furthermore, the boundary behavior of the fields in the gauge A θ = 0 around θ = 0 are (although this is not the gauge we will choose later) where : su(2) → u(k) is an irreducible embedding of su(2) into u(k), with τ as in (3.31) .
There are similar boundary conditions at θ = π. The constant term ϕ a (0) in the ϕ a -expansion is constrained to be in the commutant of embedding . With = [k] the irreducible embedding, this commutant is simply the diagonal u(1) ⊂ u(k), so ϕ a (0) is a constant diagonal matrix. This condition propagates by supersymmetry to the other fields.
The maximally supersymmetric configurations are vacua of the theory preserving eight supercharges and are given by solutions to the BPS equations We can now address the validity of the singular cylinder limit (θ) = constant. In the following we will reduce the theory on the interval and find that the dominant field configurations are given by solutions of Nahm's equations. The resulting four-dimensional theory will be a sigma model into the moduli space of solutions of Nahm's equations. It is easy to see that for arbitrary (θ) describing a smooth two-sphere metric, the same dimensional reduction will be dominated by field configurations satisfying the modified Nahm's equations (3.28). We can then reasonably expect that the reduction will lead to a four-dimensional sigma model into the moduli space of the modified Nahm's equations. However we argued at the end of section 3.2 that this moduli space is the same as the moduli space of standard Nahm's equations, so the reduction for arbitrary (θ) would lead to the same sigma model.
Finally, let us comment on generalizations of the Nahm pole boundary conditions with two arbitrary partitions 0 and π for the scalar fields at the two boundaries θ = 0, π, respectively, as described in [22]. The polar boundary behavior at θ = 0 is given by (3.43) with → 0 and the subleading constant piece ϕ a (0) takes values in the commutant of 0 (i.e. matrices commuting with the image of 0 ). These boundary conditions preserve the same amount of supersymmetry and admit global symmetry groups H 0 ×H π ⊂ SU (k)×SU (k) acting by gauge transformations at the end-points of the θ-interval. H 0 and H π are the groups, whose algebras h 0 , h π are respectively the commutants of 0 and π in su(k). These global transformations leave the 0 and π boundary conditions invariant. In the reduction to 4d, only a subgroup of H 0 × H π can be preserved (see the discussion in section 2 of [22]).
The general ( 0 , π ) boundary conditions correspond to inserting singularities or 'punctures' of the type 0 at one pole of the two-sphere and of the type π at the other pole in the 6d (0, 2) theory. All our results can be directly generalized to having general ( 0 , π ) Nahm poles at the boundaries of the θ-interval. In this case we would obtain sigma-models into a different moduli space: the moduli space of Nahm's equations with ( 0 , π ) boundary conditions.
For the sphere with two punctures labeled by two arbitrary partitions 0 , π , it is very natural to consider the metric describing a cylinder, since this is the topology of a sphere with two punctures, and the reduction, whether with the sphere or the cylinder metric, is expected to lead to the same four-dimensional theory. From this point of view, the sphere without punctures, or "trivial punctures", is simply a subcase corresponding to the specific partitions

Nahm's Equations and 4d Sigma-Model
In the last section we have seen that the 5d SYM in the background corresponding to the S 2 reduction of the 6d (0, 2) theory requires the scalars ϕ a to satisfy Nahm's equations, and the supersymmetric boundary conditions require them to have Nahm poles (3.43) at the boundary of the interval. The four-dimensional theory is therefore dependent on solutions to Nahm's equations. To dimensionally reduce the theory, we pass to a description in terms of coordinates on the moduli space M k of solutions to Nahm's equations and find the theory to be a four-dimensional sigma-model into M k with the action with X I the coordinates on the moduli space and ξ (i) , where i = 1, 2, Grassmann-valued sections of the pull-back of the tangent bundle to where S ± is the spin bundle of ± chirality on M 4 . The sigma-model for M 4 = R 4 is supersymmetric, with N = 2 supersymmetry in 4d. The coupling constant for the sigma-model is proportional to the area of the two-sphere, which is ∼ r , as anticipated.

Poles and Monopoles
Before studying the dimensional reduction to 4d, we summarize a few well-known useful properties of the moduli space M k . The moduli space M k of solutions to Nahm's equations, on an interval with Nahm pole boundary conditions given by the irreducible embedding = [k], is well-known to be isomorphic to the moduli space of (framed) SU (2) magnetic monopoles of charge k [23,24,42,43], which is 4k-dimensional and has a Hyper-Kähler structure. The metric of the spaces M k is not known in explicit form, other than for the cases M 1 R 3 × S 1 (which is the position of the monopole in R 3 and the large gauge transformations parametrized by S 1 ) and for the case where M AH is the Atiyah-Hitchin manifold [24]. A detailed description of the metric in the latter case will be given in section 6.2. Hitchin showed the equivalence of SU (2) monopoles of charge k with solutions of Nahm's equations [43] dT where T i are matrix-valued, depending on θ ∈ [0, π] and have poles at the endpoints of the interval, the residues of which define representations of su (2). Furthermore, Donaldson [23] identified Nahm's equations in terms of the anti-self-duality equation on R 4 , where T θ , the gauge field along the interval, can be gauged away and the T i are taken independent of the x i coordinates. The metric of the solution-space (modulo gauge transformations) has a Hyper-Kähler structure [44,45]. This Nahm moduli space (or monopole moduli space) takes the form [24] where R 3 parameterizes the center of mass of the k-centered monopole. A particularly useful characterization of the reduced Nahm moduli space M 0 k is in terms of Slodowy-slices. Kronheimer has shown that the solutions of Nahm's equations with no poles at the boundaries have a moduli space given by the cotangent bundle of the complexified gauge group, T * G C ≡ g C × G C , which has a natural Hyper-Kähler structure. Furthermore, Bielawski showed in [26,27], that the moduli space of solutions with Nahm pole boundary conditions for k-centered SU (2) monopoles is given in terms of where the Slodowy slice for an embedding ρ : su(2) → u(k) is Here τ ± ≡ τ 1 ± iτ 2 are the raising/lowering operators of su (2). The Hyper-Kähler metric on M k will play a particularly important role in section 6, where this will be discussed in more detail.

Reduction to the 4d Sigma-Model
To proceed with the reduction on the θ-interval to four dimensions, we take the limit where the size of the interval, r, is small. 6 The terms in the action (3.39) are organized in powers of r, and in the limit, the divergent terms which are of order r −n , n = 2, 3, must vanish separately. The terms of order r −1 contain the four-dimensional kinetic terms and lead to the 4d action. The terms of order r n , n ≥ 0 are subleading and can be set to zero. To perform this reduction we must expand the fields in powers of r, Φ = Φ 0 + Φ 1 r + Φ 2 r 2 + · · · , and compute the contribution at each order. We find that only the leading term Φ 0 contributes to the final 4d action for each field, except for the 'massive' scalars ϕ,φ and spinors ρ − p , whose leading contribution arise at order r. The final 4d action will arise with the overall coupling 1 r . Let us now proceed with detailing the dimensional reduction. At order r −3 we find the Furthermore, we choose the gauge fixing (4.14) The terms at O(r −2 ) vanish by imposing the spinors ρ The kinetic term of these spinors becomes of order r and can be dropped in the small r limit.
The fermions ρ p − become Lagrange multipliers and can then be integrated out, leading to the constraints on the fermions ρ − q ] = 0 , which are supersymmetric counterparts to Nahm's equations (3.42). We will use these localizing equations below to expand the fermionic fields in terms of vectors in the tangent space to the moduli space of Nahm's equations, M k .
Finally we drop the order r kinetic terms of the 4d gauge field and scalars ϕ,φ (which contribute only at order r), and we are left with the terms of order 1 r which describe the 4d action. The remaining task is to express this action in terms of the fields X = {X I } and the massless fermionic degrees of freedom, and to integrate out the 4d components of the gauge field A µ and the scalars ϕ,φ, which appear as auxiliary fields in the 4d action. The subleading terms (at order r) in the ϕ a expansion can similarly be integrated out without producing any term in the final 4d action, so we ignore these contributions in the rest of the derivation.
In addition one should integrate over the one-loop fluctuations of the fields around their saddle point configurations. We will assume here that the bosonic and fermionic one-loop determinants cancel, as is frequently the case in similar computations [46], and now turn to deriving the 4d action. Some of the technical details have been relegated to appendix E.

Scalars
We will now describe the 4d theory in terms of 'collective coordinates' X I , similar to the approach taken in e.g. [46] for the dimensional reduction of 4d SYM theories on a Riemann surface resulting in a 2d sigma-model into the Hitchin moduli space. Related work can also be found in [47,48]. The resulting theory is a (supersymmetric) sigma-model (4.2), where for this part of the paper we will consider M 4 = R 4 . The three scalar fields ϕ a and A θ are expanded in the collective coordinates as follows where I = 1, . . . , 4k. Here, the basis of the cotangent bundle of M k is given by where E I defines a u(k) connection ∇ I ≡ ∂ I + [E I , .] on M k . The Υ a I , Υ θ I satisfy linearized Nahm's equations The metric on M k can be expressed in terms of these one-forms as The Hyper-Kähler structure on M k can be made manifest in this formulation, by defining the three symplectic forms (see for instance [49]) Some useful properties of these are summarized in appendix E.1. Using the expansion (4.18) we obtain This will combine with terms arising from integrating out the gauge field to give the usual sigma-model kinetic term.

Fermions
The fermions satisfy the equation (4.16), which is the supersymmetry variation of Nahm's equations. The spinors therefore take values in the cotangent bundle to the moduli space M k and we can expand them in the basis that we defined in (4.18) where λ (4.24) The expansion in (4.23) can be seen to satisfy the equation of motion for the spinors (4.16) by making use of (4.19) and the gauge fixing condition (E.6). Then substituting in the kinetic term for the spinors and making use of the expression for the metric on M k (4.20), the symplectic forms ω a IJ and the constraint (4.24), we find (4.25)

4d Sigma-Model into the Nahm Moduli Space
Finally, we need to integrate out the gauge field and the scalars ϕ,φ, which is done in appendix E.2. The conclusion is that, in addition to giving the standard kinetic term for the scalars, this covariantizes the fermion action and results in a quartic fermion interaction that depends on the Riemann tensor of the moduli space. In summary we find the action where D µ λ obeying the reality conditions and the constraint (4.29) The 4d sigma-model action from flat M 4 into the monopole moduli space M k is then given The supersymmetry transformations are (4.31) We have thus shown, that the M5-brane theory reduced on an S 2 gives rise to a fourdimensional sigma-model with N = 2 supersymmetry, based on maps from R 4 into the moduli space M k of Nahm's equations (with = [k] boundary conditions).

Relation to the Bagger-Witten Model
There is an equivalent description of the sigma-model in (4.30), which relates it to the models in [28,29]. In this alternative description we make use of the reduced holonomy of the Hyper-Kähler target M k . We will consider an (Sp(k) × Sp(1))/Z 2 subgroup of SO(4k), under which the complexified tangent bundle of a Hyper-Kähler space decomposes into a rank 2k vector bundle V and a rank 2 trivial bundle S. The index I decomposes under this into i p, where i = 1, · · · , 2k labels the 2k-dimensional representation of sp(k) and p = 1, 2 is the doublet index of sp(1) = su(2) R . The map I → i p is realized by the invariant tensors f i p I [50], which satisfy The alternative description of the sigma-model is obtained by defining the fields (4.33) which can be inverted, by using the constraint on the fermions (4.29) Using this decomposition the 4d untwisted sigma-model action into the monopole moduli space M k can be re-expressed in terms of the fermionic fields (4.33) where the covariant derivative is The tensors w Ij i and W ijkl are the Sp(k) connection on V and the totally symmetric curvature tensor, respectively. These are expressed in terms of the Christoffel connection and Riemann tensor as (4.37) The supersymmetry transformations are It is natural to ask how this sigma-model can be extended to general, oriented four-manifolds

Topological Twist
Twist 1 in section 2.1 was formulated for the 6d theory. We now briefly summarize how this twist acts in 4d. From now on we switch to Euclidean signature 8 . Recall, that in 6d, we twist the su(2) ⊂ su(2) ⊕ su(2) r of the 4d Lorentz algebra with the su(2) R ⊂ su(2) R ⊕ so(2) R ⊂ sp(4) R . From the point of view of the 4d theory, we start with the R-symmetry su(2) R and twist this with the Lorentz symmetry of M 4 , which generically is so(4) L ∼ = su(2) ⊕ su(2) r , resulting in g 4d = su(2) R ⊕ so(4) L → g twist = su(2) twist ⊕ su(2) r . (5.1) In terms of 4d representations, p and (2) p are Weyl spinors of positive and negative chirality respectively. We adopt the convention that negative/positive chirality spinors correspond to doublets of su(2) /su(2) r respectively. After the twisting, (2) p has one scalar component under su(2) twist ⊕ su(2) r , which is selected by the projections where the indices a and a are identified in the twisted theory. The spinor (2) p parametrizes the preserved supercharge and can be decomposed as where u is a complex Grassmann-odd parameter and˜ p is a Grassmann-even spinor normalized so that˜ p˜ p = 1 .

(5.4)
We can associate the u(1) R charge −1 to the parameter u and consider˜ p as uncharged.
The su(2) R R-symmetry with which we twist rotates the complex structures of the target and therefore is identified with the sp(1) ⊂ so(4k) of the Hyper-Kähler target. This means that SU (2) R /Z 2 is mapped to an SO(3) isometry of the metric on M k . In order to do the twist one needs to know how the coordinates X I transform under this sp(1) ≡ su(2) R . For the monopole moduli space with charge 1 and 2, M 1 and M 2 , where the explicit metric on the moduli space is known, the coordinates split into two sets transforming respectively in the trivial and adjoint representation of su(2) R . This suggests that this property could hold for moduli spaces M k , with k > 2. Under the twist, the coordinates transforming in the adjoint of su(2) R become self-dual two forms on M 4 and the resulting theory is a sigma-model, whose bosonic fields are maps into a reduced target space and self-dual two-forms. We shall study the M 1 and M 2 cases in section 6.
A simplification occurs when the bundle of self-dual two-forms on M 4 is trivial i.e. when

Topological Sigma-Model for Hyper-Kähler M 4
The 4d sigma-model into the Nahm moduli space (4.30) can be topologically twisted for Hyper-Kähler M 4 . We now show that this reduces to the 4d topological theory by Anselmi and Frè [31], for the special target space given by the moduli space of Nahm's equations. This topological theory describes tri-holomorphic maps from M 4 into M k which satisfy the triholomorphicity constraint where the index a = 1, 2, 3 is summed over and j a and ω a are triplets of complex structures on M 4 and M k respectively, which define the Hyper-Kähler structures. We will also comment in section 5.3 on how this can be obtained by first topologically twisting the 5d SYM theory, and then dimensionally reducing this to 4d. This alternative derivation from the twisted 5d SYM theory can be found in appendix F. We now turn to the topological twisting of the 4d sigma-model into the Nahm moduli Despite the fact that the index I transforms non-trivially under the R-symmetry SO(3) R , this will not play a role in the twist for the Hyper-Kähler four-manifold M 4 : the holonomy is reduced to su(2) r and the su(2) connection that we twist with vanishes. To be even more concrete, the covariant derivatives acting on fields with an index I will not pick up any su(2) twist connection because the connection vanishes, so we may treat I as an external index. This is of course not true for non-Hyper-Kähler M 4 .
The most general decomposition of the spinors into twisted fields is given by where the Grassmann-odd fields λ I , χ I µν , κ I µ are respectively a scalar, a self-dual two-form and a one-form, valued in the pull-back of the tangent bundle of the target space X * T M k .
However the components of ξ (i)I p are not all independent as they satisfy the constraint (4.29). This constraint on the components of ξ where ω µν I J ≡ −(j a ) µν ω aI J . As the self-dual two-form χ I µν is not an independent degree of freedom we shall consider the decomposition of ξ impose upon the fermionic one-form κ I µ the constraint (5.9), which can be re-expressed as The action in terms of the twisted fields takes the form and is invariant under the supersymmetry transformations This is precisely the form of the topological sigma-model of [31] for Hyper-Kähler M 4 . The action takes a simpler form than in the model presented in [31] since the target space M k is also Hyper-Kähler (i.e. has a covariantly constant quaternionic structure).
The topological BRST transformation Q (with δ u = uQ) squares to zero Q 2 = 0 on-shell. To make the algebra close off-shell, we can introduce an auxiliary one-form b I µ valued in the pull-back of the tangent space to M k , b ∈ Γ(X * T M ⊗ Ω 1 ) and satisfying the constraint We then define the BRST transformation to be QX I = λ I Qλ I = 0 The action (5.12) can then be recast in the form where S and S T are Q-exact and topological, respectively, given by we recover the on-shell action (5.12). The term S T is 'topological', in the sense that it is invariant under Hyper-Kähler deformations, and can be written as Finally, to show that the theory is topological, meaning independent of continuous deformations of the metric (which preserve the Hyper-Kähler structure), we must check that the energy-momentum tensor T µν associated with S part of the action is Q-exact. We find where L is the Lagrangian density in (5.17). This can be expressed as Clearly it is of interest to study further properties of these theories, in particular observables, which will be postponed to future work. Some preliminary results for sigma-models that localize on tri-holomorphic maps have appeared in [31], however only in terms of simplified setups, where the target is the same as M 4 .

Relation to topologically twisted 5d SYM
The topological sigma-model (5.12) for the Hyper-Kähler case, can also be obtained by first topologically twisting the 5d SYM theory on an interval obtained in section 3, with the twist described in section 5.1. The derivation is quite similar to the analysis in section 4, and we summarize the salient points here. The details are provided for the interested reader in appendix F. There, we also discuss the topological twist 1 in the context of the 5d SYM theory. The action for the bosonic fields, and some analysis of the boundary conditions in terms of Nahm data, has appeared in [21]. The supersymmetric version has appeared in [51], albeit without the supersymmetric boundary conditions.
The topologically twisted 5d SYM theory can be written in terms of the fields B µν , which is a self-dual two-form defined in (F.3), a complex scalar field ϕ, the gauge field A µ and fermions, which in terms of the twisted fields have the following decomposition 6 Sigma-models with Self-dual Two-forms Having understood the Hyper-Kähler M 4 case, we can finally turn to the case of general M 4 . The reduction proceeds in the same way as for the Hyper-Kähler case, but the situation is somewhat complicated by the fact that part of the coordinates X I become sections of Ω + 2 (M 4 ), namely self-dual two-forms. We consider in detail the abelian case with target space M 1 R 3 × S 1 and the first non-trivial case, corresponding to the reduction of the 5d U (2) theory, with target space , where M 0 2 is the Atiyah-Hitchin manifold. In the case of an arbitrary (oriented) four-manifold M 4 , there is no Hyper-Kähler structure, only an almost quaternionic structure [52]. One could anticipate dimensionally reducing the twisted 5d SYM theory, as discussed in section 5.3 and appendix F.1. However, this requires that Nahm's equations for the self-dual two-forms B µν to be solved locally on patches in M 4 and the patching must be defined globally, according to the transformation of B on overlaps. Generically this means that part of the mapping coordinates X I will transform from one patch to the other and therefore belong to non-trivial SU (2) bundles over M 4 . A similar situation appears in [46] appendix B, when twisting the sigma-model into the Hitchin moduli space. To understand precisely, which coordinates X I become sections of SU (2) bundles on M 4 , we require a detailed understanding of the metric on M k and the action of the SU (2) isometries. In the following, we will address this in the case of k = 1, 2, where the metrics are known.
We provide here the analysis in the case of the reduction of the abelian theory, as a warmup, and then the reduction of the U (2) theory, which is the first non-trivial case. In these cases we find that the four-dimensional theory is a topological sigma-model with part of the coordinates X I on the target space transforming as self-dual two-forms on M 4 .

Abelian Theory
Recall that the dimensional reduction on S 2 of the untwisted single M5-brane theory gives a free hyper-multiplet in R 1,3 . We shall now discuss this in the context of the topologically twisted theory on S 2 × M 4 and determine the sigma-model into the one-monopole moduli space M k=1 ∼ = R 3 × S 1 , with R 3 the position of the center and S 1 parametrizing a phase angle. As the metric is known, we can identity the coordinates parametrising the position of the center as those which transform under the su(2) R and the twist gives a topological model for general M 4 . In fact, we find the abelian version of a model in [30] in the context of 4d topological A-models. The 4d field content is the self-dual two-form B µν , the scalar φ and (twisted) for the fermions, a scalar η, a vector ψ µ , and a self-dual two-form χ µν .
We begin by decomposing the target space index I → (a, φ), with a = 1, 2, 3. Under this decomposition the constraints on the spinors ξ (i)I p can be solved as leaving only ξ (i)φ q as the unconstrained fermions in the theory. Under the twist the fields become Field g 4d g twist Twisted Field X φ (1, 1, 1) ( where the twisted fermions are obtained from the decompositions The scalars X a are decomposed in terms of the self-dual two-form B µν by making use of the The action for the k = 1 topological sigma-model from flat space into the monopole moduli space M 1 is then To show that this action is topological we introduce the auxiliary field so that δP µ = 0 and δψ µ = uP µ . The action can be written as the sum of a Q-exact term and a topological term by noting that δ u = uQ For M 4 without boundary, the second term in (6.9) vanishes upon integrating by parts. This action can then be generalised to arbitrary M 4 by covariantising the derivatives, and add curvature terms The resulting theory is a (free) topological sigma-model based on the map φ : together with a self-dual two-form B and fermionic fields and is given by )) . Note, likewise one can obtain the same abelian theory starting with the 5d twisted theory for curved M 4 as discussed in section 5.3 and appendix F.1. The reduction can be done straight forwardly, integrating out the fields ψ (1) , χ (2) and η (2) , and taking the leading 1/r terms in the action. The match to the action in (6.12) can be found by defining the fields in the 4d reduction as 14) The scalar φ is actually defined in a gauge invariant way as φ = π 0 dθA θ . Moreover it takes values in iR/Z = U (1) 9 , where the Z-quotient is due to the large gauge transformations δ( A θ ) = 2πin, n ∈ Z 10 .

U (2) Theory and Atiyah-Hitchin Manifold
In this section we study the simplest non-abelian case, corresponding to two M5-branes wrapped on S 2 , or equivalently we study the reduction of the 5d U (2) theory to 4d on an interval with Nahm pole boundary conditions. The flat 4d theory is given by a map into the 2-monopole moduli space M 2 , with the action given in (4.30). For the curved space theory we find a description in terms of a sigma-model into S 1 × R ≥0 supplemented by self-dual two-forms obeying some constraints. We provide a detailed analysis of the geometrical data 9 The factor i is due to our conventions in which A θ is purely imaginary. 10 These transformations correspond to gauge group elements g = e iα(θ) with α(0) = 0 and α(π) = 2πn. The quantization of n is required for g to be trivial at the endpoints of the θ interval.
entering the sigma-model and we give the bosonic part of the topological sigma-model on an arbitrary four-manifold M 4 .
The 2-monopole moduli space has been studied extensively in the literature (see for instance [24,[53][54][55][56]), starting with the work of Atiyah and Hitchin [24]. It has the product structure where R 3 parametrizes the position of the center of mass of the 2-monopole system, and M AH is the Atiyah-Hitchin manifold, which is a four-dimensional Hyper-Kähler manifold.

(6.24)
We can express the AH metric in terms of the y i,a coordinates by using the relations (σ 1 ) 2 = 1 2 (−dy 1,a dy 1,a + dy 2,a dy 2,a + dy 3,a dy 3,a ) (σ 2 ) 2 = 1 2 (dy 1,a dy 1,a − dy 2,a dy 2,a + dy 3,a dy 3,a ) (σ 3 ) 2 = 1 2 (dy 1,a dy 1,a + dy 2,a dy 2,a − dy 3,a dy 3,a ) , where the index a is summed over. The AH metric (6.16) is then understood as the pull-back of the metric ds 2 AH = f 2 dr 2 + v 1 dy 1,a dy 1,a + v 2 dy 2,a dy 2,a + v 3 dy 3,a dy 3,a , (6.26) As already mentioned the AH manifold M AH admits three complex structures J a , a = 1, 2, 3, preserved by the above metric, and satisfying the quaternionic relations where the indices I, J, K run over the four coordinates of the AH metric 11 . Lowering an index with the AH metric G IJ (6.16), we define the three Kähler forms (Ω a ) IJ = G IK (J a ) K J . These forms can be nicely expressed as the pull-back of the forms Ω a on the space parametrized by the r, y i,a coordinates: 12 These forms can be further simplified by using the functions w 1 = bc, w 2 = ca, w 3 = ab, which obey We obtain the nice expression The pull-backs Ω a are complex structures on M AH , hence they obey dΩ a = 0. This description of the complex structures is convenient, because it is much simpler than the expression in terms of the Euler angles θ, φ, ψ, but more importantly because it makes manifest the fact that the three Kähler forms Ω a , or the three complex structures J a , transform as a triplet under the SO(3) M 2 isometry.
After this preliminary work we can express the bosonic part of the flat space sigma-model action (4.30) in terms of the new coordinates β, φ a , r, y i,a , describing the maps M 4 → M 2 . Fixing f (r) = 1 for simplicity, we obtain where the sigma-model coordinates y i,a are constrained to form an SO(3) matrix (6.23) and to obey (6.24). These constraints can be stated explicitly δ ab y i,a y j,b = δ ij , abc y 1,a y 2,b y 3,c = 1 . Their kinetic term gets covariantized by adding suitable curvature terms and we obtain The constraints (6.33) become y i µν y j µν = 4δ ij and y 1 µ ν y 2 ν ρ y 3 ρ µ = 4.
The fermionic part of the action S M 2 ,ferm that is obtained from the untwisted action (4.30), is somewhat more involved, due to the presence of the four-Fermi interaction and the constraint (4.29) on the fields ξ (i)I . From the abelian part of the U (2) theory we obtain the fermionic field content of the abelian model (6.12). In the following we describe only the fermions related to M AH . Explicitly we can define the push-forward of the fermionic fields where the index I runs over r, (i, a). In the twisted theory we identify the su(2) and su(2) R doublet indices q and q and the fermionic fields of the resulting sigma model are a vector κ µ , a scalar η and self-dual two-forms η i,a ∼ η i µν satisfying the constraints δ ab y i,a η j,b = −δ ab y j,a η i,b , j y j,a η j,b = − j y j,b η j,a . (6.37) The other fields appearing after the twisting are expect to be expressed in terms the above fields by solving the constraints (4.29). However the computation is rather involved and we do not provide an explicit expression here.
The sigma-model we obtain seems to be different from the sigma-models studied in the literature so far. It is a sigma-model with target S 1 × R ≥0 with constrained self-dual twoforms. To study this sigma-model, and in particular to show that it defines a topological theory, one would need to work out the details of the fermionic part of the Lagrangian and the action of the preserved supersymmetry (or BRST) transformation on the fields. We leave this for future work.
To conclude we can see how the bosonic action (6.35) compares with the bosonic action of the topological model that we obtained for Hyper-Kähler M 4 (5.12). More precisely we would like to know how the action (6.35) decomposes into Q-exact plus topological terms as in (5.17). For this we simply evaluate S T for the sigma-model into M 2 , using the explicit form of the Ω a (6.31). The terms involving the fields φ and b vanish upon integration by parts as in the abelian case, assuming M 4 has no boundary. When the theory is defined on an generic four-manifold M 4 , the remaining contribution is where D µ is covariant with respect to the Christoffel connection and SU (2) Lorentz rotations (in the tangent space), and "+curv." denotes extra curvature terms, which appear when we consider a general curved M 4 and covariantize S T . Replacing X I → r, y i,a we obtain From the third to the fourth line we have integrated by parts assuming M 4 has no boundary. The result on the fourth line can be recognized as containing only curvature terms (no derivatives on the fields r, y i µν ) which must cancel each-other. This is necessary for supersymmetry to be preserved (since this term must be supersymmetric by itself). We conclude that the sigma-model action (6.35) must be Q-exact, without an extra topological term. Clearly, studying topological observables and further properties of this model are interesting directions for future investigations.

Conclusions and Outlook
In this paper we determined the dimensional reduction of the 6d N = (0, 2) theory on S 2 , and found this to be a 4d sigma-model into the moduli space M k of k-centered SU (2) monopoles.
There are several exciting follow-up questions to consider: 1. 4d-2d Correspondence: Let us comment now on the proposed correspondence between 2d N = (0, 2) theories with a half-topological twist, and four-dimensional topological sigma-models into M k .
The setup we considered, much like the AGT and 3d-3d correspondences, implies a dependence of the 2d theory on the geometric properties of the four-manifold. In [13] such a dictionary was setup in the context of the torus-reduction, which leads to the Vafa-Witten topological field theory in 4d. It would be very important to develop such a dictionary in the present case. From the point of view of the 2d theory, the twist along M 4 is the same, and thus the dictionary developed between the topological data of M 4 and matter content of the 2d theory will apply here as well. The key difference is that we consider this theory on a two-sphere, and the corresponding 'dual' is not the Vafa-Witten theory, but the topological sigma-model into the Nahm moduli space.
2. Observables in 2d (0, 2) theories: Recently much progress has been made in 2d (0, 2) theories, both in constructing new classes of such theories [13,[57][58][59] as well as studying anomalies [60] and computing correlation functions using localization [17]. In particular, the localization results are based on deformations of N = (2, 2) theories and the associated localization computations in [61,62]. The theories obtained in this paper from the compactification of the M5-brane theory do not necessarily have such a (2, 2) locus and thus extending the results on localization beyond the models studied in [17] would be most interesting.
3. Observables in the 4d topological sigma-model: An equally pressing question is to develop the theory on M 4 , determine the cohomology of the twisted supercharges, and compute topological observables. For the case of Hyper-Kähler M 4 , with the target also given by M 4 , some observables of the topological sigmamodel were discussed in [31]. However, we find ourselves in a more general situation, where the target is a specific 4k dimensional Hyper-Kähler manifold. For the general M 4 case we clearly get a new class of theories, which have scalars and self-dual twoforms. The only place where a similar theory has thus far appeared that we are aware of, is in [30] in the context of 4d topological A-models. We have studied the topological sigma-models for k = 1, 2, and the explicit topological sigma-models for k ≥ 3 remain unknown. It would certainly be one of the most interesting directions to study these.

Generalization to spheres with punctures:
The analysis in this paper for the sphere reduction can be easily generalized to spheres with two (general) punctures, i.e. with different boundary conditions for the scalars in the 5d SYM theory. We expect the 4d theory to be again a topological sigmamodel, however, now into the moduli space of Nahm's equations with modified boundary conditions. Studying this case may provide further interesting examples of 4d topological field theories, which seem to be an interesting class of models to study in the future.

Reduction to three-dimensions and 3d duality:
The four-dimensional sigma-model that we found by compactification of the 6d (0,2) theory on a two-sphere, can be further reduced on a circle S 1 to give rise to a threedimensional sigma-model into the same M k target space. Similarly the twisted sigmamodel on a manifold S 1 × M 3 reduces along S 1 to a twisted sigma-model on M 3 . On the other hand the compactification of the twisted 6d (0,2) A k theory on S 2 × S 1 × M 3 can be performed by reducing first on S 1 , obtaining 5d N = 2 SYM theory on S 2 × M 3 , and then reducing on S 2 . We expect this reduction to yield a different three-dimensional theory, which would be dual to the 3d sigma model into M k , for M 3 = R 3 , or twisted sigma model, for general M 3 , that we studied in this paper. This new duality would be understood as an extension of 3d mirror symmetry [63] to topological theories. To our knowledge the reduction of 5d SYM on the topologically twisted S 2 has not been studied 13 . It would be very interesting to study it and to further investigate these ideas in the future.
cussions. We also thank Damiano Sacco for collaboration at an earlier stage of the project.    A.2 Gamma-matrices and Spinors: 6d, 5d and 4d We work with the mostly + signature (−, +, · · · , +). The gamma matrices Γ A in 6d, γ A in 5d and γ A in 4d, respectively, are defined as follows: with the Pauli matrices The 6d gamma matrices satisfy the Clifford algebra and similarly for the 5d and 4d gamma matrices. Futhermore we define and similarly for all types of gamma matrices.
The chirality matrix in 4d is γ 5 = −σ 3 ⊗ 1 2 and in 6d is defined by The charge conjugation matrices in 6d, 5d and 4d are defined by They obey the identities To define irreducible spinors we also introduce the B-matrices which satisfy The indices of Weyl spinors in 6d can be raised and lowered using the SW/NE (South-West/North-East) convention: with (C mn ) = (C mn ) = C. There is a slight abuse of notation here: the indices m, n go from 1 to 8 here (instead of 1 to 4), but half of the spinor components are zero due to the chirality condition. When indices are omitted the contraction is implicitly SW/NE. For instance with (Γ A ) n m the components of Γ A as given above. The conventions on 5d and 4d spinors are analogous: indices are raised and lowered using the SW/NE convention with (C m n ) = (C m n ) = C in 5d and with the epsilon matrices pq = pq = ṗq = ṗq , with 12 = 1. They are contracted using the SW/NE convention.

R-symmetry reduction :
In this paper we consider the reduction of the R-symmetry group The fundamental index m of sp(4) R decomposes into the index ( p, x) of su(2) R ⊕ so(2) R . A (collection of) spinors ρ m in any spacetime dimension can be gathered in a column four-vector ρ with each component being a full spinor. The decomposition is then with ρ (1) = (ρ (1) p ) transforming in the (2) +1 of su(2) R ⊕so(2) R and ρ (2) = (ρ (2) p ) transforming in the (2) −1 . So the four spinors ρ m get replaced by the four spinors ρ (1) p , ρ (2) p . From the sp(4) R invariant tensor Ω m n , with Ω = ⊗ σ 1 , and the explicit gamma matrices (A.12) we find the bilinear decompositions. For instance Another useful identity is where the background fields have been converted to sp(4) R representations with We choose to set η = 0. After inserting our ansatz, in particular T m n BCD = b A = 0, we obtain with Here, 'traces' indicates terms proportional to invariant tensors Ω m n , δ m r , δ n r . Again the background fields are converted to sp(4) R representations using (B.3).
With T m n BCD = 0, we obtain the simpler conditions . (B.14) The background we found corresponds to the twisting u(1) L ⊕u(1) R → u(1) on S 2 . It preserves half of the supersymmetries (and no conformal supersymmetries) of the flat space theory, and corresponds to the topological half-twist of the 2d theory.
C 6d to 5d Reduction for b µ = 0 In this appendix we detail the reduction of the six dimensional equations of motion on an S 1 .
This is done following [9,37] however we choose to gauge fix b µ = 0, which is possible without loss of generality.
We start by decomposing the six dimensional frame as where the 5d indices are primed. We work in the gauge b µ = 0, which is achieved by fixing the special conformal generators, K A . Note that this choice is different from the gauge fixing of b µ in [9,37], in particular α is not covariantly constant in our case. Furthermore, we fix the conformal supersymmetry generators to ensure ψ 5 = 0, which means that e (C. 2) The components of the spin connection along the φ direction are given by where G = dC, and can be derived from the six dimensional vielbein using ν] − e ρ[A e B]σ e C µ ∂ ρ e σC . (C.4)

C.1 Equations of Motion for B
The 6d equations of motion for the three-form H are given by We decompose H into 5d components We can solve the second equation of motion by setting where F µ ν is a two-form in five dimensions. Substituting this into the expansion of H and reducing to 5d we obtain The equations of motion dH = 0 imply which can be integrated to the 5d action Together with the constraint dF = 0, which identifies F with the field strength of a fivedimensional connection A, given by

C.2 Equations of Motion for the Scalars
The dimensionally reduced 6d scalar equations of motion are The 6d Ricci scalar R 6d can be rewritten of course in terms of the 5d fields. This equation of motion can be integrated to the following action where (C.17) From this we obtain the action

D Supersymmetry Variations of the 5d Action
The supersymmetry variations (3.14), which leave the 5d action (3.24) invariant, can be decomposed with respect to the R-symmetry, following appendix A.3. This decomposition will be useful in further proceeding to four dimensions. The scalar and gauge field variations are then and for the fermions we find Inserting this back in the action the terms with ϕ,φ results in The terms we obtain by integrating out A µ will be grouped into three types of terms. The first type are such that X I appear quadratically (E.13) These terms combine with terms in the scalar action (4.22) to give the usual sigma-model kinetic term Terms of type 2 are linear in X I and covariantise the kinetic terms of the spinor it can be shown that this quartic fermion interaction combines with the term (E.12) to make the Riemann tensor of the target space appear where the Riemann tensor is given by

F.1 Topological Twist
Let us first consider the topological twist 1 of section 2.1 applied to the 5d SYM theory. From now on we switch to Euclidean signature 15 . The twisted 5d theory was already considered in [21,51].
Twist 1 of the 6d N = (0, 2) theory identifies su(2) ⊂ su(2) ⊕ su(2) r of the 4d Lorentz algebra with the su(2) R ⊂ su(2) R ⊕ so(2) R ⊂ sp(4) R . Under dimensional reduction to 5d the symmetries after the twist are sp(4) R ⊕ so(5) L → g twist = su(2) twist ⊕ su(2) r ⊕ u(1) R . (F.1) The fields of the 5d theory become forms in the twisted theory, according to their transformations with respect to the g twist , as summarized in the following table: Field g twist Representation Twisted Field The fields A µ , ϕ,φ do not carry su(2) R charge and are thus unaffected. The scalars ϕ a transform as a triplet of su(2) R . In the twisted theory they become a triplet ϕ a of su(2) twist , defining a self-dual two-form B µν on M 4 : where the three local self-dual two-forms j a transforming as a triplet of su(2) twist . They can be defined in a local frame e A µ as (j a ) µν = e A µ e B ν (j a ) A B , a = 1, 2, 3, with (j a ) 0b = −δ a b , (j a ) bc = − a bc , a, b, c = 1, 2, 3 .

(F.4)
In this local frame we have B 0a = ϕ a , B ab = abc ϕ c , a, b, c = 1, 2, 3 . (F.5) The self-dual tensors j a are used to map the vector index a of so(3) to the self-dual two-form index [AB] + . The tensors (j a ) µ ν define an almost quaternionic structure, since they satisfy The spinor fields transform as doublets of su(2) R . They become scalar, self-dual two-forms and one-form fields on M 4 as indicated in the table. The explicit decomposition, is obtained using the Killing spinor associated to the scalar supercharge in the twisted theory. This Killing spinor can be found as follows. The spinor m generating the preserved supersymmetry is a constant spinor and is invariant under the twisted Lorentz algebra su(2) twist ⊕ su(2) r . As p , satisfying the projections (3.17) (1) p + γ 5 (2) p = 0 . (F.7) As explained in section 5.1, p has one scalar component under su(2) twist ⊕ su(2) r selected out by the projections (γ 0a δ q p + i(σ a ) q p ) (2) q = 0 , a a = 1, 2, 3 , (F.8) where the indices a and a gets identified in the twisted theory. The spinor (2) p parametrizing the preserved supercharge is then decomposed as where u is complex Grassmann-odd parameter and˜ p is a Grassmann-even spinor with unit normalisation. The decomposition of the spinors into the twisted fields is then given by (F.10)

F.2 Twisted 5d Action
We rewrite now the action in terms of the twisted fields and provide the preserved supersymmetry transformations. The bosonic part of this action has appeared in [21], and related considerations regarding the supersymmetric versions of the twisted model can be found in [51].
The action in (3.39) in terms of the twisted fields takes the form µ δA θ = uη (1) δB µν = uχ (1) µν δϕ = 0 δφ = 2uη (2) δψ (1) where the self-dual part of the gauge field is defined as To define the twisted action for curved M 4 , in addition to covariantising the derivatives, the curvature terms RB µν B µν and R µνρσ B µν B ρσ , (F.14) must be added to the action in order to preserve supersymmetry. These terms can be repackaged with the kinetic term for B µν changing the action for the scalars to where D is defined to be covariant with respect to the curvature connection on M 4 and the gauge connection. The 5d twisted action on curved M 4 can be written in the form S 5d = QV + S 5d,top , (F. 16) where the Q-exact and topological terms are given by where P µν and P µ are auxiliary fields. The supersymmetry transformations are Qψ (2) µ = P µ , Qχ (2) µν = P µν . (F.18) The auxiliary fields are integrated out by (F. 19) We can now proceed with the dimensional reduction to four-dimensions.

F.3 Triholomorphic Sigma-model with Hyper-Kähler M 4
We now reduce the twisted 5d SYM theory to 4d on Hyper-Kähler M 4 . We proceed similar to the analysis in section 4.2 and in appendix E, and expand all fields in powers of r and demand that the leading order terms in 1 r in the action (F.11) vanish. This sets ϕ =φ = O(r) and leads to Nahm's equations for the self-dual two-forms  Combining the information above the solutions are ϕ = 8irΦ IJ λ I λ J + 2irΦ IJ ζ I µν ζ Jµν ϕ = −8irΦ IJ κ I µ κ µJ A µ = E I ∂ µ X I − 8iΦ IJ (λ I κ J µ − ζ I νµ κ Jν ) . (F.41) Replacing the fermionic and bosonic zero modes in the action one obtains Substituting in the solution for the gauge field (F.41) we obtain three different types of terms, which we address in turn. Terms of type 1 are proportional to ∂ µ X I ∂ ν X J and combine with the terms in the scalar action to give S scalars + S Aµ,type 1 = 1 4r d 4 x |g 4 | G IJ g µν ∂ µ X I ∂ ν X J . (F.43) Terms of type 2 combine with terms from the action of the fermions to give S Aµ,type 2 = − 2i r dθd 4 x |g 4 | (δ K I g σν − ω σν I K )Tr Using the identities The final action upon combining all the above terms is The action can be further simplified by using relations between the complex structures ω µν I J and the fermions (F.33) to eliminate the self-dual two-form ζ I µν . In addition we know that the target space M k is Hyper-Kähler, which means that the three complex structures ω µν