$\mathcal{N}=2$ supersymmetric field theories on 3-manifolds with A-type boundaries

General half-BPS A-type boundary conditions are formulated for N=2 supersymmetric field theories on compact 3-manifolds with boundary. We observe that under suitable conditions manifolds of the real A-type admitting two complex supersymmetries (related by charge conjugation) possess, besides a contact structure, a natural integrable toric foliation. A boundary, or a general co-dimension-1 defect, can be inserted along any leaf of this preferred foliation to produce manifolds with boundary that have the topology of a solid torus. We show that supersymmetric field theories on such manifolds can be endowed with half-BPS A-type boundary conditions. We specify the natural curved space generalization of the A-type projection of bulk supersymmetries and analyze the resulting A-type boundary conditions in generic 3d non-linear sigma models and YM/CS-matter theories.


Introduction
The study of supersymmetric quantum field theories on rigid curved backgrounds in diverse spacetime dimensions has been a powerful source of new non-perturbative results in recent years. So far, a rather complete and systematic understanding of such results has been obtained for supersymmetric field theories on closed manifolds. Most notably, these theories can be engineered by taking appropriate rigid limits of certain supergravity theories. This framework constrains the background geometry and determines the couplings of the field theory to the curvature and the auxiliary background fields in the supergravity multiplet [1][2][3]. Partition functions and other supersymmetric observables can then be evaluated exactly with the powerful technique of supersymmetric localization providing a new window into nonperturbative physics in quantum field theory. Some of the original work in this direction in two, three, four, and five spacetime dimensions includes [4][5][6][7][8][9]. Analogous situations on manifolds with boundary, or more generally, on spaces with co-dimension-1 defects, are comparatively much less elaborated upon. There are two key aspects of this story one would like to develop systematically. The first aspect is related to the geometric properties of boundaries. Given a fixed bulk supergravity background that supports supersymmetric field theories, what restrictions should be imposed on the geometry of a codimension-1 surface to preserve a subset of the bulk supersymmetry? The second aspect is related more directly to the specific dynamic properties of the field theory in question, in particular, the boundary conditions that can be imposed on the defect.
Regarding the first point, it is immediately clear that since the commutator of supersymmetries squares to isometries on the compact manifold, the boundary should be oriented along directions parallel to these isometries to preserve the corresponding supersymmetries. Moreover, one can ask if supersymmetry puts any constraints on co-dimension-1 foliations of a compact manifold. A foliation preferred by supersymmetry could be used to decompose closed manifolds into a union of manifolds with boundary. Indeed, we will show that such a foliation exists in a general class of 3-manifolds.
As far as the second point is concerned, it is well known that the invariance of generic observables under bulk symmetries (including supersymmetries) is spoiled, in general, by boundary effects. A symmetry can be restored by cancelling these boundary effects. This can be achieved with the introduction of suitable boundary conditions and/or the introduction of appropriate boundary degrees of freedom.
In the present work we concentrate on three dimensions and develop a systematic treatment of half-BPS boundaries in N = 2 supersymmetric field theories on compact 3-manifolds. We discuss general aspects of the interplay between supersymmetry and the geometry of manifolds with boundary, and analyze a wide class of related half-BPS boundary conditions. We concentrate on the classical aspects of the problem. The main contributions of this work can be summarized as follows.
the background fields, generalize the more familiar analysis in flat space [12] [14].
The case of general (non-abelian) YM/CS-matter theories is discussed in section 8. We find boundary conditions that include the curved space generalization of holomorphic Neumann boundary conditions for Yang-Mills gauge fields and matter fields, and holomorphic Dirichlet boundary conditions for the gauge fields in CS theories.
A summary of useful formulae, and an exposition of technical details for results used in the main text are relegated to two appendices at the end of the paper.

Prospects
We conclude this short introduction with a few remarks on some of the interesting open questions raised in this work and the prospects of further related developments.
Our main motivation for the study of the classical problem in this paper is the eventual formulation of general half-BPS co-dimension-1 defects in 3d N = 2 supersymmetric quantum field theories on curved spaces, and the non-perturbative computation of observables associated with these defects.
The observables we are interested in include the partition function of N = 2 supersymmetric gauge theories on curved backgrounds with boundary. With A-type boundary conditions these partition functions are computing a class of supersymmetric wavefunctions. It would be interesting to explore the dependence of these observables on the moduli of the defects, i.e. the moduli of the boundary conditions we formulate, generalizing the bulk analysis of Ref. [15]. A preliminary computation of partition functions on manifolds with boundary in three dimensions using localization techniques has been performed in special cases in [16,17]. The results in the present paper can be used to extend known results in this direction.
Moreover, one can also attempt to use the information of supersymmetric wavefunctions to study the structure of observables on closed manifolds that do not involve co-dimension-1 defects. Hints of such a possibility come from a variety of previous results: the holomorphic block decomposition of 3d partition functions [17,18], and the analogous phenomenon in different dimensions [7,19], the recent progress in D-brane amplitudes in 2d N = (2, 2) theories [20][21][22], and tt ⋆ arguments in flat toroidal backgrounds in three, and four spacetime dimensions [23].
Boundary conditions also introduce another tool to probe dualities between quantum field theories. If two theories are dual at the quantum level, we expect corresponding boundary conditions on each side to be mapped to each other in a non-trivial way. For instance, in the case of mirror symmetry, the duality between boundary conditions can be understood in the mathematical framework of symplectic duality [24]. 3d Seiberg duality also acts nontrivially on boundary conditions. We refer the reader to Ref. [25] for a recent discussion of the relation between 3d Seiberg dualities and 2d level-rank dualities in this context. Similar problems with Wilson loops were investigated in [26], [27]. Finally, an intriguing interpretation of co-dimension-1 defects relates the expectation value of these operators to the entanglement structure of the field theory [28].
Another arena of potential applications of such computations is M-theory. The study of boundary conditions in the ABJM theory [29], which is an N = 6 Chern-Simons-matter theory, is expected to yield information about the physics of M2, M5-branes and their interactions. For instance, it is anticipated that the low-energy theory at the orthogonal intersection of M2 and M5-branes in C 4 ×Z k is a 2d theory with N = (4, 2) (or in special cases N = (4, 4)) supersymmetry. The non-abelian quantum properties of this theory are still illusive. A recent bare Lagrangian formulation of this theory in terms of boundary degrees of freedom motivated by D-brane physics in type IIB Hanany-Witten setups was proposed recently in [30]. For a study of half-BPS boundary conditions in ABJM theory see [31], [32].
Finally, there are several aspects of the general theory of supersymmetric boundaries in three dimensions that are not discussed in this paper. One of these aspects is the general curved space analog of B-type boundary conditions in 2d N = (2, 2) theories. Another aspect that is worth exploring further is the formulation of half-BPS boundaries using explicit boundary degrees of freedom and boundary actions [33]. The analysis of supersymmetric boundaries in 2d N = (2, 2) theories in [20] was performed in this manner.

Review of rigid supersymmetry on curved 3-manifolds
In the modern approach to rigid supersymmetry on curved spaces, the metric tensor g µν (or any other background field) is embedded into a certain supergravity multiplet, and the field theory is obtained by taking the rigid limit of Festuccia-Seiberg [1] (FS). With a U (1) R symmetry, the supergravity of interest in 4d is the "new minimal supergravity" of [34], and the supergravity multiplet contains an R-symmetry gauge field A (R) µ , a conserved vector V µ and the two gravitini Ψ µα , Ψ µα . Following FS, the rigid field theory of chiral and vector superfields on the curved space, is obtained from the action of off-shell supergravity coupled to chiral and vector fields, by freezing the bosonic components of the supergravity multiplet to a configuration in which δΨ µ = δ Ψ µ = 0. The advantage of this formulation is that the whole procedure can be carried out without the need of an explicit solution to the equations δΨ µ = δ Ψ µ = 0.
In 3d it is possible to perform a twisted dimensional reduction of the 4d rigid theories to infer a consistent new minimal 3d algebra [3]. At the end of this process, the background fields are, the metric g µν , an R-symmetry gauge field A (R) µ , a conserved vector V µ (as in 4d), and an extra scalar field H. The conditions δΨ µ = δ Ψ µ = 0 reduce to the following two Killing spinor equations 2) The two Weyl spinors ζ andζ have R-charges +1 and −1 respectively. In practice, given a choice of the background metric, the other background fields can be adjusted to obtain at least one solution of the Killing spinor equations. On the other hand, assuming that at least one Killing spinor exists as a solution of the equations (2.1), (2.2), it is possible to deduce what geometric structure the manifold needs to possess. In 3d, this analysis was first carried out in [3,15,35]. In sec. 2.1 we will review in some detail the relevant geometry since it will play an important role in our problem. In fact, in order to set up supersymmetric boundary conditions, it will be useful to improve slightly the way in which the relevant geometric structure is characterized. The new material is presented in section 3. Experts familiar with rigid supersymmetry on spaces without boundary (e.g. the work in [3,15,35]) may skip to section 3. We follow closely the notation of Ref. [3].
In our presentation, it will be convenient to make an explicit distinction between commuting and anti-commuting Killing spinors. In particular, we will denote the commuting spinors with ζ andζ, and the anti-commuting spinors with ǫ andǫ. Both sets of Killing spinors satisfy the same equations. The anti-commuting spinors ǫ andǫ, will provide the parameters of the supersymmetry transformations of the field theory. The commuting spinors, ζ andζ, will be used to explore the geometry of the manifold.

Geometry of M 3
The existence of Killing spinor solutions, ζ andζ, strongly constrains the geometric structure of the background fields. We will not repeat the general analysis here, but we recall two important results of [3], which will be useful for later purposes. The first states that a solution of (2.1), or (2.2), when it exists, is nowhere vanishing. 1 The second result states that given one Killing spinor, say ζ for concreteness, it is possible to cover the manifold with a transversely holomorphic foliation (THF), and write the metric in the following form By definition of the THF, the adapted coordinate τ is real, whereas {z,z} are complex. The leaves of the foliation are the submanifolds z = const., and two patches are related by transitions functions, f and h, such that z ′ = f (z) with f holomorphic, and τ ′ = h(τ, z,z) with g real. In particular, g can be put in the form h(τ, z,z) = τ + t(z,z). The origin of the transversely holomorphic foliation is an integrability constraint. The one-form η = η µ dx µ can be represented as the spinor bilinear 4) and the following field can be defined, 2 The spinor ζ c is the charge conjugate to ζ. Notice that from the properties of ξ µ and J µ ν it also follows that the Killing spinor equation of ζ, (2.1), is invariant under the shift symmetry, where the scalar k and the vector field X µ are such that J µ ν X ν = iX µ and ∇ µ (X µ +kξ µ ) = 0. After gauge fixing the shift invariance, the Killing spinor equation (2.1), implies the constraint Given the condition (2.7), the authors of Ref. [3] showed that it is possible to find the adapted coordinates {τ, z,z} introduced in (2.3). This is the THF associated to ζ. On equal footing, there exists the THF associated withζ, which is defined as in (2.4) with the substitution ζ →ζ, i.e.η µ = (ζ c γ µζ ) |ζ| −2 , and the Killing spinor equation ofζ, (2.1), remains invariant under a shift similar to (2.6).
Manifolds that admit two complex supercharges of opposite R-charge have additional properties compared to the THF. They have a nowhere vanishing Killing vector K µ , and a contact structure. The Killing vector is represented as (2.8) It solves the equation from which ∇ {µ K ν} = 0 follows. The norm of K µ is K µ K µ = (ζζ) 2 ≡ Ω 2 , and the function Ω is such that Notice that the Killing spinor equations are linear, therefore ζ and λζ, with λ an arbitrary complex number, are both solutions. Similarly forζ. However, the relationζζ = Ω breaks the arbitrariness in the normalization of ζ andζ, and only the symmetry ζ → λζ withζ → λ −1ζ remains. Eq. (2.8) is also invariant under this scaling. 2 The triple (ηµ, ξ µ , J µ ν ), with ηµ, ξ µ , and J µ ν such that ηµξ µ = 1 and J 2 = −1 + ξ η, is called an almost contact structure (ACS). This definition only requires that ηµ, ξ µ , and J µ ν , satisfy algebraic constraints. It does not require the manifold to have a metric. For Riemannian manifolds, a metric gµν is said to be compatible with the ACS if ξ µ = g µν ηµ. The ACS is then promoted to an almost contact metric structure (ACMS). Similarly to the definition of a complex structure, the difference between an almost and a contact structure, is a differential constraint. However, this constraint is not (2.7) but: dη (ξ, ·) = 0 for the contact structure, and dη (·, ·) = g (J·, ·) for the contact metric structure [36]. It is perhaps useful to mention that the condition for a contact metric structure resembles the one for Kähler manifolds in even dimensions [37].
When the Killing vector is real, the geometry can be further characterized by the orbits of K µ . Two cases can be distinguished: either the orbits of K µ are periodic, or they do not close. The first case consists of manifolds with the topology of an S 1 -bundle over a 2d Riemann surface. In the second case, it can be proved that there exists another independent Killing vector, transverse to K µ , and that the isometry group of M 3 is at least U (1) × U (1) [38].
In section 3 we will supplement the above results on 3-manifold geometry with a further new refinement that facilitates the introduction of boundaries preserving a subset of the bulk supersymmetries.

Supersymmetric multiplets and transformations
Rigid supersymmetric field theories exist on any curved background M 3 , equipped with the two Killing spinors ǫ andǫ. Their Lagrangians are obtained by exploiting the multiplet calculus of 4d new minimal supergravity [34] and its 3d version (see appendix of [3]).
By multiplet calculus we mean the collection of all the supersymmetry transformations of the components of a generic multiplet S. The total number of independent degrees of freedom in S is 16 bosonic plus 16 fermionic. They are organized as follows: (2.13) The R-charges are (0, −1, +1, −2, +2, 0, 0, +1, −1, 0) relative to the bottom component C.
The supersymmetry transformation rules δ ǫ S +δǫS are summarized in appendix A. The set of all these transformations realize an algebra on the space of fields. Denoting with ϕ (r,z) a field of arbitrary spin, R-charge r, and central charge z, the supersymmetric algebra is represented by 14) The symbol L K is defined in [3] as a modified Lie derivative along K The covariant derivative associated to L K will be denoted as Here the background gauge field C µ is related to the background conserved vector V µ by the µ , but the two are related by a redefinition is. Accordingly, it is convenient to express (L K + ǫǫ(z − rH)) as In what follows, we will mostly use D µ , as defined above, since we adopt the notation of Ref. [3]. Sometimes, however, it will be convenient to consider A (R) in the covariant derivative. When this happens we will be very explicit.
For the benefit of the reader we list here two standard short multiplets S that will play a dominant role in the main discussion. The shortening of the multiplets is obtained by imposing restrictions on its components.

Chiral and the anti-chiral multiplets
Chiral (anti-chiral) multiplets are obtained by imposing the conditionsχ α = 0 (χ α = 0). This implies that not all components of the generic multiplet are independent. A chiral multiplet Φ, with independent components {φ, ψ α , F } is organized as follows, (2. 19) In the above formula, R[φ] = r is the R-charge of φ, and z is the central charge. The transformation rules of {φ, ψ, F } are (2.20) The shorthand notation for Φ will be Φ = {φ, ψ, F }. The case of the anti-chiral multipletΦ is analogous. The independent components are { φ, ψ α , F } and the supersymmetric transformation rules are where the R-charge of φ is R[ φ] = −r and its central charge is −z.

Real and gauge multiplets
A real multiplet Σ arises by imposing on S the conditions M =M = 0, and r = z = 0. The subset of independent components can be defined by α , j µ , σ (Σ) }, and Σ is organized as follows The vector field j µ is a conserved current, ∇ µ j µ = 0. The supersymmetric transformations rules are (2.23) An abelian gauge multiplet V is a generic multiplet S subject to the gauge freedom δV = Λ +Λ, where Λ is a chiral multiplet. After the standard procedure of Wess-Zumino gauge fixing the independent fields reduce to {A µ , σ, λ α ,λ α , D}. Notice that an abelian gauge multiplet becomes a real multiplet under the identification: where f νρ is the field strength of A µ . This parametrization will be particularly useful in later sections.
In the case of non-abelian gauge multiplets the supersymmetry transformation rules have extra terms compared to (2.23). The complete set of transformation rules in the non-abelian case is (2.25) F µν is the field strength of A µ , and D µ is the non-abelian gauge covariant derivative (5.28).

Curved D-and F-terms
So far we have not specified whether S is an elementary or a composite multiplet. The supersymmetric transformations are, of course, valid regardless of this distinction. Once elementary multiplets are defined, any composite multiplet K of the form K = (K, χ (K) ,χ (K) , M (K) , . . .) is generated by the multiplet calculus. In practice, given the definition of the bottom component K, as a function of the elementary fields C I , the other components in the multiplet are obtained in a step-by-step procedure: varying K(C I ) with the use of δC I one reads off the definitions of χ (K) andχ (K) , and so on. From the composite multiplets it is then possible to construct kinetic terms for the elementary fields and thus generic supersymmetric Lagrangians whose variation is a total derivative.
Such Lagrangians can be understood as follows. Given a generic multiplet S with r = 0 and z = 0, its D component almost transforms as a total derivative. Terms that are not total derivatives are proportional to background fields, and the flat space result is recovered when these vanish. In curved space the correct combination transforming into a total derivative is [3] curved D−term : (2.26) The result for the F (orF ) component of a chiral Φ (or anti-chiralΦ) multiplet of R-charge r = 2 (or r = −2) and central charge z = 0 is the same as that in flat space. The F-term is curved F−term :

Manifold decomposition for curved A-type backgrounds
In this paper we focus on a class of background geometries introduced in [10], that we call "A-type". 3 By definition, these backgrounds admit two supercharges related by charge con-jugation. The charge conjugate spinors, ζ c ≡ +iγ 2 ζ ⋆ andζ c ≡ +iγ 2ζ ⋆ , 4 solve the equations In general, given a Killing spinor, say ζ, its complex conjugate ζ c is an independent spinor that does not solve any of the Killing spinor equations (2.1) and (2.2). However, if the background fields A (R) µ and V µ are real, and H is purely imaginary, then ζ c solves the same Killing spinor equation asζ. Therefore, for an A-type background, ζ and ζ c are the two Killing spinors of opposite R-charge. Now we are going to show that it is possible to understand any A-type background in terms of a supersymmetric foliation in which the leaves are topologically tori. As a mathematical statement about irreducible orientable closed 3-manifolds, it is certainly well known in the literature that such a toric foliation exists, however we will use supersymmetry and the Killing spinors ζ andζ to re-derive this result. Very explicitly, the geometry of the foliation will be characterized by a distribution of orthogonal vector fields built out of the Killing spinors. One of these vectors will be the Killing vector K µ , and we will construct another vector N µ that: 1) is orthogonal to K µ , and 2) can be used to define a proper orthogonal submanifold.
The use of vector fields, instead of the adapted coordinates of the THF, will be essential in the formulation of boundary conditions preserving a subset of the bulk supersymmetry. With such a foliation in place, we will be able to decompose the compact manifolds by placing a boundary (or a co-dimension-1 defect) along any leaf of the foliation. Our main purpose will be to formulate rigid supersymmetric fields theories on the resulting spaces with boundary that are topologically solid tori. Since the metric is part of a supergravity multiplet, the decomposition of the manifolds should be combined with certain extra conditions on the remaining background fields. We will discuss concretely how the manifold decomposition is carried out in the rest of this section. In the final subsection 3.5, we revisit some of the well-known examples of compact 3d manifolds, and re-discuss them from the perspective of this decomposition.

Normal vector
Let us consider how the existence of the Killing spinors ζ and ζ c determines the geometry of A-type manifolds. By fixing the normalization ofζ to beζ = ζ c , we show that supersymmetry provides a "refinement" of the THF in which a special orthogonal direction to K µ is selected out.
The starting point of our treatment is based on the use of a Fierz identity for commuting spinors that allows us to show that the real vector N µ , defined as is orthogonal to K µ , i.e. K µ N µ = 0. The same result about N µ can be obtained by noticing that where e µ 2 is an unspecified vielbein. Hence, the real part of (3.4) gives N µ = 2ε µνρ e 2 ν K ρ , which is manifestly orthogonal to K µ . The tangent space T M 3 can then be spanned by the following orthogonal vectors: K µ , N µ , andK µ ≡ ε µνρ N ν K ρ . By construction, we also have It is, therefore, convenient to choose a reference frame, {e 1 , e 2 , e 3 }, such that K µ = e µ 3 and K · e 2 = 0. For such a frame we deduce from (3.4) that e µ 2 ∝ ε µ νρ N ν e ρ 3 , and N ν ∝ e ν 1 . By consistency, we have to prove that the inverse metric g µν can be written in terms of the bilinears K µ K ν , N µ N ν andK µK ν . Indeed, from the Fierz identity applied to K µ K ν , and from the very definition ofK µK ν , we obtain the relation Our adapted dreibein fields are with ||K|| = ||N || ||K||. The norms of K µ and N µ are The generic form of the metric on M 3 was given in the previous section We can now compare it with (3.6). For A-type manifolds, the knowledge of (2.12) implies that K µ = Ω 2η and e 3 = Ω (dψ + x 1 dy 1 ). On the other hand, from (2.4) andζ = ζ c , we get η = Ωη = e 3 . Then, we can make use of the coordinates {z,z}, instead of {x 1 , y 1 }, by implementing the contact structure condition on the function h. As a result, h is ψindependent, and since ∂ ψ Ω = K µ ∂ µ Ω = 0, the function Ω is also ψ-independent. The metric (3.9) takes the final form [3] ds 2 = Ω (z,z) 2 (dψ + h(z,z)dz + c.c.) 2 + c(ψ, z,z) 2 dzdz . (3.10) Because of (3.6), we also know that there exists a real parametrization of the plane dzdz in terms of the vectors, N µ andK µ . In terms of the contact structure,η, the vectors N µ and K µ are understood as the distribution H = kerη. Recalling Frobenius' theorem, the defining property of a contact manifold, namelyη ∧ dη = 0, implies that the distribution H = kerη is not integrable. Instead, we will now study under what conditions the distribution generated by K µ andK µ is integrable. This will provide a regular foliation of the A-type manifold.

Integrability condition
Frobenius' theorem guarantees that the distribution E, generated by K andK, is integrable if the commutator [K,K] belongs to E [39]. 5 This commutator is equal to The second and third terms in (3.11) can be manipulated by using the equation of K ν , given in (2.9). We obtain the expression It is given by the more involved expression Substituting (3.13) into ∇ α N ν , we get several contributions In the adapted frame (3.7), and after some algebra, we can show that Then, the commutator becomes, The vector N µ does not belong to the distribution E, thus the distribution is integrable iff the commutator [K,K] vanishes. The explicit expression given in (3.16) can be rearranged by noticing thatζ (3.17) 5 The normalizations of K andK are not important in the argument. Even though they contribute to the commutator, through the termsK µ (K α ∂α 1 ||K|| ) and K µ (K α ∂α 1 ||K|| ), these contributions belong to E . Thus the statement of Frobenius' theorem remains unchanged.
The final form of the constraint coming from [K,K] = 0 is e 3 σ ε σµν e α 3 ω α ab e a µ e b ν + ∇ µ e 3 ν = e 3 σ ε σµν e α 3 ω α ab e a µ − ω µ 3b e b ν = 0 . Given an A-type manifold such that (3.18) is satisfied, Frobenius' theorem [39] then implies the integrability of the distribution E and the existence of the foliation. As an example, all metrics of the type with F θ , F ϕ , ψ-independent, and g θθ , g ϕϕ generic, satisfy the constraint (3.18). Locally, where N µ dx µ = dθ, the 2d submanifolds θ = const. define the foliation generated by E. The leaves of the foliation will be denoted by M ′ 2 . 6 We refer to M ′ 2 as a supersymmetric leaf of M 3 . This terminology follows from the observation that the algebra of supersymmetry involves the Killing vector K µ in the Lie derivative L K . In the simplest case, since the commutator of two transformations δ ǫ and δǫ squares to a translation along the orbit of the Killing vector K µ , M ′ 2 preserves supersymmetry because K µ belongs to T M ′ 2 .

Topology and manifold decomposition
We can show with a simple argument that the topology of M ′ 2 cannot be genus zero, i.e. the leaves of the supersymmetric foliations are not spheres. The reasoning goes as follows. M ′ 2 contains the orbits of the Killing vector K µ , and K µ is nowhere vanishing because, as we mentioned in section 2.1, the Killing spinors are nowhere vanishing. If M ′ 2 was a sphere, K µ would correspond to the U (1) isometry of the sphere, which is unique. However, this cannot be the case since the U (1) isometry of the sphere vanishes at the north and south poles.
The topology of M ′ 2 is a torus. We showed that it cannot be genus zero, but also it cannot be a higher genus surface either, because a 2d Riemann surface of genus g > 1 would not have a Killing vector. Thus, an A-type background is topologically a torus fibered over a closed interval. The example of the round three-sphere is very instructive: S 3 does admit a genus zero topological (Heegard) decomposition as the union of two 3-balls [40], however the supersymmetry that we are considering rules out this possibility and allows only for g = 1 decompositions. A similar phenomenon has been noticed in 4d. Ref. [2] showed that a 4d supersymmetric manifold for which [K,K] = 0 is topologically Σ × T 2 , where Σ is a 2d Riemann surface.
The results we have obtained so far can be summarized by the statement that any supersymmetric compact space M 3 of A-type admits a toric foliation. We now pick one leaf M ′ 2 of the toric foliation, and slice M 3 along its volume. As a result, we obtain two manifolds T 1 and T 2 , which share a common boundary, the leaf M ′ 2 , such that T 1 #T 2 ∼ = M 3 . Generically, we will refer to T as a solid torus, borrowing the terminology from surgery theory. The solid torus is the analog of the hemisphere in 2d, and the tip of the hemisphere corresponds here to the shrinking of one of the two boundary cycles. Following the analogy with the lower dimensional case, another interesting 3d manifold is represented by the "cylinder", which topologically would be a torus fibered on the interval with both a left and a right boundary. We note the obvious fact that when a boundary is inserted the homotopy properties of the manifold change.
Since the metric belongs to the supergravity multiplet, whose components include the Rsymmetry gauge field A (R) µ , the vector field V µ , and the scalar H, any manifold decomposition should be consistent with the profile of these background fields. Being a scalar field, H is not constrained by the manifold decomposition. However, a condition on V µ follows from the fact that V µ is a conserved vector, and therefore we should require n µ V µ ≡ V ⊥ = 0 at the boundary. As a further simplifying, but not necessary, assumption in some of the examples that will be analysed below we will also consider n µ A (R) µ = 0.

Clifford algebra and bilinears at the boundary
The frame fields k µ ,k µ and n µ , split the algebra of the γ matrices into a 2d "parallel" Clifford algebra, which lives on M ′ 2 , and an orthogonal matrix γ ⊥ = n µ γ µ . As a consequence, all possible spinor bilinears obtained from ζ andζ are classified in terms of scalars and tensors on T M 2 . One obvious example is K µ =ζγ µ ζ, which is a vector on T M 2 , and has no scalar component because n µ K µ = 0.
It will be useful for later purposes to have the explicit decomposition for all spinor bilinears. Since we have an expression for n µ in terms of the Killing spinors, we can use Fierz identities to bring the bilinears in a simple form. It is enough to consider a generic bilinear with at most two γ matrices; higher order bilinears would not be independent, because of the identity γ µ γ ν = g µν + i ε µνρ γ ρ . For notational convenience we use the indices ν for directions parallel to M ′ 2 . The bilinears of interest are 7 As a technical remark, we observe that the right column of (3.21) can be obtained by complex conjugation of the left column usingζ = +iγ 2 ζ ⋆ , and γ µ⋆ = −γ 2 γ µ γ 2 . The norm of the normal vector N µ = (ζ ⋆ γ µ ζ) + c.c. was given in (3.8). However, by using the symmetries of the commuting spinors, we can also write N µ N µ = N µ [(ζγ µ ζ ⋆ ) + c.c.], and from the Fierz identity we obtain Therefore, We conclude that the only new geometric information needed, in order to parametrize the bilinears (3.21), is a phase The phase ̟ can be calculated explicitly, given the Killing spinor ζ and the norm of the Killing vector. In general, we expect ̟ to be coordinate dependent: ̟ = ̟(ψ, z,z) if we are using the THF parametrization. We will present examples in section 3.5. Finally, we can ask how the bilinear ζγ µ ζ decomposes in the basis {k µ , n µ ,k µ }. The answer is again obtained by using Fierz identities and reads Consequently, we also find that the metric can be written equivalently as

Twisting and phases
So far we have discussed several of the characteristic properties of A-type backgrounds. The appearance of the phase ̟ is one of the properties that will play an important role in the subsequent analysis and as such it deserves some further elaboration. A constant shift of ̟ can be understood as part of the U (1) invariance that is built into the relations Ω = ζζ andζ = ζ c , as we discussed in sec. 2.1. The coordinate dependent part of ̟(ψ, z,z) is due to the non-trivial profile of the background fields and is closely related to the explicit solution of the Killing spinor equations. The choice of the frame fields, and therefore the definition of the curved γ matrices becomes important when we discuss the Killing spinor equation. We fix possible ambiguities in the choice of vielbein by working in the preferred frame {n µ , k µ ,k µ }. The relation between the coordinate dependent phase, ̟(ψ, z,z), and the Killing spinor, that we discuss here is made in this frame. 8 We make the following observation. Given a metric g µν with corresponding background fields and a generic non-trivial ̟(ψ, z,z) we can consider a U (1) R gauge transformation that 8 The reader familiar with the S 3 geometry may notice that by using the Maurer-Cartan forms two out of four Killing spinors of the S 3 are constant (see for example [41]). As we emphasize in the next section, U µ is always well defined and so is ̟. It can be explicitly checked that the phase ̟ = ψ/2 will show up in U µ , even in the Maurer-Cartan formulation. In the frame {nµ, kµ,kµ} the phase will appear in the Killing spinors.
sets it everywhere to zero. As a result of this operation, the new background R-symmetry, in which the phase is constant, is where dΛ is a flat connection (in the simplest case a non-zero constant).
Globally, the addition of a non-trivial flat connection can lead to interesting phenomena. Even though we expect the details of the manifold to become important at this point, we know for sure that the leaves of the supersymmetric foliations are tori, and therefore we can make the following general comments: • When π 1 (M 3 ) is trivial, the two cycles of a generic leaf M ′ 2 will shrink in the bulk, identifying the location of the north and south pole.
new is inserting a singularity, effectively changing the topology. For example, it inserts punctures at the north/south pole.
• When π 1 (M 3 ) is non-trivial, e.g. π 1 (M 3 ) = Z, the new flat connection will generically decompose into a combination of an holonomy and a singularity (if both are non vanishing).
• When the manifold has a toric contact structure, the Killing vector K = ∂ ψ is a combination of the vectors ∂ φ 1 and ∂ φ 2 , where φ 1 and φ 2 are 2π-periodic coordinates on the leaves. The effect of a constant A (R) ψ will result to the insertion of a vortex loop at the north and south pole, together with an holonomy along the corresponding non-shrinking cycles.
From the point of view of the Killing spinor equations, the addition of a flat connection, from A new , is twisting the original solution. 9 Indeed, assume that in the old background the spinor is of the type e i̟ η 0 , with η 0 a constant spinor and ̟ = ̟(ψ, z,z). Then, in the new background η 0 is the spinor and the Killing spinor equation becomes ∂ µ η 0 = 0.
Returning to the coordinate system (3.19) we further observe that the ψ dependence of the Killing spinor is always constrained to be a phase. This is due to the fact that k = ∂ ψ , and the fact k µ ∂ µ (ζζ) = 0 that follows from the Killing spinor equations. The generic ansatz for a solution of the Killing spinor equations is then According to this ansatz, for generic f neither ζ norζ are scalars under translations along the Killing vector, however, the ψ dependence can always be solved by considering a gauge transformation of A (R) old such that k µ ∂ µ ζ = k µ ∂ µζ = 0. By using the integrability condition (3.16) we can prove that old can be thought as a twisting of a theory with no A (R) [22]. Here we are saying something slightly different, in particular we identify A old as a gauge transformation.
To prove this equation contract the Killing spinor equation of ζ with k µ andζ. The same result follows by considering the Killing spinor equation forζ. We come back to this relation in sections 7 and 8, where it will used as an input to solve for boundary conditions preserving a subset of the bulk supersymmetry.
As a final remark, we would like to emphasize the following fact. The field theories defined by the rigid limit of a supergravity theory depend explicitly on the background fields. As a result, for a given field theory in flat space, the corresponding rigid theories coupled to {g µν , A new , V }, are generally different theories with different Lagrangians. It would be interesting to understand via localization how the partition functions of the two theories are related. We leave this question to future work.

Examples: spheres and their squashings
Important examples of A-type backgrounds include: the round three-sphere S 3 , the ellipsoid S 3 b , the SU (2) × U (1) squashed spheres of [42], and geometries of the type S 2 × S 1 . Round and squashed spheres were the first manifolds on which the use of supersymmetric localization made possible the exact computation of the partition function of N = 2 theories [5,10,43,44]. Our main interest here will be to calculate the triple of vectors {n µ , k µ ,k µ } for the round sphere and its deformations. We will also mention the case of S 2 × S 1 which admits both an A-type and a different "non-real" structure. In the context of squashed spheres, the distinction between these two structures has been also emphasized in [45].

Ellipsoid
Our first example is S 3 b , defined as the set of points (z, w) ∈ C 2 , with the property |z| 2 ℓ 2 + |w| 2 ℓ 2 = 1. The squashing parameter b is usually defined as the ratio b 2 =l/ℓ. The parametrization z =l sin θ e iφ 1 , w = ℓ cos θ e iφ 2 , gives the metric where f (θ) 2 = ℓ 2 sin 2 θ +l 2 cos θ 2 . The coordinates take values in the range θ ∈ [0, π/2] and φ i ∈ [0, 2π] for i = 1, 2. They are toric, and make manifest the U (1) × U (1) symmetry of the geometry. The north pole at θ = 0, and south pole at θ = π/2, are conventionally defined by the shrinking of the corresponding S 1 cycles. The precise form of f (θ) is not important and all of the following calculations will be valid for a generic regular function g θθ (θ). 10 The background fields can be taken to be (in a gauge where V µ = 0) as 10 Regularity means any function that asymptotes tol, ℓ at θ = 0 and θ = π/2, respectively.

Notice that
± φ 2 → 0 at the south pole. 11 There are solutions to the Killing spinor equations with both + and − signs. It is then convenient to distinguish between positive and negative Killing spinors, respectively.
Our immediate task is to obtain the Killing spinors, ζ ± ,ζ ± , and calculate the vector fields K µ , N µ andK µ . Notice that (θ, φ 1 , φ 2 ) are not the adapted coordinates introduced in the previous section, but since we have coordinate-independent expressions for K µ , N µ and K µ , the choice of coordinates is not an issue. In the frame the explicit expression of the Killing spinors is with the matrix M given by (3.34) In (3.33) we have chosen a normalization such thatζ ± = ζ c ± . In fact, since the curved background is real, we are guaranteed that ζ c solves the equation ofζ. The Killing vector associated to ζ ± is 35) and the novel vectors, It is interesting to write the metric in the adapted frame {k µ , n µ ,k µ }. In the case of positive Killing spinors, the metric takes the form (3.19) The background A (R) of [42] is recovered by the substitution φi → −φi. The difference in the sign is due to our choice of γ matrices that differs from the one in [42].
For the round three-sphere ℓ =l and we recover well known results. The coordinates {ψ, θ H , ϕ} coincide with the familiar Hopf coordinates, for which S 3 is seen as a U (1) fibration over the two-sphere dθ 2 H + sin 2 θ H dϕ 2 . 12 The interpretation of the Killing spinors is manifest, K + = ∂ ψ and sits along the Hopf fiber, whereas K − = ∂ ϕ generates the U (1) isometry of S 2 . Furthermore, in the example of the round three-sphere written in Hopf coordinates, we can write the S 2 at the base of the fibration as CP 1 , and exhibit the THF of S 3 where τ = (ψ + ϕ)/2 and z = tan(θ/2) e iϕ . It is worth emphasizing that the Killing spinors ζ − andζ − , which generates K − = ∂ ϕ , become the standard spinor of the S 2 , after a change of frame. In the next section we will make this statement more precise. The Killing spinors ζ ± in (3.32) have non-trivial dependence on θ and ̟. As can be seen from evaluating M [θ,̟] η , the ̟ dependence reduces to a phase. Following the discussion in the previous section, we can twist away the phase by performing a gauge transformation on the background R-symmetry connection. To see how this works in practice, let us observe that we can indeed decompose A ± new in which the spinors ζ ± are constant along the direction of the Killing vectors, can be obtained either by an explicit computation, or by solving the general relation µ old above. The latter strategy implies that ± new becomes well defined on the ellipsoid S 3 b with punctures at the north and south poles.

On U (1) fibrations and non A-type geometries
Another class of interesting real curved spaces are the SU (2) × U (1) squashings of the round three-sphere of [42]. We will consider a slightly more general class of backgrounds, whose metric is given by (3.43) 12 Notice that when ℓ =l the periodicities of ϕ and ψ are different from those of the round three-sphere. In where 2R 2 = ℓ 2 +l 2 and b = (l 2 − ℓ 2 )/(l 2 + ℓ 2 ).
When u(θ) = cos θ, g θθ = 1, and ℓ =l we recover the Hopf fibration of the S 3 . When ℓ =l, the U (1) fiber of the round sphere gets squashed, and the metric only preserves the SU (2) × U (1) subgroup of the original SO(4) isometry group. We may also take u(θ) = u 0 constant, and for the particular value u 0 = 0 we recover the metric of S 2 × S 1 . It will be useful to define the parameter β =l/ℓ. It measures the squashing for geometries that are deformations of S 3 , hereafter S 3 β . Also, it measures the inverse temperature for geometries of the type S 2 × S 1 . By a global rescaling we can set ℓ = 1. We will work with the dreibeins The background scalar field H is taken to be purely imaginary, and we turn on In this setup, the metrics (3.43) admit two Killing spinors of opposite charge From these Killing spinors we calculate the frame {n µ ,k µ , k µ }, and find We also recognize that the dreibeins The phases ±iψ of the spinors ζ andζ in (3.47) can be re-absorbed by twisting A (R) . The corresponding gauge transformation leaves A Observe that for S 2 × S 1 geometries, the function u(θ) is trivial, and therefore By taking H = 0 this S 2 × S 1 background becomes the topologically twisted background of [47]. The reasoning that led to the background fields (3.45) and (3.46) is based on simple observations, which we now elucidate. First of all, A (R) and V are real when H is imaginary, hence the family of backgrounds is of the A-type. For example, considering the round threesphere, β = 1, u = cos θ, we find A (R) µ = 0, and V 3 = −iH − 1, thus in the gauge H = +i, the spinors ζ andζ correspond to the positive Killing spinors (3.32) and (3.33) calculated in the new frame (3.44). The more general background fields, (3.45) and (3.46), are obtained by solving the Killing spinor equation for A (R) and V , upon insisting that ζ in (3.47) is a solution. By writing the Killing spinor equation in the following form ψ from the other two equations. Some of the details of this calculation can be seen explicitly in the cases of S 3 β and S 2 × S 1 geometries. The equations (3.51) become where p = 1 for S 3 β , and p = 0 for S 2 × S 1 . For the round three sphere β = 1, H = +i, V 3 = 0 and the r.h.s of (3.52) and (3.54) vanish identically. The use of the frame fields (3.44), compared to the toric frame of the previous section, makes the computation of the positive Killing spinors particularly simple: two out of three equations can be trivially satisfied, and the remaining one, ∂ ψ ζ = − i 2 γ 3 ζ, is solved by (3.47). For the SU (2) × U (1) squashing S 3 β , the background field V 3 is tuned in such a way that the r.h.s of (3.51) becomes a projector, as one can check from (3.52). Then, the positive Killing spinor of the round sphere is promoted to a Killing spinor of the squashed sphere. 13 In this case, the R-symmetry background is proportional to A (R) 3 , and it is aligned with V 3 . The analysis of S 2 × S 1 geometries follows the same logic. Before we move on, let us notice that when we consider the negative Killing spinors of the round three-sphere, a different simplification takes place in the equations (3.52)-(3.54): the trivial equation becomes ∂ ψ ζ = 0, whereas the equations along θ, and ϕ become effectively those of the S 2 in its standard parametrization [7], (3.56) 13 Squashings whose Killing spinors reduce to the negative Killing spinors of the round sphere, have been studied in [44]. In this case, the ansatz for Killing spinors need to be slightly modified. 14 These S 2 Killing spinors can be uplifted to S 3 , as explained in [48]. In two dimensions, γ3 anti-commutes with γ (2d) µ . Therefore, the positive Killing spinors of the S 2 are proportional to γ3ζ, with ζ given in (3.55).

S 2 × S 1 and non A-type geometry
Metrics of the type S 2 × S 1 are interesting for a second reason: they are perhaps the simplest 3d example admitting both a real and a non-real structure. The non-real structure is obtained by considering the following background fields The Killing spinor equation (3.53) becomes trivial: ∂ ψ ζ = 0. After A (R) and V have been subtracted, the equations on the S 2 base, (3.52), and (3.54), become The Killing spinor equations forζ are not obtained from (3.58) by charge conjugation. Indeed, the background is not real. Instead, from the original Killing spinor equation (2.2) we find, The equations (3.59) appear in the same form in [8,17] for g θθ = 1. The explicit solutions are proportional to the following four spinors If the background fields are all purely imaginary (as in (3.57)), it follows from (3.1) that the charge conjugate spinor ζ c is independent of ζ and solves the same equation. For example, ζ 2 = ζ c 1 in (3.60). The same statement applies toζ andζ c . We conclude that if a background admits two Killing spinors of opposite R-charge, and all the background fields are purely imaginary, by construction it supports N = 4 supersymmetry.

Boundary effects in theories with rigid supersymmetry
Given a supersymmetric field theory on a compact manifold M 3 , defined by an action it is not guaranteed that the action will remain supersymmetric when we insert a boundary along M ′ 2 , and restrict the fields to the solid torus T . In fact, for any symmetry δ acting on the fields, the Lagrangian is locally invariant up to a total derivative, δL = ∇ µ V µ , hence the action restricted on T will be invariant under the symmetry δ iff where in the last step we used the divergence theorem. Typically, the condition (4.2) is solved by imposing appropriate boundary conditions such that n µ V µ = 0, or by adding appropriate degrees of freedom on the boundary. In this paper we consider only the first possibility. In the case of supersymmetry V µ is both a function of the anticommuting Killing spinors, ǫ and ǫ, and the fields of the theory. Therefore, in order to solve (4.2), one generally synchronizes the boundary conditions on the fields with certain conditions on the spinors. For example, if we assume that a certain projection on the spinors realizes a specific sub-algebra of the bulk supersymmetry, we can insert this knowledge into n µ V µ to simplify the problem and deduce definite boundary conditions for the fields of the theory. For example, in the case of boundary conditions in two-dimensional N = (2, 2) theories on a strip [12,49], one can consider two different types of 1 2 -BPS boundary conditions, called A-and B-type. They are characterized by the spinor projections ǫ ± andǭ ± are the complex components of the 2d Weyl spinors ǫ andǭ, andǭ is the complex conjugate of ǫ. The phase α is an arbitrary constant and the minus sign is a convention. An N = (2, 2) theory has 4 real supercharges and the 1/2-BPS projections preserve (1, 1) or (2, 0) supersymmetry, for A-type or B-type, respectively. Such conditions play an important role in D-brane physics described by setups with N = (2, 2) worldsheet supersymmetry. In 3d theories with N = 2 supersymmetry similar projections (and corresponding boundary conditions) have been formulated in flat space in [13].
When one attempts to apply this standard logic to a theory on a curved background, as in this paper, one encounters inevitably some obvious difficulties. Most notably, on curved backgrounds many of the simplifications of constant flat space spinors are absent. The Killing spinors ǫ,ǫ are, in general, non-trivial functions of the coordinates and an A-or a B-type projection cannot be imposed in the simple standard flat space form written above.
In what follows we will describe how to impose a direct generalization of the A-type condition on the anticommuting spinors ǫ andǫ in a generic three-dimensional A-type background. We will do so by introducing a "canonical" formalism that builds on the observations of the previous two sections. We anticipate that a similar generic formulation exists also for B-type projections. However, in this paper we will focus exclusively on A-type boundary conditions leaving B-type projections and B-type boundary conditions to a separate treatment in future work.

Generalized A-type projections on supersymmetry
Out of the commuting spinors ζ andζ we construct two natural projectors, P and P It is simple to check that P 2 = P, P 2 = P and P + P = I. Since the Killing spinors ζ,ζ are nowhere vanishing these projectors are everywhere well defined. Moreover, both P and P are invariant under the symmetry ζ → λζ,ζ → λ −1ζ , with λ ∈ C. By acting with P and P on both ǫ andǫ we formulate the generalized A-type conditions Defining the parameters The restrictionζ = ζ c (which is possible in A-type backgrounds) is imposed here because the scalar product (ǫ ζ) = (ζǫ) alone does not enforce a relation between ǫ andǫ. Indeed, by rescalingǫ → αǫ and ǫ → βǫ, with arbitrary α, β ∈ C, it is always possible to find two representatives of the commuting spinors, λζ and λ −1ζ , for which the relation (ǫ ζ) = (ζǫ) is satisfied. The conditionζ = ζ c is needed to break the invariance under the rescalings by λ ∈ C to a residual U (1).
We emphasize that the curved space version of the above A-type condition is, by construction, compatible only with A-type curved manifolds, for whichζ = ζ c . The projections (4.7), (4.8) reduce the amount of supersymmetry by one half.
In sections 6-8, we will demostrate how the input of the projections (4.7), (4.8) affects the (in)variance of a generic N = 2 field theory under supersymmetry, and we will study corresponding general A-type boundary conditions on N = 2 supersymmetric gauge theories that preserve half of the bulk supersymmetry at the boundary.

Bulk A-type supersymmetries and BPS equations
Having understood how to project the anticommuting Killing spinors of a generic A-type background, we now go back to the supersymmetry transformations of chiral and vector superfields, and reformulate them accordingly. First we spell out the supersymmetry transformations with generic ϑ and ϑ, and then we study what happens upon enforcing the projection ϑ = ϑ.
Before entering the details we point out that we can decompose any spinor ψ as Moreover, we notice the useful identities (4.10)

Chiral and anti-chiral multiplets
In (2.20) and (2.21) we wrote down the supersymmetric transformation rules for chiral and anti-chiral multiplets for generic Killing spinors. When we further specialize the supersymmetry to an A-type background we obtain the following expressions.
• For a chiral multiplet: (4.11) • For an anti-chiral multiplet: (4.12) It is clear, in particular, that the fixed point (BPS) equations, in which the fermions are set to zero and the bosons satisfy δf = 0 for any fermion f of the multiplet, depend on the assumption we make about ϑ and ϑ. For the A-type projection, ϑ = ϑ, we obtain Further assuming the reality conditions φ = φ ⋆ and F = F ⋆ , these equations reduce to We obtained the last equation using the property U = −U ⋆ . In the case of arbitrary ϑ and ϑ we would have instead F = F = 0 and U µ D µ φ = 0 independently. In the presence of a superpotential, we should integrate out F a in favor of g ac ∂c W . Recalling that U µ = e i̟ (n µ − ik µ ), we see that the equation iU µ D µ φ a − F a = 0 becomes the natural 3d generalization of the domain wall equations in two dimensional (2, 2) theories.

Real and gauge multiplets
The supersymmetric transformation rules for the gauge field were discussed in subsection (2.2.2). There we made a connection between the real multiplet and the gauge multiplet: Here we use the real multiplet parametrization for the fermions, and write the field strength F in terms of the vector a µ . The supersymmetry transformations on an A-type background then takes the following form.
• For the ǫ variation of the bosons (4.17) • For theǫ variation of the bosons (4.18) • For the fermionic fields The fixed point equations ψ Σ = ψ Σ = 0 and δψ Σ = δ ψ Σ = 0 are Let us notice that, from its definition (4.16), j µ (and thus a µ ) is imaginary if the gauge field A is real, and real if A is imaginary. The solutions to (4.20) and (4.21) include the 'Coulomb branch' solution 22) and the solution The equations (4.23) and (4.24) generalize to arbitrary A-type backgrounds those of [46] [47].

N = 2 Lagrangians
With all the geometric prerequisites in place we need one more element before we can start discussing concretely how to treat N = 2 supersymmetric field theories on A-type curved backgrounds with boundaries. We need to collect all the surface terms that arise in the supersymmetric variation of explicit Lagrangians. This is the main purpose of this section.

N = 2 non-linear sigma models
In this subsection we study first the most general (classical) N = 2 theory of chiral superfields on A-type curved manifolds. In flat space such theories are characterized in standard fashion by an action governed by a Kähler potential K and a superpotential W . The curved space generalization of this action is straightforward. We spell out the details for a non-linear sigma model (NLσ) of s elementary chiral superfields {φ a , ψ a α , F a }, and their conjugate { φc, ψc α ,Fc}, with a generic superpotential. As far as we know, some of the following calculations are not listed in the literature.

General Kähler interactions
In flat space, supersymmetry turns a generic target space into a Kähler manifold. This continues to be true in curved space. In addition, the Lagrangian contains a set of new couplings between the dynamical fields and the background fields H and V µ . By following the strategy outlined in the review section 2, the Lagrangian of the curved non-linear sigma model is obtained from the curved D-term combination (2.26) evaluated on the composite multiplet where R is the curvature of the background manifold and we have defined: Kcn am ψc ψnψ a ψ m , L bos H = + Hrc − zc (Hr a − z a ) φcKc a φ a − H 4 (Hr a − z a ) K a φ a + Hrc − zc Kc φc + 3H 4 z a K a φ a + zcKc φc , (5.4b) In (5.3) we are using the covariant derivatives As in flat space, the function K defines a Kähler potential for the metric G ac ≡ K ac . Consistency of the supersymmetric transformation rules requires K to be a quasi-homogeneous function of vanishing R-and central charge. 15 Collecting the fields φ a and φc under the variable C I = (φ a , φc), the two conditions on K are I r I C I K I = 0, r I = (r a , −rc), These extra conditions on the Kähler potential arise from coupling the theory to the background field H.

Superpotential interactions
Superpotential interactions are introduced as F-terms for a chiral multiplet Ω W = (W, ψ (W ) , F (W ) ), where W is a holomorphic function of the chiral fields φ a . The resulting Lagrangian in components is Invariance under supersymmetry requires W to be a quasi-homogeneus function of the φ a of degree 2 In a similar way W is quasi-homogeneous of degree −2. The R-charges of ∂ a W and ∂ c W are The most general Lagrangian for a set of chiral superfields is then specified by the two functions K and W , and by the assignment of charges. Schematically, from (5.3) and (5.9) we find L N Lσ = L K + L W . (5.12)

Variation under supersymmetry
Given L N Lσ , the object of interest for us is the total derivative that arises in a supersymmetric variation δL N Lσ +δL N Lσ = ∇ µ (V µ N Lσ ). (5.13) The supervariation can be obtained either by varying the action explicitly or by evaluating (2.26) and (2.27) for the multiplets K, Ω W and Ω W . The result in both cases is Kcn a ( ψc ψn) . (5.14) The equations of motion of the auxiliary fields F a andFc are K ac Fc + 1 2 K acn ( ψc ψn) = ∂ a W , F a K ac − 1 2 Kc am (ψ a ψ m ) = ∂c W .

(5.15)
Integrating out F a andFc we obtain the final expression √ 2 V µ N Lσ = +ǫ γ µ γ ν ψ a D ν φc − (rcH − zc) γ µ ψ a φc − iV µ ψ a Kc − iγ µ ψcW a K ac −ǫ γ µ γ ν ψc D ν φ a − (r a H − z a ) γ µ ψc φ a + iV µ ψcK a + iγ µ ψ a Wc K ac , (5. 16) where we have defined the vectors The R-charges of these vectors can be deduced from (5.6) and (5.11): R[W a ] = r a − 2, R[ Wc] = 2 − rc, and so on. Observe that the bilinears appearing in V N Lσ , are the most general bilinears of vanishing R-charge with the correct index structure built out of ǫ andǫ, ψ and ψ, and the corresponding bosonic fields. For example, it is obvious that derivatives of the superpotential W a only couple to ǫγ µ ψ, not to ǫγ µ ψ.

Digression on target space geometry
In differential geometry, a Kähler manifold is defined as a symplectic (real) manifold (N , ω), equipped with a complex structure J such that G(·, ·) ≡ ω(·, J ·) is a Riemaniann metric on T N . The last condition is called ω-compatibility [37]. In a local description with coordinates (φ a ,φc) the metric is represented as and the two-form ω is represented as ω αc ∝ G ac dφ a ∧ dφc. The target space of the non-linear sigma model, listed above, is such a Kähler manifold. For many of the explicit computations in the following sections, a different parametrization will turn out to be especially useful. This involves the change of variables φ a = Φ a + iΦ a+s ,φc = Φc − iΦc +s , where a,c = 1, . . . , s. When the reality conditionφc = φ c ⋆ holds, the fields Φ I are real. However, in general, we may consider φ andφ as two independent complex variables. Then the fields Φ I are also complex and (5.19) is a standard change of variables in GL(2s, C).
Collecting the labels of the type (a, a + s) into one index I = 1, . . . , 2s, the matrix that represents the change of variable is where the symbol δ a i+s stands for a diagonal matrix in the off-diagonal blocks of M, and is defined to be δ a i+s = 1 (or 0) if a = i (or a = i), as is clear from (5.19). The metric changes accordingly  16 The second one in (5.22) is precisely the condition of ω-compatibility, which is part of the definition of N . By construction, two types of "products" exist on a Kähler manifold, one is the symplectic product defined from the tensor ω IJ and the other one is the metric. In components, we find for any pair of vectors V I , W J . The formulae (5.23) and (5.24) will be useful in several occasions. Here we mention one simple application regarding the kinetic energy, which in the new variables Φ I is the sum of both the metric and the symplectic product. Because of the following identity valid for any connection a µ , it is possible to introduce the analog of the covariant derivatives (D µ φ a , D µφc ) acting on Φ I . In particular, we define where the bold symbol Φ represents the vector Φ I and the matrices R and Z are given by Notice the absence of negative signs in the right bottom corner of R (Z), corresponding to rc (zc). The bold symbols Ψ, W, and K, will be used to describe the vectors corresponding to Ψ I , W I and K I , that indeed appear in the supervariation (5.16).

YM and CS theories
Next, consider a vector multiplet V = {A µ , λ,λ, σ, D} valued in the Lie algebra g of a gauge group S, possibly non-abelian. The field strength F µν of the gauge field and the covariant 16 When we write matrix products we always understand row by column multiplication, from right to left.
derivatives of the various fields in the vector multiplet are (5.28) In three dimensions a gauge field admits both Yang-Mills (YM) kinetic terms and Chern-Simons (CS) kinetic terms. For abelian theories the supersymmetric Lagrangian is obtained as the curved D-term of the composite multiplet − 1 e 2 Σ 2 , where Σ is the real multiplet associated to V, and e is the coupling constant. The non-abelian Lagrangian is the standard generalization of this construction, and the result in components is For CS theories the supersymmetric Lagrangian is Finally, if the gauge group contains a (product of) U (1) factors we can add for each abelian factor the corresponding FI term

Matter couplings
Matter can be added both to CS and YM theories by coupling the vector multiplet to chiral and anti-chiral superfields in arbitrary representations of the gauge group S. We consider matter superfields Φ a and Φc labelled by a bold index which collectively indicates both the color index a and flavor index m, i.e. a = (a, m). The color indices are contracted in scalar products defined in the appropriate representation of the chiral and anti-chiral fields. Similarly, the components of the gauge multiplets act on the matter fields according to their representation, and the covariant derivatives contain both the background and the gauge fields, D µ ϕ (r,z) = D µ ϕ (r,z) + irV µ ϕ (r,z) − iA µ ϕ (r,z) for any field ϕ (r,z) . The gauge invariant interactions among different flavors are fixed by a choice of Kähler potential and superpotential. For the simplicity of the presentation, we will consider a canonical Kähler potential. Each flavor may also have different background R-charge r m and central charge z m . Assuming that chiral and anti-chiral superfields have opposite charges, it is convenient to define the diagonal matrices of Rand Z-charges. The Lagrangian is L matter = L K + L W , (5.33) where L W contains a gauge invariant superpotential, (5.9), and L K is given by In this formula R is the curvature of the background manifold and Equipped with the precise form of L Y M , L CS and L matter it is possible to write down the most generic quiver gauge theory. In this case, the gauge fields will be also labelled by a bold index of the type, m = (a, m), where m labels the nodes of the quiver theory, and a labels the generators of the gauge group S m at the node m. Considering normalized generators for the gauge groups, the CS coupling κ is promoted to a matrix of the form κ mn = δ ac ⊗ κ mn , with κ mn a symmetric tensor.

Variation under supersymmetry
The supersymmetric variation of the actions L Y M , L CS and L matter has the following properties. Let us begin with the non-abelian YM theory. The change in the action under a supersymmetric transformation is given by the total derivative of whereF ρ = ε µνρ F µν . In the real multiplet parametrization, we can rewrite V µ Y M in a more compact form as follows For the CS action (5.31) the variation under supersymmetry gives The case of the FI Lagrangian (5.32) is straightforward, and we obtain Finally, the variation of the matter action generates The contraction of the color and flavor indices is packaged into G ac . Notice that in the last line σ, ψ Σ and ψ Σ act appropriately on color indices.

Boundary conditions: a preview
In the previous sections we made precise two key elements of our initial discussion: we decomposed any compact A-type background M 3 into the union of submanifolds with boundary, called T , and we wrote down supersymmetric field theories for N = 2 chiral and vector superfields on M 3 , explicitly calculating the expressions for the supersymmetric variation V µ . When these field theories are restricted on T , the action can only be invariant under a subset of the bulk supersymmetries if there are boundary conditions solving the corresponding constraints V ⊥ = 0. In addition, a well-defined classical problem requires appropriate boundary conditions that annihilate all the surface contributions in the Euler-Lagrange variation of the system. Schematically, given a field Φ, and a bulk action S = M 3 L [Φ] the equations of motion of the theory require δS = 0, where On a space with boundary, one demands simultaneously A priori, the boundary equations B = 0 are a set of on-shell equations. In what follows, some of these boundary equations will be required to hold also off-shell and will be used to find solutions of V ⊥ = 0, which is our main goal.
It is convenient to rewrite both terms in a uniform way. Defining the doublet δΨ = (δψ a , δ ψc) and Ψ = (ψ a , ψc), we obtain the expression The form (6.5) also covers the case of (6.3). It is convenient to use ψ Σ instead of λ. If the generators of the Lie algrebra {t a } are normalized so that Tr[t a t c ] = δ ac , the corresponding metric K is the identity. In the real notation of subsection 5.1.4 both (6.3) and (6.4) can be written in the compact form with a general (possibly field-dependent) matrix M that has the property The boundary condition (6.9) respects the R-symmetry whatever r-charge is assigned to Ψ.

Vectors
There are two possible actions for a vector field A µ in 3d: L CS , and L Y M . The Euler-Lagrange variation with respect to A µ , yields the boundary terms F is the full non-abelian field strength and a ρ is defined in eq. (5.37). For a given set of generators {t a } of the gauge group, we can write A = A c t c and a = a c t c . Then, both (6.11) and (6.12) can be expressed in terms of the tensor with V ρ = A ρ for CS, and V ρ = a ρ for YM. We introduced bold indices m and n to describe general quiver gauge theories. Specifically, m = (a, m) is a double index where m labels the nodes of the quiver and a labels the generators of the gauge group S m , that refers to the node m of the quiver. Considering orthonormal generators, the matrix G is G mn = δ ac ⊗ κ mn . In the orthogonal frame {k µ ,k µ } on T M ′ 2 , we can further decompose V and δA along k andk to obtain (6.14) We used ε ⊥ρνk ρ k ν = −1. When M ′ 2 is endowed with a complex structure, the action of Sp(2, C) has a natural interpretation. By construction, these solutions are valid both for CS and YM gauge theories.
We point out that an additional interesting solution of B v [V, δA] = 0 is available in the case of CS theories. In general, the tensor κ mn is symmetric, but need not be positive definite. In that case, it may have isotropic subspaces. On this subspaces B v [V, δA] vanishes automatically, independently of the coordinate dependence of V and δA. For example, given an isotropic vector v m such that v m κ mn v n = 0, we may consider boundary conditions δA = v m δA m µ dx µ and V = v m V m µ dx µ with arbitrary components δA µ and V µ on M ′ 2 . For a general treatment of such boundary conditions in CS theory we refer the reader to [50].

Scalars
In the non-linear sigma model, the variation of L scalar with respect to φ a , φc, yields the result 17 The term proportional to V ⊥ does not contribute, because V ⊥ = 0 at the boundary. The first term can be written in compact notation as where G is the target space metric and Φ the vector of scalars, introduced in section 5.1.4. We can set (6.17) to zero by assuming that the two vectors δΦ and D ⊥ Φ = 0 are orthogonal. The standard way to do this, is to consider Dirichlet, δφ a = 0, or Neumann, D ⊥ φ a = 0, boundary conditions (and similarly for the scalars φ). Notice that in general D ⊥ contains non-vanishing normal components of a gauge connection. In supersymmetric YM theories, the gauge multiplet contains a real scalar σ in the adjoint representation of the gauge group. The variation δσ of the action yields the boundary term Tr (δσD ⊥ σ). This term is similar to (6.17), and can be set to zero in the same way.

Path integral and closure under supersymmetry
We conclude this section with an additional remark. In the ensuing sections 7 and 8 we solve the equations V ⊥ = 0 to obtain half-BPS boundary conditions for general supersymmetric gauge theories. This is sufficient for the purposes of the classical problem.
In the quantum problem we are integrating over generic field configurations in a path integral. In the presence of a boundary the integration is further restricted to configurations with specific boundary conditions. Consequently, in this context the invariance of the path integral with respect to a given symmetry requires that the boundary conditions are also invariant under the symmetry in question. In general, this is not automatic and it may lead to further restrictions on the boundary conditions.
Although we are mainly interested in the classical problem in this paper, we will partially address the issue of the closure of boundary conditions under supersymmetry in the following sections.

Boundary conditions I
In this section we address the precise form of A-type boundary conditions in general threedimensional non-linear sigma models. A good prototype for this exercise are A-type boundary conditions in 2d N = (2, 2) non-linear sigma models on the strip that define D-branes in a Kähler target space X . In that case we know, [12], that the solution of the A-type boundary conditions is describing D-branes wrapping Lagrangian submanifolds in X . We will describe how similar solutions arise in three-dimensional theories. We work out first the case of a flat space background, and then explain how things are modified when the 3d theory is placed on a general curved A-type background.

General equations
The action S of a supersymmetric non-linear sigma model is specified by a Kähler potential K, a superpotential W , and finally the R-charges and central charges of the chiral superfields. In this subsection K is generic (a flat Kähler potential for chiral superfields will be considered in the ensuing section 8). We continue to call the target space X .
In section 5 we calculated the variation of S under supersymmetry, and found a generic expression for V µ N Lσ . Here we are interested in solutions of the equations The indices a,c run from 1 to s, where 2s is the real dimension of the target space X . It is convenient to use the identity γ µ γ ν = g µν + γ µν and rewrite + ǫγ ⊥ γ ν ψ a D ν φc = +ǫψ a D ⊥ φc + ǫγ ⊥ν ψ a D ν φc , In equations (7.1)-(7.3) we recognize the combinations which appeared in the analysis of B s [Φ, δΦ] (6.16). This is expected because on-shell we can always use the Noether current to rewrite V ⊥ .
Following the discussion in section 6, we require V ⊥ N Lσ = 0. The analysis of this equation reduces naturally to the study of four types of terms: In order to obtain explicit boundary conditions for the fields that appear in these equations we have to disentangle the spinorial and target space structures. The reader can find the details of this computation in appendix B. Here we outline the main steps. Firstly, the anticommuting spinors are decomposed in components using the projectors P and P. As a result, all the geometric information can be packaged into the bilinears (3.24) Secondly, we impose the A-type projection on the spinors ǫ andǫ, ζǫ =ǫζ . (7.10) Finally, we impose the boundary condition (6.9) on the spinors, i.e. e i̟ (ζΨ) I = M I K (Ψζ) K . These manipulations introduce the orthogonal matrix M and the phase ̟ in V i . At the end, the V i depend only onζǫ and Ψζ. Hence, a bilinear ǫΨ common in all terms can be factorized out, and the result for V ⊥ N Lσ can be understood as a condition on the bosons. This is nicely expressed in the matrix notation Φ and Ψ of section 5.1.4. As a simple example of these manipulations, we obtain The complete result is where the matrix P The matrix R[̟] is the matrix of local R-symmetry.

Solutions in flat space
Having obtained the general formula (7.11) we are now in position to study solutions to equation V ⊥ N Lσ = 0. Flat space is of course a special case of our discussion. It is instructive to exhibit first how Lagrangian 'D-branes' come out of (7.11) for a theory defined on a euclidean 3d half-plane. In this case the boundary leaf M ′ 2 is a 2-plane. In flat space the profile of the background fields is trivial, and the covariant derivative D µ reduces to the standard partial derivative ∂ µ . In what follows we will also set, for convenience, Z = 0 for the central charges. The role of Z in (7.11b) is the same as that of a real mass obtained by giving a vev to the bottom component of a real multiplet coupled to Φ. We will consider such masses in relation to YM and CS theories in section 8.
Before going into the details of the solution, it is worth emphasizing two simplifying special properties of flat space: 1) There is always a choice of coordinates, say {θ, x,x}, such that the frame {n µ , k µ ,k µ } is precisely {∂ θ , ∂ x , ∂x}. The boundary is placed at a fixed value of θ.
2) The phase ̟ appearing in M [̟] is a constant.
Both of these features are generically absent in curved space because of the background curvature.
Focusing on the vanishing of the components (7.11a)-(7.11b), we obtain the conditions  18 The effect of the matrix R[̟] is to change the orientation of the Lagrangian submanifold by a constant angle ̟. The Lagrangian submanifold just described contains Φ(M ′ 2 ), the image of M ′ 2 under the maps Φ. Both M and the derivatives of Φ are objects in T X . The solutions (7.16)-(7.17) transform correctly under a change of coordinates in the target space. Locally, we may take a chart such that the Lagrangian submanifold is described by mixed Dirichlet and Neumann boundary conditions. We impose Neumann boundary conditions along the directions parallel to the submanifold, and Dirichlet conditions along the directions transverse to the submanifold.
In the simplest situation, in which X is an affine vector space and the Kähler potential is canonical, the Neumann and Dirichlet boundary conditions can be seen explicitly by solving The case with ̟ = 0, is solved by rotating the fields accordingly with the projector. The latter can be written as In this case the projector depends on M T R[−̟], in agreement with R-symmetry considerations. The vector W was defined in section 5.1.4, and in the complex basis it has components W a = K ac ∂c W , Wc = Kc a ∂ a W . Since W = Re W + iIm W is a holomorphic function of the fields, the Cauchy-Riemann equations imply the relations 18 For the convenience of the reader we remind that a Lagrangian submanifold L (defined on a symplectic manifold (N , ω), where ω is the symplectic form) is characterized by the two conditions: When the symplectic manifold N is Kähler, the Riemaniann metric GIJ can be used to characterize L, and the definition just given is equivalent to the condition The quantity G W is where ∂ i is shorthand notation for ∂ i = ∂/∂Φ i . Implementing the rotation Φ = R[−̟/2]Φ ′ , we obtain from (7.21) the projection equations where ∂ ′ = ∂/∂Φ ′ . Because Ker(1 ± M ) span the tangent space T M, and J is a bijection between these two kernels, we can understand the boundary condition ( which translates into the statement that ∂ ′ I ImW (Φ ′ ) has no component along the span of {Jv i } s i=1 and therefore ImW (Φ ′ ) is constant along the wordvolume of the submanifold L.

Solutions in curved space
In the previous section, we solved the equations V ⊥ N Lσ = 0 relying on two special features of flat space: the fact that the phase ̟ is constant, and the fact that there is a coordinateadapted orthogonal basis in T M 3 . In curved space we do not expect in general these two features to hold.
For example, in the case of the ellipsoid in toric coordinates with background fields (3.30) we find ̟ ± = ψ with frame vectors Consider now a more general manifold M 3 in toric coordinates (θ, φ 1 , φ 2 ), in similar notation to the one above for the ellipsoid. By definition, the Killing vector k = 1 Ω ∂ ψ is expressed as a combination of ∂ φ 1 and ∂ φ 2 , and ̟ is only a function of ψ. The triple of vectors (k µ , n µ ,k µ ) takes the form The functionsf , f n and v µ depend on the details of the background, however, the integrability condition (3.18) implies [v µ ∂ µ , ∂ ψ ] = 0. M 3 is decomposed in solid tori as before, M 3 ∼ = T 1 #T 2 , and the fields are restricted on one of the solid tori, call it T for simplicity. In this case, the general solution of V ⊥ = 0 has (see (7.11a), (7.11b)) In the first line we used, for illustration purposes, the simplifying assumption A (R) ⊥ = 0, which clearly holds for the example of the ellipsoid (7.26). The covariant derivatives in (7.32) arẽ new . In that case, the term that appears inside the parenthesis on the r.h.s. of equation (7.34), with the substitution new , vanishes because of the condition we found in (3.28). Consequently, we obtain as in flat space The analysis ofk µ D µ Φ requires more detailed knowledge off andk µ A (R) µ new . To be concrete, in the case of the geometries introduced in section 3.5 we obtain the following expressions: • For the ellipsoid, A (R) µ new and its scalar product withk µ , given in (7.29), are The functionf is also proportional to cot(2θ).
• For the circle bundles of section 3.5.2, we find The functionf is proportional to u(θ)/ sin θ.
We notice that the boundary condition fromk µ D µ Φ simplifies when the boundary is placed at the equator of the corresponding geometries, because at that pointk µ A (R) µ new = 0. When this happens, the covariant derivativek µ D µ Φ becomes a combination of partial derivatives, and again we can solve the boundary conditions as in flat space. Namely, we impose Neumann boundary conditions along the directions parallel to the submanifold, and Dirichlet for the directions transverse to the submanifold. The value off at the equator is not important in this statement. When the boundary is placed away from the equator a more complicated boundary condition (7.32) has to be imposed.
In more general setups, a background M 3 exhibits a coordinate-dependent phase ̟. In that case the boundary equations (7.31)-(7.32)

Real multiplets
Before we tackle general gauge theories, there is another comparatively simple example we would like to discuss. It is well-known that in 3d flat space there is a simple duality between a chiral superfield and an abelian gauge field. 19 We expect the corresponding boundary conditions to be mapped trivially under this duality. With this in mind, in this subsection we present A-type boundary conditions for N = 2 theories of s abelian vector superfields interacting via a constant target space metric G.
The supersymmetric variation V µ is expressed most conveniently in terms of the real parametrization of the abelian vector superfields in (5.38): We can further rearrange V ⊥ real by borrowing results from the study of V ⊥ N Lσ in the previous section. In particular, let us define the two complex combinations: ∂ ρ φ Σ ≡ a ρ − i∂ ρ σ and Im ϕ Σ ≡ (D + (H)σ), Re ϕ Σ ≡ 0. Then, we can rewrite V ⊥ real as and with an obvious change of variables, it is clear that we have obtained an expression that is essentially the sum of V 1 , V 2 and V 3 , given in (7.5), (7.6), (7.7), respectively. Consequently, the surface term V ⊥ real takes the suggestive form where the matrix M fixes the spinor boundary conditions e i̟ζ Ψ Σ = M Ψ Σ ζ. From the definition of a µ = −j µ − σV µ , and the fact that V ⊥ = 0, we finally obtain the boundary conditions The last condition is correctly invariant under the shift symmetry (2.6). Assumingk µ V µ = 0, the boundary conditions for j µ and ∂ µ σ are arranged as those of a neutral chiral multiplet.

Closure under supersymmetry
As we noted in subsection 6.2 the boundary conditions may transform non-trivially under supersymmetry. We would like to know if the boundary conditions that were formulated above are invariant under the A-type supersymmetries, and if not, whether invariance can be restored by imposing further constraints. Since the boundary conditions on the fermions are algebraic, it is immediately possible to examine how things work in some generality. In particular, when the matrix M is field independent we find that supersymmetry invariance of the fermion boundary conditions does not impose any new constraints. In contrast, the analysis of the transformation of the boson boundary conditions is more involved and case-dependent. Since the boson boundary conditions involve derivatives of the bosons, their transformation leads to expressions that involve derivatives of the corresponding fermions. The details of the resulting expressions depend on the specifics of the differential operators and, in general, have to be analyzed case by case. For that reason, and in order to keep the discussion as generic as possible, in what follows we will concentrate mostly on the transformation properties of the fermion boundary conditions.

Chiral and anti-chiral multiplets
The supersymmetry transformations of the fermions (ψ, ψ) in a chiral multiplet are For A-type supersymmetries, we set θ = θ. We want to examine how A-type supersymmetry transforms the boundary conditions e i̟ζ Ψ = M Ψζ. Assuming for simplicity that the matrix M is invariant we only need to consider the bilinears δΨζ,ζδΨ. Straightforward manipulations yield the scalar products Consequently, the condition e i̟ζ δΨ = M δΨζ holds if the following equations are satisfied In these formulae we recognize the boundary conditions that we derived previously for the bosons. As a minor difference comparing (7.49) to the original boundary condition (7.11a), we notice a sign change in front of the term Jk µ D µ Φ. This sign difference, however, is irrelevant in the final boundary conditions, since the two terms in the second equation in (7.49) have to vanish independently. We note that both n µ D µ Φ and Jk µ D µ Φ belong in Ker(1 + M [̟]) and their relative normalization is not fixed by the boundary conditions.
We conclude that the supersymmetry invariance of the fermion boundary conditions does not impose any new constraints when M is separately invariant. In the more general case of a field dependent M one needs to consider extra contributions from the supersymmetric variation of M .
Finally, regarding the variation of the bosons at the boundary, it is possible to prove in complete generality the orthogonality condition δΦ G D ⊥ Φ = 0. From the A-type supersymmetry, the definition of δφ and δ φ, and the boundary condition on the spinors e i̟ζ Ψ = Ψζ, we deduce the boundary variation

Real multiplets
The supersymmetry transformations δψ Σ and δ ψ Σ in a real multiplet are very similar to the ones of the chiral fermions δψ and δ ψ. The only difference is the contribution of the D-term Repeating the evaluation of e i̟ζ δΨ Σ = M δΨ Σ ζ we obtain results similar to the chiral multiplet case. Also in this case supersymmetry invariance of the fermion boundary conditions does not impose any further constraints.

Boundary conditions II
In this section we study A-type boundary conditions in general (non-abelian) N = 2 supersymmetric CS/YM-matter theories. The corresponding actions on curved backgrounds and their supersymmetric variations V µ were obtained in section 5.2. Our analysis recovers previously known results in special cases, e.g. flat space, and extends them to general A-type backgrounds T with a solid torus topology.
We discuss first the conditions arising from the supersymmetric variation in the gauge sector. The corresponding conditions in the matter sector are presented separately.

Description and summary of results
From the supersymmetric variation of the Yang-Mills and Chern-Simons actions, respectively, we obtain the boundary terms In the gauge sector analysis we will also include a term coming from the vector-matter couplings {t a } is a basis for the generators of the gauge groups in play, and φ t a φ denotes the action of the adjoint fermions λ = λ a t a andλ =λ a t a in the representation of each of the matter fields φ and φ. Recall that we use bold indices a to describe general quiver gauge theories.
In the multi-index a = (a, m) m labels the nodes of the quiver theory, and a the generators of the gauge group S m at the node m. For any set {t a } of generators we set Tr[t a t c ] = G ac , and κ ac = G ac ⊗ κ mn . For canonically normalized generators G ac = δ ac . The expressions in ( To analyze the supersymmetric variations V µ gs Y M and V µ gs CS we need to disentangle the geometric and spinorial structures. This can be achieved, as before, by using the projectors P, P, and the A-type projection on ǫ andǫ. On the anti-commuting spinors Ψ Σ = (ψ Σ , ψ Σ ) we impose the general boundary condition Supersymmetry will soon fix some of the properties of the matrix M , as we found for the non-linear sigma model in sec. 7. Nevertheless, the case of the non-linear sigma model and the case of the general gauge theory discussed here exhibit conceptually different properties. Let us highlight the origin of these differences. In the boundary condition (8.4), the spinors of the vector multiplets ψ Σ , ψ Σ have been arranged conveniently as a doublet Ψ Σ . The same doublet can be formed in non-linear sigma models out of the fermions in the chiral superfields. In that case we can also form naturally a corresponding doublet of bosons Φ = (φ, φ). This is also possible for abelian real superfields, where the role of Φ is played by the complex combination of the dual photon and the real scalar σ. In the case of a non-abelian gauge theory, however, there is no obvious natural bosonic Φ that we can associate to Ψ Σ . As a result, we cannot proceed identically to the non-linear sigma model case thinking in terms of a generalized target space structure on the gauge indices.
An alternative approach is suggested by the 2-form that appears in the on-shell boundary value problem for vectors. In (8.5) both V and δA are 1-forms on the boundary. For example, B v appears in the Euler-Lagrange variation of CS theories, (6.13), as well as in the supersymmetric variation V ⊥ CS , (5.39), In this equation B v couples the two components of A ν in the boundary directions to a combination of the spinors. It is therefore natural to think in terms of doublets distinguished by the spacetime indices of vectors parallel to the boundary. Similar manipulations can be employed in V ⊥ Y M using the identity γ µ γ ν = g µν + γ µν to rewrite the kinetic terms as follows The first two terms in (8.7a) and (8.7b) have the same structure as B v in (8.5) (up to a difference in ± signs). Introducing the notation we show in the next subsection (see eqs. (8.25), (8.28)) that where P U is a certain projector depending on a matrix U that has only gauge indices and satisfies U T GU = G and U 2 = 1. The interplay between A-type supersymmetry and the geometry of the form B v fixes the relation between U and M by setting Note that unlike the boundary conditions in the non-linear sigma model case, (6.9), (7.15), in (8.9) the gauge indices of ψ Σ and ψ Σ do not mix. With these boundary conditions and the standard, by now, manipulations on spinor bilinears we arrive at compact expressions for V ⊥ gs CS and V ⊥ gs Y M . In order to keep the notation simple and most transparent, let us quote the pertinent results in the case of a single gauge group. In Chern-Simons-matter theories we obtain where P U A =k µ A µ + iU k µ A µ . In the case of Yang-Mills theories where When the gauge group has an abelian component, a FI term can also be added to the Lagrangian. Since the variation of this term is of the type we can easily include the FI parameters in (8.11) by considering the shift D → D − ξ.
In summary, without assuming any further constraints on the spinors Ψ Σ other than (8.9), the most generic boundary conditions on the bosonic fields of the gauge multiplet are CS − theories : As a special solution, one can further impose P U V = 0 both in CS and YM theories. In the next subsection we show that this is equivalent to requiring I ± = 0. This projection, which is natural from the point of view of the Euler-Lagrange variations in Chern-Simons theory (6.15), selects a Lagrangian submanifold of B v , as we explain in the next section. The remaining conditions yield: • In the case of CS theory, (8.15) reduces to the algebraic equation of motion of the auxiliary field D, • In the case of YM, the conditionF ⊥ = 0 translates into ε ⊥µν F µν = 0, where the free indices are constrained to run over the boundary indices by anti-symmetry. Then, F ⊥ = 0 is satisfied if the non-abelian connection is flat at the boundary, namely F = 0 at the boundary. In components, the boundary condition on σ becomes

Technical details
Let us elaborate further on the details that led to the above boundary conditions. The key quantity is I ± (Ψ, V) defined in (8.8). We re-express this quantity using the A-type projection on ǫ andǫ. Leaving the label Σ of the spinors implicit, the resulting expression is 20) or equivalently in matrix notation (with ε ⊥νρk ρ k ν = +1) .

(8.21)
It is clear that I ± is similar in form to B v evaluated on specific complex combinations of the components of V and the spinors. We mentioned in sec. 6.1.2 that the most general solution to the equations B v = 0, are the Lagrangian submanifolds of the two-form B v . In special cases the general A-type boundary conditions (8.15)- (8.17) are solved by these Lagrangian submanifolds. We proceed to examine this aspect more closely. Starting with I − , which appears in the CS case, we notice that we can rewrite the fermions in (8.21) as follows ǫ The last expression can be written as a projector P − M acting on ǫΨ with The matrix M , which acts on the doublet Ψ = (ψ m , ψ m ), is of the general form M = R (2×2) ⊗ U , where U acts on the gauge indices and R is a 2-by-2 matrix. P ± M is a projector only if R = ±1. Choosing R = +1 for concreteness, (the R = −1 choice is very similar), P M becomes 24) and the matrix U is required to be orthogonal with respect to G, and to satisfy U 2 = 1. The quantity I − takes the final form Since U is orthogonal with respect to G, the condition I − = 0 can be achieved by setting Notice that the dependence on the phase ̟ has disappeared in the above manipulations.
The calculation of I + proceeds along similar lines. In this case, the relevant combination of spinors in (8.21) can be recast as yielding the final expression In conclusion, with R = +1, both conditions I + = 0 and I − = 0 lead to (8.27). Considering instead the choice R = −1 would lead to V ñ k − U n c iV c k = 0. Clearly, this choice is equivalent to the substitution U → −U .
We noted in the previous subsection that by setting I ± = 0 in V ⊥ CS or V ⊥ Y M we are led to a special solution of the boundary conditions (8.15)- (8.17) where P U V = 0: • In CS theories, where V = A, this is equivalent to a single boundary condition on the gauge field • In YM theories, where V =F, Dσ, one obtains two separate boundary conditions: one on the non-abelian field strength and another one on D µ σ These equations are natural covariant generalizations of corresponding boundary conditions in flat space that set components parallel to the boundary of the dual field strengtĥ F µ and D µ σ to zero.

Matter Sector
Next we focus on terms that arise from the supersymmetric variation of the matter sector of the gauge theory. These terms are functions of the spinors ψ and ψ of the chiral and anti-chiral multiplet. The relevant boundary contributions can be summarized in the expression The effects of a gauge invariant superpotential W can be incorporated, as already done in (5.14), by considering the on-shell relations G ac Fc = ∂ a W , F a G ac = ∂c W . (8.32) In (8.31) the chiral and anti-chiral superfields transform in arbitrary representations of the gauge group. In the bold multi-indices a = (a, m), a is a color index and m a flavor index. The metric G is the scalar product in the combined flavor/color index space. In non-abelian theories σ acts on φ ( φ) and ψ ( ψ) according to their representations. We will make a slight abuse of notation where the specifics of this action are suppressed. By making use of the standard identity γ ⊥ γ ν = g ⊥ν + i ε ⊥νρ γ ρ and the fact that V ⊥ = 0 at the boundary, V ⊥ matter can be rewritten in the form: The analysis of V ⊥ CS and V ⊥ Y M selected boundary conditions in the gauge sector on the basis of the two form B v . Even though V ⊥ matter can still be thought of as V ⊥ N Ls , on the basis of the flavor indices, in this subsection we will not follow the approach of section 7. Instead, we will explore the extension of the manipulations of the previous subsection 8.1 to the matter sector. Accordingly, we assume from the start the following boundary conditions on the matter fermions The matrices S and S act on the representation space of the matter. They are required to have the properties S 2 = S 2 = 1, and SG S = G. M acts diagonally on the doublet Ψ = (ψ a , ψc). This is the same type of ansatz that emerged in the gauge sector. Here two possibly different S and S are allowed because of the two chiralities. With standard manipulations of the spinor bilinears, we recast V ⊥ matter in terms of two independent spinor components ǫψ andǫ ψ, The projectors P S and P S are the analog of P U in the gauge sector. For matter charged under the R-symmetry, we see that the terms in V ⊥ matter proportional to the R-charges, ±r(iH), correctly combine with the covariant derivatives along the Killing vector.
The expression (8.35) allows us to read off the following general boundary conditions on the matter sector chiral : iP S D φc + D ⊥ φc − Scn(zn − qnσ) φn + ie −i̟ Fc = 0 , (8.38) anti − chiral : iP S Dφ a − D ⊥ φ a − S a m (z m − q m σ)φ m + ie i̟ F a = 0 . (8.39) A special solution of these boundary conditions is obtained by imposing the Lagrangian condition P S Dφ = P S D φ = 0. Setting to zero the remaining terms in (8.38), (8.39) we obtain The term (z − qσ) in (8.40) corresponds to the standard real mass. By taking S = 1 in flat space, the Lagrangian condition P S Dφ = 0 would become D + φ = 0, with D + a covariantized holomorphic derivative along the coordinates of the boundary, which in that case would be a plane. Thus, P S Dφ = P S D φ = 0 are the natural generalization to curved space of such boundary conditions. Comparing P S and P S with the projector P (̟,±) M in a non-linear sigma model, we observe that the boundary conditions in (8.38) and (8.39) are rather different from those describing a Lagrangian brane in the target space. Also, the projector P (̟,±) M was found to be ̟ dependent, whereas ̟ plays no role in P S and P S ; it only enters (8.38) and (8.39) through F and F .
The covariant derivatives in P S and P S contain both the dynamical gauge fields A and the R-symmetry connection A (R) µ . As simple illustrating cases, consider the following examples. The covariant derivative normal to the boundary, D ⊥ , simplifies under the additional assumption A (R) ⊥ = 0, and reads D ⊥ = ∂ ⊥ − iqA ⊥ . For the covariant derivatives along the boundary, D k and Dk, we can borrow part of the discussion in section 7.1.3 to understand their precise form. In the case of A-type backgrounds with twisted spinors, the twisted R-symmetry gauge field is such that D k becomes For the ellipsoid and the manifolds with SU (2) × U (1) symmetry that we introduced in section 3.5, we may also usek µ A (R) µ = 0 at the equator to obtain Dk in the simplified form In that case the boundary condition P S Dφ = 0 reads A similar result holds for P S D φ = 0. 20 20 The action of S on (Aµφ) m and of S on (Aµ φ)n should not be confused with the separate action of U that was defined in the gauge sector.
The expressions (8.41)-(8.43) hold under a set of simplifying assumptions for the background fields. For a generic A-type background, the full convariant derivatives, including the R-symmetry gauge fields, should be considered.
Finally, let us obverse that the two boundary conditions P S Dφ = 0 and P S D φ = 0 are genuinely complex, and the reality condition φ = φ ⋆ would impose either D k φ = Dkφ = 0 or S ⋆ = − S. In the latter case, the boundary conditions in (8.40) would decompose further into (z − qσ)φ = 0 and D ⊥ φ − ie i̟ F = 0.

Closure under supersymmetry
We conclude the analysis of the above boundary conditions, both in the gauge and the matter sector, with a study of their transformation under supersymmetry. We already looked at this problem when we discussed the boundary conditions of Lagrangian branes, and similar comments continue to apply here. In particular, the variation of the boundary conditions on the fermions are algebraic, and it is immediate to check whether they are closed under supersymmetry or not. For CS theories, the variation of the boundary conditions on A µ and σ are also simple and both turn out to be algebraic. In YM theories, the boundary conditions on the bosons are boundary conditions on the derivatives, hence their analysis requires specific information about the details of the background.

Matter sector
The supersymmetric variation of the matter fermions δψ and δ ψ under the A-type supersymmetry, θ = θ, is The conditions we want to check are in this case e i̟ζ δψ a Σ = S a m δψ m Σ ζ , e i̟ζ δ ψc = Sc n δ ψ n ζ . Compare these formulae with the boundary conditions (8.38) and (8.39). The two sets of conditions do not coincide, because several signs do not match. Requiring that they hold simultaneously requires This restricted set of conditions is consistent with the reality condition φ = φ ⋆ under the assumption S = −S ⋆ .