Defects in Chern-Simons theory, gauged WZW models on the brane, and level-rank duality

We consider Hanany-Witten setups of 3- and 5-branes in type IIB string theory that realize N=(1,0),(2,0) and (1,1) gauged WZW models in 1+1 dimensions. The gauged WZW models arise as theories residing on the boundary of D3 branes ending on D5 branes. From the point of view of low energy dynamics the D5 branes play the role of half-BPS co-dimension-1 defects (domain walls) in 3d N=1 or N=2 Chern-Simons theories. Extending the analysis of previous works on the subject of boundary conditions in (supersymmetric) Chern-Simons theory, we discuss in detail the field theory construction of a large class of Chern-Simons domain wall theories and its embedding in open string dynamics. Finally, we exhibit how standard brane moves that result to 3d Seiberg duality, translate in our setup to a generalized level-rank duality in gauged-WZW models.


Introduction
We are interested in Hanany-Witten (HW) brane configurations [1] that preserve one or two real supersymmetries and realize at low energies three-dimensional supersymmetric Chern-Simons (CS) theories with a half-BPS co-dimension-1 defect (boundary or domain wall). The CS theories reside on D3-branes suspended between two stacks of 5-branes and the co-dimension-1 defect lies at the intersection of the D3-branes with D5-branes.
Half-BPS domain walls and boundaries in 3d supersymmetric gauge theories are interesting from a quantum field theory point of view (see [2,3,4,5] for a sample of recent discussions), but also because of the information that they carry about D-and M-brane dynamics in string/Mtheory (an example of prominent interest involves M2-branes ending on M5-branes [6,7]).
In the present work the 3d theories of interest are topological CS theories with N = 0, 1, 2 supersymmetry. CS theories on spaces with boundary have been studied extensively in the past starting from the seminal work of Witten [8]. It is well known, in particular, that with suitable boundary conditions the boundary theory is a Wess-Zumino-Witten (WZW) model [9]. Our main contribution to this story can be summarized as follows: (a) We give a boundary degrees of freedom reformulation of the boundary conditions [10] that realize gauged WZW models. In the process, we find it useful to formulate an extended class of domain wall theories in CS theory that has a natural embedding in brane setups.
(b) In N = 1, 2 supersymmetric CS theory we consider half-BPS domain wall theories that are given by N = (1, 0), or N = (1, 1), (2, 0) gauged WZW models. The boundary effects of supersymmetry are treated using the formalism of Ref. [11], however, our approach differs from previous work on this subject in two ways. First, we treat gauge invariance differently from [12] by suitably incorporating boundary degrees of freedom along the lines of a generalization of [13,14]. Second, compared to [14] we describe the case of 3d N = 2 supersymmetry in N = 2 superspace formalism without going to the so-called Ivanov gauge [15].
The relevant constructions in field theory are discussed in sections 2-4. Section 2 analyses a class of bosonic prototype cases. It captures the gist of the construction without going into the subtleties of supersymmetry. Domain walls in N = 1 CS theory are discussed in section 3. The case of N = 2 CS theory is explained in section 4.
The more interesting case of Chern-Simons-matter theories can be considered with similar methods by incorporating matter to our discussion. The relevant construction of boundary actions in the context of the ABJM theory [16] and the orthogonal M2-M5 intersection will be discussed elsewhere.
As we mentioned already, suitable brane configurations in string theory can provide an interesting perspective on the physics of domain wall theories in supersymmetric gauge theories. Along these lines: (a ′ ) We present brane setups that realize at low energies half-BPS domain walls in supersymmetric CS theories, and we argue that the corresponding domain wall theories are the ones analyzed in sections 3, 4.
(b ′ ) A standard brane deformation that exchanges the position of 5-branes acts on the lowenergy 3d theory as Seiberg duality [17,18]. We argue that the corresponding effect on the domain wall theories is a generalized level-rank duality.
The precise properties of open string dynamics in the brane configurations of interest will be considered in sections 5, 6. In the rest of this section we would like to give an up front summary of the main configurations that will be discussed.

Brane setups
We consider brane configurations in type IIB string theory that involve D3, D5, NS5 and (1, k)5- This table lists the number and type of branes involved and the directions along which they are extended. The notation |6| denotes that a brane is oriented along the direction 6 in a finite interval. Here the D3-branes are suspended between the NS5-brane and the (1, k)5-brane which are separated along the direction 6. The notation D3 ± and 2 ± denotes that the D3-branes are 1 We use conventions where (p, q)5 denotes a fivebrane bound state with p units of NS5-brane charge and q units of D5-brane charge. Without loss of generality we will henceforth assume that k > 0.
extended along the half-line x 2 > 0 for +, and x 2 < 0 for −. Both sets of D3-branes end at x 2 = 0 at N + M D5 branes. Finally, the notation 5 9 θ denotes that a brane stretches in the (59)-plane along a line at angle θ from the 5-axis. The quoted supersymmetry is preserved when the angle θ is fixed in terms of k and the string coupling constant g s via the relation tan θ = k g s .
In appendix A we show that the above configuration preserves only one real supersymmetry realizing a 2d N = (1, 0) theory at the intersection along the (01)-plane. Away from the overlapping N + M D5-branes the low energy theory at the (2 + 1)-dimensional intersection is pure N = 1 CS theory [19]. The D5 intersection is a co-dimension-1 defect from the point of view of the CS theory. Across the defect the rank of the gauge group of the CS theory changes.
The common direction 3 of the 5-branes gives a classically massless N = 1 scalar superfield in the 3d bulk theory, but as was pointed out in [20] quantum effects make the non-abelian part of this multiplet massive and irrelevant for the deep infrared (IR) physics.
1.3 1/16-BPS configuration: d = 2, N = (1, 1) Finally, by changing the orientation of the D5-branes we can obtain a non-chiral d = 2 N = (1, 1) theory at the defect The supersymmetries preserved by this configuration are verified in appendix A.
Our main interest in all of the above cases is the low-energy domain wall theory that describes the dynamics at the two-dimensional D3-D5 intersection at x 2 = 0. Before we analyze this theory it will first be useful to revisit separately the issue of domain walls and boundaries in CS theory.
We proceed to discuss three relevant examples with increasing amounts of supersymmetry, N = 0, 1, 2.

Chern-Simons domain walls
Our prototype for the more involved supersymmetric theories that follow is standard bosonic Chern-Simons theory in 2 + 1 dimensions (parametrized by coordinates (x 0 , x 1 , x 2 )) in a slightly uncommon situation where the gauge group jumps abruptly on a (1 + 1)-dimensional defect located, say, at x 2 = 0. To be specific, consider on the left (x 2 < 0) CS theory at level k with gauge group H, and on the right (x 2 > 0) CS theory at level k with gauge group G. We assume H ⊂ G. The total bulk action of the system at x 2 = 0 is where we define the 3-form A is a gauge field in the adjoint of G and B is a gauge field in the adjoint of H.
With a parity transformation hence in an equivalent form In this form our theory has been reformulated as a boundary problem. In what follows, we will mostly work with the boundary formulation (7).
It is well known that the CS theory has a gauge anomaly on spaces with boundary [9]; under a gauge transformation the action changes by a non-vanishing surface contribution. There are two standard ways to cancel this surface contribution. We can either impose suitable boundary conditions on the gauge field, or add explicit new degrees of freedom at the boundary that transform under a gauge transformation.
We proceed to describe a specific example that shows how boundary degrees of freedom deal with the gauge anomaly in the context of the bulk action (7). Then, we demonstrate the equivalence of this construction with an alternative formulation in terms of a suitable set of boundary conditions.

Boundary degrees of freedom
Following a slight generalization of the discussion in Ref. [13] we consider boundary degrees of freedom g that are group elements in the larger gauge group G. For these degrees of freedom we postulate the boundary action We use notation where D A µ , D B µ are gauge covariant derivatives with respect to the bulk gauge fields A and B respectively, Π H is a projector of elements of the Lie algebra of group G to elements of the Lie algebra of H, and is the g gauge transformation of A (similar expressions apply to B).
Both lines on the RHS of eq. (8) are supported on the boundary x 2 = 0. In particular, the term on the second line, which arises from the subtraction of two three-dimensional contributions, is a total derivative [13]. In the definition of the second line the domain of the group elements g is extended in the bulk, but since the final result has support only on the boundary this bulk extension is not unique. For instance, we can use different bulk extensions of g for the gauge fields A and B.
Notice that the first term on the RHS of eq. (8) is obviously gauge invariant. It introduces kinetic terms for the degrees of freedom g that lead to a natural two-dimensional CFT on the boundary. Clearly, the boundary theory is not unique, and these terms are part of the choice we are making.
With this construction it is obvious that the total bulk-boundary action is invariant under gauge transformations in the group G of the form The bulk action has originally a G × H gauge anomaly at the boundary, and the second line in the boundary action (8)  Before we move on, it is instructive to consider the more explicit form of the total action (10). Introducing light-cone coordinates x ± = x 0 ± x 1 and expanding out the boundary part (8) we find S W ZW,G k denotes the standard action of the WZW model at level k with group G On the other hand, the bulk part (7) can be recast into the following form after integrating by F µν and G µν are respectively the field strengths of the non-abelian gauge fields A µ and B µ .
When we add together (12) and (14) to obtain S total the boundary term Tr cancels out and the gauge field components A + , B + appear as Lagrange multipliers. Integrating them out we obtain F 2− = 0, G 2− = 0 that we solve by setting Then, employing the Polyakov-Wiegmann identity, and setting [21] Π we find that S total is the vector G k /H k gauged WZW action with gauge fields

Boundary conditions
For later purposes it will be useful to know if the same final result can be obtained by using appropriate boundary conditions. It is known [10] that the bulk CS action (7) admits the following boundary conditions These conditions set the boundary term Tr (14) to zero and then by standard manipulations analogous to the ones performed in the previous subsection they lead to the vector G k /H k gauged WZW action (17). We conclude that the boundary conditions (19), (20) are equivalent to the boundary action (8). Notice in particular that, as in the case of (19), (20), the axial part of H A × H B is broken explicitly at the boundary by (19), but the vector part is preserved.
In the following sections we will choose to formulate domain walls and boundaries in supersymmetric CS theories using the approach of boundary degrees of freedom and boundary interactions. This approach provides a flexible uniform prescription for many cases that is convenient for the resolution of issues related to supersymmetry and gauge invariance.
Note. The Euler-Lagrange variation of the action imposes additional on-shell boundary conditions. We will not discuss these conditions explicitly here. The relevant details can be found for example in Ref. [13].

An extended class of boundary actions
The domain wall theory (8) is by no means unique. The basic building block of (8) is the boundary action [13] written here for a single bulk gauge field A in the Lie algebra of a group G, and g an element of the same group. In section 5 we will encounter a generalization of this construction that arises naturally from open string dynamics. It will be useful to describe this extension here in a simplified non-supersymmetric, bosonic context.
The crucial feature that makes (21) work is the fact that g transforms as a field in the fundamental representation of the bulk gauge group (under the left action of the group). As is evident from (11) the simultaneous left action of the group on g with a bulk gauge transformation is enough to render the combination A g invariant. The passive role of the right action of the gauge group in this manipulation suggests a natural generalization, where instead of considering boundary degrees of freedom g in the bi-fundamental of G L × G R , 2 we consider them in the bi-fundamental of the general product G L × G ′ . G ′ can be different from G. Accordingly, in this more general case, we will denote the Hermitian conjugate of g byḡ, instead of g −1 .
With these specifications, we can construct an extended class of boundary actions where A g in the Lie algebra of G ′ denotes the combination g is defined asg with the property gg = 1 G ′ . We are using bold fonts for A g to distinguish it from the standard With these definitions, (22) continues to exhibit the nice features of (21). It is a boundary action for g independent of its bulk extension. Moreover, A g , as well as the total bulk-boundary are invariant under the combined bulk-boundary gauge transformation as desired.
In section 5 the boundary action (8) will be recovered from this more general construction as a special case where extra massive boundary degrees of freedom are integrated out to produce naturally the terms that involve the projection on the subgroup H.

N = 1 Chern-Simons theory
We proceed to describe the N = 1 supersymmetric version of the previous discussion. Compared to the bosonic case, where we had to worry only about the gauge symmetry, here we also have to consider what happens to the supersymmetry. In general, the co-dimension-1 defect breaks the bulk supersymmetry, and since we are interested in half-BPS defects some additional care needs to be taken to ensure that the appropriate amount of supersymmetry is restored on the defect by suitable boundary interactions.

Details of N = 1 supersymmetry
It is convenient to work in 3d N = 1 superspace formalism with coordinates (x µ , θ α ). Our conventions are summarized in appendix B.
The supersymmetric multiplet that contains the gauge field can be packaged in a spinor superfield Γ α that contains a Majorana spinor χ α , a real scalar M , the gaugino λ α and the The N = 1 CS theory at level k with gauge group G takes the form Specific expressions for the spinor superfield Ω α in terms of Γ α are provided in appendix B.
Supersymmetry restoring boundary interactions. We will introduce boundary degrees of freedom and boundary interactions that restore half of the bulk supersymmetry along the lines of [11,12]. More specifically, it has been shown [11] that the general bulk-boundary action preserves the supersymmetry generated by the supercharge Q ∓ . In what follows, we choose, by convention, to preserve the supersymmetry generated by Q − . The domain wall theory is a 2d Applying the prescription (29) to the N = 1 CS theory (28) we obtain the action We notice that the auxiliary field M (see eq. (27)) is absent from this action.

N = (1, 0) gauged WZW models
We are now in position to formulate the N = 1 supersymmetric version of subsection 2.1. In the bulk (x 2 > 0) we have N = 1 CS theory with gauge group G at level k and N = 1 CS theory with gauge group H ⊂ G at level −k. We will denote the corresponding spinor superfields Γ G α and Γ H α . With the inclusion of the supersymmetrizing boundary interactions we denote S bulk is invariant under Q − , but gauge symmetry is broken at the boundary.
In complete analogy to the bosonic case of subsection 2.1 we restore the G H × H vector part of the gauge symmetry by introducing boundary N = 1 scalar multiplets g valued in the larger with the boundary action Γ g denotes the gauge transformed spinor multiplet where ∇ α is a super-gauge-covariant derivative defined in appendix B. For the component fields (χ α , A µ , λ α ) of each of the spinor multiplets Γ G and Γ H that appear in (33) the gauge transformation (34) acts as follows Finally, in (33) S kin,W ZW k [g, Γ] denotes the gauge-covariant kinetic term that appears in the N = (1, 0) WZW model [22,23]. Denoting bŷ the boundary N = (1, 0) projection of the group superfield g, we can write ∇ α denotes the boundary version of the super-gauge-covariant derivative, where all contributions are evaluated at the boundary and projected on the appropriate chirality [14].
Putting these formulae together we obtain the N = 1 supersymmetric version of subsection 2.1. The total action provides the N = (1, 0) supersymmetric completion of the vector G k /H k gauged WZW model.

N = 2 Chern-Simons theory
The extension to N = 2 CS theory can be performed in a similar fashion. With N = 2 supersymmetry there are two types of half-BPS domain walls: the first type preserves N = (2, 0) supersymmetry on the defect and the second type N = (1, 1) supersymmetry.

Details of N = 2 supersymmetry
Our conventions for N = 2 supersymmetry are summarized in appendix B. Before we delve into the details of our construction it will be useful to highlight a few well known details of N = 2 supersymmetry that one should be aware of.
The N = 2 superspace has two sets of N = 1 Grassmann variables, (θ 1α , θ 2α ), that can be combined into the complex Grassmann variables Accordingly, the component fields of an N = 2 vector multiplet can be arranged in an N = 2 superfield V that can be built out of two N = 1 scalar superfields A, B and an N = 1 spinor superfield Γ α as The N = 2 CS theory at level k with gauge group G has a four-dimensional formulation of the form [15] D α is the N = 2 superspace covariant derivative (96).
with g an N = 2 chiral superfield valued in the group G, can be used to shift away the N = 1 superfield A in (41) and to go to a convenient gauge where A = 0 and the N = 2 CS action (42) simplifies to In this gauge (the so-called Ivanov gauge) the B superfield is auxiliary and the N = 2 CS action is essentially identical to the N = 1 CS action.
For an abelian gauge group G the integral over s in (42) can be performed explicitly in the general gauge and the N = 2 CS action can be expressed simply in three dimensions in terms of the N = 1 superfields A, B, Γ α [15]. For non-abelian G, however, there is no such simple expression and it is common to work instead in the Ivanov-Wess-Zumino gauge where A = 0 and χ α = M = 0 in Γ α .
In our case, the presence of the defect obstructs this passage to the Ivanov gauge, unless we choose to break explicitly part of the N = 2 super-gauge invariance. In what follows we will opt to keep the full N = 2 symmetry. That means we will have to work with the complete four-dimensional actions (42) without implementing the Ivanov gauge.
Supersymmetry restoring boundary interactions. The prescription of Ref. [11] can be applied to the general N = 2 action (see also [12]) to restore either N = (2, 0) or N = (1, 1) supersymmetry on the two-dimensional boundary. It is not hard to verify that the action preserves the supersymmetries generated by (Q 1∓ , Q 2− ). We will use the notation S (±) SUSY,G k [V ] to denote S (±) in the case of N = 2 CS theory.

N = (2, 0) gauged WZW models
In the bulk (x 2 > 0) we consider N = 2 CS theory with gauge group G at level k and N = 2 CS theory with gauge group H at level −k. We will denote the corresponding N = 2 vector multiplets V G , V H . Including the boundary interactions that restore the (Q 1− , Q 2− ) supersymmetries we set We restore the gauge symmetry that leads to the 2d N = (2, 0) G k /H k coset on the boundary by introducing N = 2 chiral multiplets g valued in the larger gauge group G with boundary action S bdy = S The gauge covariantized N = (2, 0) WZW kinetic terms S (2,0) kin,W ZW k [g, V ] have the following form [22,23]. On the boundary we project the bulk group N = 2 superfield g intô and define WZW kinetic terms. For quick reference we summarize the pertinent formulae: with the boundary projectionĝ

1/32-BPS brane setups and level-rank duality
After the long technical introduction, we are finally in position to discuss more explicitly the properties of the brane configurations (1). Our main goal is to identify the low-energy gauge theory that resides on the semi-infinite D3-branes and to formulate the domain wall theory on D3-D5 intersections.

Identification of the low-energy theory
Brane configurations with D3-branes suspended along the direction 6 between NS5 and (1, k)5branes, without the D5-branes across (013456), have been studied extensively in the past [24,19,25]. The low-energy 3d theory on N suspended D3-branes is N = 1 CS theory at level k with gauge group U (N ) coupled to a classically massless N = 1 scalar multiplet in the adjoint representation that captures the classically free motion of the D3 branes in the common fivebrane whereP is the Hermitian conjugate of the matrix P , and the hat denotes the boundary projection. When P acquires the vev (55) only the h modes of the form

Bulk Seiberg duality and level-rank duality on the defect
Further evidence in favor of the above proposal can be obtained by analyzing the effects of other deformations of the brane setup (1). We will focus on a deformation that slides the (1, k)5brane along the 6-direction through the NS5-brane. This operation is commonly performed in HW setups to argue in favor of Seiberg duality for the low-energy gauge theory on the brane configuration. Characteristic applications of this deformation include [27] in the context of type IIA HW setups and 4d N = 1 super-QCD theory, or [17] in the context of type IIB HW setups and 3d N = 2 Chern-Simons-SQCD theory. In Ref. [26] the exchange of the (1, k)5-brane with the NS5-brane along the 6-direction in the brane configuration (1) without any D5-branes was used to exhibit the IR equivalence (Seiberg duality) between the 3d N = 1 CS theory at level k with gauge group U (N ) -in short, U (N ) k CS theory-and its dual, the U (|k| − N ) −k CS theory. 3 In the presence of a domain wall, or boundary, we expect to see the bulk Seiberg duality to translate on the two-dimensional defect to a corresponding duality symmetry. With a gauged WZW model on the defect the anticipated duality is some version of level-rank duality. Indeed, this expectation is borne out correctly by the above-proposed description of the brane setup (1).
With N + M > 0 D5-branes the following effects are taking place. As we pass the (1, k)5- With the common assumption that the IR physics of the brane configuration are insensitive to the above brane operation we are led to the following equivalence between N = (1, 0) gauged WZW models on the domain wall Borrowing a standard language from Seiberg duality, we will call the LHS of this equivalence the 'electric theory', and the RHS the 'magnetic theory'.
This equivalence passes a number of immediate tests: (a) The global U (M ) symmetry matches on both sides.
(b) The central charges match non-trivially: We remind the reader that the central charge of the U (N ) k supersymmetric WZW model is (c) In the special case with N = 0 we obtain the duality relation which is a well known level-rank duality relation following from the triviality of the coset Superficially, the generalized level-rank duality (60) follows from (63) by applying it separately on the numerator and denominator of the coset (d) The brane configuration suggests that the Witten index of the model is The above formula is a consequence of the s-rule. It corresponds to the various ways the Repeating the brane argument for Seiberg duality of subsection 5.2 we recover verbatim the N = (2, 0) or N = (1, 1) version of the generalized level-rank duality (60).

Outlook
We described in field theory a general class of domain wall theories in (supersymmetric) CS theories. Some of these theories realize gauged WZW models. We demostrated that these construc- Chern-Simons-matter theories undergo a Giveon-Kutasov duality [17]. The domain wall theories in our setup will exhibit a corresponding level-rank duality in the presence of additional matter fields and bulk-boundary couplings that is interesting to analyze further.
Another interesting, potentially related, aspect of our work appears in the context of the brane setups (1). In the absence of the D5-branes that produce domain walls it was pointed out in [26] that the low-energy 3d N = 1 gauge theory is very closely related to the Acharya-Vafa (AV) theory [29] that has been argued to describe the low-energy theory on the domain walls of the four-dimensional N = 1 super-Yang-Mills (SYM) theory. Ref. [26]

A Brane supersymmetries
The supersymmetries preserved by brane configurations of the type (1)-(3) without the D5branes have been studied extensively in the past [24,19,25]. Here we summarize the less studied effects of the N + M D5-branes, that play the role of the domain wall in the low-energy gauge theory.
Brane setup (1). We obtain the projection equations The angle θ obeys the relation It is easy to check that the combined D3, NS5, (1, k)5 equations preserve only 2 supersymmetries. With the use of these equations the last one coming from the D5-branes becomes reducing the supersymmetry by a further 1/2. This results to a 2d chiral spinor.
Without the D5-branes the configuration preserves four supersymmetries. A little manipulation shows that the 16-component spinor ε 1 is projected twice The D5-branes reduce supersymmetry by a further 1/2 giving again the projection equation (71). We obtain d = 2 N = (2, 0) supersymmetry.
Brane setup (3). The only difference compared to the previous configuration is the D5 orientation that results to the spinor projection equation D5 : Γ 016789 ε 1 = ε 2 .

B Supersymmetry conventions
For quick reference and the convenience of the reader in this appendix we summarize our conventions for N = 1 and N = 2 supersymmetry in three dimensions.

Spinor index manipulations
We use small letters from the beginning of the Greek alphabet, α, β, . . ., to denote spinor indices.
Small Greek letters from the middle of the Greek alphabet, µ, ν, . . . , are reserved for the 3d spacetime indices.

SUSY algebra, covariant derivatives and superfields
The supercharges Q α of N = 1 supersymmetry and the corresponding superspace derivatives D α obey The two basic multiplets of N = 1 supersymmetry can be formulated with the use of the off-shell superfields Spinor : The spinor superfields, in particular, are used to package the gaugino λ α and the gauge field A µ . The Majorana spinor χ α and the real scalar M are auxiliary fields. In the formulation of the N = 1 CS theory (28) it is also useful to define the related superfields In the case of gauge theories it is also convenient to form the super-gauge-covariant derivative The conventional gauge-covariant derivative, which is the (γ µ θ) α component of ∇ α , is denoted as in the main text.

B.2 N = 2 supersymmetry
For N = 2 supersymmetry we sometimes use the complex Grassmann variables For Grassmann integrations Accordingly, we define the N = 2 superspace covariant derivatives as which can be decomposed to the N = 1 superspace derivatives

Superfields
The main superfields of N = 2 supersymmetry are vector superfields and chiral superfields.
In the so-called Ivanov gauge one sets A = 0 and uses the Wess-Zumino gauge (χ = M = 0) for Γ α .
In the presence of a gauge symmetry the chiral N = 2 super-gauge-covariant derivative takes the form