Four Dimensional Supersymmetric Theories in Presence of a Boundary

In this paper, we study $\mathcal{N} =1$ supersymmetric theories in four dimensions in presence of a boundary. We demonstrate that it is possible to preserve half the supersymmetry of the original theory by suitably modifying it in presence of a boundary. This is done by adding new boundary terms to the original action, such that the supersymmetric variation of the new terms exactly cancels the boundary terms generated by the supersymmetric transformation of the original bulk action. We also analyze the boundary projections of such supercharges used in such a theory. We study super-Yang-Mills theories in presence of a boundary using these results. Finally, we study the Born-Infeld action in presence of a boundary. We analyse the boundary effects for the Born-Infeld action coupled to a background dilaton and an axion field. We also analyse the boundary effects for an non-abelian Born-Infeld action. We explicitly construct the actions for these systems in presence of a boundary. This action preserves half of the original supersymmetry.


Introduction
The action for most renormalizable quantum field theories in four dimensions, including supersymmetric theories, is at most quadratic in derivatives.So, variation of the action for such theories produces a bulk term as well as a total derivative term.For manifolds without a boundary, the total derivative terms vanish due to the absence of a boundary.However, in presence of a boundary, such total derivatives give rise to boundary contributions.The presence of a boundary breaks the translational invariance of the theory, and this in turn breaks supersymmetry.In fact, supersymmetric variation of a supersymmetric action is known to be a total derivative.Thus, in presence of a boundary, the supersymmetric variation of an action, which is supersymmetric in flat space, produces a nonvanishing boundary term, this in turn breaks supersymmetry.
It is possible to restore supersymmetry on-shell by imposing some boundary conditions [1]- [2].There are various constraints generated from supersymmetry on the possible boundary conditions [3]- [7].However, this does not resolve the problem with the surface terms off-shell, since these boundary conditions are only imposed on the on-shell fields, and the supersymmetry is still broken offshell.Since most supersymmetric theories are quantized using path integral formalism which uses off-shell fields, it is important to try to construct actions which preserve some supersymmetry off-shell.
Here we show that it is possible to construct an action which preserves half the original supersymmetry off-shell.This can be done by modifying the original action through the addition of boundary terms.The new boundary term added to the original action exactly cancel the boundary contribution generated from the supersymmetric variation of the original bulk action.This procedure has been applied in three dimensions for N = 1 supersymmetric theories [8].The results thus obtained have been used for studying a system of multiple M2-branes ending on M5-brane [9]- [12].As the gauge sector for the action of multiple M2-branes comprises of Chern-Simons theories, and the gauge transformation of Chern-Simons theories in presence of a boundary also generates a boundary term, new boundary degrees of freedom had to be introduced on the boundary of the M2-branes.The gauge transformation of the action for these new boundary degrees of freedom exactly cancels the boundary contribution generated from the gauge transformation of the bulk action.A system of M2branes intersecting with M5-branes have also been analysed in the supergravity regime using a fuzzy funnel solution [13]- [18].
Apart from the M2-branes, the supersymmetric theory in presence of a boundary has also been used for analyzing non-anticommutativity in presence of a boundary for a three dimensional theory with N = 2 supersymmetry [19].By suitably combining the boundary effects with non-anticommutativity, a three dimensional theory with N = 1/2 supersymmetry has been constructed.In fact, the coupling of a three dimensional super-Yang-Mills theory to background flux has been studied on a manifold with a boundary [20].The BRST symmetry for this system has also been analyzed.However, all this work has been done in three dimensions.
It may be noted that just like M2-branes can end on M5-branes, D3-branes can also end on other objects in string theory.Such systems can be studied using fuzzy funnel [21]- [22].In fact, a system of D3-branes ending on other D3-branes have been analysed using fuzzy funnel [23].The fuzzy funnel have also been used to describe a system of D3-branes ending on D5-branes [24], and a system of D3-branes ending on D7-branes [25].It would be interesting to apply to extend first develop a formalism for analysing four dimensional supersymmetry in presence of a boundaries, and then using such a formalism for studying D3-branes ending on various objects in string theory.As the four dimensional super-Yang-Mills theory can be thought as a low energy limit of D3-brane action, we will analyze a four dimensional super-Yang-Mills theory in presence of a boundary.The construction of four dimensional supersymmetric theories in presence of a boundary can find several other applications, and we are going to mention some of them in conclusion section of this paper.
The remaining paper is organized as follows.In section 2, we discuss the general formalism for analysing N = 1 superfields in presence of a boundary, and construct a supersymmetric Lagrangian which preserves half of the original supersymmetry in presence of a boundary.In section 3, we discuss the transformation of bulk and boundary superfields and supercharges in presence of a boundary.In section 4, we apply this formalism to super-Yang-Mills the-ory.Finally, in section 5, we summarize our results and discuss some possible applications of the results of this paper.

Boundary Superfields
Let us start with a four dimensional theory in N = 1 superspace.This superspace can be parameterized by two supercharges, Q a = −i∂ a − (γ µ ∂ µ θ) a , and It is also possible to define superderivates which commute with these generators of These superderivatives can be represented as D a = ∂ a + i(γ µ ∂ µ θ) a , and Da = ∂a + i(γ µ ∂ µ θ) a , and satisfy Now we can write the Lagrangian for a supersymmetric theory with N = 1 supersymmetry as It may be noted that a linear combination of θ a and θa can be used to represent the four dimensional supersymmetry, where x ij are complex numbers such that, x 11 x 22 − x 12 x 21 = 0. We can write the original Lagrangian using these new coordinate as where J is the Jacobian for transformation.It is possible absorb the Jacobian for transformation, using field redefinition, Φ(θ is the original superfield.We shall assume this to be this case and neglect the numerical factor coming from the Jacobian.Now we choose x ij , such that in the new coordinates, the superderivatives take the form, and satisfy The generators of N = 1 supersymmetry corresponding to these superderivatives are given by and they also satisfy, These supercharges also commute with these superderivatives,{Q 1a , We also define, We can also define , and so, a Q 2b , we can write the bulk charges Q 1a and Q 2a as Using the super-derivative which commutes with the generator of N = 1 supersymmetry, we can write a Lagrangian for a supersymmetric theory with N = 1 supersymmetry as where the N = 1 superfield has been decomposed as It may be noted that p 1 (θ 1 ), p 2 (θ 2 ), q 1 (θ 1 ), q 2 (θ 2 ), r 1 (θ 1 ), r 2 (θ 2 ) are superfields in their own right depending only on θ 2 or θ 1 .Under the supersymmetric transformations generated by Q 1a and Q 2a , they transform as Thus, under these supersymmetric transformations generated by Q 1a this Lagrangian transforms as ), and under these supersymmetric transformations generated by Q 2a this Lagrangian transforms as ).So, the action is invariant under the supersymmetric transformations generated by Q 1a and Q 2a , in absence of a boundary, However, in presence of a boundary, the supersymmetric transformations generated by Q 1a and Q 2a produce boundary terms.Thus, if we assume that a boundary exists at x 3 = 0, then the supersymmetric transformations of the Lagrangian can be written as ).The presence of these boundary terms will breaks the supersymmetry of the resultant theory.We can perverse half the supersymmetry of the resultant theory by either adding or subtracting a boundary term to the original Lagrangian.Now if L 1b and L 2b is the boundary term added or subtracted from the bulk Lagrangian with N = 1 supersymmetry, then we have where q 1± (θ 1 ) = (1 ± γ 3 )q 1 (θ 1 )/2 and q 2± (θ 2 ) = (1 ± γ 3 )q 2 (θ 2 )/2.Hence, the Lagrangian L± L 1b preserves the supersymmetry generated by ǫ 1∓ Q 1± , and the the Lagrangian L ± L 2b preserves the supersymmetry generated by ǫ 2∓ Q 2± .It is not possible to simultaneously preserve both the supersymmetry generated by , in the presence of a boundary.However, in the presence of a boundary, we can construct the Lagrangian which preserves the supersymmetry generated by ǫ 1∓ Q 1± and ǫ 2∓ Q 2± as It may be noted that this Lagrangian preserves only half of the supersymmetry of the original Lagrangian.This is because if we preserve the supersymmetry corresponding to ǫ 1∓ Q 1± and ǫ 2∓ Q 2± , then we will break the supersymmetry corresponding to ǫ 1∓ Q 1∓ and ǫ 2∓ Q 2∓ .

Transformation of Boundary Fields
In this section, we will analyse the decomposition of the supercharges for a four dimensional theory with N = 1 supersymmetry.We can write the bulk supercharges as The boundary supercharges given by where s is the index for the coordinates along the boundary, i.e., the case µ = 3 has been excluded for a boundary fixed at x 3 .Now by definition Q 1± , Q 2± , are the generators of the half supersymmetry of the bulk fields and Q ′ 1± , , Q ′ 2± , are the standard supersymmetry of the boundary fields.We can now express the boundary superfields in terms of bulk superfields as follows, The original superfield also gets decomposed as follows, where Now we will analyse the superalgebra for the four dimensional theory with N = 1 supersymmetric theory, in the presence of a boundary.The non-vanishing part of the superalgebra is given by It may be noted that {Q 1± , Q 2± } = {D 1± , D 2± } = 0, and Now contracting these equation and using (P − ) a a = 1, we obtain the following result, Thus, we can see how the Lagrangian with the measure preserves the right amount of supersymmetry on the boundary, because we can write We can write it in terms of boundary superfields as where ′ means the quantity is evaluated at the boundary and The boundary measure only contains and ǫ 2± ′ Q ′ 2∓ act as independent supercharges.Thus, we can add a boundary Lagrangian to the original theory, which will still preserve half the supersymmetry of the original theory, where L t is the total Lagrangian for the bulk and the boundary theory, L is the Lagrangian for the original theory and L b is the Lagrangian for the boundary theory.Thus, we can add the following terms to Lagrangian Here ω ′ 1±2± are only defined on the boundary, where λ ′ 1±2± can be purely boundary Lagrangian.We can take a suitable gauge invariant coupling between this purely boundary fields and the bulk fields.It may be noted that on the boundary only the supersymmetry generated by 4 Super-Yang-Mills Theory In this section, we will write the action for super-Yang-Mills theory as using a vector field V A T A , where T A are the generators of the gauge symmetry, We can write the Lagrangian for the super-Yang-Mills theory as using a vector superfield where the superfield strengths are given by W a = −i D2 (e −V D a e V )/4 and Wa = −iD 2 (e −V Da e V )/4.Here the potential V(Φ, Φ) is a function of Φ and Φ.Even though this action looks like a non-local action, the component action in the bulk will be a local action.This is because it is another way of writing a local action.It may be noted, as we were only interested in analysing the amount of supersymmetry preserved, we will did not need the explicit form of super-Yang-Mills action in real superfields.It may be noted that even though the expression for it would involve an complicated expression containing the non-local operator, the component action would be local.This is because it can be transformed back into the local action.However, it is not clear if the resultant boundary action is local or not, as it cannot be transformed into any local action.So, we will express this Lagrangian into an alternative formalism, and in that formalism we will be able to obtain a local action for the super-Yang-Mills theory even in presence of a boundary.
The gauge transformations of the superfield V transforms are given by e V → e i Λe V e −iΛ , where Λ and Λ are chiral and anti-chiral gauge parameters.So, it is possible to write a covariant derivative which transforms under gauge transformation as ∇ a = e −V D a e V → e iΛ ∇ a e −iΛ and ∇a = Da → e iΛ ∇ a e −iΛ , and another covariant derivative which transforms under gauge transformation as ∇a = D a → e i Λ ∇a e −i Λ and ∇a = e V Da e −V → e i Λ ∇a e −i Λ.However, it is also possible to define another covariant derivative which transforms under a real gauge parameter u as ∇ a → u∇ a u −1 and ∇a → u ∇a u −1 [26].Now we can express this covariant derivative in terms of θ 1a and θ 2a rather than θ 1a and θ 2a .We can absorb the Jacobian using field redefinition, and then use the modified measure on the boundary.However, it would be more convenient to express the original covariant derivative in terms of the real spinor superfield and then work out the modification by the boundary theory.So, we define two gauge valued spinor superfields Γ 1a = Γ A 1a (θ 1 )T A and Γ 2a = Γ A 2a (θ 2 )T A , and use them to construct gauge covariant derivatives for matter fields Φ(θ 1 , θ 2 ) and Φ(θ 1 , θ 2 ), These fields transform under the gauge transformation as, Γ 1a → u∇ 1a u −1 , and Γ 2a → u∇ 2a u −1 , and so the covariant derivatives transform as ∇ 1a → u∇ 1a u −1 and u∇ 2a u −1 .It may be noted if we define ∇ a and ∇a as a linear combination of ∇ 1a and u, then we will get to correct transformation for the original covariant derivatives.This is because where x ij are complex numbers.We can also construct the field strengths as follows, Now these field strengths transform as W 1a → uW 1a u −1 , and W 2a → uW 2a u −1 .Now we can write the action for super-Yang-Mills theory as where V[Φ, Φ] is a potential term which is given by product of superfields Φ and Φ.Now we can write the Lagrangian for super-Yang-Mills theory which preserves various supercharges as follows, This result can be also be written as It is thus transparent that this modified Lagrangian only preserves half the original supersymmetry.

Conclusion
In this paper, we have analysed the restoration of half the supersymmetry for a four dimensional theory in N = 1 superspace formalism, on manifolds with a boundary.We first use the fact that a total derivative term is obtained from the supersymmetric variation of a Lagrangian for a four dimensional theory with N = 1 supersymmetry.This total derivative term vanishes in absence of a boundary.However, in presence of a boundary, this total derivative term generates a boundary term, which breaks half the supersymmetry of the original theory.However, half of the original supersymmetry can be preserved by adding new boundary terms to the original Lagrangian.The supersymmetric variation of these new boundary terms exactly canceled the boundary terms generated by the supersymmetric transformation of the original bulk Lagrangian.We explicitly constructed such boundary terms for the four dimensional theory with N = 1 supersymmetry.We also related the bulk supercharges to the boundary supercharges.The bulk supercharges behaved as two independent supercharges on the boundary.However, the inclusion of the new boundary terms only preserved the supersymmetry only with respect to one of these projections.Thus, it was demonstrated that only half of the supersymmetry of the original theory was preserved.This analysis was done using the real superfields, and the Jacobian of transformation was absorbed in field redefinitions.We finally applied our results to the super-Yang-Mills theory.We explicitly constructed the Lagrangian which preserves half the supersymmetry of the original theory.It is possible to generalize this present analysis to theories which have higher amount of supersymmetry.In fact, the analysis of this present paper can be used for analysing various aspects of the AdS/CF T correspondence [27]- [30].This is because according to the AdS/CF T correspondence type IIB string theory on AdS 5 × S 5 is dual to the N = 4 super-Yang-Mills theory on its conformal boundary.Thus, the theory which describes the low energy limit of the action for a stack of D3-branes on AdS 5 ×S 5 is the N = 4 super-Yang-Mills action with the gauge group U (N ).The four would-volume coordinates of the D3-branes become the Minkowski coordinates, and six transverse coordinates to the D3branes give rise to the six gauge valued scalar fields of N = 4 super-Yang-Mills theory.This theory also contains eight gauge valued fermions, and a gauge field.It would be interesting to analyse N = 4 super-Yang-Mills theory in presence of a boundary.It is again expected that the N = 4 super-Yang-Mills theory in presence of a boundary will preserve only half the supersymmetry of the original theory.
It has been demonstrated that using the Horava-Witten theory, one of the low energy limits of the heterotic string theory can be obtained from the eleven dimensional supergravity in presence of a boundary [31]- [34].In this construction, it has been possible to obtain a unification of gauge and gravitational couplings.It would be interesting to analyse the connection between the Horava-Witten theory and the boundary supersymmetry discussed in this paper.In order to do that, it might be interesting to first generalize the results of this paper to five dimensions.This is because, motivated by Horava-Witten theory, a five dimensional globally supersymmetric Yang-Mills theory coupled to a four dimensional hypermultiplet on the boundary has been constructed [35].
It may be noted that in Randall-Sundrum models our four dimensional universe is thought to be located on a three-brane in a five-dimensional spacetime with negative cosmological constant [36]- [37].These models provide a geometrical solution to the electroweak hierarchy problem.A supersymmetric generalization of such models have also been analysed in [38]- [39].In fact, it has been argued that for the supersymmetric Randall-Sundrum models to be consistent, the issue of supersymmetric boundary conditions has to be analysed [40].We are hoping to generalize such a procedure to help constructing supersymmetric Randall-Sundrum models.