Deformation of ${\cal N}=4$ SYM with varying couplings via fluxes and intersecting branes

We study deformations of ${\cal N}=4$ supersymmetric Yang-Mills theory with space-time dependent couplings by embedding probe D3-branes in supergravity backgrounds with non-trivial fluxes. The effective action on the world-volume of the D3-branes is analyzed and a map between the deformation parameters and the fluxes is obtained. As an explicit example, we consider D3-branes in a background corresponding to $(p,q)$ 5-branes intersecting them and show that the effective theory on the D3-branes precisely agrees with the supersymmetric Janus configuration found by Gaiotto and Witten in arXiv:0804.2907 . D3-branes in an intersecting D3-brane background is also analyzed and the D3-brane effective action reproduces one of the supersymmetric configurations with $ISO(1,1)\times SO(2)\times SO(4)$ symmetry found in our previous paper arXiv:1710.09792 .


Introduction
As it is well-known, the effective theory on D3-branes in flat space-time becomes N = 4 supersymmetric Yang-Mills (SYM) theory in the field theory limit (α ′ → 0). If the background has non-trivial fluxes, the effective theory on the D3-branes will be deformed accordingly. This is one of the useful ways to obtain 4 dimensional gauge theories with less (or no) supersymmetry (SUSY). In fact, various deformations realized in this way have been investigated, for instance, in [3,4,5,6] in the context of flux compactifications. In these works, because the main motivation was to obtain a model beyond the Standard Model, the deformations were assumed to preserve 4 dimensional Poincaré symmetry ISO (1,3). One of the main purposes of this paper is to generalize the deformations to the cases where ISO(1, 3) is explicitly broken. In particular, the couplings * in the action may depend on the space-time coordinates.
In our recent paper [2], we wrote down the conditions to preserve part of the supersymmetry in deformed N = 4 SYM with varying couplings and found various non-trivial solutions. † Though the motivation was in string theory, the analyses in [2] were purely field theoretical. In this paper, we try to realize such systems in string theory by putting probe D3-branes in supergravity backgrounds with fluxes and find a map between the couplings in the action of the deformed N = 4 SYM and the fluxes in the background.
One way to obtain a theory with varying couplings is to consider a background corresponding to D-branes (or other branes) intersecting with the probe D3-branes. A typical example is a system with D3-branes embedded in a background with [p, q] 7-branes that appear as codimension 2 defects in the D3-brane world-volume [10,11,12]. ‡ In this system, it is known that the complex coupling (2.9) is not a constant but depends holomorphically on a complex coordinate which is a complex combination of 2 spatial coordinates transverse to the 7-branes. Our results for the D3-brane effective action can be applied to this system as well as various other intersecting brane systems. We demonstrate it in two explicit examples; intersecting D3-(p, q)5-brane and D3-D3 systems. In the former example, we show that the effective action on the probe D3-branes precisely reproduces the action for the supersymmetric Janus configuration found in [1]. The latter example reproduces one of the supersymmetric solutions with ISO(1, 1) × SO(2) × SO(4) symmetry obtained in [2].
These two examples correspond to the realization of the deformed N = 4 SYM studied in [1] and [2] in string theory. In Section 5, we conclude the paper with various discussions on the present work and further applications. In addition, two appendices are included. In Appendix A, we summarize our conventions used for the supergravity fields. In Appendix B, we show the explicit calculations to obtain the effective action of D3-branes given in Section 3.

Deformations of N = 4 SYM with varying couplings
In this section, we review some of the results obtained in [2]. Following [2], we use a 10dimensional notation, in which N = 4 SYM is regarded as a dimensional reduction of 10dimensional N = 1 SYM. The 10-dimensional gauge field A I (I = 0, . . . , 9) is reduced to a 4-dimensional gauge field A µ (µ = 0, . . . , 3) and 6 scalar fields A A (A = 4, . . . , 9), and the 10- Let us consider the following deformation of the N = 4 SU(N) supersymmetric Yang-Mills theory: where I, J = 0, . . . , 9; µ, ν = 0, . . . , 3; A, B = 4, . . . , 9. F IJ is defined as 3) (2.5) * In [2], the fermions are chosen to have negative chirality (minus sign in the right hand side of (2.1)). Here, we choose the chirality to be positive, in order to match the convention used in [16]. One way to relate our convention here and that in [2] is to use a transformation x 9 → −x 9 , which induces Ψ here = Γ9Ψ there , A here for I ′ = 9, and similar sign changes for the parameters d IJA , m AB and m IJK .
The covariant derivatives on the fermion field Ψ are defined as where the indicesμ,ν = 0, . . . , 3 are flat indices, Γνρ ≡ 1 2 (ΓνΓρ − ΓρΓν) and ωνρ µ is the spin connection. We assume that the metric g IJ has the form and g IJ denotes its inverse. The 4-dimensional Levi-Civita symbol ǫ µνρσ is defined such that We also introduce a vielbein eÎ I satisfying eÎ I eĴ J ηÎĴ = g IJ and its inverse e Î I . The gamma matrices with the curved indices are defined by Γ I = e Î I ΓÎ. The quantities a, c, d IJA , m AB are real parameters and M is a 32 × 32 real anti-symmetric matrix. All of them may depend on the space-time coordinates x µ . In this paper, we call these parameters as "couplings", though m AB and M are related to masses. The couplings a and c are related to the gauge coupling g YM and the theta parameter θ as follows: It is useful to define the complex coupling τ in terms of these quantities: The parameters d IJA and m AB exhibit the following symmetries whereas the most general form for M is given by where m IJK is a real rank-3 anti-symmetric tensor and The ansatz for the SUSY transformation is where ǫ is the SUSY parameter represented as a 10-dimensional Majorana-Weyl spinor and B A is a 32 × 32 real matrix. Both ǫ and B A may depend on space-time.
Then, the invariance of the action (2.2) under this SUSY transformation implies the following equations: where F is a real 32 × 32 matrix acting on the spinor indices and The condition (2.14) has a trivial solution, e IJK = 0, which is equivalent to Using the symmetries of the deformation parameters (2.10), the latter is written as Further discussions on the nature of these equations and their solutions, we refer to [2]. Let us summarize a few explicit solutions that are relevant for our discussion.
The case with ISO(1, 2) × SO(3) × SO(3) symmetry is analyzed by Gaiotto and Witten in [1]. (See also section 3.4 in [2]) It is a solution of the SUSY conditions (2.14)-(2.18) with the parameters depending only on x 3 . ISO(1, 2) is the Poincaré group acting on x 0,1,2 and SO(3) ×SO(3) acts on x 4,5,6 and x 7,8,9 . The metric is assumed to be flat and the non-trivial components of the couplings in the action are given as follows: where τ 0 and D are real constants and ψ is an arbitrary real function of x 3 with 0 < ψ < π/2 assuming D > 0.
The case with ISO(1, 1) × SO(2) × SO(4) symmetry is given in section 4.1 in [2]. Here, ISO(1, 1), SO(2) and SO(4) act on x 0,1 , x 4,5 and x 6,7,8,9 , respectively, and the couplings in the action may depend on x 2,3 . The metric (2.7) is assumed to be where the indices are α, β = 0, 1; m, n = 2, 3; a, b = 4, 5 and p, q = 6, 7, 8, 9. In this case, the complex coupling (2.9) turns out to be an arbitrary holomorphic (or anti-holomorphic) function of a complex coordinate z ≡ 1 √ 2 (x 2 + ix 3 ) with Im τ > 0 and ϕ in the metric (2.27) is an arbitrary real function of x 2,3 . M is of the form: and α m and β m are determined by τ and ϕ as where s = ±, ǫ nm = ǫ n m ′ g m ′ m is the Levi-Civita symbol for the x 2,3 -plane and Λ is an arbitrary real function. ‡ The non-trivial components of d IJA and m AB are g mn q m q n − g mn ∂ m q n + 8g mn β m β n − 4s∂ m β n ǫ mn δ ab , (2.32) m pq = − 1 2 g mn q m q n − g mn ∂ m q n δ pq , (2.33) † Our convention is slightly different from that in [1]. The solution shown here is taken from section 3.4 in [2] with b 0 = 2 and l(z) = −iz/ √ 2. We also made a transformation x 9 → −x 9 . (See the footnote in p.3.) ‡ Λ can be absorbed by a local SO(2) rotation of the x 4,5 -plane. See Appendix C.2 in [2].
In this case, (ϕ − log Im τ ) is a harmonic function on x 2,3 -plane satisfying This is the case studied in [10,11,12]. (See also section 3.3 in [2]) It is related to the effective theory on the D3-branes embedded in a 7-brane background as mentioned in the introduction.

D3-branes in curved backgrounds with fluxes
In this section, we study the effective action of D3-branes in curved backgrounds with fluxes and try to relate the couplings in the action (2.2) with the supergravity fields. As reviewed in Appendix B.1, the effective action of Dp-branes in general backgrounds is known, at least, to the extent needed for our purpose. (See (B.1) and (B.13) for the bosonic and fermionic parts, respectively.) However, the expression of the effective action reviewed in Appendix B.1 is not convenient for a direct comparison with the action (2.2) used in the field theoretical analysis.
To find a relations between couplings in (2.2) and the fluxes in the supergravity background, we expand the D3-brane effective action with respect to α ′ = l 2 s and keep only the terms that survive in the α ′ → 0 limit, assuming that the background fields are of O(α ′0 ).
We consider N D3-branes embedded in a 10 dimensional space-time parametrized by (x µ , x i ) with µ = 0, 1, 2, 3 and i = 4, . . . , 9. We use the static gauge, in which the world-volume of the D3-branes is parametrized by x µ (µ = 0, 1, 2, 3). The scalar fields, which are related to A A (A = 4, . . . , 9) in the previous section, are denoted here as Φ i (i = 4, . . . , 9). The scalar field Φ i describes the position of the D3-branes in the x i direction. Assuming that the D3-branes are placed at x i = 0 when Φ i = 0, the relation between the position of D3-branes and the value of scalar fields is given by Φ i = λx i with λ ≡ 2πα ′ = 2πl 2 s . (See (B.9) for the precise meaning of this identification for N > 1.) To simplify the analysis, we assume that (µ, i) components of the metric g µi vanish everywhere, and all the components of the Kalb-Ramond 2-form fields and all the R-R fields, except the R-R 0-form C 0 , vanish at x i = 0 (i = 4, . . . , 9) * : Note that unlike in (2.7), the (i, j) component of the metric g ij may have non-trivial x µ dependence.
Here, we simply state our results on the D3-brane effective action and leave the details to Appendix B. Neglecting the O(α ′ ) terms in the action (B.1) and (B.13), we obtain: where the upper (lower) signs correspond to the case of D3-(D3-) branes, D µ denotes the 4 dimensional covariant derivative defined in (2.6) and (B.10), and ω µîĵ is the (µ,î,ĵ) component of the spin connection † related to the vielbein e î j as 2) are defined in (B.59), (B.60) and (B.54). See also Appendix A for our conventions for the supergravity fields.
The quantity M ± in (3.3) is given by The potential V (Φ i ) in (3.2) has two contributions: 6) * It is generically possible to choose a gauge such that B 2 | x i =0 = 0 and C n | x i =0 = 0 (n = 0) (at least locally) provided the components H µνρ and F µνρ vanish at x i = 0. Obviously, the reason for considering a non-vanishing C 0 is that we want to capture the theta parameter θ in the SYM action (2.2). † The hatted indices are the flat indices as in the previous section. We assume e î µ = 0 and e μ i = 0 without loss of generality under the assumption (3.1). where All the supergravity fields and their derivatives in the action (3.2) and (3.3) are evaluated at x i = 0. The first term in V DBI (3.7) can be discarded in the comparison with the field theory results, because it doesn't depend on Φ i . If we require Φ i = 0 and A µ = 0 to be a solution of the equations of motion, the linear term in (3.6) has to vanish: which is the condition that the force due to NS-NS and R-R fields cancel each other. In this case, we can safely take the l s → 0 limit.
Since the metric used in the action (2.2) is assumed to be of the form (2.7), we introduce a new metricḡ where µ, ν = 0, . . . , 3; A, B = 4, . . . , 9 and ω is a real function. Here, we put a factor e 2ω in the 4-dimensional metric, because it is often convenient to make a Weyl transformation to get a metricḡ IJ that can be identified with g IJ used in the previous section. ‡ In addition, we redefine the scalar fields as where e A i is the vielbein for the transverse space eˆi i with the identification A =î = 4, . . . , 9. so that the kinetic term can be written as in (2.2) with the metricḡ IJ defined in (3.14).
Then, discarding the total derivative terms, the bosonic part of the D3 brane action (3.2) becomes etc., and m AB is defined as The fermionic part (3.3) is rewritten as where we have defined and Here, Γ I (I = 0, 1, . . . , 9) are the gamma matrices satisfying Now, we can readily find the correspondence between the couplings and the supergravity fields. By comparing the action (2.2) with (3.16) and (3.19), assuming (3.13), we obtain and Note that the first equation of (3.25) can be written as Then, the relations (3.23)-(3.27) imply (2.20) and (2.22), which is equivalent to the condition e IJK = 0 that solves one of the SUSY condition (2.14) as discussed in the previous section.
In the following section, we are going to check these identifications by explicitly inserting some particular backgrounds in the effective action for the D3-branes and comparing with the supersymmetric deformations of the N = 4 SYM reviewed in Section 2.

(p, q) 5-branes and Gaiotto-Witten solution
In this subsection, we consider D3-branes embedded in a background with (p, q) 5-branes. * The brane configuration is summarized as The effective action on the D3-brane world-volume can be written down by using (3.2) and (3.3). As we will soon see, because the (p, q) 5-branes are not extended along the x 3 -direction, the gauge coupling and the theta parameter of the D3-brane action depend on the coordinate x 3 . This brane configuration is related to the supersymmetric Janus configurations considered in [1]. We will show that the action obtained by using (3.2) and (3.3) is indeed consistent with that obtained in [1], which provides a consistency check of our results in section 3.
The dilaton and R-R 0-from combined into a complex scalar field τ ≡ g −1 s (C 0 + ie −φ ) can be written as where are real constants and ψ(r) is a real function satisfying Here, we have assumed p, q, χ 0 are all positive and 0 < ψ < π 2 . The complex scalar field (4.9) evaluated at x i = 0 corresponds to the complex coupling (2.9). In fact, the expression in (4.9) agrees with the complex coupling obtained in [1]. (See (2.23).) Note here that ψ| x i =0 can be chosen to be a generic real function of x 3 , because as mentioned above, h(r) in (4.11) can be replaced with an arbitrary positive harmonic function on R 4 transverse to the (p, q) 5-branes.
where we have defined , (4.15) and introduced new coordinates y µ (µ = 0, 1, 2, 3) satisfying Then, using the coordinates y µ , the bosonic part of the effective action (3.2) becomes In order to compare with the action (2.2), it is convenient to rescale the scalar fields as Then, the kinetic term of the scalar fields can be rewritten as where A, B = 4, . . . , 9 and the non-zero components of m AB are Here, the prime denotes the derivative with respect to y 3 ,e.g., ξ ′ = ∂ y 3 ξ. These expressions precisely agree with (2.24) Note that the non-zero components of (G R ± ) ijk are and one can show the following relations: Using these, the last term of (4.17) can be written as In summary, the bosonic part is written as with φ and C 0 given by (4.9).
The supergravity solution corresponding to n D3-branes placed at x m = 0 and x p = x p 0 , in the string frame, is where g s is a constant and h(r) is given as The metric evaluated at the position of the probe D3-branes, i.e. x i = 0 (i = 4, . . . , 9), is where we have defined andḡ µν dx µ dx ν ≡ η αβ dx α dx β + e ϕ δ mn dx m dx n . It is again easy to see that both V DBI and V CS in the potential (3.6) are flat, because F 3 = H 3 = 0, F µνρσi = 0 and L DBI defined in (B.31) is a constant. Then, the bosonic part (3.2) becomes whereǭ µνρσ is the Levi-Civita symbol withǭ 0123 ≡ 1/ √ −ḡ. In order to compare with the results in [2], we redefine the scalar fields as Then, the kinetic terms for the scalar fields become  The non-zero components of ( * 4 F 5 ) µ ij are whereǭ 23 = −ǭ 32 = ḡ 22ḡ33 = e −ϕ . The last term in (4.39) becomes where ε 45 = −ε 54 = 1. This gives Collecting all these results, (4.39) becomes The fermionic part (3.3) for this configuration is As in the previous subsection, we rescale Ψ, M ± and the gamma matrices by The rescaled gamma matrices satisfy Then, we obtain whereǭ n m ≡ǭ nn ′ḡ n ′ m . This gives

Conclusions and outlook
In this work, we have complemented the study of deformations of N = 4 SYM with varying couplings that we initiated in [2] by showing that some of these gauge theories can be realized on the probe D3-branes in curved backgrounds with fluxes. In particular, we obtained the effective action on the D3-branes for general backgrounds satisfying (3.1) and gave an explicit map between the couplings in the deformed N = 4 SYM and the fluxes of the curved background on which the D3-branes are embedded.
As a check, we explicitly showed that the effective action on the D3-branes in a background with (p, q) 5-branes (see (4.1)) reproduces that of the supersymmetric Janus configuration found in [1]. We also studied D3-branes in a background with another stack of D3-branes intersecting with them (see (4.31)) and found that the action agrees with one of the solutions of SUSY conditions with ISO(1, 1) × SO(2) × SO(4) symmetry found in [2].
On the other hand, in [2], we found a lot of solutions of SUSY conditions, for which the realization in string theory is not known. Our results in (3.23)-(3.27) suggest that it is possible to extract some information of supergravity fields from the couplings in the deformed N = 4 SYM. Indeed, it is now easy to know which fluxes have non-trivial profiles for the brane configuration that realizes the deformed N = 4 SYM. For example, for the cases with ISO(1, Such configurations, assuming that they can be realized in string theory, should have non-trivial ( * 4 F 1 ) 012 , ( * 4 F 1 ) 013 , (G R ± ) 456 and (G R ± ) 789 fluxes. Despite we have not shown this explicitly, this fact suggests that such a configuration corresponds to D3-branes in a background with (p, q)5and [p ′ , q ′ ]7-branes: It would be interesting to see this more explicitly.
Finally, we want to stress that we didn't use the equations of motion for the supergravity fields in our analysis in Section 3. That is to say, some additional constraints are imposed on the couplings from the supergravity equations of motion. In this respect, some works has been done in [3,20], where it has been shown that the couplings have to satisfy some algebraic equations obtained from the supergravity equations of motion. Furthermore, if we require SUSY, the background as well as the D3-brane configurations should satisfy BPS conditions. It would be interesting to see whether such conditions agree with the SUSY conditions found in [2]. Actually, there is a logical possibility that the deformed N = 4 SYM action (2.2) could have some additional SUSY solutions which are not necessarily related to backgrounds satisfying the equations of motion in supergravity. It would be important to study the correspondence in more detail and clarify this issue.

A Conventions for supergravity fields
We follow the conventions for the supergravity fields used in [21]. The bosonic part of the type IIB supergravity action in the string frame is where and |ω n | 2 for an n-form ω n is defined as In our convention, the dilaton φ vanishes asymptotically and κ is related to the Newton's constant G N , string length l s and string coupling g s as In addition, we have to impose the self-duality condition Here, the Hodge star * is defined by * (dx I 1 ∧ · · · ∧ dx In ) = 1 (10 − n)! ǫ I 1 ···I 10 g I n+1 J n+1 · · · g I 10 J 10 dx J n+1 ∧ · · · ∧ dx J 10 , where ǫ M 1 ···M 10 is the 10-dimensional Levi-Civita symbol with ǫ 01···9 = 1/ √ −g.
It is useful to define F n with n > 5 by φ, B 2 , C 0 , C 2 and C 4 are related to those used in [22], denoted with superscript "P", as The metric in the Einstein frame is defined as The action can be written as where |ω n | 2 E is defined as in (A.3) with the metric in the Einstein frame, This action is invariant under the SL(2, R) transformation: with κ, g E M N and C 4 + 1 2 B 2 ∧ C 2 kept fixed.

B Derivation of the D3-brane effective action
In this appendix, we show the detailed derivation of the action (3.2) and (3.3).

B.1 Dp-branes in curved backgrounds (review)
For convenience, we first review the effective action of Dp-branes embedded in general backgrounds in Appendix B.1, following [21] and [16] for bosonic and fermionic parts, respectively.

B.1.1 Bosonic part
In this subsection, we review the bosonic part of the effective action on Dp-branes embedded in a curved background with fluxes following [21].
The 10-dimensional space-time coordinates are denoted as x I (I = 0, 1, . . . , 9). We choose the static gauge in which x µ (µ = 0, 1, 2, 3) are identified as the coordinates on the Dp-brane world-volume and x i (i = 4, . . . , 9) parametrize the transverse directions. The bosonic sector of the effective theory contains a U(N) gauge field A µ (µ = 0, . . . , p), (9 − p) scalar fields Φ i (i = p + 1, . . . , 9), which belong to the adjoint representation of the gauge group U(N). The reference position of the Dp-brane is chosen to be x i = 0 and small deviations from it is described by the values of the scalar fields. where the Dirac-Born-Infeld (DBI) and Chern-Simons (CS) terms are given by Here, the parameters T p , µ p and λ are given by where l s is the string length, g s is the string coupling, and the upper (lower) sign appearing in µ p , which is proportional to the R-R charge of the Dp-brane, corresponds to the case of Dp-branes (Dp-branes). The quantities M µν and Q i j are given by where F is the field strength of the gauge field A living on the brane, and φ is the dilaton field, g IJ is the background metric, B 2 = 1 2 B IJ dx I ∧ dx J is the Kalb-Ramond 2-form field and C n (n = 0, 2, 4, 6, 8) are the Ramond-Ramond (R-R) n-form potential. The hat " " on the background fields indicates that they are evaluated at the position of the Dp-branes placed at x i = λΦ i , which is defined via a Taylor expansion as, e.g., The symbol P[· · · ] in (B.3) and (B.5) denotes the pull-back of the bulk fields over the Dpbrane world-volume, in which the ordinary derivative ∂ µ Φ i is replaced by the covariant derivative D µ Φ i : (B.10) For example, the pull-back of E µν is given by 3) denotes the interior product by a vector (Φ i ), e.g., The symbol Str{· · · } in (B.2) and (B.3) denotes the symmetrized trace, which means Φ i in the expansion (B.9), F µν , D µ Φ i and [Φ i , Φ j ] are symmetrized before taking the trace.

B.1.2 Fermionic part
In this subsection, we write down the fermionic part (quadratic terms with respect to the fermion fields) of the effective action on a Dp-brane embedded in any supergravity background following [16]. Here, we consider the cases with a single Dp-brane in type IIB string theory.
The action, after fixing the κ-symmetry, is given by where ψ is the fermion field (dimensional reduction of the 10-dimensional positive chirality Majorana-Weyl spinor field), Γ µ ≡ ΓÎeÎ I ∂ µ x I is the pull-back of the 10-dimensional gamma matrices, M µν is the Abelian version of (B.5): and other quantities are defined as follows.
The covariant derivative ∇ (H) ν is the pull-back of the 10-dimensional covariant derivative including the H-flux: where ωĴK I is the spin connection and H IJK is the field strength of the Kalb-Ramond 2-form field.

B.2 D3-brane effective action
In this appendix, we consider the particular case of D3-branes under some simple and relatively general assumptions (3.1). We will study the expansion of the full action to leading and sub-leading orders that survive in the field theory limit and establish a relation between the backgrounds fields and the couplings in the action (2.2) of the deformed N = 4 SYM.

B.2.1 DBI action
In this section, we present the extended calculations of the expansion of the DBI term (B.2) with respect to λ. Let us first consider the quantity M µν defined in (B.5). The pull-back of the first term of (B.5) is given in (B.11) and it is expanded as where we have used the assumptions in (3.1). The expansion of Q i j in (B.6) is , it is easy to see that under the assumptions (3.1) and we can discard such higher-order contributions.
On the other hand, using the formula for general matrices X and δX, we get The first term in (B.29) gives the DBI part of the scalar potential. Let us define The derivatives are Evaluating these quantities at x i = 0, we obtain Then, the final expression for the DBI action is (B.37)

B.2.2 CS term
Let us study now the expansion of the CS-term of the D3-brane action, (B.3).
First, we define K ≡ n:even Note that it satisfies dK = n:odd For an n-form ω n , we define an m-form with (n − m) indices in the transverse directions (ω n ) m,i 1 ···i n−m as For example, Under the assumptions (3.1), the CS-term (B.3) is expanded as Expanding the first term, we get where [· · · ] 0 denotes the pull-back on the world-volume at x i = 0 (obtained by setting x i = 0 and dx i = 0), DΦ j is a 1-form defined as and we have used the notation (B.41).
The trace of the second term in (B.44) can be rewritten as (B.46) * In this section, we often omit the symbol "∧" in the products of differential forms.

Using this equation and the identities
which are valid under the assumptions (3.1), we obtain (B.48) The second and third terms in (B.43) are rewritten by using respectively.
Plugging these results in (B.43), the CS-term becomes It can also be written as where the potential V CS (Φ) is and we have defined and used the relation where we have set for D3-branes and D3-branes, respectively, and defined where we have used the notation (B.54) and defined ( * 4 F 1 ) νρσ ≡ ǫ µνρσ F µ = ǫ µ νρσ ∂ µ C 0 . (B.74) For the non-Abelian case, the covariant derivative ∇ µ should be replaced with where D µ Ψ is defined in (2.6) and we have assumed g µi = 0 and ω µνî = −ω µîν = 0.
Then, our final expression for the fermionic part of the action is where (G R ± ) iµν and (G R ± ) ijk are defined in (B.59) and (B.60), respectively.