Supersymmetric Gauge Theory with Space-time Dependent Couplings

We study deformations of ${\cal N}=4$ supersymmetric Yang-Mills theory with couplings and masses depending on space-time. The conditions to preserve part of the supersymmetry are derived and a lot of solutions of these conditions are found. The main example is the case with $ISO(1,1)\times SO(3)\times SO(3)$ symmetry, in which couplings, as well as masses and the theta parameter, can depend on two spatial coordinates. In the case $ISO(1,1)$ is enhanced to $ISO(1,2)$, it reproduces the supersymmetric Janus configuration found by Gaiotto and Witten. When $SO(3)\times SO(3)$ is enhanced to $SO(6)$, it agrees with the world-volume theory of D3-branes embedded in F-theory (a background with 7-branes in type IIB string theory). We also find the general solution of the supersymmetry conditions for the cases with $ISO(1,1)\times SO(2)\times SO(4)$ symmetry. The cases with time dependent couplings and/or masses are also considered.

However, in string theory, they are obtained as the values of background fields and, in this sense, there is no conceptual difference between putting QFT in a curved space-time and making the couplings space-time dependent. In fact, there are a lot of examples. The Newton constant and gauge couplings are proportional to the value of dilaton field φ. In type IIB string theory, it is combined with RR 0-form field C 0 , which corresponds to an analog of the theta parameter, to have a complex coupling τ = ie −φ + C 0 . One way to make it vary in space is to include D7branes, or more generally [p, q] 7-branes. Regarding τ as the modulus of a torus, we can think of uplifting the 10-dimensional space-time with 7-branes to an elliptic fibered 12-dimensional space-time, which is called F-theory [1]. Another interesting example of varying coupling is the so-called Janus configuration, in which the coupling depends on one of the spatial coordinate [2,3].
In this paper, we consider deformations of N = 4 supersymmetric Yang-Mills (SYM) theory with varying couplings as typical examples of QFT with space-time dependent parameters. In particular, we investigate the conditions to preserve part of the supersymmetry (SUSY). It is a natural extension of the works to find curved space-times preserving SUSY, for which systematic methods using, e.g., topological twist [4] or supergravity [5] have been developed. We are of course not the first one to consider this class of theories. Supersymmetric Janus configurations were studied in [6,7,8,9,10,11,12]. Their generalization to the configurations with couplings depending on more than one directions were also investigated in [13,14]. SUSY configurations in N = 4 SYM with varying couplings have been studied for example in [15,16,17,18,19,20,21] (see also [22,23] for earlier works). Time dependent couplings have also been considered for example in [24,25,26,27,28,29,30].
Our approach here is perhaps the most rudimentary one. We write down all the possible deformations in the action of N = 4 SYM and the SUSY variation, and find the conditions to preserve part of the SUSY by performing the SUSY variation. Although the calculation is a bit tedious, it is straightforward and easy to understand the details. We hope it will provide a useful guide for further analyses and generalizations to other SUSY QFT.
Many of the examples considered in this paper can be realized as the world-volume gauge theory on probe D3-branes embedded in some non-trivial backgrounds in type IIB string theory. This system is related by duality to M5-branes wrapped on a torus with varying modulus in M-theory. This gives a 6-dimensional description of the 4-dimensional QFT with its complex coupling identified with the modulus of the torus that corresponds to the extra two dimensions, which is analogous to the idea of F-theory mentioned above. The SUSY condition for the deformed N = 4 SYM should be related to the conditions to preserve SUSY for the D3-branes in the supergravity background. In this way, our field theoretical analysis contains some information of the supergravity background. As we will mention in Section 3.3, it is possible to extract part of the equations of motion, including Einstein equation, of supergravity for the case of an F-theory configuration.
The organization of the paper is as follows. In Section 2, after fixing our notations and ansatz for the action and SUSY variation, the conditions to preserve SUSY are derived. The details for the calculation are summarized in Appendix D. It turns out that one of the SUSY conditions only has a trivial solution if we impose certain symmetry. Appendix E provides an explanation for this fact. In Sections 3 and 4, we demonstrate how our formalism works by examining some explicit examples. We will analyze the case with ISO(1, 1) × SO(3) × SO(3) symmetry in detail in Section 3. We show in Section 3.2.1 that the SUSY conditions reduce to two simple equations (3.54) and (3.62). As shown in Section 3.2.2 and Section 3.2.3, large classes of solutions of these equations are found. An F-theory configuration corresponding to the case with ISO(1, 1) × SO(6) symmetry and the supersymmetric Janus configuration found in [12] are obtained as special solutions. The case with ISO(1, 1) × SO(2) × SO(4) symmetry is studied in Section 4.1, in which the general solution for this case is found. Some examples with time dependence are discussed in Section 4.2. In Section 5, we summarize the paper and discuss some future directions.

SUSY conditions for N = 4 SYM with varying couplings 2.1 Notations and Ansatz
In this paper, we consider N = 4 SU(N) supersymmetric Yang-Mills (SYM) theory in a curved background with space-time dependent gauge coupling g YM (x µ ) and theta parameter θ(x µ ). The leading order action is where I, J = 0, 1, · · · , 9 and with µ, ν = 0, 1, 2, 3 and A, B = 4, · · · , 9. Here we are using the 10-dimensional notation, * in which the gauge field A µ and 6 adjoint scalar fields A A are combined into a 10-dimensional gauge field A I , though it depends only on the 4-dimensional space-time. The 10-dimensional metric g IJ is assumed to be of the form and g IJ is its inverse. The indices are lowered or raised by this metric and its inverse. ǫ µνρσ is the 4-dimensional epsilon tensor with ǫ 0123 = 1/ √ −g, where √ −g ≡ − det(g IJ ) = − det(g µν ). * Our notation is similar to (but, not exactly the same as) that used in [12].
The fermion field Ψ is written as a 10-dimensional negative chirality Majorana-Weyl spinor, which is equivalent to 4 Weyl spinor fields in 4-dimensional space-time. It is a real 32 component spinor satisfying where Γ (10) ≡ Γ0Γ1 · · · Γ9 is the 10-dimensional chirality operator. † Here, the gamma matrices ΓÎ (Î = 0, 1, · · · , 9) are 10-dimensional gamma matrices which are realized as 32 × 32 real matrices satisfying {ΓÎ, ΓĴ } = 2ηÎĴ , where (ηÎĴ ) ≡ diag(−1, +1, · · · , +1) is the Minkowski metric (see Appendix B for our notations and useful formulae for the gamma matrices). The hatted indices (Î,Ĵ, · · · ) are those for the local Lorentz frame which can be converted to the curved indices (I, J, · · · ) by contracting them with vielbeins e Î I as Γ I = e Î I ΓÎ. The Dirac conjugate Ψ is defined by Ψ ≡ Ψ T Γ0. Gamma matrices with more than one indices are anti-symmetrized products of the gamma matrices defined as 3) is the spin connection: (2.7) In our notation, D µ denotes the covariant derivative including gauge field A µ , spin connection ωνρ µ and Levi-Civita connection Γ ρ µν , depending on the field it acts. When the metric is flat and the couplings g YM and θ are constant, we know that the action is invariant under the supersymmetry (SUSY) transformation with 16 independent SUSY parameters. ‡ If the metric and/or the couplings are not constant, SUSY is in general completely broken. In order to maintain a part of SUSY, we have to add additional terms to the action (2.1) and the SUSY transformation has to be modified accordingly. The action that we consider is where a, d IJA , m AB are real parameters and M is a real 32 × 32 matrix that may depend on the space-time coordinates x µ . a and c are related to g YM and θ in (2.1) by (2.9) † This should not be confused with Γ 10 = Γ 1 Γ 0 . ‡ In this paper we will not consider the special conformal supersymmetry.
We also use the complex coupling defined by We impose the following conditions for the parameters d IJA and m AB , which can be done without loss of generality: (2.11) The second condition in (2.11) can be imposed because a term with d µ(AB) tr(F µA A B ) can be converted to a mass term for A A via integration by parts. We can further assume that the matrix (m AB ) is diagonal, which can be realized, at least locally, by using local SO(6) R transformation.
Note that ΨMΨ is non-vanishing only when Γ0M is an anti-symmetric matrix that commutes with Γ (10) . The most general form of such matrix is where m IJK is a real rank three totally anti-symmetric tensor.
The action (2.8) is constructed by adding operators of dimension ¶ less than 4 into the leading order action (2.1). Although it is not the most general one, it is general enough to cancel the SUSY variations of the leading order action, as we will see in Section 2.2 and Appendix D.
The ansatz for the SUSY transformation is where B A are real 32 × 32 matrices acting on the spinor indices that commute with the chirality operator Γ (10) . and B A ≡ −Γ0(B A ) T Γ0. The SUSY parameter ǫ is a 10-dimensional negative chirality Majorana-Weyl spinor. B A and ǫ may also depend on the space-time coordinates. § Square and round brackets on indices indicate antisymmetrization and symmetrization of the indices, respectively. For example, . ¶ Our analysis in this paper is classical and the anomalous dimensions are not taken into account. For example, an operator like tr(A (A A B A C) ) is not included.

SUSY conditions
The goal of this section is to determine the conditions under which the action (2.8) is invariant with respect to the SUSY transformations (2.14) for a non-zero ǫ. The approach that we follow in this work is straight. We firstly calculate the variation of the deformed action (2.8) with respect to the deformed transformations (2.14). Then, by imposing the vanishing of the variation, we obtain several constraints on the deformation parameters and the SUSY parameter ǫ. * * Here, we provide the outline of the derivation and leave the details to Appendix D.
Applying the SUSY variation (2.14) to the action (2.8), we get where M ≡ M − 1 2 Γ µ ∂ µ log a. In this expression, d JKI and m IA can be non-zero only if I = 4 ∼ 9, and ǫ IJKL can be non-zero only for I, J, K, L = 0 ∼ 3.
It can be shown after some calculation that (2.14) vanishes (up to surface terms) if and only if the parameters satisfy the following two conditions (see Appendix D.1 for more details): Conditions (2.15) and (2.16) correspond to cancellation of terms with dimension 7 2 and 5 2 operators, respectively. Following [12], we call them first order and second order equations, respectively.
Further algebra shows that the first order equation (2.15) is equivalent to the following con- * * A similar analysis with a more elegant approach using supergravityà la Festuccia-Seiberg [5] has been given in [18]. However, a detailed comparison of the results still remains.
ditions (see Appendix D.2 for the derivation): where F is a real 32 × 32 matrix acting on the spinor indices, and is a projection, as it satisfies In addition, for an arbitrary G I , we have The equation (2.20) can be solved when the integrability condition (2.20) can be formally solved as where ǫ 0 is a constant spinor, "P exp" denotes the path ordered exponential (the ordering is taken from left to right) and x 0 is a fixed position. The integrability condition (2.26) guarantees that (2.27) is well-defined in a neighborhood of x 0 .
(2.17) has a trivial solution e IJK = 0. In fact, in all the examples we consider in the following sections, one can show that e IJK = 0 is the only solution of (2.17) that is compatible with the imposed symmetry. When e IJK = 0, (2.18) is simplified as  In order to demonstrate how to solve the SUSY conditions obtained in the previous section, we consider the cases with ISO(1, 1) × SO(3) × SO(3) symmetry. Here, ISO(1, 1) is the Lorentz symmetry acting on x 0,1 and the first and second SO(3) act as rotation of x 4,5,6 and x 7,8,9 in the 10-dimensional notation, respectively. We also impose translational symmetry along x 0,1 directions. Although our analysis is purely field theoretical, our motivation is in string theory. Consider a D3-D5-D7 system in the following table: symmetry. If we regard the D5 and D7-branes as a supergravity background and the D3-brane as a probe embedded in it, the low energy effective theory on the D3-brane is expected to be a deformation of the N = 4 SYM that we have discussed. We may replace the D5-brane and D7-branes with (p, q) 5-brane and [p, q] 7-branes, respectively, to have more complicated configurations preserving SUSY.
The case with D3-branes in the (p, q) 5-brane backgrounds corresponds to the supersymmetric Janus configuration considered in [12]. D3-branes in the [p, q] 7-brane backgrounds can be generalized to D3-branes in F-theory configurations, which was recently analyzed in [17,19,20] (see also [23]). These configurations appear as special cases in our example as we discuss separately in Sections 3.4 and 3.3, respectively.

Ansatz and SUSY conditions
We decompose the coordinates in four sectors: α, β = 0, 1; i, j = 2, 3; a, b, c = 4, 5, 6; p, q, r = 7, 8, 9. The metric and the couplings are assumed to depend only on x i and preserve the Using the general coordinate transformation and Weyl transformation * with appropriate rescaling of the fields, we can assume e(x i ) = 1 and g ij ( The form of M consistent with the symmetry is The non-zero components of d IJA , m AB are We also define The non-trivial components of e IJK defined in (2.22) are In this case, one can show that the condition (2.17) implies e IJK = 0 (see Appendix E.1) and hence The integrability condition (2.32) for the first equation of (3.7) is The non-zero components of the spin connection are and A µ defined in (2.23) is The condition (2.20) with µ = 0, 1 implies Using this the equation, A i in (2.20) with µ = i = 2, 3 can be replaced with The integrability condition (2.26) for (3.16) is The second order equation (2.16) for this case is Using (3.14)-(3.16), one can show that (3.18) and (3.19) are equivalent to Therefore, equations that we have to solve are (3.8), (3.13), (3.17) and (3.21). Once we find a, α j , β, γ, Φ and r − r satisfying these equations, the other parameters can be easily obtained by (3.7), (3.14), (3.15) and (3.20). It can be easily checked that these SUSY conditions reduce to those given in [12] when the symmetry is enhanced to

Solutions of the SUSY conditions
In this section we are going to elaborate on a prescription for the SUSY parameter ǫ which simplifies the study of solutions. In addition, we give two examples of solutions, where the latter is a generalization of the former.

More on SUSY conditions
Let us first try to solve (3.17). Decomposing ǫ as with ǫ ± Γ 01 = ±ǫ ± , (3.17) can be written as If g kj ∂ k ∂ j Φ = 0, both ǫ + and ǫ − can be non-zero. However, if g kj ∂ k ∂ j Φ = 0, it has a non-trivial solution only if are satisfied. Therefore, when Φ is not a harmonic function, the unbroken SUSY is inevitably chiral in 2-dimensions. In the following, we impose (3.24), though we do not assume g kj ∂ k ∂ j Φ = 0. The general solution for the case with g kj ∂ k ∂ j Φ = 0 can be easily obtained by taking a linear combination of a solution with ǫ = ǫ + and that with ǫ = ǫ − .
(3.24) can be solved (at least locally) if and only if there exists a function ϕ ± satisfying Then, the solution of (3.16) is where ǫ 0 ± is a constant spinor satisfying The second condition in (3.27) follows from (2.5).
This ǫ 0 ± belongs to the 8-dimensional Majorana-Weyl representation of the SO(8) subgroup of the 10-dimensional Lorentz group. Since the operators acting on ǫ in (3.13) and (3.21) commute with SO(3) × SO(3) generators Γ ab and Γ pq , it is convenient to decompose it to two (2,2) representations of the SO(3) × SO (3) as is the basis of the spinors satisfying (3.27). They can be constructed explicitly as follows. Let us define Since all the operators acting on ǫ 0 ± in (3.13) and (3.21) do not mix the spaces with different index v, we can assume with fixed v. (The general solution is just a linear combination of this type.) Note that Γ23 = Then, (3.33) can be written as As a consequence, the number of remaining SUSY is 4 in general, corresponding to the choice of v = 1, 2, 3, 4. This agrees with what we expect from the brane configuration (3.1). If the phase ξ can be chosen freely without changing the action, the number of SUSY is enhanced to 8. As mentioned above, if Φ satisfies the Laplace equation g kj ∂ k ∂ j Φ = 0 and both ǫ + and ǫ − are allowed, the number of SUSY is doubled. We will see some examples with the SUSY enhancement later.
Since ξ can be absorbed by the shift of ϕ ± in (3.26), we set ξ = 0 in the following. Then, using (3.25), (3.26) and (3.35), one can show that the SUSY condition (3.13) is equivalent to and (3.21) is equivalent to It is convenient to write these equations using a complex coordinate z ≡ 1 where we have used (3.40) in the last step. β, γ and (r − r) are determined by (3.40) and the real part of (3.41). The imaginary part of (3.41) gives a non-trivial constraint: This equation can be solved if there exists a real function f (z, z) satisfying This is equivalent to The complex conjugate of this equation is The sum of these two equations gives which is equivalent to This equation can be solved if there exists a real function g(z, z) satisfying Inserting this into (3.44), we obtain Note that (3.49) can also be written as This equation can be solved if there exists a real function h(z, z) satisfying It shows that the gradient of g and h are orthogonal to each other. Conversely, if we are able to find real functions g and h satisfying (3.54), f is obtained as can also be written as In addition to these equations, we should also solve (3.8). Using (3.25), (3.8) can be written as which is equivalent to This equation can be solved if there exists a real function k(z, z) satisfying From the first equation of (3.7) and (3.25), we see that this k is proportional to the theta parameter as up to an additive constant.
Using (3.48), (3.59) becomes This equation can also be written as In summary, the SUSY conditions are now reduced to a problem of finding real functions h, g, a and k satisfying (3.54) and (3.62). Then, the real function f is obtained by (3.55) and other parameters are determined by (3.48) (or (3.56)), (3.50), (3.51) and (3.60). Although we have not been able to find the general solution of the SUSY conditions (3.54) and (3.62), a lot of non-trivial solutions have been found. In the following subsections, we show some of the explicit solutions.

Solution 1
First, we introduce new coordinates (y 1 , y 2 ) defined as and l(z) is its complex conjugate. They are related to the original coordinates (x 2 , x 3 ) by a conformal transformation on the 2dimensional plane. Note that our ansatz explained in Section 3.1 is compatible with the conformal transformation and hence our results in the previous subsection are valid in the coordinates (y 1 , y 2 ) as well. In fact, it is easy to see that the equations (3.54) and (3.62) are invariant under the conformal transformation. Once one finds a solution, one can generate new solutions by the conformal transformations. In order to emphasize this point, we write down the solutions that work for any choice of the holomorphic function l(z), rather than using this degrees of freedom to simplify the equations.
Since the condition (3.54) is equivalent to the statement that the gradient of g and h are orthogonal to each other, it is clear that it can be solved when g and h are of the form: where G 1 and H 2 are real functions. The subscripts of these functions suggest which of the coordinates (y 1 or y 2 ) they depend on. Inserting these into (3.62), we obtain where the prime denotes the derivative, e.g., G ′ 1 ≡ ∂ y 1 G 1 (y 1 ). One can check that the following ansatz gives a solution of (3.65): where K 1 and K 2 are real functions satisfying with real constants κ i (i = 1, 2) and A is a real function defined as L(y 1 , y 2 ) ≡ κ 2 G 1 (y 1 ) − κ 1 H 2 (y 2 ) .

Solution 2
Let us generalize the solutions in the previous subsection by considering the following ansatz: where, G i and H i (i = 1, 2) are real functions. Then, (3.54) can be solved when these functions satisfy where c 0 is a real constant. Note that (3.55) implies where we have defined

81)
A ≡ aW + 2c 0 k , Note that in the Taylor expansion of (3.78) with respect to c 0 , the leading term is trivially satisfied and the O(c 0 ) term reproduces (3.65).
Here, we try to solve (3.78) using the following ansatz: where A i and B i are real functions. Inserting this ansatz into (3.78), we obtain These equations can be solved when where κ i (i = 1, 2) are real constants. Then, we obtain where K i = K i (y i ) (i = 1, 2) are real functions satisfying (3.67), and a 0 and b 0 are real constants.
(We can set a 0 = b 0 = 0 by absorbing them in the constant parts of G 1 , H 2 and K i , but we will keep them for convenience.) Then, by the definition of A and B, we get One can check that this solution reduces to the one given in the previous subsection in the c 0 → 0, σ → 0 limit.
3.3 F-theory configuration: ISO(1, 1) × SO (6) As an explicit example of the solutions obtained in Section 3.2.2, let us consider In this case, (3.66) gives and hence is a holomorphic function of z. This implies that the complex coupling τ ∝ i(a ± ik) defined in (2.10) is holomorphic or anti-holomorphic depending on the chirality of the unbroken SUSY.
Another immediate consequence is that the combination Φ − log a ∝ Re(log ∂ z l) is a harmonic function satisfying the Laplace equation in 2-dimensions: Actually, in this case, (3.69)-(3.74) imply β = γ = 0 and r = r, and the symmetry SO(3)×SO (3) is enhanced to SO (6).
Though this solution is a special solution, the properties (3.101) and (3.102) hold for general solutions with ISO(1, 1)×SO(6) symmetry. In fact, it is not difficult to find the general solutions for the cases with ISO(1, 1) × SO(6) symmetry. If we impose β = γ = 0, (3.50) implies g = constant or f = constant. However, in order to have non-singular solution of (3.48), g cannot be a constant. Then, f has to be a constant and (3.49) implies that g is a harmonic function satisfying ∂ z ∂ z g = 0. Then, (3.48) and (3.61) imply that and a + ik are holomorphic functions. Therefore the general solution can be written as with holomorphic functions l(z) and m(z).
Note that all the parameters in the Lagrangian are invariant under a constant shift of the imaginary part of m(z), but ϕ ± is shifted. This shift induces a shift of ξ in (3.35) through (3.26) and hence we can choose any value for ξ without changing the action. Therefore, as explained below (3.35), the number of preserved SUSY is enhanced to 8.
This solution corresponds to the D3-brane probes in the F-theory configurations (background with 7-branes in type IIB string theory) and the equation (3.102) is interpreted as the Einstein equation [31]. To see this explicitly, note that type IIB supergravity action for dilaton φ, RR 0-form C 0 and gravity is given by where τ ≡ C 0 + ie −φ . Assuming that τ only depends on x 2 and x 3 , and using the ansatz (2.4) for the metric, the Einstein equation and the equation of motion for τ become 108) Then, it is easy to see that τ = τ (z) (a holomorphic function of z) solves the equations of motion for τ and then (3.107) becomes which agrees with (3.102) with the identification τ ∝ i(a + ik).

Gaiotto-Witten solution: ISO(1, 2) × SO(3) × SO(3)
The supersymmetric Janus configuration found by Gaiotto-Witten in [12] can be obtained as a special solution of the solutions obtained in Section 3. To see this, let us consider the case with κ i = 0, K i = 0 (i = 1, 2) and H 2 = y 2 . In this case, Φ is a harmonic function and, as discussed in Section 3.2.1, we may have solutions with both ǫ + and ǫ − being non-zero. To obtain such solutions, we should make sure that the parameters in the Lagrangian are consistent with the SUSY conditions for both ǫ + and ǫ − simultaneously. To this end, we choose   In this case, we regard the D3-brane extended along x 0∼3 directions as a probe embedded in the supergravity background corresponding to the other D3 and D7-branes. The 4 dimensional gauge theory with varying couplings is realized on the world-volume of the probe D3-brane.

Ansatz for deformation
First, we set the metric and other deformation parameters consistent with the global symmetry. The ansatz for the metric (2.4) is the same as that used in Section 3.1: where the indices are α, β = 0, 1; i, j = 2, 3; a, b = 4, 5 and p, q = 6, 7, 8, 9. The non-trivial components of the spin connection are given in (3.9). Parameters M, d IJA and m AB consistent with the symmetry are of the form where ǫ ab is the epsilon tensor satisfying ǫ 45 = −ǫ 54 = 1. Here, α i , β i , v i , r and r are functions of x i .

Solutions of the SUSY conditions
First, let us apply the above ansatz to the first order equations (2.17)- (2.20). It is easy to show that all the components of e IJK vanish. The components of e IJK that are allowed by the symmetry are e 01i = a −1 ∂ j (ac)ǫ ji − 4α i , (4.5) The equation (2.17) for (I, J) = (6, 7) implies Because Γ23 does not have a real eigenvalue, we conclude that e 01i = e i45 = 0 to have non-zero ǫ. Therefore, we obtain In this case, (2.28) gives where q i ≡ ∂ i log a. When we decompose ǫ as ǫ = ǫ + + ǫ − with ǫ ± Γ 01 = ±ǫ ± , (4.10) becomes Note that the second equation of (4.9) implies that the complex coupling (2.10) satisfies (4.14) where notation for complex coordinates is give in Appendix A.1. From this we find that the complex coupling τ is holomorphic or anti-holomorphic when q i = 4α i or q i = −4α i , respectively. This is the same situation as that observed in section 3.3. For the case with (4.13), the gauge coupling and theta parameter are constant.

From (2.23) we obtain the components
We find that (3.11) is also valid in this case. Then, (2.19) implies and (2.20) with µ = i = 2, 3 can be written as The integrability condition (2.26) for (4.17) is equivalent to In order to solve this equation, we decompose ǫ into the eigenspaces of Γ 01 and Γ23 45 as where ǫ s t (s = ±, t = ±) satisfy ǫ s t Γ 01 = t ǫ s t , ǫ s t Γ23 45 = s ǫ s t , ǫ s t Γ (10) = ǫ s t . (4.20) (4.18) implies that ǫ s t can be non-zero only if is satisfied. This equation can be solved when there exists a function ϕ s t satisfying Then, the solution of (4.17) is given by where ǫ 0s t is a constant spinor satisfying the conditions in (4.20). Then we distinguish the following four cases: • (C1): α j = 0 and ǫ ij ∂ i β j = 0.
(4.12) and (4.21) imply that only one combination of the signs (t, s) can have non-zero ǫ s t .
Next, let us consider the second order equation (2.16). In this case, it can be written as Inserting (4.16) into these equations and using (4.17), we obtain where we have used the relation in (4.9), that is valid for both cases (4.12) and (4.13). Then, from (4.28) we obtain where we have used (3.21) in order to have non-zero ǫ s t . In summary, we can construct a generic solution of the SUSY conditions by the following steps. First, pick a holomorphic (or anti-holomorphic) function τ (z) such that Im τ > 0. Then, a and c are obtained from (2.10) and α i is given by (4.12). Next, choose arbitrary real functions Φ and ϕ s t . Then, β i is determined by (4.22). If one wants to find a solution with ǫ ij ∂ i β j = 0, (the case (C2) above), Φ is determined by solving (4.24). More explicitly, the solutions of (4.24) are obtained by choosing a holomorphic function l(z) and setting Φ = log a + l + l .
The number of unbroken SUSY is 4 for the generic case (C1), 8 for the cases (C2) and (C3), and 16 for (C4). The case (C4) is a trivial solution that is related to the undeformed N = 4 SYM by a coordinate transformation and a field redefinition. In fact, because Φ is harmonic, it can be eliminated by a conformal transformation on the z-plane. β i is a pure gauge configuration and it can be eliminated by a local SO(6) rotation (see Appendix C.2). One can guess that the cases (C1), (C2) and (C3) correspond to the D3-D3-D7, D3-D7, D3-D3 systems, respectively [32].

Time dependent solutions 4.2.1 ISO(3)
Let us consider the case in which the couplings depend on time. First, as a trial, let assume that spatial translational and rotational symmetry ISO(3) are preserved. It turns out that the time dependent solutions of the SUSY conditions with this symmetry can always be mapped to a system with a constant gauge coupling and theta parameter by general coordinate transformations and field redefinitions. * Note that the metric (2.4) can be chosen to be flat. This is because the general form of metric preserving ISO(3) symmetry (flat FLRW metric) can be written as where i, j = 1, 2, 3 and η is the conformal time defined by dη = e −Φ/2 dt, and the overall factor e Φ in the 4-dimensional metric can be eliminated by the Weyl transformation † (g µν → e −Φ g µν ) without loss of generality.
Then, it is particularly easy to show that the gauge coupling cannot depend on time to preserve SUSY. To see this, note that (2.18) may be written as Because Γ IJK Γ 0 is an anti-symmetric matrix for all I, J, K = 0, . . . , 9, the right hand side of this equation is ǫ times a real anti-symmetric matrix. However, since a real anti-symmetric matrix cannot have a real non-zero eigenvalue, the only possibility is that both left and right hand sides are zero:

52)
The integrability condition (2.26) for (4.56) is which implies The solutions (4.49) and (4.50) satisfy the first equation of (4.58) and the general solution of the second equation is 59) § The solution (4.50) was discussed in [13].
where f ± are arbitrary real functions of x ± . Then, the equations (4.56) can be integrated as where ǫ 0 ± is a constant spinor satisfying ǫ ± Γ01 = ±ǫ ± . Using the above results, the second order equation (2.16) becomes simply 0 = ǫ m AB Γ B . (4.61) Multiplying this equation by m AC Γ C , we find m AB = 0.

Conclusion and outlook
We As we mentioned in the introduction, it is commonly faced situation that gauge couplings are not constant when one tries to engineer a gauge theory using D-branes in string theory. Therefore, it is natural to ask whether there are string theory realizations of the solutions we found in this paper. We hope to address this question in our forthcoming paper [32].
Our analysis in this paper is classical and it would be important to take into account the quantum effects. Since these theories preserve SUSY, we may be able to use some techniques, such as localization, to calculate some physical quantities exactly. Since our system can be realized as D3-branes embedded in type IIB string theory, we may be able to study S-duality and holographic dual. It would also be interesting to consider how the BPS solitonic objects, such as dyons or instantons, behave when the couplings depend on space-time.
As mentioned in Section 2.1, our ansatz for the action in (2.8) is not completely general, even if we restrict the deformation to be of dimension less than 4. As an obvious extension of the analysis, one may try to include all the terms that are compatible with renormalizability. It would also be interesting to consider the case with U(N) gauge group and include the terms like tr F IJ , tr A A . The U(1) part will be important to consider the configurations of D3-branes, when the system is embedded in string theory. Since our strategy should work for arbitrary supersymmetric theory, further generalization would also be possible.
The following formulas are useful. Γ I 1 I 2 ···In Γ J 1 J 2 ···Jm = Γ I 1 I 2 ···In J 1 J 2 ···Jm + (−1) n−1 nm δ where D is the number of dimension. These formulas work for any D, though we are mainly interested in the case D = 10.

C Useful local transformations C.1 Weyl transformation
Let us consider the transformations Then the following quantities entering the action transform as: According to the definition where the spin connection is these objects transform as follows: The action (2.8) is invariant under the Weyl transformation (C.1), if we also transform the couplings as (C.10)

C.2 SO(6) R transformation
Let us consider a local SO(6) R transformation (6) and O is the corresponding SO(6) element in the spinor representation of SO (1,9) acting on the fermion. This implies Then, the action (2.8) is invariant if we also transform the couplings as Since d B µA behaves as the gauge field of the SO(6) symmetry, it is useful to define the covariant derivative including the SO(6) gauge field: which transform as The action (2.8) can be written as First, the variation of the action (2.1) is given by with the understanding that d JKI and m IA can be non-zero only if I = 4 ∼ 9, and ǫ IJKL can be non-zero only for I, J, K, L = 0 ∼ 3. Here, we have used and defined The SUSY variation of the action with respect to the transformations (2.14) is given by where we have used the identity * We would like to find a condition under which the integrand of (D.5) is a total derivative. Our ansatz for the total derivative is where A IJK and B IA are 32 × 32 matrices with A IJK = −A IKJ to be determined. This total derivative term can be expanded as Comparing (D.5) and (D.8), we obtain the following conditions: aǫ(−Γ JK Γ I ) = ǫA IJK , (D.10) In (D.9), E IJK is a 32 × 32 matrix which is totally antisymmetric with respect to I, J, K, which can be added because of the Bianchi identity: (D.14) (D.10) and (D.12) determine ǫA IJK and ǫB IA , respectively. It is easy to see that (D.9) is satisfied with E IJK = aΓ IJK , using the identity Then, using (D.10) and (D.12), (D.11) becomes 16) which is equivalent to This is the condition (2.15). Similarly, (D.13) becomes which is equivalent to This gives (2.16).

D.2 Derivation of (2.17)∼(2.20)
Using the identity we rewrite (2.15) as Inserting (D.21) back to (2.15), we obtain where we have defined and with arbitrary G J . Therefore, if ǫ C IJ can be written as satisfies (D.27). It can also be shown, using (D.26), that v I ′ J ′ P IJ Then, using the property (D.26), one can easily show Therefore, (D.23) can be written as (D. 30) In this equation, M can be replaced with M, because of the relation (D.26).
When M is expanded as Therefore, (D.30) is equivalent to This is (2.17).
Here, we prove the following statement: Suppose e IJK with mixed indices such as e αβa and e αab are all zero. Then, the condition (2.17) implies that all the components of e IJK vanish when d = 3, 7. It also holds for the case with d = 7, which will be shown separately in Appendix E.2.