Supersymmetric Janus solutions of dyonic ISO(7)-gauged \( \mathcal{N} \) = 8 supergravity

Journal of High Energy Physics, Apr 2018

Abstract We study supersymmetric Janus solutions of dyonic ISO(7)-gauged \( \mathcal{N} \) = 8 supergravity. We mostly find Janus solutions flowing to 3d \( \mathcal{N} \) = 8 SYM phase which is the worldvolume theory on D2-branes and non-conformal. There are also solutions flowing from the critical points which are dual to 3d SCFTs from deformations of the D2-brane theory.

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Supersymmetric Janus solutions of dyonic ISO(7)-gauged \( \mathcal{N} \) = 8 supergravity

HJE ISO(7)-gauged = 8 supergravity Minwoo Suh 0 1 2 0 supergravity. We mostly 1 Daegu 41566 , Korea 2 Department of Physics, Kyungpook National University We study supersymmetric Janus solutions of dyonic ISO(7)-gauged N = 8 nd Janus solutions owing to 3d N = 8 SYM phase which is the worldvolume theory on D2-branes and non-conformal. There are also solutions owing from the critical points which are dual to 3d SCFTs from deformations of the D2-brane theory. AdS-CFT Correspondence; Supergravity Models 1 Introduction 2 Dyonic ISO(7)-gauged N = 8 supergravity 3 The SU(3)-invariant Janus solutions 4 The G2-invariant Janus solutions 5 Conclusions A The equations of motion B The SO(4)-invariant truncation well studied in N = 4 super Yang-Mills theory. The Janus eld theories from N were constructed in type IIB supergravity in [5{8] or in gauged N = 8 supergravity in ve dimensions in [9] and uplifted to type IIB supergravity in [10]. Although Janus eld theories from ABJM theory are not known, the Janus solutions are proposed in eleven-dimensional supergravity [11{14]. The ABJM Janus solutions are also studied in gauged N = 8 supergravity in four dimensions, and some of them are uplifted to eleven-dimensions [15, 16]. There are more examples of Janus solutions in four-dimensional gauged supergravity [17{19]. Lately, Janus solution was studied in F (4) gauged supergravity in six dimensions, and was proposed to be dual to codimension one defect in 5d superconformal eld theories [20]. In this paper, we study Janus solutions dual to 3d SCFTs from deformations of D2brane theory via dyonic ISO(7)-gauged N = 8 supergravity. The well-known examples of the AdS/CFT correspondence on D3-, M2- and M5-branes involve anti-de Sitter spacetime as a near horizon geometry of the corresponding branes [1]. There are corresponding large N conformal eld theories dual to the AdS geometries, and they are N = 4 super YangMills theory, ABJM theory and 6d ( 2,0 ) theory, respectively. When it comes to D2-branes, the near horizon geometry of a stack of D2-branes is not an AdS spacetime, and its dual gauge theory is 3d N = 8 supersymmetric Yang-Mills theory which is non-conformal. However, by adding a Chern-Simons term, 3d N = 8 SYM ows to a conformal xed { 1 { point with N = 2, U( 1 )R SU(3) symmetry [21, 22]. Otherwise, by adding a mass term to one of the chiral scalars, it ows to another conformal xed point with N = 3, SO(4) symmetry [23]. Recently, gravity duals of these 3d SCFTs from D2-branes were discovered in dyonic ISO(7)-gauged N = 8 supergravity [24]. Dyonic ISO(7)-gauged N = 8 supergravity is a consistent truncation of massive type IIA supergravity on six-sphere [25{27]. The Romans mass corresponds to the magnetic gauge coupling of dyonic ISO(7)-gauged N = 8 supergravity. The scalar potential of the theory has four known superymmetric critical points, and they are N = 2 SU(3) U( 1 ), N = 1 SU(3), N = 1 G2, and N = 3 SO(4) critical points. The N = 2 SU(3) U( 1 ) and N = 3 SO(4) critical points are dual to the 3d SCFTs from the deformations of D2-brane theory discussed in the previous paragraph. The xed point AdS solutions were uplifted to massive type IIA supergravity [26], and holographic RG ows between critical points were studied [28]. The gravitational free energies [25, 29, 30] and spin-2 spectrum [31, 32] were calculated and matched with the eld theory results. In the usual supergravity theories, most of the RG ows and Janus solutions are attracted to critical points which are dual to conformal eld theories e.g. N = 4 super Yang-Mills theory, ABJM theory, and 6d ( 2,0 ) theory. On the other hand, in dyonic ISO(7)gauged N = 8 supergravity, most of the RG ows and Janus solutions are attracted to a non-conformal phase which is dual to 3d N = 8 SYM on the worldvolume of D2-branes. Therefore, as we will see, usual Janus solutions in dyonic ISO(7)-gauged N = 8 supergravity do not exhibit AdS-behavior. On the other hand, if we ne-tune initial values of the scalar elds, we obtain Janus solutions staying at a critical point for a while and then moving away to the non-conformal 3d SYM phase. In section 2, we review the N = 1, Z2 SO(3)-invariant truncation of dyonic ISO(7)gauged N = 8 supergravity. In section 3, we study Janus solutions in the N = 2, SU(3) U( 1 )-invariant truncation. In section 4, we study Janus solutions in the N = 1, G2-invariant truncation. We conclude in section 5. In appendix A we present the equations of motion from the truncations we consider. In appendix B, we present the N = 1, SO(4)-invariant truncation. 2 Dyonic ISO(7)-gauged N = 8 supergravity We begin by considering the N = 1, Z2 SO(3)-invariant truncation of dyonic ISO(7)gauged N = 8 supergravity. This truncation was studied in detail in appendix A of [25], and we review it in this section. There are three complex scalar elds, 1 , 2 and terms of canonical N = 1 formulation, the scalar action is given by S = 1 1)2 1 + 2 ( 2 2)2 2 + 6 3)2 P : (2.1) K = 3 log i( 1 1) log i( 2 2) 3 log i( 3 3) ; (2.2) { 2 { N = 3 N = 2 N = 1 N = 1 SU(3) U( 1 ) SO(4) G2 SU(3) 31=2 24=3 31=2 2 51=231=2 27=3 31=251=2 22 1 21=3 0 1 27=3 31=2 22 Then, we de ne the complex superpotential,1 and the real superpotential, The scalar potential is obtained from W = p2eK=2V ; W 2 = jWj2 : P = 2 " 4 3 1) 2) The scalar potential has all four known supersymmetric critical points of dyonic ISO(7)gauged N = 8 supergravity [25], and they are listed in table 1. The ratio of electric and magnetic gauge couplings, g and m, respectively, is denoted by c = m=g. The N = 2 SU(3)-, N = 1 SO(4)-, and N = 1 G2-invariant truncations are obtained as sub-truncations by identifying the complex scalar elds as, [25], SU(3) truncation: SO(4) truncation: G2 truncation: 1 = t ; 1 = 1 = 3 = t ; 2 = 3 = t ; 2 = 2 = u ; 3 = u ; where we introduce a parametrization in real scalar elds, t = + ie ' ; u = + ie : 1There is an additional factor 2 in the complex superpotential de ned in (2.3) of [28] compared to (3.27) is at t; u ! +i 1, which corresponds to ; = 0 and '; ! When obtaining a scalar potential of the sub-truncations, one has to rst perform the di erentiations in (2.7), and then identify the scalar elds by (2.8). Otherwise, one has to employ the scalar potential formula from the canonical N = 1 formulation, [25], h P = eK K p q (D p V)(D p V) i 3VV ; where the Kahler covariant derivative is (2.10) (2.11) t0 = u0 = i i ; { 4 { 3 The SU(3)-invariant Janus solutions The scalar action for the SU(3)-invariant truncation, [25, 28], is obtained by (2.8), S = 1 The Kahler potential is and the holomorphic superpotential is The scalar potential is obtained from K = 3 log i(t t) 4 log [ i(u u)] ; V = 2g t2 + 6tu2 + 2m : P = 2 " 4 3 (t The scalar potential has N = 2 SU(3) U( 1 )-, N = 1 SU(3)-, and N = 1 G2-invariant critical points, and more non-supersymmetric critical points. For Janus solutions, we consider the AdS3-sliced domain wall for background, ds2 = e2A(r)d2AdS3 + dr2 : Then, we solve the supersymmetry variations of fermionic elds on the curved background, as it was done on the Lorentzian AdS3-domain walls in [15] or on the Euclidean S3-domain walls in [33]. The supersymmetry equations for the complex scalar elds, (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) A0 = r W 2 e 2A l 2 ; are obtained where l is the radius of AdS and = 1. The constant, = 1, is not related to the signs in (3.7). Reversing the sign of merely generates a solution re ected in the r-coordinate. In terms of the real scalar elds, the supersymmetry equations are (3.7) (3.8) '0 + 0 + Now we numerically obtain supersymmetric Janus solutions of the SU(3)-invariant truncation.2 We closely follow the method employed in [15]. We set l = 1, g = 1, m = 1, and = +1 for all numerical solutions in this paper. As the supersymmetry equation for the warp factor, (3.7), involves a square root, it involves a branch cut. In order to obtain smooth Janus solutions, we should choose the upper sign for r > 0 and lower sign for r < 0, [9, 10, 15, 20]. In order to avoid dealing with branch cuts, instead of the rstorder supersymmetry equations, we solve the second-order equations of motion which are presented in appendix A. To numerically solve the second-order equations of motion, we should specify the initial conditions for the scalar elds, ', , , , the warp factor, A, and their derivatives. We impose a smoothness condition at the interface, Because of the smoothness condition, if we x initial conditions for the scalar elds, f'(0), (0), (0), (0)g, initial condition for the warp factor, A(0) is automatically determined through the supersymmetry equation for A(r) in (3.7). Similarly, f'0(0), 0(0), 0(0), 0(0)g are also determined through the supersymmetry equations for the scalar elds in (3.8). Finally, we have only four free parameters, f'(0), (0), (0), (0)g, and when they are xed, the rest of the parameters are xed with the smoothness condition, (3.9), and the supersymmetry equations. When we obtain numerical solutions of the equations of motion, we numerically check whether they satisfy the supersymmetry equations. 2In SO(8)-gauged N = 8 supergravity, the SU(3) U( 1 ) U( 1 ) sector was considered to study Janus solutions in [15]. In dyonic ISO(7)-gauged N = 8 supergravity, one can turn o the scalar eld, , and obtain the SU(3) U( 1 )-invariant truncation [24, 25]. However, it is not consistent to turn o the scalar eld, , and there is no SU(3) U( 1 ) U( 1 )-invariant truncation. One can see this by examining the equation of motion for . Moreover, it is group theoretically impossible as ISO(7) does not have an SU(3) U( 1 ) U( 1 ) subgroup. A0(0) = 0: (3.9) { 5 { and N = 1, SU(3) critical points are denoted by yellow and green dots, respectively. In the left plot, the 3d N = 8 SYM phase is located at plot of the superpotential. Following [34], in order to have both N = 2, SU(3) U( 1 ) and N = 1, SU(3) critical points depicted on contour plots, we replace some scalar elds by or = 2 + ( 1 = 2 + ( 1 '2 2) '2 1 , 2 are the values of the scalar elds at the N = 2, SU(3) critical point, and '1, 1 , 1 at the N = 1, G2 critical point. In gure 1, the contour plot with (3.10) is presented on the left and (3.11) on the right. Regular, singular and critical-point Janus solutions are depicted by blue, green and red lines, respectively. First class of solutions are regular Janus solutions. Typical initial conditions give solutions attracted to 3d N = 8 SYM phase which is non-conformal.3 Therefore, warp factors of the solutions do not show AdS-behavior of constant slope. To the best of our knowledge, they are the rst examples of non-conf ormal Janus solutions. We plot a typical regular Janus solution by blue lines in gures 1 and 2. The more interesting class of solutions arises as we approach the critical point. As we have four scalar elds, we x initial conditions of three scalar elds, (0) = 3This is the main di erence between the solutions here and the known Janus solutions of four- and vedimensional gauged supergravity where the solutions are mostly attracted to the conf ormal points which are dual to 3d ABJM [15] and 4d N = 4 SYM [9, 10], respectively. They show constant slope behavior of AdS solutions all along the ow. { 6 { -40 -40 40 r 40 r -40 -20 -40 -20 φ -1 -2 -3 -4 f'(0) = ' + 2:9 , by their critical point values, and we have only one initial condition to x, '(0). When '(0) < 'cr, we obtain the rst class of solutions, i.e. regular Janus solutions. We found the value of 'cr to be ' + 1:6 When we ne-tune '(0) = 'cr, numerical solution starts approaching the critical point, stays at the critical point for a while, and then move away to the 3d SYM phase. While the solution is staying at the critical point, the warp factor exhibits the AdS-behavior with a constant slope. The warp factor is given by A(r) W r; (3.12) (3.13) where the slope, W , is the value of the superpotential at the critical point. We plot a typical critical-point solution by red lines in gures 1 and 2. On the contour plot in gure 1, this solution takes the steepest descent from the SU(3) U( 1 ) critical point to the SYM phase.4 Regularity of the solution owing from the SU(3) U( 1 ) critical point can be seen from two aspects. First, as one can see from gure 2 where the solution is plotted in red, the numerical solution is well-de ned though all range of r. The singular solution we are about to present is not well-de ned at some nite r and diverges. Second, on the contour plot of the superpotential in 1 [28].5 Any other direction at in nity corresponds to a singularity. gure 1, the point dual to 3d N = 8 SYM phase is located at The solution ows to the location of 3d N = 8 SYM phase, which is regular. 4The three classes of solutions we found are very much analogous to the solutions found in SO(8)-gauged N = 8 supergravity in gures 7 and 8 of [15]. Unlike our case, in SO(8)-gauged N = 8 supergravity, the critical points exist in pairs: they are Z2-symmetric to each other. 5In the parametrization of [28], the 3d N = 8 SYM phase corresponds to z ! 1, 12 ! 1. { 7 { denoted by green lines in gures 1 and 2. Singular solutions diverge at nite r. On the contour plot in gure 1, they ow to in nity which is not the 3d SYM phase. We have also obtained analogous solutions around the N = 1, SU(3) critical point. However, as they are similar to the solutions we obtained, we do not present them in the paper. The analysis of the G2-invariant Janus solutions parallels the one in the previous section, and we will be brief on details. The scalar action for the G2-invariant truncation, [25], is obtained by (2.8), Now we obtain numerical solutions in the same way we obtained the SU(3)-invariant solutions. As we will see in detail, there are three classes of solutions: regular Janus { 8 { S = 1 The Kahler potential is and the holomorphic superpotential is The scalar potential is obtained by K = 7 log i(t t) ; V = 14gt3 + 2m : P = 2 " 4 7 (t 3W 2 : The scalar potential has N = 1, G2-invariant critical point, and more non-supersymmetric critical points. The 3d N = 8 SYM phase is at The supersymmetry equations for the complex scalar eld, and for the warp factor, are obtained where = 1. In terms of the real scalar elds, the supersymmetry equations are t0 = i e A l A0 K A0 = r W 2 e 2A l 2 ; 0 + 4 7 l k k = 0 : (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) is denoted by a yellow dot. The 3d N = 8 SYM phase is located at g (Red), and f'(0) = ' + 2:0 solutions, singular solutions, and as a special case of regular Janus solutions, there are Janus solutions owing from a critical point. The solutions are depicted on contour plot of the superpotential in gure 3. First class of solutions are regular Janus solutions. Typical initial conditions give solutions attracted to 3d N = 8 SYM phase which is non-conformal. We plot a typical solution by blue lines in gures 3 and 4. The more interesting class of solutions arises as we approach the critical point. We x (0) = , the value at the critical point. When '(0) < 'cr, we obtain the rst class of solutions, i.e. regular Janus solutions. We found the value of 'cr to be ' + 1:4 stays at the critical point for a while, and then move away to the 3d SYM phase. While the solution is staying at the critical point, the warp factor exhibits the AdS-behavior with a constant slope. The warp factor is given by where the slope, W , is the value of the superpotential at the critical point. We plot a typical critical-point solution by red lines in gures 3 and 4. On the contour plot in gure 3, this solution takes the steepest descent from the G2 critical point to the SYM phase. Lastly, when '(0) > 'cr, we start getting singular solutions. A singular solution is denoted by green lines in gures 3 and 4. HJEP04(218)9 A(r) W r; (4.9) 5 Conclusions In this paper, we numerically obtained supersymmetric Janus solutions in dyonic ISO(7)gauged N = 8 supergravity. Unlike the Janus solutions known in four-and ve-dimensional gauged supergravity, our Janus solutions are mostly attracted to the non-conformal 3d N = 8 SYM phase, which is the worldvolume theory on D2-branes. We also discovered a number of solutions which stay at a critical point for a while and then move to the SYM phase. In dyonic ISO(7)-gauged N = 8 supergravity, there is also the N = 3, SO(4) critical point. Interestringly, this critical point is non-supersymmetric in the SO(4)-invariant truncation we consider. It is because the SO(4)-invariant gravitino becomes massive at the critical point and breaks supersymmetry. If we consider the full N = 8 supergravity, there are three gravitinos outside the SO(4)-invariant sector, and hence the critical point is N = 3 supersymmetric, [25, 28]. Therefore, this critical point is not captured by the superpotential or the supersymmetry equations of our truncations. For this reason, we could not study Janus solutions of this critical point. It will be interesting to study Janus solutions of this critical point by employing a larger truncation preserving supersymmetry of this critical point. We present N = 1, SO(4)-invariant truncation in appendix B for completeness. Acknowledgments We are grateful to Hyojoong Kim and Krzysztof Pilch for helpful discussions and communications. We would like to thank Adolfo Guarino for explaining his works, and Hyojoong Kim and Nakwoo Kim for collaboration on related subjects. We also would like to thank Adolfo Guarino and Krzysztof Pilch for comments on the preprint. This work was supported by the National Research Foundation of Korea under the grant NRF2017R1D1A1B03034576. The equations of motion We present the equations of motion in this appendix. For the SU(3)-invariant truncation, the equations of motion are given by 2A00 + 3A0A0 + 2 3A0A0 + 2 1 3 l l e 2A = e 2A = 2 2 ) 3 2 2 ) The SO(4)-invariant truncation The scalar action for the SO(4)-invariant truncation, [25, 28], is obtained by (2.8), S = 1 and the holomorphic superpotential is The scalar potential is obtained from6 K = 6 log i(t t) log [ i(u u)] ; V = 2g 4t2 + 3t2u + 2m : P = 2 " 2 3 (t 4(u The scalar potential has N = 1 G2- and N = 3 SO(4)-invariant critical points, and more non-supersymmetric critical points. 6We have corrected some misprints in appendix C.1 of [28]. 3 2 7 2 1 3 l l e 2A = e 2A = For the G2-invariant truncation, the equations of motion are given by 7 2 2 ) 2A00 + 3A0A0 + 2 3A0A0 + 2 2 7 '0'0 + This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Theor. 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Minwoo Suh. Supersymmetric Janus solutions of dyonic ISO(7)-gauged \( \mathcal{N} \) = 8 supergravity, Journal of High Energy Physics, 2018, 109, DOI: 10.1007/JHEP04(2018)109