JHEP10(2017)159 Springer October 10, 2017 October 23, 2017 September 8, 2017 : : eories on curved : Accepted Published Received e relation between the in- ur-dimensional field theories, linearly realized supersymme- om a type II perspective. etries of the field theory. We , supersymmetries are coupled to e e Published for SISSA by , CNRS, CEA, https://doi.org/10.1007/JHEP10(2017)159 and Hagen Triendl [email protected] , a,c,d . 3 aie Prins Dani¨el 1707.07002 a,b The Authors. c D-branes, Flux compactifications, Supergravity Models

We discuss non-linear instantons in supersymmetric field th , [email protected] Piazza della Scienza 3,Department I-20126 of Milano, Physics, Italy ImperialLondon College SW7 London, 2AZ, U.K. E-mail: School of Physics, Korea Institute85 for Hoegiro, Seoul Advanced 130-722, Study, Korea Dipartimento di Fisica, Universit`adi Milano-Bicocca, Piazza della Scienza 3,INFN, I-20126 sezione Milano, di Italy Milano-Bicocca, Institut de physique Universit´eParis th´eorique, Saclay Orme des Merisiers bat. 774, F-91191 Gif-sur-Yvette, Franc [email protected] b c e d a Open Access Article funded by SCOAP Abstract: Ruben Minasian, Supersymmetric branes and instantonsspaces on curved ArXiv ePrint: supergravity backgrounds in a similartries, fashion and as provide those details with on how to derive suchKeywords: couplings fr spaces arising from D-branes.we Focussing on derive D3-branes the and supersymmetry fo stanton conditions solutions and and show thedemonstrate the non-linearly that intimat field realized theories supersymm with non-linearly realized JHEP10(2017)159 3 5 7 8 1 3 6 31 37 11 15 21 23 25 25 27 29 30 38 12 r interest for the realized in a super- rsymmetric field theory on sible to gain deeper insights tion function on curved spaces = 8 supergravity background = 2 supergravity background N N – 1 – = 4 field theory to= an 1 field theory to an N N = 4 = 1 N N ] for a review on the subject). There are two questions of majo symmetric fashion? that curved space? 1 C.1 C.2 4.3 The dilatino4.4 variation Instantons on curved backgrounds 3.3 Instantons on D5-branes 4.1 Coupling 4.2 Coupling 2.1 Supersymmetric2.2 D-branes in type Linear IIB and flux non-linear backgrounds supersymmetries 3.1 Hermitian Yang-Mills 3.2 Non-linear instantons 1. On which curved spaces can a supersymmetric field2. theory What be are the classical supersymmetric solutions of a supe localization programme: 1 Introduction In recent years newinto localization supersymmetric techniques field have theories made by(see it computing [ pos their parti D Breaking supersymmetry using D7-branes with flux B Squaring linear supersymmetry C Derivation of the Killing spinor equations 5 Conclusion A Conventions and identities 4 Non-linear supersymmetries on curved spaces 3 Instantons on D3-branes Contents 1 Introduction 2 D-brane field theories JHEP10(2017)159 l lways, ]. The ]. 14 13 ] for the case ] and explains 12 16 , ]. In more general 15 19 hat they admit addi- orldvolume gauge field f supersymmetric field persymmetric solutions ux compactifications of ground bulk fluxes and e worldvolume fields, and ise in string and M-theory, ry become non-dynamical. field theory coupled to off- ns seems a priori unlikely. holonomy manifolds shows persymmetric field theories ding order by the worldvol- tions on curved spaces, we solutions of supersymmetric calibration condition on the r. ies, and the geometric prop- x compactifications. eaking [ = 4 supergravity [ sible four-dimensional spaces ux background. Embeddings dvolume. In this way, a single s into separate constraints on s in the supergravity multiplet of string theory. rance of curvature terms in the ized supersymmetries and lead tuning the background fields of hout bulk fluxes. The study of eal non-perturbative properties d non-linearly on the worldvol- N sting for various reasons. Brane he brane worldvolume [ ] for early works on this subject. olume flux [ 10 – 2 ] and further studied in [ 11 ]. It also naturally leads to non-commutative – 2 – 17 ]. These non-linearly realized supersymmetries are crucia 18 , this worldvolume flux is constrained by supersymmetry. As a in the field theory. F F As we will see, in the presence of general worldvolume flux, su Since the supersymmetry conditions on branes depend on the w One major feature of brane systems in string and M-theory is t Supersymmetric field theories on curved spaces naturally ar The relationship between the two setups, i.e. the branes in fl on the worldvolume naturallyto involve the non-linear non-linearly instantons. real In order to understand these solu It is theworldvolume aim flux of this paper to understand the interplay of back flux backgrounds such splittinggeometry of supersymmetry of condition wrapped branes and on gauge-theoretic conditio systems within string theory [ for the understanding of supersymmetric brane systems in flu via its field strength fact that these supersymmetryume transformations is are reminiscent realize ofthe spontaneous non-linear partial nature supersymmetry of br the DBI action [ of M5-branes. In theume limit action where of physics the is brane,These are determined the then bulk to naturally fields lea identified with ofcoupled the to the background the string field supersymmetric or field M-theo stack theory of living D3- on the or worl M5-branes naturally couples to conformal theories on a given space, which will be the focus of this pape The second problem naturally leads to the study of instanton to background off-shell supergravities andthe by supergravity subsequently multiplet in order tosupersymmetry compensate conditions. the appea A fairlypreserving thorough four study or of less admis supercharges is available: see [ The first problem has been extensively studied by coupling su supersymmetric branes withthat worldvolume the flux supersymmetry inside conditions special submanifolds usually decompose wrpapped into by a branes,a, generally which non-linear, is instanton independent equation of for th the worldv the situation is better understood in compactifications wit tional supersymmetries that are non-linearly realized on t of the field theory.theories realized as This brane work systems aims in at flux compactifications astring deeper and understanding M-theory o andshell the supergravity, has description been of recently a clarified supersymmetric in [ usually realized by systemsof of supersymmetric calibrated field branes theories in insystems a string offer given theory a fl are UV-completion intere forerties supersymmetric and string field theor dualities of these brane systems often rev JHEP10(2017)159 ˆ ε ˆ n ε, = 8 n P (2.2) (2.1) (2.3) 1 P − N n M 2 Γ 1 − ...M n 1 2 (8) R-symmetry M ∗ Γ ) such that the ...M 1 1 − n M 2 Γ 1 we will discuss linear − have positive chirality. ...M = 4 worldvolume field ) n 1 3 2 2 g, φ, F, H M 2 M ε , n e details on the latter can the full non-linear theory, ising form D3 branes, we ds. A (bosonic) type IIB near and non-linear super- N F Ψ ...M and brane supersymmetries ! , 1 ill study this problem for a try can also be analyzed in quation for the worldvolume try condition for the appendices. n ux background and show that 1)! 0 M and ˆ ymmetry, the resulting system Conventions, technical details 1 M 2 F − n 1 − ε n on between non-linear instantons 1) 6 0 1 ly. In that case the SU 1)! (2 = (Ψ − − 1 ( n n M

X (2 φ we review the supersymmetry conditions e = 2 n 8 1 n X − P φ Employing this language, we then show that ˆ ε ) e , we discuss the coupling of field theories with 2  1 4 – 3 – ), where both ˆ 1 16 2 P , χ ˆ ε , 1 + , ˆ χ 1 ε ! ε P MNP 1 = ( Γ − = (ˆ NP χ 1 0 0 ε Γ

MNP H = MNP 1 12 H P . 1 8 + A ˆ ε + ˆ ε M M )Γ ∇ φ = M ˆ = 8 background supergravity. ε ∂ ( M  N ˆ D = ˆ ε = ˆ D M This work is organized as follows. In section = The case of Lorentzian D3-branes can be discussed analogous Ψ 1 ˆ ε χ δ ˆ ε supersymmetry variation of the gravitino doublet Ψ Here vanish for some spinor doublet ˆ and the variation of the dilatino δ supersymmetric flux background is a profile for the IIB fields ( be found in appendix 2.1 Supersymmetric D-branesSince in type our IIB main fluxshall focus backgrounds concentrate will on be supersymmetric on type four-dimensional IIB theories flux ar backgroun 2 D-brane field theories In this section, we(linear shall and briefly non-linear) review and the set basics up of some the of bulk our notations. Mor symmetries of the worldvolumeand theory non-linear on instanton solutions, theand and brane. non-linear clarify supersymmetry. the In In relati sectio non-linear section supersymmetries to supergravityand backgrounds. the relation to Licherowitcz theorem is relegated to the of D-branes in type II backgrounds, with an emphasis on the li the supersymmetry conditions still split into a supersymme group is replaced by its compact counterpart SU(8). background supergravity fields andflux. a If non-linear solutions to instanton bothis e equations overall preserve the supersymmetric. samesimilar supers fashion Solutions using with intersecting brane less systems. supersymme including the non-linearlyEuclidean realized D3-brane supersymmetries. in a generalwe We (not w can necessarily compact) describe fl theory that to system by coupling the corresponding have to analyze the coupling of supergravity backgrounds to JHEP10(2017)159 = +1 n ) as and m (2.6) (2.8) (2.4) (2.5) (2.7) (2.9) P γ ), and S 2.2 curved -brane. : , p } need not 2.6 M and ), ( 9 +1 s S n 2 1 2.1 P P , s ,..., to the D-brane P 0 ...m M and there is a minus 1 d by the D ˆ i ∈ { D m which live on mensional spacetime − γ . l η 2 -brane does not fill out the y the dimension of the n lly refer to ( p P 1 rs tions of motions. In the − orldvolume read l +1 ions 2 n n ch squares to one, and for a 2 M,N,... 1) F ) by cient and from now on we shall − ... here the D 2.2 . , 2 , phrase our discussion in terms of n ) as the open string supersymmetry ε 1 n ] = 2( = 0 = 0 2.6 F ˆ ε +1 s ) and ( n ˆ , D P ...m S ε , , 2.1 1

are defined only locally. We shall return to n ε m = ˆ l P ξ 2 [ , ε = ˆ – 4 – -brane (also referred to as kappa-symmetry) is , Dε ; normal directions will be labeled by indices p Γˆ ε S ...n 1 and = 0 n = 0 ǫ , ˆ ε ) of type IIB supergravity. Note that η l ε . Trivially, we do the same for the algebraic operator R M S m !2 +1 S 1 | , s ˆ n ! D D ] l M P to are flat worldvolume indices, ˆ 21 D ε } +1 , -branes, Γ comes with an additional factor of p p = = ] = 2 20 are the pullbacks of the spacetime gamma-matrices Γ s + 1 X n M + l P m 2 . We shall slightly depart from the standard analysis of ( D , γ . We similarly restrict the differential operator } ) ε P 9 [ ) F which live on the space transverse to the brane. Of course, , . . . , p g = 1 ξ + ε g ,..., so that det( ∈ { 6 , -brane in type IIB string theory takes the form: S p 5 det( p , i ) to Γ 0 be the submanifold of the ten-dimensional spacetime wrappe p 2.6 S ∈ { . In order to differentiate between the two, we will occasiona m, n, . . . we can work locally and split the tangent bundle of the ten-di ) = M Let The supersymmery condition of a D F S Γ We will only consider the Euclidean scenario in this paper, w 2 , so that the closed string supersymmetry conditions on the w defined as the restriction of ˆ M Γ( m ˆ a, b, . . . into tangent and normal directions to On this point later. For ourrestrict purposes, ( the local analysis is suffi “internal” spinors be a spin manifold, and hence both E ε and the amount ofspace the of unbroken its supersymmetry solutions.Euclidean is D Γ determined is b a traceless Hermitian matrix whi described by the equation [ which can then be decomposed as a product of “external” spino anti-commute with each other and obey the commutation relat Generically we will notfollowing we require will a abbreviate background the to vanishing of obey ( the equa rather than working with the ten-dimensional spinors ˆ the closed string supersymmetry conditionsconditions. and to ( respectively. where spacetime indices, and generate the symmetry group SL(2 worldvolume sign in front of the determinant. D time direction. For Lorentzian D JHEP10(2017)159 (2.16) (2.14) (2.10) (2.13) (2.11) (2.12) (2.15) reads + ε + 1)-dimensional ) as the super- ) describes the p ) simplifies dras- 2.6 2.10 2.7 ion ( , while ( 0 espect to quation. In total, the su- calars. We can express this uation ( , l gravitino variation, which is 2 / ..., .... description of the supersymmet- 1) + − + vior under Γ p ε ( 3 − P ε 1) } ε . ... λ , ..., are preserved by the brane, while the ) ween the two notions lies in the appearance of 1 − into − +1) 0 . + ) p δ + + ε Γ − ( ε F + ) + + iγ ε ± , the matrix Γ given in ( ε = 0 F λ )] F (1 ε + )(Γ( – 5 – F Γ( a δ 2 1 0 FP

), describing the external gravitino and dilatino , Γ 0 D Γ( = = = , Γ 2.9 = 0) = − 0 { λ ± . After appropriate gauge choice for kappa-symmetry δλ 2 1 ] [Γ ε F + λ (1 λ δ 1 2 − 4 1 21 − Γ( , = = = = ≡ 20 λ λ 0 λ 0 + − δλ Γ . δ δ 0 Γ -brane and given by in Γ obeys [ p 2 / λ are broken. = 0 the supersymmetries +1) − p ( ε F P as ± ε ) matrix R , On the worldvolume we can understand the calibration condit For a stack of D3-branes, the supersymmetry variation with r Note that here we use subscripts for the transformation beha 3 where supersymmetries We shall mostly concentrate here on ( where the dots suppress termsin depending terms on of the worldvolume s and we can split the supersymmetry generators symmetry variation of the gaugino For the case of vanishingtically. worldvolume We flux find variations, since these are theric equations gauge pertinent theory for the onembedding curved of the spaces. brane into The the internal full gravitino type2.2 eq IIB background. Linear and non-linear supersymmetries We see that for the physical gaugino persymmetry conditions of typeperpendicular IIB to also the include D the interna We will refer to the first equation as the external gravitino e chirality will be denotedthe by sl(2 superscript. The difference bet JHEP10(2017)159 ε − . ε ε = 0, . We being (2.17) F F ε -branes p = 4 vector ] for in-depth = 0 for some N 6 upersymmetries , 5 δλ es actions and the .... + = 0, and show that he variation under − − ε orldvolume theory of an ε ensional Killing spinors ˆ ), which means the Yang-  , in which case one should e condition for preserving mmetries preserved by the cal analysis in terms of is only linear for D ) iation of the or the vacuum with S erized by worldvolume. persymmetry variation with .e. s equation. Subsequently we s are given by the D3-brane F + shall not worry about 2.16 1 to the brane. The rest of the , wrapped by the brane, allows ε dicular to the brane may have ons on the D3-brane worldvol- supersymmetries are crucial for − S -brane in a supersymmetric flux g + δ . This is the case for U(2)-holonomy S det( . We will close the section by comments ε . We shall refer to e.g. [ p 4 correction. We will discuss instantons for the + S ′ is not an SU(2)-structure manifold are studied α – 6 – which can be written as a product of external (4) . There are two ways in which our decomposition S γ 4 ε S . mn F (4) spinors on 3.3 c mn ) , they allow for more general, non-linear instanton solu- F Spin ∗F and some non-zero profile for the worldvolume flux ( ε 4 1 − 2  5 an additional cubic term appears in ( = ≥ this variation clearly is non-zero, indicating that these s λ p − F δ into “charged” , as − ε ]. These supersymmetry variations are particular to D-bran ε 4. For 17 , ≤ 16 p At various points in this section, we will decompose ten-dim Note that the supersymmetry variation with respect to are spontaneously broken in the vacuum on the worldvolume. T case of D5-branes below in section corresponds to the non-linearume supersymmetry [ transformati supersymmetry is given by thewill ordinary discuss Hermitian the Yang-Mill instanton equation for general this is exactly the case of linear instantons, namely when th For vanishing supersymmetry parameter will discuss four-dimensional field theoriesbackground. related We to will a first D3 instanton consider are the those case where linearly the realized supersy on the worldvolume, i discussion of these issues, and from now one carry on with a lo 3 Instantons on D3-branes Let us now turnEuclidean our D-brane. attention to Supersymmetric instanton instantons solutions are for charact the w Mills supersymmetry variation receives an and internal spinors. Our discussion is purely local, and we discussion will be phrased in terms of into non-vanishing components parallel and perpendicular which corresponds to the standard linear supersymmetry var multiplet. In contrast,respect the to gaugino transforms under the su on the instanton equation for D5-branes. tions on the worldvolume. Thusunderstanding these the non-linearly spectrum realized of supersymmetric states on the for general configurations of with Kalr manifolds.(K¨ahler) Secondly, the spinorslocal along zeros, or as perpen is the case for D3-branes on in the literature andwrapping a are generic manifold well-understood. K¨ahler or Notable example globally well-defined. We tacitly assumefor that the a manifold local SU(2)-structure. Cases when DBI action. Although these supercharges are always broken f can go wrong. Firstly,decompose one need not have a spin-structure on JHEP10(2017)159 = 0, (3.8) (3.7) (3.2) (3.6) (3.1) (3.3) (3.4) (3.5) Since as − ε ξ 5 is independent + η he solutions to each . We will solve for a , with in our case of ε ± α ymmetries to the bulk η = chiral spinors of Spin(4) . ε ± α 0 } mpose the ten-dimensional η 2 ese couplings. We will come , metries are linearly realized. 1 . = 0. We will consider ε ined by . ∈ { , = + ξ ε − α . 0 = 0 η . ) . − 1 + ⊗ = Γ , α leads to some constraints on the decomposed . η ε mn ε mn − 1 ε α F

J ξ = 0 = 0 or (4) P ε , = 0, such that Γ ⊗ ε iγ + = = Ω − + c.c. (4) = ξ such that we have 16 supercharges. − − − η η + − α α mn ε ε ξ iγ , η η } γ – 7 – 1 + 4 mn mn will yield a copy of the instanton equation, but + (4) ⊗ = ε ⊗ γ γ mn 0 − c α Spin(4) as + e ηγ ± α + α ξ e η F η Γ η = ( ξ i defines on the worldvolume an SU(2)-structure via × 1 2 + ,..., F

− + 1 = + + α ) implies that we can rewrite the Killing spinor 5) ε ⊗ η η , 1 + ) that supersymmetry implies the Hermitian Yang-Mills 5) perpendicular to the brane, ε + ⊗ 2.6 ∈ { ∼ , , each ξ + α . η α ) reads ξ 4 α 2.16 3), charge conjugation does not change chirality, hence = Spin(1 , 3.2 1 + ε → 9) , and note that the instanton solution is the intersection of t via Spin(1 4 chiral spinors of Spin(1 α 1 + . Thus the (Majorana-Weyl) reality condition of ε ± α − ξ η For now we will ignore the coupling of the worldvolume supers are independent for each This can be derived straightforwardly from imposing Γ Note that unlike for Spin(1 4 5 ± α specific with respect to a possibly different SU(2)-structure determ seperate equation. Thus, ( ξ In other words, let us assume for now that equation fields and the possible supersymmetryback to breaking this induced by point th in section 3.1 Hermitian Yang-Mills Let us start by studying instanton solutions whose supersym The normalized chiral spinor The supersymmetry condition ( such that which only admits a non-trivial solution for on the brane. Generically, D3-branes, Then we see already from ( In order to solvespinor the Hermitian Yang-Mills equation, we deco with spinors, i.e., we could also have written from JHEP10(2017)159 6 ]. 19 (3.9) ) and (3.11) (3.14) (3.12) (3.10) (3.13) reduce to 3.2 ε wing [ = ε . ε ) ) has to obey  F (4) 3.2 γ are non-zero for the mn , ε − F ε ) and ( . = ements imply the third: mn ε F -linear instantons that are 0 2.6 ! into a doublet. In this way, and . 1 4 -branes, and study the case e brane. + − + − ε ε rsymmetry being linearly real- − ε ε mn 1 ), while the third term vanishes

, F − = = 0. , 3.2 and ) mn ε = 0 ! + F F F

ε 1 = 0 1 4 + − − ε ε y. The general solutions of Γ( mn ], where one finds non-linear instanton equations λ solving both ( g ). As this equation is highly non-linear in Ω , 19 − with respect to the metric defined by the ε . ε ε δ + 2.6 ! mn ] δ = = F } + 0 F 0 4 ε ε ) = 1 + – 8 – Γ λ ⋆ ) 0 Γ ) is that the DBI-like term simplifies to , = F + , det( Γ F 1 δ Γ = [Γ 3.7 but also setting to zero a phase, which is responsible for the − p mn = g  −{ F J F ) to show that the linear instanton equation ( + + δλ ] mn δ } − ε 0 0 F Γ P 3.11 Γ = 0 we would have found an anti-selfdual worldvolume flux. , = 0. For this, note 0 , det( Γ Γ [Γ + − ). { p ξ ε F

2.6 − , we call these configurations non-linear instantons, follo 1 2 ε F 1) − not only by dropping ) that 0 ε -brane we can compose the spinors p 2.6 = ε = (Γ ), hence we find 0 ε ) reduces to 1) together imply ( 3.10 , or in other words fulfills the instanton equation Γ( satisfies the Hermitian Yang-Mills equation corresponds to a linearly realized supersymmetry, i.e. Γ 3.2 ε ε − ε ε ε Conversely, we can show that any spinor For any D In total we have shown that any two of the three following stat = = (Γ Note that this does not contradict the conclusions of [ 1. 3. 2. 6 ε ε 0 0 supersymmetric instanton configuration on the Euclidean D3 The second term is obviously zero due to selfduality ( we find from ( In similar fashion we can use ( which is equivalent to stating that on calibrated sumbanifolds of manifolds of special holonom the worldvolume flux Γ due to ( Γ Had we instead considered 3.2 Non-linear instantons We now want to turn to the more general case where both Thus linear instantons are linkedized to on the the corresponding worldvolume. supe related to In general the supersymmetries following on we the will worldvolume discuss of th non which corresponds to the general solution to ( SU(2)-structure. One consequence of ( Thus ( solutions of Γ deformation of the HYM equations. JHEP10(2017)159 is . and (3.16) (3.20) (3.19) (3.17) (3.15) (3.18) + ! η + − ε ε of Spin(4) non-trivial.

± − = ψ η , ! leads to two sets ⊗ + − ! − ε ε ± ), but may possibly ξ + − η ε ε (4) γ !

, e relevant when coupling and he internal spinors in the to = (4) we consider + γ ling spinor equations (as we η ! pq + − ⊗ F ∧ F y the relation between F . ε ε is multiplied by wedge products. 2 1 ion in the case where we consider + − α − α ξ mn F η ζ F ! make an appearance in the gravitino ⊗ ⊗ ) . FP

of Spin(4), any spinors ✘ P ✘ − c c − − α α c F . ) η η ξ ξ mnpq ✘ η η ) = 1 + η . ✘ + ∗ bη ǫ F ∗ ζ a m 1 8 ✘ F + b + ✘ + γ and ✘ + + + α cosh( α + − ✘ m η ζ aη c sinh( + 1 ✘ and aη bη η cosh( – 9 – ⊗ ⊗ = = + P − ) = = ) + + η α α + − ✘ ✘ ✘ ξ ξ (4) F F + + ψ ψ γ ✘ ✘ ζ η , = = pq ✘ ✘ 1 1 F ) ) F sinh( into account, the inclusion of both cosh( ✘ ✘ + − ε ε − mn α ) = ( ( F

FP −

F

) − F ) mnpq g ǫ sinh( + 8 1 g should be equivalent. We thus consider ) only involves gamma matrices on the brane worldvolume, and det( 1 + ± det( ε

3.15 s ) F ) simplifies to ) g + ), this reduces to g 3.15 det( 2.7 det( Note that ( Given a normalized positive-chirality spinor . Such solutions are a subset of the ones analyzed here, and ar s − may locally be written as so that ( Let us for the moment restrict our attention to the case where For a D3-brane we have This is the general non-linear instanton equation. Using ( where in the power series of cosh and sinh the argument field theories to curved backgroundswill as discuss in determined the by next the section) Kil since both of equations. The two sets are similar (up to sign changes due just a single term with external spinors variations. Having said this, we will demonstrate the solut involve different SU(2)-structures, which are determined b η As when taking the sum over Unlike for the HYM case, solutions do exist with both therefore diagonal on thedecomposition normal of bundle. As a consequence, t JHEP10(2017)159 ≃ , we ] for ˜ η (3.26) (3.25) (3.23) (3.22) (3.21) (3.24) 4 19 uations and vol , ure. One can c ˜ η   4   bgroup. See [ F ∧ F F ∧ F F ∧ F F ∧ F 2 1 . 1 2 1 2 iθ 2 1 − ie + , − + 4 4 = c is invariant under Spin(4) 4 4 . SO(3) iα pinors by considering  ∈ 1)vol SU(2) 1)vol , e − ∈ 2 − α ) ) + 1)vol ) . ) + 1)vol ,R F F F ∧ F sin F 1 2 F 1 1 2 ) 1 1 − − iα )    b − ,X g − 1 2 ai g b + g ai g ( − 4 + ! + e J + 2( + − c δ ), this solution is not the most general. By δ = η δ Re Ω Im Ω η δ vol 1 – 10 – ] via is due to a different choice of chirality. 

   = det( 3.20 det( 2 19 k , b det( det( R X k F ] for occurrences of this phenomenon for K3 surfaces p 2 p α = ( p ( p ( 23 ( → →  J  Ω = 0 , cos   ∗ ∗ ) and multiplying both sides by both vol !    a a b b iα 22 c 1 2 η F ∧ η F ∧ = = − 3.17 J ∗ ∗ ie

Ω = Ω = Re Ω Im Ω Ω Ω =    a F ∧ F ∧ b F ∧ F ∧ ) into ( a ) are exactly the four-dimensional non-linear instanton eq ∗ ∗ b a − + 3.20 + J 3.22 − J . The sign difference for J J α ), ( ), we have essentially “gauge fixed” a choice of SU(2)-struct F ∧ F ∧ i i ∗ F ∧ ∗ F ∧ 3.18 b = tan a bi 3.20 ai k SU(2) frame transformations, of which the above SU(2) is a su − − 7 × ]. As it stands, due to the ansatz ( Our variables may be related to those of [ 19 7 Note that such a rotation doesSU(2) not affect the metric; the metri or, equivalently, at the level of the forms by considering The conditions ( in the context of flux compactifications. more details, or compare with [ writing ( obtain another SU(2)-structure, either at the level of the s of [ We have defined here which can be simplified to find such that Inserting ( find the following set of equations: JHEP10(2017)159 3) , (3.32) (3.31) (3.30) (3.29) (3.28) (3.27) ] for analysis 1 − . ε 24 ) − ε (6) γ  Q tor Γ for a D5-brane is + 1) ) ) + + 3 ε F (6) we refer to [ tanton equation is actually eed to be present to ensure 1  γ + Γ − at the derivation of both the 1) . Q eeding analogously to the D3- g -brane rather than a D3-brane. 2 ean D5-brane. Similarly to the leads to instantons to supergravity back- . − rs in(6) as well as those of Spin(1 + ε ) + F − 1 − − δ + Γ (6) ε η ζ F pq = 1 ) γ 1 pq η defines an SU(3)-structure via F ε ⊗ ⊗ − γ det( (6) g η + Γ η , − − γ mn mn − Q p ξ ξ γ mn 0 + ) F c F δ e + + η mnp (Γ F i the external Spin(6) spinors of the Euclidean 1 ) + + (1 + e − ηγ − ± ζ η F − R

F det( = 0 as the linear instanton equation. However, g − ζ mnpq 1 – 11 – mnpqrs = = ) , ⊗ ⊗ ǫ − − p such that , + = = ( ± F 3 ε g + + 1 0 δ η ⋆ 2 mn ξ ξ 1 + 1 + ( mnp Γ + ( aη + J ε ε 1 0 0 1 8 3! 1 ) Ω = = 3 δ P Γ Γ P 1) = det( − − ) P (6) 1 − 1 + + − ε ε γ (6) (6) p = = (6) η det( (6) γ γ FP

γ

(6) Q p R + ( i γ γ 3) spinors, P , ≡ ≡ − ≡ Q ≡ R F − R

) then yields the non-linear instanton equations ( 0 1 2 3 (6) Γ = − Q F − R

1 ± Γ Γ Γ Γ γ ( ) ε  F (6) 1 = − iγ g F − R

± ( , + .  1 ± δ F ε det( = ( the internal Spin(1 ± p ( ε ± ξ Using similar notation as before, the kappa-symmetry opera change chirality. We normalize D5-brane. Unlike for thereality, due to D3-brane, the terms fact that of under conjugation both spinors of chirality Sp n with brane case. First, we decompose the Killing spinors as Let us now solve these non-linear instanton equations, proc Using D3-brane case, we shall refer to taking of such six-dimensional supergravity“linear” backgrounds. and the We non-linear repe instanton equation for a Euclid Inserting these into the supersymmetry constraint Γ we will find thatnon-linear for in the D5-brane, the resulting “linear” ins with In this section, we will takeAlthough a the slight derivation detour of and the consider coupling agrounds of D5 from six-dimensional string theory is beyond the scope of this paper, 3.3 Instantons on D5-branes given by JHEP10(2017)159 ] 19  (3.33) (3.35) (3.36) (3.34) J ead to . However F F ∧F ∧ 1 2 + 6  , are truly indepen-  ) does not yield the J ) leads to the following 3.35 = 4, due to the presence s formalism will include make a clear distinction not linear in d )+1)vol uaring it one finds a well- 3.30 n the supersymmetric cou- F ∧F ∧F F ntons can only be supersym- ymmetry appear to be more . Let us consider the linear he gravitational background. 1 . 1 . 3! nton equations originating for F ∧ F ∧ − 6 2 1 ike in g − = 0 = 0 = 0 J + + vol δ ∧ tion for six-dimensional Yang-Mills. 6 J − 0) , J (2 = vol det( F ∗ ,  F ∧ p ). Ω k 1 2 ( ∧   = 0 = F ∧ F ∧ F ∧ F ∧ F . ), and leads to 3.36 b b Ω 1 2 ai ai 0) 1 3! , 3 – 12 – + = = (2  3.21 − = 0. Imposing this on ( i 6 J Ω=0 F 2 J − − , one may find another justification for our definition ε vol ∧  F ∧ B J F ∧ F ∧ F = = 0) in the non-linear solution ( 1 F ∧F ∧ 3! 3 F ∧F ∧F k 1 2 J 1 F ∧ 3! − 1 − 3! 1 2 J − 6 = 0 ( J ∧ b vol ∧ J  J ). Once more, the formula matches the deformed instanton of [ 1 2 ) − Ω) satisfy . Using this, we find that the non-linear instanton equations l F ∧ b F ∧ ) ai 2 1 1 2 ( bη J, F 1 − = − g + (1 ζ + / δ b ai = 4 case the two equations, defining the linear supersymmetry = det( d “linear” supersymmetry yields instanton equation that are k p As we shall discuss in appendix  R We also set with So far we have onlythe discussed D3-brane the field linear theory. and Onmetric non-linear a insta curved if background the these corresponding insta We supersymmetry will is now preserved discuss bypling the t of supersymmetry the conditions that D3-brane gover field theory to a curved background. Thi of “linear” supersymmetry givendefined here (albeit containing from higher-deriative couplings) the ac fact that by sq 4 Non-linear supersymmetries on curved spaces full system and misses the first equation in ( (see eq. (3.28)).between the However, notions due ofcase, to which linear this for and us exercise is non-linear defined weset as supersymmetry of are setting equations: able to where the forms ( restrictive than those of six-dimensional HYM equations. L As we see, in general the constrains imposed by linear supers This is the six-dimensional analogue of ( dent. Notice that taking of unlike JHEP10(2017)159 ) (2) me- (8)- 4.2 ∗ ∗ (4.1) (4.2) In order = 1 “old 8 N ]. The connection 28 e more general than – = 4 SCFT can be coupled. 26 ackground. In the case N worldvolume theory, but so consider the variations ear supersymmetry of the to construct a field theory d non-linear R-symmetries n these ideas. tric backgrounds are deter- mmetries are associated to 16/16 supergravity (rather commutator, it is necessary round does not differentiate n-linear supersymmetries to vitino; in particular this was percharges from the SU rcharges, whereas the gravity ry to a supersymmetric back- onents of fluxes appearing in symmetric fluxless Euclidean dean backgrounds by treating d to the vanishing of the grav- e point of view of the worldvol- mensional way. Doing so, ( auxiliary fields to be complex. ]. 27 , = 4 conformal supergravity. , = 1 supergravity [ N = 0 ε = 1 supersymmetric field theories on Rie- ε N } = = 0 0 N ε ε Γ 0 – 13 – m , = 4 Killing spinor equations can be derived from Γ (8) R-symmetry group of the supergravity back- = 1, use was made of a pair of D7-branes. In fact, m D ∗ d D N { = 4 off-shell supergravity theory to which the field theory = 4 supergravity backgrounds to which d N = 1 supergravity (in their case, use was made of both = 1 by using D7-branes in the background and obtain an SU ) is a necessary but not a sufficient condition for the supersym N N 4.2 ], it was shown that a pair of ] for an investigation of ], it was shown how to construct 2 11 25 = 1 Killing spinor equations obtained in this way turn out to b In [ In [ The equation ( N See [ 8 is being coupled. This formalismspinors can of also opposite be chirality applied independently to and Eucli taking the minimal” and “new minimal” supergravity).mined by Such a supersymme pair ofitino Killing variations spinor for equations, the which specific correspon R-symmetry group in the supergravity sector. Let us expand o type IIB supergravity byD3-brane requiring with the compatibility external of supersymmetrydone of a by the super type noting IIB that gra try of the stringto theory consider background. also In theof addition the commutator. to internal the gravitino Furthermore, anti- and one the should dilatino. al The specific comp to break the supersymmetry down to between the low-energy effectivethan action minimal supergravity) of was string already theory made early and on [ corresponds to the variation of the gravitino of the mannian manifolds. In particular,Lagrangian on it a curved was space, shown oneground that should of couple in an the order field off-shell theo imply which can then be explicitly written in a manifestly four-di background preserves 32. Thus,of we the will D3-brane see enhance that to the the linear full an SU ume theory. In fact,between we them. will find This that issupersymmetries the to that gravitational be are backg expected: brokenneed from the not the non-linear be point supersy brokenwith of maximal from view supersymmetry, the the of D3-brane point preserves the 16 of supe view of the supergravity b both supersymmetries that are linear and non-linear from th ground. The coupling ofsupergravity field can theories be obtained with by less projecting linear out and some no of the su symmetric theory. WeD3-brane will down exemplify to this by breaking the lin minimal supergravity, corresponding to 16/16 JHEP10(2017)159 , we (4.4) (4.3) − ε . A brief ), and both equa- 4.3 , but otherwise both ε brane with non-trivial es are linear and which er tting the generator of the ment occurs. Furthermore neously in order to obtain dvolume flux. As we have = 1 instantons, coupling to ring theoretic point of view w to construct a full string gnored in our analysis. This ese two equations decouple. equivalent of the work done ticular, the anti-commutator modified) dilatino variations dered as new minimal super- t imposing the first constraint n the rest of this section, we dimensional dilatino variation. N ocus now on the supersymme- ersymmetries on the D3-brane. her fermions in the background nstanton configurations of such of the supersymmetry variation . background. Since the details of Hence the equations arising from − linear supersymmetries couples to = 0. For general supersymmetric − ε ε } ± ] − 0 ε . N 0 ε Γ Γ ± , , m = m ) = 0 D ) holds in addition to ( D ± ). By not imposing the vanishing of { − ε ε 0 = = [ 4.1 Γ 3.15 + – 14 – + + ε ε + ] } ε 0 ( 0 = 8 supergravity, and Γ Γ supersymmetries. We will explicitly demonstrate this m , , D m N ), our starting point will be m N D D [ 4.1 { ) generalizes ( 4.3 , we will discuss how the breaking to 1/4 supersymmetry works 4.2 ) now has a non-trivial right-hand side, that is, we find 4.2 ). Thus, D3-branes with a field theory with = 4 instantons, coupling to ], except that we keep explicit track of which supersymmetri Let us consider how ( In this section, we generalize this construction from the st In particular, rather than ( 4.1 N 29 = 2 supergravity. configurations on the braneworldvolume flux, we the will instanton keep equation in ( mind that on the D3- find that ( conditions are completely independent.try Therefore condition we imposed will by f the gravitational background. the gravitino variations ofwill the do supergravity so background. explicitly.in [ In I a sense, this is the (Euclidean) IIB As we discussed, inThis the is case no of longer only the linear case, supersymmetry, and th one needs to solve both simulta are not. We will thenin demonstrate how subsection the R-symmetry enhance tions have to be satisfied for the same supersymmetry paramet The case of onlynon-linear linear supersymmetries supersymmetries to can zero, be i.e. obtained by by se taking shown in the previous section,a the flux existence is related of to non-linearIn the i other presence words, of we non-linearly willof consider realized gravity sup ( backgrounds withou will lead to the vanishingsupergravity. of supersymmetry For variations example, of ot 16/16gravity supergravity coupled to can a be chiralof consi matter the multiplet; fermion the of vanishing this multiplet can be obtained from theby ten- allowing a Euclidean D3-brane to carry a non-trivial worl contains the internal rather than thethe external anti-commutator connection. may bebackground considered containing as our data four-dimensional supergravity specifyingthe ho field theory on the D3-braneis are however not not affected, these the can case be for i the dilatino variation since the ( the commutator and the anti-commutator are different. In par N for a background supergravity with 2 JHEP10(2017)159 ns → (4) into ∗ (4.6) (4.5) 9) ± , ε minology 5) spinors , (4). This is ∗ k fields should his is given by 9 SU = 2 supergravities ≃ N 5) linear supersymmetries , inear and non-linear re- . However, for symmetry hould combine the linear ± We will eventually combine λ = 10 Killing spinors into a spinor viewed as the representation = 8 and ated complexified Lie algebra. , and allow for both positive ing spinor equations must ad- equations originating from the linearly, enhancing the SU . We decompose Spin(1 st the supersymmetry variation ± D α persymmetries are indistinguish- N ζ , ± l λ − α − α as the “external spinors”. From the η ζ (8) representations. In the following ± ∗ α ) reduce to dictate the supersymmetry ⊗ . ⊗ η 2 supercharges (with 2 components for − − = 8 supergravity background α α 4.4 ! × ξ ξ = 4). We will refer to the Spin(1 2 ± ± N + + ζ η d × + + α α

η ζ – 15 – = ⊗ ⊗ ± + + α α (8)-covariant form. ξ ξ λ ∗ when no confusion can arise. = = α 1 − 1 + ε ε (8) symmetry that acts on , and the link to the previous section on non-linear instanto ∗ . 4.3 4.4 and the non-linear supersymmetries , since there are 16 = 4 = 4 field theory to an } ± α 4 ) into a pair of Killing spinor equations for η = 4 four-dimensional Riemannian backgrounds. We use the ter N 4.4 N ,..., Spin(4) as 1 (8) symmetry, and the components of the ten-dimensional bul SU(4), in the Euclidean case, we instead have Spin(1 ∗ = 4 R-symmetry group; in the (externally) Lorentzian case, t ⊗ ∈ { ≃ 5) N α , The background supergravity does not distinguish between l We will suppress spinor-sum indices as the “internal spinor”, and the Spin(4) spinors = 4 to refer to the number of linear supersymmetries; the non- 9 ± α nothing more than a different choice of real form of thealizations associ of supersymmetrysupersymmetries on the D3-brane. Therefore we s four-dimensional point of view,of the the internal spinors can be ξ Spin(6) both positive and negative chirality spinors in symmetry of the linear supersymmetries.mit an Therefore SU the Kill reassemble in four-dimensional auxiliary fields in SU we will show this explicitlyexternal by gravitino variation rewriting in the an Killing SU spinor Killing spinor equations fromthe the equations external ( gravitino. Since for the (Euclidean) background supergravityable, these su we should find an SU Spin(1 where conditions for N will double this amount. We consider the splitting of the is made in subsection 4.1 Coupling We can now examine how the doublet equations ( examination of the dilatino variations of four-dimensiona Killing spinors parallel andas well perpendicular as to negative the chirality. D3-brane Let us reiterate the specifics purposes, it turns out that it will be convenient to use not ju will follow in subsection JHEP10(2017)159 . ± ± λ (4.9) (4.8) (4.7) λ (4.10) metries ! The fi- !
11 . cdef npq C m = 0. We have F F φ m mab ] ie . F ab [ φ + will have enhanced m abcdef ω ǫ ie F = 0 = 0 ± φ φ npq ± ie − + ie H ,K λ λ 1 4! ∓ 2 2 ± ± mab . P P + − + H ed to appendix ,T a ). They are defined in terms m m λ λ on the internal components of ± γ γ 2 2 Ψ iders the modified gravitino from the mab metry variations of the gravitino δ + − A 4.8 P P mnpq a ω ǫ m m 2 Γ 1 removed by a superconformal transforma- 3! γ γ + Φ + Φ . m + −
γ 10 − + nsiders the (anti-)commutators of each term K K 1 2 ), the ten-dimensional supersymmetry cdef is given by λ λ = 0 npq n n m + + − ± m ± α ]. 4.5 γ γ F F λ − + φ A 11 mab δχ λ λ curved index, such that + mn − mn (6) a ie F m m m d ∇ φ − γ γ γ m We have chosen a gauge for the vielbeine such abcdef – 16 – = + Ω + Ω ǫ ie 2 1 F (8) representations, whereas the other terms do a 6 np np φ ∗ φ ± α + − γ γ npq 12 ∓ ξ + α ie λ λ ie + − np np H 1 4! (4) m m ± ]. No truncations have been performed, nor have any   T T ± mab ∇ Ψ 1 4 4 1 can be chosen such that − and 32 δ am am H – a a ± d − − can be ignored in the case of only coupling the linear supersym ε = ω ω 29 mab ) + − 2 2 1 1 mnpq m ω λ λ ǫ 2 4.7

1 − − is a 4 3! δψ ( . We will instead consider + m − m

φ φ µ m

A A m m ⊗ ∂ ∂ ⊗ + + ±   ± ξ m m ξ 2 2 1 1 ab 1 4 ∇ ∇ ˇ γ − − 1 8 − = 0, where = = 8 supergravity [
m α e ± N λ = ± ξ µ a e = 4 conformal supergravity, as has been done in [ = 4, Let us now give explicit expressions for all fields in ( We have, however, assumed that There are two ways to perform the computation. Either one cons Note that the additional terms are spin-1/2 terms that can be ± m d N 12 11 10 A start and then considersseparately. (anti-)commutators, We have or done one the latter. first co These equations are precisely the vanishing ofof the supersym of the fluxes, asthe eigenvalues of Killing the spinors. following The Clifford composite action connection assumptions been made onthat the metric. equations imply nal result that we find is that, making use of ( tion. Therefore the redefinition written the equations in such a way that the first terms The computation is rather lengthy, and is therefore relegat to of the ten-dimensional Ψ R-symmetry and furnish non-trivialnot; SU we will come back to this later. JHEP10(2017)159 (4.14) (4.16) (4.12) (4.11) (4.15) (4.13) tructure is defined by . . ± mn ∓ , T ∓ ∓ λ λ ∓ λ ! λ ! ! ! ) an case, existence of abc mna ey can be written as a F F 1234 φ mna φ a 13 ω ie . F abc ie ∓ amn orbed into the definition of 0 λ 5). In a practical sense, this + xactly of the opposite sign. ± ∓ + ω W ± − , ξ 2 λ ! a ± ± . ) ξ ± abc F abc a amn ∓ ( mna bc ˇ γ H ω λ φ W ˇ γ H n ± a mnabc abc ie γ − ± ˇ γ 0 F − W mna φ φ abc ± mn abc a ) 0 ω F , which refers to the fact that ie 1 T ∂ ˜ mna φ 3!8 W ( 0 1 ± ] for details. 3! + F = ie 1234 is selfdual), which is also enforced by the abc = 0) requires the space to be flat. In the + + φ ⊗ a 29 ∓ − , and hence satisfies W F ie ± ± ∓ amn amn λ − ξ ξ ξ abc mn m ± 0 ω ω ± a a + a m – 17 – mnabc γ T abc 2 ˇ ˇ γ W γ γ γ ]. a F

. Note that there will certainly be constraints on 0 a a is exact; in this case, a Weyl rescaling can be found H np ± φ F np 34 ± = ˇ α γ ( a = ⊗ ± mna W W γ ie ξ φ
a H 1 ∓ W given by 3!

± np = = ∓ ie ξ ± T λ W a − ⊗ . Group theory gives us the decomposition of the product ±

1 4 ˇ ± γ ± mn

ξ

∓ ξ 2 1 T abc ⊗ ξ − ⊗ (6) a ⊗ ∓ ∓ ± abc ∓ ∇ ,W ξ ∓ Φ = a ξ a ˇ γ a ξ a ˇ γ
ˇ γ W ˇ γ 1 abc 8 1 ∓ defined by 3!8 1 4 ˇ γ λ − ± + 1 8 of the vector and a chiral spinor of SO(1 ± − can be expressed in terms of torsion classes and the (local) s is anti-selfdual (and ξ is given by = 4 ] = ±
abc + ⊗
mn 33 ∓ ± mn K T ∓ 6 W λ rather than to the (anti-)selfduality condition, which is e Ω λ ± ± ∓ ξ ξ λ and ± ± ξ a ± mn K W T , as a naive degree of freedom counting shows. In the Riemanni instead, when working with curved indices, by noting that th We will generically assume that Finally, we have the fields We hope no confusion will arise from the superscript abc ± 13 covariantly contant spinors (i.e., Lorentzian case, this is not necessary [ W contraction with the gamma matrices We have that group defined by the Killing spinors which is not selfdual.K These spin connection termsderivative acting can on be the abs curved vielbein. See [ representation comes down to [ Here, acting on Let us explain the terms Next, the field This is in contrast to the term such that the term Φ can be trivialized. JHEP10(2017)159 = × ck (8) ∗ N (4) (4.20) (4.22) (4.21) (4.17) (4.18) (4.19) ∗ (4) algebras, ∗ , selfdual and su m ± . A 0 . ) s discussed above, we 1 . , ) = 0 = 0 1 by ( an − + imensional spin connection a ravity does not distinguish ⊕ λ λ ω 2 2 hem on equal footing. The 2 . mnp 1 2 e to the dilatino comes about , P P − inear supersymmetries to ω ) m m − , i.e. 4 γ γ , (8) to representations of SU (8) connection !+ , φ SU(8). ab + − ∗ ∗ !+ ¯ 4 1 n γ ) ( ∂ K K ± x → ( . The two commuting ( , enhances to the R-symmetry group ⊕ ) for example. + + 1 0 0 1 + ab µν y 1 1 1 mn +2

( − + ˜ g T R γ ) ) ± 6 λ λ U(1) x 2 1 g ¯ 4 ⊗ ( ×

) m m , . We will now study their respective SU φ × ). The R-symmetry group of the linear and x γ γ 4 − 2 ab ⊗ ± ( (4) factors is just the geometric Lorentz group ˇ γ − (4) np np ∗ 2∆( e np K ∗ Let us comment on the representation theoretic ab – 18 – γ γ ⊕ * 4.19 e γ ˇ γ 0 + − np np ) SU(4) = SU * ) = T T ) + mnp x 4 4 1 1 × × 15 = ( ˜ ω of generators , diag 1 4 − − 1 ± x, y µν R ( ( (4) g , are then of the form − + (4) 4 ∗ = (4) ∗ ⊕ g λ λ ± ∗ 0

and a scalar np su ) = (4) su − m + m (8). In the case, of a Lorentzian D3-brane, this would be the γ 1 ∗ ∗ ± , np A A 10 g T su is exact, it can be similarly removed. This is the case for blo mnp + + 15 ( ω m m an 4 1 a → ∇ ∇ ω 63 is ), which differs from the usual spin connection ˜ denotes the span over + i R 4.8 ab × T h in ( = 8 supergravity, SU (8) R-symmetry enhancement. We note that the decomposition of the adjoint of SU Finally, let us also note that we make use of a rescaled four-d Thus, under the assumption that we can do the Weyl rescaling a (4) ∗ ∗ N mnp First, the diagonal subgroup of the two SU acting on the normal bundle, so it is generated by the ˇ SU where which we will denote by non-linear supersymmetries, SU ω In terms of the four-dimensionalthrough metric, the the warping modification du of diagonal warped metrics These equations govern the coupling of 16 linear and 16 non-l more familiar enhancement of SU(4) and provided that interpretation of the fields appearing in ( SU representations. 8 background supergravity. Asbetween we linear can see, and background non-linearauxiliary superg supersymmetries fields and appearing treats in t this equation are the SU anti-selfdual two-tensors end up with JHEP10(2017)159 , +2 ) ¯ 4 (4.24) (4.25) (4.23) (4.27) (4.28) (4.26) , 4 . 0 that com- . The two- ) aints: their 4 ¯ 4 + 0 , ) R ¯ 4 ¯ 4 ( , two components ¯ 4 ⊕ ( . ⊕ +2 form a basis of four- !+ !+ ) } . +2 1 7)-form in the SU(8) case. ) 10 1 (6) (6) , , !+ 1 , ˇ ˇ γ γ 6 − 1 , 1 ab etric. We find that − − !+ ( ˇ γ 1 − (8), as well as their complex 1 !+ ( ∗ { 1 ⊕ − 1 (6) (6) (4)-factors, and the number of !+ ⊕ 1 1 1 2 representations. We want some − ˇ ˇ γ γ ∗ 2 − − 1 1 1 1 2 1 1 1 ) or from the right as a .

− su −

1 ) ) − 1 1 1 1 ¯ 4 , representations. The 1 ⊗ ⊗ ) representations: 4 ⊗ ⊗

, , 1 1 0 !+ 6 10 ¯ 4 ) 10 ⊗ ⊗ ( ( , , , abc abc a a only acts from the right (left) on ( ˇ ˇ 1 γ γ 15 ! ! ˇ ˇ γ γ → → , 0 1 1 0

±

1 1 and ( 1 (6) (6)

¯ ¯ (6) (6) − 28 36 ˇ γ γ (6) (6) (4) ˇ ˇ − γ γ ⊗ +2 ˇ ˇ ∗ γ γ − ) – 19 – ± ± 0)-form and the primitive (1 (6) ¯ 4 and ( ± ± su (6) , 1 ˇ representations of SU ˇ γ , 1 1 , , 1 (6) ˇ γ 1 1 0 4 ˇ γ − ) ) and the ( * 0 36 1 1 1 1 2 2 . Last but not least both representations should have 1 1 )

, , (4) indices. The matrices 3!2 3!2 2 (4) factors only act from the right or left, but in total = 4 0 ∗ ± ± ∗ − , ⊗ ⊗ ) ± ± , is generated by 15 10 ) R 4 * * 4 0 4 ( ab ab * * ) , , ˇ ˇ γ γ = = 1 4 can act from the left as a ¯ 4 ⊕ = = ( and the , 2 2 * * 2 1 2 2 ab ± ∓ ⊕ − γ ) ) ± ∓ = = 28 ) 2 ) ) 1 6 2 1 , , − ) 10 − +2 , 10 6 1 ) ) , ( ( 6 , 4 ¯ 4 , 1 10 1 ( , , ) representations can be determined by the following constr ( ( 1 ( ¯ 4 4 6 ( ( ⊕ , ⊕ 1 +2 ) +2 charges, which means that their pair of off-diagonal two-by- ) 1 , 14 1 + , ) and ( 6 10 R 1 ( ( , 6 → → We will also need the Now let us identify the remaining ( Specifically, these would be the primitive (2 -charge, the fact that they commute with one of the 14 + 28 36 conjugates: and are a basis of generators that do the job. and from the left (right) on ( and their pair of diagonal components each must be anti-symm by-two matrices then have to ensure that definite representations where the two SU by-four matrices on which ˇ these representations need two SU degrees of freedom. We thus find R mutes with them, i.e., the ( Similarly, we also construct the ( These are thus respectively the ( The ( JHEP10(2017)159 17 (4)  ∗ ) as 2 oint, l (8). + , (4.29) (4.31) (4.32) (4.33) (4.30) ) ∗ k ¯ 4 , , . 4 su ab ( − + T  on ( of λ . λ . 2 − + − + ) ξ ξ 63 2 28 ¯ 4  , ) that we found. − !+ !+ 4 )  i ( 4 2 2 , T + − ¯ 4 ab ) ) 4.19 ( 1 0 1 0 0 1 0 1 6 6 − T , , 1 1 − − 2  a a ( ( − We see that

) T T ions ( 4 , and the generators of the Lie − −  mab ⊗ ⊗ 16 ¯ 4 0 2 2 ( 15 F ) ). + − T φ ± s. mn ) ) tly that of the abc abc 15  1 1 ct the representations T , ˇ ˇ , , ie γ γ to the left and right action of SU 1 , which drops outs due to the chiral 6 6 ab 2 a 1 a

4.14 ( ( ( . ) ¯ npq T T T 28 (6) (6) +   F ¯ ¯ ˇ γ γ − 28 36  4.29 0 2 is given exactly by the ( − ) 0 0 ∼ ∼ + amn amn 1 ) ) ) , 1 + ˇ 1 mnpq ¯ 4 4 ω ω ¯ 4 ǫ , , + , mn − 2 2 − 15 on. 4 ¯ 4 φ mn 4 ab a a ), explicitly in terms of the generators we , and the two components of the represen- onto ( ab ( ( T ( , given in ( T K 1 1 − − ie − T : T T T ± 3!2 3!2    1 ± 3! ¯ mn 2 2 4.10 , , + 28 (4) (4) + T − mna mna ∗ ∗ ) ) 2 – 20 – − ! ! mab 6 F F 6 − , , 1 1 ) φ φ  1 1 63 28 36 SU H a a 4 ( ( 2 , − − ie ie 2 1 + ¯ 4 T T ab × ∼ ∼ ∼ ) ( 2 2 ¯ 4 . + + T , 10 0 10 0 + + − − + ± m 2 4 2  +  0 0 ( mn

 + − ) ) A 0 K ) ) ) T T (4) ) ¯ 4 4 , given by ( , , 1 1 fills out the ∗ ⊗ ⊗ 1 , , , 15 4 ¯ 4 m ± cdef abc abc , + 6 6 ( ( a a 1 a a ( ( 1 ( ab A T T 2 ˇ ˇ ( γ γ m − T T mn T − T

F )   T 4 , is the composite connection, and should therefore be the adj + (6) (6) ¯ 4 npq ˇ ) γ γ ( mnabc mnabc mna mna ± m 1 T , F F H abcdef . − H H A φ φ  ǫ ) l 15 , ab 1 1 + ˇ φ ( − − ie ie k (8). Let us generically denote generators of a representati m ( T h ∗  ie mnpq F T 2 2 1 1  ǫ 1 8 8 1 φ 1 2 * * 3! 4!2 ie = =
. mab
= = − − − of SU } ω + − 0 0  λ λ ) ) abc 1 8 − ˇ γ + ¯ 4 4 63 , ξ , , ξ a = ¯ 4 4 ˇ γ . We then have that (suppresing representation indices) ( ( { Now let us apply this group theory to the Killing spinor equat Let us repeat the process for the field ) − ± m l + In fact, note that if one prefers, one might as well add the We hope no confusion will arise between the supergravity field In order to see this, compare the action of SU mn mn , k T A T 17 16 15 ( projection operators in the generators of the representati algebras labeled by T onto have written above. tation correspond to the symmetric and anti-symmetric part Our claim is that i.e. the These form bispinors under SU As expected, we find that Finally, by noting that the algebra should close, we constru We see that the representation decomposition is indeed exac with anti-selfduality implicit. We thus see that First, we spell out the field JHEP10(2017)159 (4.36) (4.34) (4.35) , so let , which and the D ±   36 K 2 2 + − ) ) , . 10 10 , , 1 1 abc abc + + ( ( λ λ T T he D7-branes with of the worldvolume − + + + ξ ξ i i 2 2 0 0 sformation, + − ) ) ) ) ¯ 4 4 1 1 , , , , trate in appendix 4 ¯ 4 re Lorentzian, one of them ( ( 10 10 abc abc T T atural than any other choice. ( ( that the kappa-symmetry re- directions. The difference be- e respectively the f freedom of the Killing spinor ately, one is left with exactly ) ) hould be to obtain the appro- T T an be coupled to background 7 e supersymmetry of non-trivial = 1 non-linear supersymmetry. , rsymmetric field theory and its   6 = 1, we will introduce a pair of 1234 1234 ges when starting with sixteen. A N X = 4 field theory with an additional abc abc a a F F ε . W W N = 8 supergravity. In general also field + − = + + → N a a = 4 Euclidean supersymmetric field the- . We consider the following possibilities: = 2 supergravity background ε N   F F 3 d 2 F 2 ( ( + − N = 4 ) ) φ φ = Γ ie ie . Then supersymmetry of all three branes is 10 10 , , 7 ˜ 2 2 N ε 1 1 abc abc ( ( ˜ 7 – 21 – − − T T ,Γ 0 0 7 ) ) − − = Γ ¯ 4 4 , , 2 2 ε ,Γ 4 ¯ 4 abc abc + − ( ( 7 3 ) ) T T Γ 1 1 , , abc abc 10 10 abc abc ( ( F F T T φ φ   ie ie abc abc + 2 + 2 H H h h = 1 field theory to an directions, the other localized in the 1 1 16 16 9 , − − 8 N X = = = 4 supersymmetry to background

+ − = 1 linear supersymmetry and an additional N λ λ − + N ξ ξ flux of the Euclidean D3-brane. Both D7-branes carry no worldvolume flux. Both D7-branes carry worldvolume flux, which is the pullback − + In order to break the supersymmetry We denote the kappa-symmetry operators of the D3-brane and t Finally, there is the field that comprises the conformal tran • • K K . , such that, if the projection operators are chosen appropri ¯ It turns out that the compatibilityworldvolume of flux the on first the option D3-brane with is th problematic, as we demons worldvolume flux as respectivelyequivalent Γ to enforcing tween the two cases lies in the worldvolume flux us discuss the second option instead. ory with priate projections. There appearIn to be both two of candidates these morelocalized n cases in that the we will describe, both D7-branes a 1/4 of the orginal supercharges,priori, i.e. it four is linear not superchar clear what the properties of the D7-branes s Again, selfduality is implied. theories with less linearsupergravity. To and illustrate non-linear this, let supersymmetriescoupling us to c supergravity now of reduce the the last supe section to an As was remarked before, we see that these representations ar and D7-branes that fill out thequirements entire of the D3-brane. D7-branes The project logic outε a here number is of degrees o 36 4.2 Coupling In the previous section we worked out the coupling of decomposes as non-linear JHEP10(2017)159 . , ) × − 45 ε γ 4.19 (2) (4.41) (4.40) (4.42) (4.37) (4.38) (4.39) differs ∗ = 1 R- 05 N γ ) to construct cer- , generated by ˇ 3 envalue of ˇ 4.38 , U(1), as the eigenvalues logue would be breaking × kappa-symmetry operator theories, that comes down = 2 supergravity (see for . to U(1) f generality. We will now factors are projected to = 0 = 0 = 0, this procedure leads to 2) ± rators we should act on ( + − N − . η η ε !+ (4) m m ∗ + γ γ (6) , which then enhances to SU 1 = 1 case spelled out explicitly. ˇ − + γ (6) + ˇ γ − . M M N R . , ⊗ 3 3 1 − − , it follows that ± ± (6) × = 1 supergravity. More generally, the Γ Γ ± + − ε ! ˇ γ = 2 R-symmetry group for non-trivial − + ξ iξ iξ ξ + 1 η η =

N ± ± ± R 0567 0589 ) ) N ± 068 ε iγ iγ γ m m = = = 1 1 = 4 R-symmetry group down to the γ γ , 1 (6) − − – 22 – ± ± ± ± n n = ˇ ˇ γ ξ ξ ξ ! N γ γ = = − n n (4) 05 67 89 ξ 7 ˜ 7 1 (6) γ * ˇ ˇ ˇ γ γ γ down to iV iV Γ Γ ) is a subgroup of these two factors. Furthermore, the ˇ γ − . In essence, we will make use of ( ≡ = + − − + C ε R ± 4.23 m m ) U(1), which then enhances to U(2). Specifically, the SU(4) 1 (6) × ˇ R γ (10) iA iA × ( − + (4) ∗ * = 1 equations. The derivation will be somewhat messy and will m m ∇ ∇ SU ( ( N are projected to × 2 − generated by ( (4) ) ∗ U(1) to U(1) 4 will lead to the gravitino variations of an + , × R ¯ 4 − ε ]), where we have the embedding of the . Let us discuss the adjoint first. The two SU 35 89 and ( SU(4) γ are imaginary. In the case of a Euclidean field theory, the eig × +2 ) The projection operators break the (6) and ˇ . In the more familiar case of Lorentzian field theory, the ana ¯ 4 γ , + 67 4 ˇ of ˇ symmetry group, which then enhances to the a breaking of SU In our case, where we examine Euclidean backgrounds for field SU(4) Note that the R γ which are the gravitino variations of 16/16 acting on the normal bundle would be broken by the projection In total this gives in the Lorentzian case the gauge group SU( presence of ( to obtain the desired Killing spinor equations. In the case tain projection operators, and then figure out with which ope for the Lorentzian D7 branes is given by Taking into account our ansatze for the worldvolume flux, the Using this and the fact that Γ up to aproceed constant to prefactor obtain which the we trivialize without loss o This leads to the conclusion that therefore be relegated to appendix example [ JHEP10(2017)159 = 8 (4.46) (4.44) (4.43) (4.45) (4.47) N ). This .  ) 4.17 .  = 0 = 0  + ) 078 ) + − R λ λ 078 , iW 078 × (2) (2)  iF + ) ǫ ǫ iH ) j ∗ j (2) + ) σ ∗ σ + 078 e IIB supergravity and 069 j ) j , npq e are given by variance. + 578 − ) 069 mn F 069 iW npq F M iF M mn = 2 supergravity, as given H + 0589 ckgrounds for field theories, iH m F m + + ns of four-dimensional γ m mnpq γ eld theory is being coupled to N + + xactly as given in ( 579 iǫ + iF + 069 579 mnpq 1 W 3! 579 − F + 569 iǫ − . mn λ λ H − = 4, 1 + 3! − F ± mn ( (2) λ (2) d i − F 0567 89 ǫ ǫ + 568 ( 568 i ! m m m + m ) F 89 W 568 γ F γ ( ( + iF 89 m , and so the gauge group is complexified. (2) and H 1 0 np 579 np + ∗ ( ∓ m 0 1 ∓ − H γ 89 γ − 079 ω γ 67 mn ∓ + − np + np 578

578 F m + mn T – 23 – T 6789 F 67 F 1 4 iF 578 1 4 = − 67 m iW and ˇ m = 2 contain, in addition to the gravitino, another + F m − + ± − − iH H + ω 568 λ 67 05 − + − N + γ + 068 569 m λ + λ (2) mn 569 m ǫ F 05 iF   iF F mn 05 569 j ( ( ∗ j m iW F m ∓ φ φ σ + σ  iH e e + j m iω iH j m φ 1 1 4 4 ( ( + 079 iǫ iA 1 1 2 4 = 8 and − − iA F 079 8 1 079 + − = = = = − W − N H 0 m 0 m 2 m 0 m 3 m 1 m − = 068 − A A A A iA iA F has real eigenvalues and so the gauge group is SU 068 ± mn  from the one for ˇ + − 068 T φ W i (6) φ φ H ie  γ m  m 1 8 8 1 i ∂ ∂ 8 1 2 1 − ± 2 1 are the Pauli-matrices generating SU = = = − − j = 2 supergravities from the supersymmetry conditions of typ 1 3 2 = 4 fields are expressed in terms of the string theory fields as σ m m ] (see eq. (8.24)). All terms exhibit manifest R-symmetry in ± ± ± d N The Killing spinor equations we obtain by the above procedur ∇ ∇ 35 M M M   and of D3-branes. In orderthe to variations of fully all determineneed fermions supersymmetric of to ba the vanish. supergravity that Both the fi and The Here, We have discussed so far how to obtain the gravitino variatio matches precisely the variation of the gravitino in 4.3 The dilatino variation in [ Furthermore, ˇ by a factor of and Finally, the spin connection in is shifted by the dilatino, e JHEP10(2017)159 ], × 29 (4.48) (4.50) (4.49) SU(4) , and so ): roughly → : such matrix 4.14 0 2 matrix acting × in ( ˆ , B is the same field as ˆ A g ± , one should construct ± ˆ mn C = 0 = 0 T osition SU(8) ˆ iation of the dilatino and doublet space. However, T B lete the derivation of the ˆ of SU(8), and the vanishing ), one is able to arrive at a A . However, the problem for cture with Γ ...... l transformations and with quires a different recipe for χ C ith the four-dimensional field ndices (i.e., the indices of the logous terms in IIB would be case of (Lorentzian) maximal cing a full IIB version of [ e (i.e., the 2 + + 56 n of , ct, there is no way to compare theory perspective, it would be imensional supergravity. While ˆ ˆ on + D D , : any spinor transforming as the a ] λ − ˆ D C λ m χ Ψ γ m ˆ D ˆ ]) γ B ] D a ] ˆ ˆ ˆ A a C D C a [ C 32 ˆ ˆ Γ C B – Γ ˆ Ψ ˆ A B
ˆ 29 − A AB m [ AB + (4) a a [ m P γ Γ Γ P ), but with coefficients of the string fields such +

– 24 – · · ] + ˆ ] 1 + C 4.48 ˆ + C transforms under the − (10) (10) . λ λ ˆ Γ Γ C ∼ ± ˆ B ˆ mn ± ± mn T C ˆ corresponds to an SU(4) index tensored with a doublet A ˆ γ γ B 1 1 χ ˆ ˆ ˆ ˆ B A B A ˆ ˆ ˆ A A χ [ [ having some given expression in terms of the string fields. − mn + mn are SU(4) indices. Taking a linear combination of the two and ± T T 2 P 2 √ √ being fundamental SU(8) indices. Here, 3 4 = 2 supergravity, the dilatino does not suffer from this issue 4 3 } − 8 − ) and N A, B, . . . 4.14 ,..., 1 ∈ { U(1), an SU(8) index × ). Therefore, in order to construct (the variation of) ˆ ± In the case of From an M-theory point of view, this can be achieved by settin We believe the reason for this is as follows. Under the decomp matching ( λ ˆ = 2 supergravity is that, since the fields appearing in the var = 8 supergravity, the dilatino ˆ B,... ± ˆ in theory one should beN able to construct itthose using appearing our in methods thewhich variation ten-dimensional of object the leads gravitino tofundamental are the of distin “right” the dilatino R-symmetry group would do. necessary to understand how to obtainT the above equations, w appearing in theconstraints gravitino on variation. four-dimensional backgrounds from Thus, a in string order to comp with the ellipsis representing terms that correspond to Wey on of its variation corresponds to (compare with [ A, where the indices our methods fundamentally rely on acting on the doublet stru an object with three SU(4) indices which is a three-tensor in with the Spin(7) indices beingthe lifted building to blocks SU(8). The naive ana multiplication cannot produceobtaining a the three-tensor. desired dilatino Thus,completing variation such one of a maximal re construction four-d may be interesting for produ this stays outside the scope of our paper. speaking, the two SU(8) indicesgamma-matrices) correspond tensored to the with two a SU(4) two-tensor i in doublet spac variation of the same generalthat form one as cannot ( quite reconstruct N fermion, which we will refer to here as the “dilatino”. In the proceeding as before for the gravitino (as outlined in secti SU(4) structure; for example, this can be observed in the definitio JHEP10(2017)159 ) give ), which 4.19 3.17 etric fashion, one ) for its coupling to ral instanton config- itions in a way that linear and non-linear ). pace and are therefore ry with four linear and we see that background 4.43 ackgrounds with a non- symmetries by projecting ound. For instance, con- y condition ( 4.19 y, and the condition for a the brane world volume to non-linear supersymmetries tions can be used to express round, these theories couple e way as linear supersymme- e supersymmetries that are IB. We showed that the ap- s not violate these non-linear ll supergravity multiplet. In erent amounts of linear and ut with supersymmetric non- ion ( und to which the field theory o non-linear supersymmetries. metric field theories on curved ce non-linear supersymmetries round fields can be engineered. alized on the worldvolume, are ) or ( nd does not distinguish between y the possibilities for localization 4.8 ) and their simplified version ( 4.8 – 25 – (8)-covariant description those couplings. In particular ∗ We can obtain theories with less linear and non-linear super In order to put the field theory on a curved space in a supersymm In summary, the supersymmetry conditions for theories with Combining both supersymmetry conditions, we can study gene out some of thefour supersymmetries. non-linear supersymmetries We the found supersymmetry for condit instance a theo supersymmetries. In otherin words, question the is supergravity coupled backgro isAs also we showed supersymmetric here, with the respectthe ten-dimensional t supersymmetry supersymmetry variations condi ofparticular, the the fermions supersymmetries in which that are off-she non-linearly re needs to ensure that the coupling to the background fields doe linear and non-linear realizationscan on be the coupled worldvolume. to background Hen tries. supergravity in Only exactly when the embedded sam to in different a bulk ten-dimensional fields. string backg supergravity couples universally to all supersymmetries a backgrounds that originate frompearance D3-brane of theories in non-linearspontaneously type instantons I broken is in tied thenon-linearly to realized vacuum in existence of the the of worldvolume field theory. th theory on flat s 5 Conclusion In this work we discussed non-linear instantons of supersym figurations such that onlyare linear coupled supersymmetrically combinations to ofThis the linear gravitational should and backg leadsupersymmetric to vacuum supersymmetric and no field linearlinear theories instanton instanton solutions. solutions, on b It curved wouldtechniques be for b interesting such to backgrounds. stud can be interpreted assupersymmetric coupling the to gaugino background variation supergravity ( of the fieldurations theor for a supersymmetric field theory on a curved backgr background supergravity. Innon-linear general supersymmetries also can be theories studied. with diff an SU We have shown howdescribes to the coupling reformulate of thebackground the supergravity. bulk supersymmetric supersymmetry The field equations cond theory in on ( 4.4 Instantons on curved backgrounds supersymmetries are given by the worldvolume supersymmetr JHEP10(2017)159 tric in- ε = ε Γ . non-linear 1 NLI coupled to general SUSY: off-shell SUGRA instanton (NLI) ). Examples of such ensional theories, we ε nton configurations in 2.6 th non-linearly realized ing from a D3-brane in a = inear dependence on the dard warped Calabi-Yau here also linearly realized instantons might be of in- ε nch connecting Klebanov- ackground with imaginary ration will in general not ino variation. Similarly we e non-linear instantons dis- tion ( ,Γ en in the vacuum, as in that in a covariant form, by mod- found in table pendent of which supercharge ε The supersymmetry condition r a D5-brane. Here even the rgravity background has effec- kground in that case. However correspondence between linear -brane configurations are found ld theories. = n turn may lead to applications In principle non-linearly realized ories. So far localization compu- ariation to give rise to additional Hermitian tantons can however also be used ε 0 Γ off-shell SUGRA Yang-Mills (HYM) only linear SUSY: HYM coupled to ]. ]. All linear supersymmetries of the world- 36 39 – ε – 26 – 37 = ε 0 Γ FT coupled to twisted FT FT from D3: topologically off-shell SUGRA bulk special geometry generalized holonomy geometry in the variation of the fermions. Consequently, supersymme F FT = 0 ε = 0 . A sketch of the different cases of a field theory (FT) originat m ε m ∇ A particular example of a non-supersymmetric field theory wi The non-linearly realized supersymmetries and the related While the paper is largely devoted to D3-branes and four-dim Naturally one would like to find examples of non-linear insta D pure geometry: with fluxes: SUGRA of supersymmetric localization for non-supersymmetric fie supersymmetries is give byselfdual an anti-D3 three-form brane flux, in asvolume a discussed theory Calabi-Yau in are b [ explicitly broken by the supergravity bac supersymmetries are present the computationis should used be inde for localization,and which non-linear should instantons. lead Localization toin on a non-linear field non-trivial ins theories wherecase supersymmetry all is supersymmetries spontaneously are brok non-linearly realized. This i terest in various applications fortations supersymmetric have field only used the linearlysupersymmetries, realized supersymmetries. leading tocussed the above, theory can that also localizes be on used th for localization. In the case w backgrounds can for instanceStrassler be and Maldacena-Nunez found backgrounds along [ the baryonic bra also discussed supersymmetry variationssupersymmetries and that instantons are fo unbrokenworldvolume in flux the vacuum have a non-l Table 1 string theory. Apreserve the D3-brane same withorientifold supersymmetries non-linear compactifications instanton that of type configu are IIB.in unbroken more Instead general in these flux D3 stan backgrounds, compatible with the projec coming from the externalifying gravitino variation it can by be thesuspect written the dilatino dilatino and variation and thesupersymmetry the conditions. trace internal gravitino An of v overview the of internal our discussion gravit is given supergravity (SUGRA) background. linearly realized on thetively supergravity double background. the amount The of supe supercharges of the field theory. stantons in six dimensions are naturally non-linear. JHEP10(2017)159 . N ] (A.7) (A.6) (A.1) (A.2) (A.5) (A.3) (A.4) k M δ 1 − k ...M 1 M ] [ Γ /1 (H.T.). k ] QRS ] + Q ] ] N N . N N Γ Agence Nationale de la k ) Γ Γ ) nt Agreement n. 307286 d M P d [ [ ( ( M M P ...M δ [ [ P [ even satisfies γ on-linear supersymmetries. γ KIAS for hospitality during 1 δ δ ) tanding of the field theories ) lized supersymmerty and the k 4 16 8 d k M . , Valentin Reys and Dan Wal- − − − − − d k d ( -dimensional space is ( . =Γ l . γ ]= s which have been made use of are γ ...n d d 1 . d + ] = n ⋆ ⋆ NP N k ] = α Γ ...M Γ 1) 1) k P P QRS PQ 1 k − − Γ Γ Γ M k , , , k = +1 ...n ( ( in a γ 1 ...M k k d MNP 1 n 1) is given by ... 2 1 1 2 l α MN MN MN 1 ω M − 1) 4 1 m 1) [Γ [Γ [Γ Γ ...M ǫ ...m 1 − − 1 + , d ( − – 27 – M ( m ǫ 1) -form ǫ 1) ! M = ! 1 − k d − 1 l ∂ d d ( NP ... ( ) gamma-matrices with NP d 1 d = = = d 2 1 2 1 m ) ǫ α d M 1) ( 1) l MQR MQRS γ ∇ − + − ( k NP 2Γ ⋆ − +2Γ − = ( MQ ) M = k Γ ) MS ( 2Γ k γ Γ ( − NP QR γ NP δ NP QR M δ NP QM 12 δ − =2Γ =4 = =24 } } } } M Γ QM , QRM Γ , QRSM Γ , NP Γ , Γ NP { NP Γ { NP Γ { Γ { Relevant (anti-)commutator identities for gamma-matrice The sign we use for the Lorentzian Levi-Civita is The chirality matrix for either signature is defined as Our definition for the Hodge-dual of a The Euclidean Hodge star for Spin( while the Lorentzian equivalent for Spin(1 A Conventions and identities The connection is given by We thank Thomas Dumitrescu, Charlie Strickland-Constable dram for useful discussions. R.Mthe and course D.P. would of like this toRecherche work. thank under This the work grant(XD-STRING) was 12-BS05-003-01 (D.P.) supported (R.M.), and in by by part ERC EPSRC by Gra Programme the Grant EP/K034456 Hence we may hope thatmethods the better highlighted control in of this non-linearlyon work rea anti-branes might in lead flux to backgrounds. better unders Acknowledgments the coupling to the supergravity background preserves the n JHEP10(2017)159 - n as (A.9) (A.8) (A.12) (A.14) (A.15) (A.13) (A.10) (A.11) (A.18) (A.16) (A.17) (A.19) in 2 spacelike n 2 n C t. When splitting on matrix rmitian timelike gamma . . dentities: M conjugation of spinors by s not change chirality and is . etrized product of m ion does not change chirality, ion does not change chirality, Γ Majorana conjugation of Weyl a γ ˇ γ 4 10 6 C C C − . − = . ∗ = . . . T ) = ∗ ) ) T . ± ± ± . ) T (4) M ± θ θ . . ψ ) m † ± γ m ( Γ a ψ − − γ ε γ ( ˇ C γ C ⊗ 10 4 6 0 ⊗ = = C = = 1 = 1 a C = C C c c ( ( c 1 γ ( ] ] 1 = ) c c – 28 – 2 2 (4) (6) = Γ − ) ) c ± = = ˇ ˇ γ γ ) ε c ± ± C a ( ) ± θ θ m , , Γ , ± ψ Γ [( [( ( 4 ψ 6 10 ( C C C − − = = T 6 = T 4 C T 10 C Spin(4), we decompose the ten-dimensional gamma-matrices C × 5) , The charge conjugation matrix satisfies The charge conjugation matrix satisfies Spin(1 The charge conjugation matrix satisfies → 5). 9). 9) , , , Spinors are pseudoreal, innor the is sense it that involutive: Majorana conjugat Spin(1 Spinors are pseudoreal, innor the is sense it that involutive: Majorana conjugat The chirality matrix satisfies For Lorentzian even-dimensional spaces, we define Majorana spinors by We work with hermitianmatrices. spacelike For Euclidean gamma even-dimensional matrices spaces, we and define anti-he For the specific dimensions we require, weSpin(4). find the following i Spin(1 Note that these are both dimension and signature independen gamma matrices. It satisfies Spinors are real, ininvolutive: the sense that Majorana conjugation doe Our spinor conventions are as follows. The charge conjugati The chirality matrix satisfies Spin(1 dimensional spacetime is given (up to sign) by the anti-symm JHEP10(2017)159 . 1 − YM g (B.2) (B.3) (B.4) (B.5) (B.1) ditions . Then F

≡ ) leads to a Q 3.30 Spin(6), ⊗ ] respectively. . 42 . ]; for recent applications 40 (4) = 0 = 0 γ . While the details vary e merits of the proposal . The instanton equation, ] and [ Spin(3) 1 + 1 + pq . ε ε sed in analysing supersym- 3.3 41 alculations can be involved, on, is given by F As we shall argue here, that 3.1
→ F



supersymmetric theories, one = 0. Instead, what we should equations schematically as + Dirac operator and the covari- ing higher-derivative terms). on mn . However, spinor calculus (or 18 d best known of these theorems. η (4) . 2 F εε | γ on of supersymmetry. These con- torsion see [ ⊗ trace of the Einstein equation. As = 4 linear supersymmetry. Taking = ˜ e backgrounds a subset of equations mn Qε + ∼ L d | ε Spin(9) θ mnpq F c 2 ǫ ˜ ε 4 1 Q → mn ⊗ g ) d M-theory see [ 2 , + 9) ! F π iθ , ⋆ 8 1 0 nq ( g = 0 1 4

r = 0 to see which equations of motion follow au- mp – 29 – ε = − g Qε given by the linear generators in ( → 2 ˆ 1 ε pq g Q Q − ⊗ does have a non-trivial norm, which we fix as F ) ε ! mn F 0 1 1 F − 1 2

g = = + , since ˆ 2 2 1 + δ | , the two equations are equivalent. Let us consider , we would like to consider ε ˆ ] section 3.1 for details. We rescale the metric ε F

Q (4) , the choice of 3.1 44 | = det( γ ) is satisfied, and end up with a set of (hopefully) simpler con F

2 p Q  B.1 sin Q ] section 4.1 or [ 43 Let us start with a warm-up exercise and consider For extension of the Lichnerowicz formula to connection with = 0 we expect to end up with HYM as described in section 18 − do is consider the spinor decomposition Spin(1 representation theory) leads to the conclusion that one finds that ε As described in section In order to get rid of Then we can consider (involving higher derivative terms) to heterotic strings an to which we refer in this context as the supersymmetry equati can expect that, modulodifferently vanishing from on-shell terms, good six-dimensional Lagrangian for the gauge field (involv ditions universally involve Bianchi identities, and inof som motion. Lichnerowicz formula is probablyFor the Levi-Civita simplest connections an the differenceant of derivative is squares proportional of to the the the vanishing Ricci of scalar, the i.e. action the is often an equation of motion for that ensure the solution of the full theory given preservati widely depending on the detailsthe of basic the idea backgrounds, is and fairly the simple: c denoting the supersymmetry See [ tomatically when ( The integrability theorems aremetric among flux of compactifications. the standardfor We tools six-dimensional can u linear use supersymmetry this made tool in to subsecti argue th B Squaring linear supersymmetry one wants to use the ensuing condition JHEP10(2017)159 . 3 3 is a  need  (B.7) (B.8) (B.9) (B.6) ε (B.10) − mn mn . Then, = ε F f F ε 0 mn (6) mn and γ . F ) (or equiva- F 1 2 + 1 2  . = 2 supergrav-  ε 4.3 = 6. The linear 6 sed string super-  ( pq F d N 1 F − R

3!

YM F 1 g Tr ], where Γ mmetry were equiva- ≡ + 6 1 3! mn 11 e to F − Q − 1 4 )) and , mnpq in the decomposition of the − red in order to construct a F ) γ mnpq ǫ equations of four-dimensional is situation is in contrast with F pq 4.19 Tr = 0 = 0 2 1 2 ibed in [ F π − onsider 1 + 1 + F g ε ε YM mn 1 4 ) + iθ g YM 1) δ F g (6) − 1 4 − γ 3 32 is not a scalar. However, we now make ) det(  2 (6) 2 − p γ F mn Q (  F mn F − R

1 = 4 Yang-Mills Lagrangian by squaring the 3! that is strongly inspired by kappa-symmetry. ( − Q mn F  mn d ) – 30 – ± F F ε mn + F 2 1 1 mn F  − F 4 g 2 1 1 + 1 YM F + ) to rewrite this term. We then find g  1+ 2 Tr δ  4 1 + B.7 ). In particular, it can be shown that − = 6 Lagrangian, which includes higher derivative terms, may − det( mn d 2 mn ) p L ≡ − 3.28 F ( 2 ) (which leads to the simplified ( F = F mn ( mn 4.8 ˆ ε F 2 1 2 F ) = 1 2  Q ): the internal spinors in the decompositions of both F c 1 = YM 1 ˜ ˆ ε + g − 4.5 2 2 2 imposed by the D3-brane. ) − g defined in ( ε F + R δ )). As can be seen, the starting point is almost purely the clo L≡− ), starting from the string theory supersymmetry condition F − R

=

and 4.4 = 1 + ˆ ε det( 2 = 4 (the D3-brane), the two equations defining linear supersy = ( Q 4.43 Q Encouraged by this result, let us try to extend this procedur d c 2 = 8 supergravity ( ˜ ˆ ε Q − we find that with lent. Here, thiscombination is yielding not only the scalar-like case, quantities. and Let both us operators c are requi supersymmetry equations are given by constraint on For The only place where D3-braneKilling data spinors enters the ( computation is In this section, weN give the derivation of the Killing spinor be interesting for further study. C Derivation of the Killing spinor equations As the second term includes gamma-matrices, ity ( lently, ( symmetry condition, albeit in the basis We would propose that this supersymmetry conditions of the D3-brane. use of the second equation of ( to be identical to allowthe for situation non-trivial without worldvolume non-linear flux. supersymmetries Th as descr We thus have managed to construct the to find JHEP10(2017)159 e , − . + ε ε − (C.4) (C.3) (C.2) (C.1) ], and ε P  + P 11 . It will ]. As we  ε n  F γ m 29  m γ γ m abc n γ ˇ abc γ → − n γ ˇ γ γ  F abc  ) makes use of the mnabc F pqr . ab φ F ˇ F γ 4.4 1 − and ie 1 + 1 3! ε sole difference of using ε ) 1 ∓ as an intermediate step, , ) 12 − npqr 0 ε 0 ǫ cdefn art by demonstrating the Γ m te the (anti-)commutators ∓ m m ator associated to the susy 3 1 F + was worked out in [ unmodified ten-dimensional γ ˜ = 4 linear supersymmetries, d R − a a − ke a linear combination at the ˇ + m ˇ by careful study of [ γ γ ←→ eory to work out, and to know N ≡ (1
n L (4) (1 + Γ terested in the modified external ± abcdef γ  γ ∓
,..., ǫ ε (4) ε ).  P ab ] γ ). However, ( m P 1 ) 0 ˇ γ 48 a (4) 2.1 Γ (4) 4.4 (4) γ 1234 γ + m nab γ a m m F for the left- and right-hand side for con- ˜ and ˜ D F pq d [ d (4) . This requires one to realize in advance  a γ 1 2 + ∓ m m φ + m F + n a R ie F D F = P P , 2 1  by switching m ± m – 31 – ± m  mnpq c φ ε c + ǫ φ + L m , . . . , d } ie 1 2 + ˜ ab 0 + m ie R 1 8 ˇ γ Γ a 1 4 − , (4) (4) − γ + m γ mab and m ab m D is given by mna H (4) b ˜ b ˇ { γ − m γ F + 1 2 − a m ( + L + φ ˇ γ mab R np m ≡ n m ie γ ω ˜ a a γ 2 1 1 4 ± m  = 4 Killing spinor equations with L mnp ). There are two ways one can proceed. The first way is to comput − + = = d mna H ω m − m + m 4.7 (4)  − γ L ∇ R 1 8   2 1 − mna = = H + − ε ε } ]  = 4 0 0 4 1 Γ Γ , , N + m m In all three cases, the computation is very similar, with the D D [ { 1 2 1 2 a different operatorunmodified for external the gravitino. commutation We consider relations. Let us st Furthermore, one can obtain venience. We can work these out by making use of ( prove to be convenient to introduce defined by leads to did not, we willfermions. instead demonstrate the computations using unmodified external gravitino variation. Instead,gravitino we variation, are ( in whereas the right-hand side where we have introduced the notation In order to obtain the C.1 the starting point is the pair of doublet equations ( that such a shift willthe be correct coefficients necessary beforehand, for which the would representation be th possible end. The second wayvariation is of to the first modifiedusing construct this the external differential differential gravitino, oper operator and rather then than compu the (anti-)commutators of the unmodified shifts, and then ta JHEP10(2017)159 (C.5) (C.8) (C.6) (C.9) (C.7) (C.10) m γ ) is equivalent to . n γ − + − + , C.1 ζ ζ η η  − + n − + m ab ξ ξ ξ ξ γ γ m ˇ γ st and second by taking m   ) ) n γ  γ n γ m m m m . b b ˜ ˜ n d d γ  abc ) ) m cdefn a ˇ H γ − +  γ − + ˇ i γ m m F | | − + 2 2 m m  m m pqr + ˜ ˜ a a c c ), such that ( 1234 . iR iR F   abc mnabc m a n m abcdef ˇ | γ F γ + ( + ( F ± ± ǫ H + 2 + + φ i φ npq φ npqr + −  abc m m + − ǫ | | ie ,L a ie η η ie H − ζ ζ F − + 1 1 φ ˇ γ 1 8 m + − φ 1 + − | 48 R R m ie ξ ξ + 1 ξ ξ ( ( − 1 ie γ pqa ) ) − L 1 1 1 2 2 mnpq   n  48 4!2!8 1 m m F  48 γ m m iǫ b b  − ˜ ˜ + ), it easily follows that (suppressing spinor-sum d d = (  1  + 4! – 32 – ) = ) = + −  n + + − ab + m mnpq + γ 4.5 m m m − | | m m ǫ + ˇ γ  L γ + 2 − 2 n m m φ a a  ), we find the following expressions: = ˜ ˜ c c γ ab  a  ie abn iL iL ), we find the following four equations; again, we note ˇ   γ ˇ γ a ab m  F ˇ 1 γ C.3 a ˇ γ ± ± 16 φ ) = ( ) = ( a H ˇ γ C.8 mab m mna ) = ) = F m m ie m m − | | | | γ φ mab ω H + 1 − 1 m m + 2 + 2 n | | 1 a 1 2 8 16 1 ω mna − − 2 2 ie L L γ ˇ γ iL iL ( ( 1 4 F 1 8 n ) and ( − − − 1 1 2 2 φ iR iR F + −     ) into ( + φ mna ie + − C.2 m m 1 8 m = = = = | | ie H + 1 + 1 m m 1 8 | | 4 1 ∇ − C.10 m m m m − − 1 1 L L ˜ ˜ b c ˜ ( ( ˜ d a −    R R 1 2 2 1 ( ( = = = = 1 2 2 1 ) with ( m m m m ) and ( b c d a C.4 C.9 and the explicit tensor product) α and Inserting ( where we have introduced Using the Killing spinor decomposition ( that the third and fourth equation can be obtained from the fir indices the equations as well as We denote the doublet components as Comparing ( JHEP10(2017)159 ; ) ) , ± ξ − + η η

(C.12) (C.13) C.11b C.11b n n (C.11c) (C.11a) (C.11b) (C.11d) γ γ − + ) ) ), ( ζ ζ .

+ mn − mn ∓ n n ξ γ γ Υ Υ ∓ i i ) ) npq C.11a ± ξ abc + − a F ξ , ˇ γ ∓ npq + mn − mn − + ˇ γ ξ ab ± ∓ ζ ζ Υ Υ + mn − mn H a − + ξ ˇ ξ γ i i

ˇ γ η η a Σ Σ mnpq n n ab 1234 a + −

ˇ ǫ γ mnabc a ˇ γ γ − − γ ) = 0 ) = 0 n n mab F φ ) ) F mnpq F + − φ γ γ F + mn − mn φ ie φ ) ) f these only admit trivial +( +( iǫ λ λ mna φ + − mab ie e mn mn n n ie e 1 for the internal spinors m m 1 + − 1 1 4! 8 8 48 Y Y iω H mn mn γ γ γ γ 1 1 +(Σ +(Σ Y Y ) ) 1 2 48 8 16 1 − − + n − n − − − − ˜ ˜ + − m m ) that vanish. Thus, all fields + + A A ======γ γ + − mn mn Θ Θ ) ) i i − − + − mn mn ± ± ± ± ± ± X X m m + − ( ( ξ ξ ξ ξ ξ ξ + − − + Λ i i X X Θ Θ H ± ± λ λ ( ( + − ± m ± m i i C.11d = 0), the r.h.s. of the ( i i + − ± mn ± mn + − Ξ Π Θ ∆ + − ) into two doublet matrix equations. We (Π (Π + − Σ m m Υ ˜ ˜ F K K + − γ γ − − ( ( m m + − γ γ m m (Π (Π t t C.11 + − γ γ − − ) and ( t t − − + − − − = 4 conformal supergravity is exactly given by n n + + + + η η + − γ γ – 33 – n n

− + ζ ζ N γ γ λ λ m m + −

C.11c ± mn mn n n γ γ ξ + − m m V V mn mn n n γ γ γ γ ab V V − − γ γ n n ) ) ˇ γ + − mn mn γ γ n n ˜ ˜ ) ) ) which leads to the first doublet equation, while ( T T + + + + n n H H + − + − ∓ cdef i i + + ξ η η ζ ζ H H 19 i i m − + a



+ − ˇ ∓ γ F ∓ + n − n C.11c λ λ − + m m m m ξ ) ) ξ γ γ γ γ Ξ Ξ pq + n − n a i i n n n n + m − m a ˇ γ ∓ Ξ Ξ abc γ γ γ γ ± i i − + F ξ A A ) ) ) ) ˇ γ . abcdef ξ a n n n n n n + − ǫ ˇ + + γ F mna ab ) lead to another. These doublet equations are given by Λ Λ φ n n abc m is is is is ˇ γ F mnpq m m − − F ie F = 0 (which is implied by taking φ ǫ + − + − φ φ ˜ ˜ mna ∇ ∇ φ e ie + − + − → − 1 − n n n n ( ( mab +(Λ +(Λ +( +( ie e ε 384 8 8 1 1 C.11d ω iH U U U U m m m m F 1 1 ( ( 1 48 4 4 16 1 − − − H H H H +( +( − − i i i i ) together with ( ======− − + − + − and 0+ m 0 m 0+ m 0 m ], eq. (3.6), (3.7) for comparison. ± ± ± ± ± ± m + m − m + m − m A A A A ξ ξ ξ ξ ξ ξ s 11 ± ζ ∆ ∆ ∆ ∆ ± ± ± m + + + + C.11a ± ± t ± 0 m mn mn mn U In the case Next, let us recombine the four equations ( m m m m Y V See [ ↔ A − − − − X ∇ ∇ ∇ ∇ 19 ± = = = = depending on the internalsolutions. manifold, it may well be that some o Note that most fields are determined by eigenvalue equations together with ( are trivial, whereasdecouple. it The is Killing the spinor equation l.h.s. of of ( take ( the first two lines in this case. η with fields JHEP10(2017)159 . or ∓ + . ε R D  , P  (C.18) (C.15) (C.17) (C.19) (C.14) (C.16) ± mn  L mn γ γ m a a γ ˇ γ . ˇ γ − ) for npq ε either by mna mna F P . C.4 H F m !  . a ) D ! ˇ γ mnpq fields. + 3 ± m + 3 ! ǫ e internal gravitino is Ξ ) ± ) mn mn i + abc abc ± m m γ ˇ iY γ ˇ γ ± Ξ  (4) is ) i ± mn γ ∓ m abc abc m ± m ± m en by equation. F m mna H is ! γ ± ± mn + Σ m  γ m  F φγ s. For the dilatino, we find that ab ± ), replacing V ± φ U φ m m ˇ γ 1 + 2(Λ Θ 12 F ± mn ie ± ∂ ie 2( − m i ± φ C.1 m + (Λ Υ t U + 1 − − i − mab 12 ( ± ie ± H mn − m F ± i (4) − ± = = = = ± 3 0 m ˜ γ H ˜ ˜ a b d c ˜ iX Π  i ∓ A which are defined as in ( φ ± ∓ ± m ± ˜ mna d ie ) Θ ± H i mn + ± m ) − Ξ ± iY ∓ i ± m (4) – 34 –  t ) a, . . . γ ± mn Ξ ± ) ∆ ± as defined in ( ± i m m m Σ m Π +3 m γ γ γ is ∓ m ± ± − mn − and ˜ ) to obtain the four equations that are the analogue is   abc R ). V ± m (Λ npq

ˇ γ ab ab ± , ± mn ± C.8 ˇ m + γ ˇ  γ 2.2 − H ± ± = m Υ U 2(Λ L − i ( m abc U ± ± ε mn mab ), ( mab − ± F H ˜ a, . . . , d m mnpq F K H i φ + ǫ iX γ m φ 2.1 ε + 2( ∓ ie  ± + a ie H + 3 i ˇ ± + γ

m 0 m

(4)  γ ∓ γ + 3 (4) A a = npq = ab γ m F ± m ˇ F γ ± m F φ npq ˜ ∆ abc ± φ mn A ie ˜ H T H ie mab

+ mnpq − + ). H ǫ = a 3 (4) φ φ ˇ γ mnpq   a , as defined in ( ± m m ie ǫ a φγ a ˇ ˜ γ ∂   C.11 A 1 1 F a 12 12 φ D φ ∂ − a a 1 1 + + 12 12   ie ∂ Γ Construct the analogues of ( Compute the analogue of ( Recombine these four into two doublet equations. Read off the Read off definitions of = = The way to construct the dilatino equation and the trace of th and = = = = • • • • + − b c d a L R The results of each of these intermediate steps are as follow From this, we see that entirely analogous to the procedure above: This is the final result for the unmodified external gravitino The terms appearing in the conformal transformation are giv and with fields given by JHEP10(2017)159 + ε (C.25) (C.24) (C.26) (C.20) (C.22) (C.23) (C.21)  a ˇ γ  + . (4) ε ! γ + − − + P ) η η ξ ξ  mn ± m m m a m m γ Ξ ˇ γ γ γ γ γ i ) ) m ) ) bcd ∓ γ ˇ γ m m m m  m is is is is appear in the external . e above equations then 8 8 (4) m γ mnbcd ! + 8 − + 8 − . ternal gravitino equations. ! is F − + 4 φ φ φ φ npq ± + 2(Λ 1 ± 4! mn η η ± m m ± m m + − F t V = 0 = 0 m m m + ∂ ∂ ξ ξ ∂ ∂ 2 b γ γ n by H + − m m − − − − ˇ γ i −

∓ + 2 + 4Π γ γ λ λ
mnpq 0 ξ ǫ m m ±

m m ± ± + ( + ( + ( + ( 3 1 . ± γ γ mn m m H H abc 1234 D ± m − + i i + − iG b − i ∓ ˇ − m + m γ H H η ξ η ξ F ξ iX ∆ ± i i bc − + ∓ A A



a ˇ γ abc + ± ) ˇ γ + − − m + m ± + + ± ± m mn mn mn mn φ (4) t Ξ Ξ iH + m − m mbc a ± γ γ γ γ Ξ i i γ + − mn 4Π ) ) ) ) i b 1 F Ξ Ξ 2 4! λ λ i∂ V i i F  + 2 2 – 35 – − 1 2 2 ∓ − + − − + − + 2 mn mn mn mn φ mn mn ±  ± − m − + 2 ie γ γ m m = = 0 φ ) yields iX iX iX iX µ µ 4 1 D iG m m i m ie ± ± − + ± 2Λ 2Λ mn − + + − mn mn − 2(Λ 8 1 ± ξ ξ ∓ is C.1 ± ± 4 − − − iX (4) − + − − +

mn mn mn mn γ − T − T G D ∓ + 2Λ + 2Λ ± − m + m V V V V m mn m + − 2 2 + m − m ∆ ∆ γ γ

H λ λ = 2 b − − i ∆ ∆ ˇ − + γ + (2 + (2 = ± ± amn K K + ( + ( + − K + 2 + 2 ω abm ± ± m mn 1 4 − + + 2 + 2 + − H iG iG T ∆ η η + 2 − + − iG iG

ξ ξ

2  (6) a − + 2 − +

1 4 ∇ + 2 − + − + − D D t t i i + 2 +  − + D D 4 4 2 t t i i φ = − 4 − 4 2 + 2 − m + ε ∂ − + 2 − + } = = = = 0 + − = Γ 8Π 8Π , ± m a − 8Π − 8Π D A { 1 2 Lastly, let us nowThe give ten-dimensional the internal gravitino expressions equations for are the give trace of the in and where we have introducedgravitino: the following fields that did not yet The explicit expressions for the fields can be read off to find and Combining the first andleads to third, equations and the second and fourth, of th Inserting these in the analogue of ( JHEP10(2017)159 . ), m − γ ε ab C.13 P ˇ (C.30) (C.27) (C.29) (C.28) (C.31) (C.32) γ − ε  a  cdef ˇ γ a ) with fields m ˇ γ (4) F . m γ ), we add the . 4.8 ± γ ∓ ξ 
ξ b  ! ˇ γ abcdef ! mn C.29 ǫ (4) ± γ φ γ , mn a ± Ξ mn ie ˇ γ γ i m fg V ∓ ˇ γ 1 is ∓ ξ 4!8 ravitino equation ( = 0 = 0 3 mna mnb + 2 m ! − + F − ). We will not require the bcde H ± 0 ) λ λ m m 1 1 + ± ± ± mn m γ + 3 mn F σ σ b 4.12 Θ U + 3Λ γ i ± ± ), denoted with a subscript Ψ to ˇ γ abc iX a 6 6 − a ˇ bcd γ 2 1 iG ± m ˇ γ ∓ ˇ γ m ˇ 3

γ ∇ ∇

± ∆ bcdefg γ abc ± ± C.19 ǫ − + bcd m mna H abm ± ± F F F ω 1 + − ). The result is given in ( mn ), which is equivalent to making use of Ξ φ φ 4!2  i 1 1 2 8 4 (Π λ λ V m m ie ie 2 φ γ 4.7 m m ) + ± − − 1 4 4 3 1 2 is γ γ ie ± − 3 , m m = = = = + m − m 0 1 Θ 3! m F Ψ Ψ ˇ ˇ i Ψ Ψ ∓ A A ˜ ± mn ˜ ˜ d c b a γ ˜ 3Λ  – 36 – + 2 1 ∓ iG ± m + + φ − 3 iX − ± U 1 2 − + ie mn , ± ± m λ λ a 1 8 γ ± ˇ γ ∆ m 4 (Π + mn mn

γ

− amn γ γ 1234
+ a (4) H

ab + − γ mn mn F ± + 2 ˇ γ ˇ ˇ φ T T + ζ m = 1 ). ± ) and the trace of the internal gravitino ( ie γ bc + + ξ σ b ± mab 1 2 b ˇ a γ ξ ˇ γ − + H ˇ γ ˇ γ + ± 4.11 λ λ a C.22 abc ˇ ˇ γ + − abm K + 2 H mn ), ( amn ω ˇ ˇ K K γ  ω 2 1 m abm a npq 4 1 γ 8 1 ˇ γ ω − 4.14 F 1 2 ab ” is symbolic for the terms discussed in ( ∓  + ˇ γ ± amn a 6 ), ( = = = ˇ ω γ mnpq

∇ mab a ǫ

1 4 ± ± − φ F F ξ ζ ε 4.10 ] φ φ + ie ± m 0 ± 6 ie ie mn ˇ Γ A ˇ 1 8 8 2 1 1 T ∇ ,

a Having computed these (anti-)commutators of the external g = = = = D [ Ψ Ψ Ψ Ψ b c 2 1 d a and with the fields defined as follows: Taking the gamma-trace, we obtain the analogue of ( The resulting doublet equations can be expressed as: and the dilatino equation ( three of them with relative weights (1 given in ( traceless part of the internal gravitino equation. make the distinction clear: and The term “ the variation of the modified external gravitino ( JHEP10(2017)159 . 7- . D  

− + − + (C.34) (C.35) (C.33) (C.36) ζ ζ η η ravitino − + − + ξ ξ ξ ξ ] ] = 8 fields ] ] , 89 89 89 89 ˇ ˇ γ γ ˇ ˇ γ γ N ± , , , , both l.h.s. and ] ] ξ ] ] 67 67 67 67 068 ˇ ˇ γ γ ˇ ˇ γ γ ˇ γ , , , , +  m m ζ m m − ) b b ˜ ˜ n d d ζ fied external gravitino. γ n − + 078 − + γ − ore, construct sets of four mn m m 578 m m iX a a T + c c mn = 1 field theory coupled to iX T + ively, or one can just straight − + [[˜ + [[˜ + [[ + [[ + N ± − − . ξ 069 ζ + − + − ) + ) ζ ζ η η 569 ζ  iX 67 + − 89 + − ˇ γ m ξ ξ  ) for ξ ξ + − i iX + X γ } } } } η η m n γ ± + 89 89 + 89 89 m m γ n ˇ ˇ 579 γ γ 4.43 ˇ ˇ γ γ γ γ , , γ , , 67  + − X 079 )(1 } } } } n  X X 89 67 67 − M M 67 67 n H ˇ ˇ ˇ γ γ γ i ˇ ˇ γ γ + 4 of the supersymmetry. In order to do so, we ) we deduce that i − , , H , , − − / i 568 – 37 – + m m 05 m m ± ). We thus see precisely how the + − ˜ ˜ C.9 d d b b X 2 n − 068 η η ( iX (1 A 2 n

X + − + − ( 4 1 4.38 1 2 ∓ i  A m m ). The latter is the more efficient way. m m m m γ γ 1 2  2! 3! ˜ ˜ c c a a ≡ n n = 2 fields with Poincar´esymmetry broken by the 4.8  ± − γ γ + {{ {{ ± {{ {{ n n   π + = 0 = = N 1 m iV iV 4 4 1 1 1 1 4 4 1 m ± ± ± iA ξ ξ ξ − − − − + − ] ] } iA − 0 m 0 m ) for unmodified external gravitino, dilatino and internal g = = 89 89 = =   89 ˇ ˇ γ γ ˇ γ , ,     iA iA − − , ] ] ) ) ) ) } C.11 + − 67 67 m m m m = = | | | | 67 ˇ ˇ γ γ + 2 + 2 − − 2 2 ˇ m m , γ , , a iL iL ∇ ∇ iR iR ˇ γ abc ab a ˇ γ + − ˇ γ + − ) to construct projection operators with which we will act on ). In particular, we define the projection operators X ab m m m m abc [[ | | | + 1 + 1 − − 1 1 X 4 1 = 1 X L C.8 4.38 L ( R R [[ ( − {{ ( ( 1 2 1 4 1 2 N 1 4 1 2 2 1  −  −   Nevertheless, we do wish to show the equations for the unmodi + = 2 supergravity, we need to break to 1 − − + π π π π and The (anti-)commutators can be worked out by making use of trace and then take theaway project proper the linear result combination. given Alternat in ( They are given by r.h.s. of ( Using these projection operators, from ( should be projected down to In order to derive the Killing spinor equations ( C.2 which are obtained making use of ( branes. Using this, oneequations can just go as in through ( the same motions as bef N will use ( JHEP10(2017)159 ]. 11 ), it = 0), (D.4) (D.3) (D.1) (D.2) (D.5) orcing (C.37) (C.38) D.3 ± ζ . + η = 0 (i.e., − n η γ − n ε supersymmetry of the γ ut worldvolume flux as − mn ). T + mn kappa-symmetry operator = 1 is problematic in the . th non-trivial worldvolume engthy equations is to draw T + ± ± 4.43 etting ± N + . − χ iχ iχ η 1 + , and our assumption ( pergravity as was found in [ . ± η ε  ± ∓ ∓ − m  α M − α = = = = ζ γ η m ε . ± ± ± − n − + . γ ε ⊗ ζ ζ χ χ χ γ ⊗ n = 3 0 3 0 , that is to say, we assume = . m m (6) − α 3 0 γ 05 67 89  Γ Γ ε − α γ γ ˇ γ ˇ ˇ ˇ γ γ γ ± χ 3 n ξ ±  ), and where we have relabeled + − ε n H (4) + + 0567 0589 i , we denote the kappa-symmetry operators γ M M H = = Γ + α i + iγ iγ α 4.47 + ζ ε + + η ± ≡ 4.2 − − ˜ 7 0 – 38 – 2 n − ε ⊗ + − ), ( ,M 7 0 ⊗ ε A 2 n = = ζ ζ Γ + α 2 1 + α A 3 m

= Γ 7 0 ˜ 7 0 ± ± ξ χ ± (10) 1 2 Γ Γ − A 4.46 m m ε ξ iξ iξ 7 0 γ γ 2 1  = = − ± ± ± n n Γ ), (  1 − 1 + γ γ + = = = = ε ε n n + 1 m m ± ± ± iV iV 4.45 V ξ ξ ξ 1 m iA − + 05 67 89 iA − ˇ ˇ ˇ γ γ γ 0 m m 0   iA iA . Then supersymmetry of all three branes is equivalent to enf − − as follows: ˜ 7 0 = 4 Killing spinor equations down to + − = = 1 ± ,Γ ε N m m 7 0 ∇ ∇ ,Γ 3 Using similar notation as in section one recovers precisely the gravitino variations of 16/16 su Then making use of the fact that Γ Let us decompose The reason why we feel theattention need to to pester the the reader left-hand with sides more l of the first two equations: s Taking into account ourfor ansatze the for Lorentzian the D7-branes worldvolume is flux, given the by In order toD7-branes make is computations ‘aligned’ doable, with the we projection will along assume Γ that the follows that respectively Γ In this appendix, weflux will demonstrate to why break using the D7-branes wi of the D3-brane with worldvolume flux and the D7-branes witho Thus, one now sees how this is embedded inD our final result ( Breaking supersymmetry using D7-branes with flux with the fields defined as in ( presence of non-trivial worldvolume flux. JHEP10(2017)159 , ± , 3.2 χ = 4 (D.6) ]. ]. 6= N , 11 ± ommons ξ , ]. e , SPIRE (2013) 072 IN (2016) 162 01 ][ , 11 SPIRE (2015) 025 IN flux. For 11 ][ (2014) 577 JHEP , JHEP Riemannian space Supersymmetry in , d , 327 4 The geometry of the spinors must satisfy JHEP quired the internal compo- , redited. anton equations in section ]. ffaroni, arXiv:1309.5876 odski, [ SPIRE arXiv:1207.2785 [ and so we will set them to one without IN ]. ]. ]. ]. ]. − ][ , contradicting our ansatz for the + η ± Rigid supersymmetric backgrounds of ξ Exploring curved superspace , ± χ (2014) 124 ξ ± 068 68 ζ SPIRE SPIRE SPIRE SPIRE SPIRE 6= γ γ 01 IN IN IN IN IN Supersymmetry on curved spaces and holography Commun. Math. Phys. ± (2012) 034 = ˇ = ˇ – 39 – , ][ ][ ][ ][ ][ χ A landscape of field theories ± − 10 ξ ξ JHEP Exploring curved superspace (II) , Rigid supersymmetric theories in JHEP arXiv:1608.02952 ]. ]. ), which permits any use, distribution and reproduction in [ , Rigid supersymmetric theories in curved superspace ]. ]. = 0. This thus leads to the case already analyzed in [ SPIRE SPIRE − SPIRE SPIRE Rigidly supersymmetric gauge theories on curved superspac ε arXiv:1109.5421 arXiv:1205.1062 arXiv:1205.1115 arXiv:1105.0689 arXiv:1203.3420 IN IN [ [ [ [ [ IN IN ][ ][ ][ ][ ⇒ CC-BY 4.0 Localization techniques in quantum field theories (2017) 440301 This article is distributed under the terms of the Creative C = 0 = Supersymmetric branes on curved spaces and fluxes (2012) 139 (2012) 061 (2012) 141 (2011) 114 (2012) 132 F A 50 to be equivalent in order to admit a non-trivial worldvolume 04 08 08 06 05 ± ε arXiv:1509.02926 arXiv:1512.03999 arXiv:1209.5408 arXiv:1207.2181 supersymmetric partition functions Lorentzian curved spaces and holography [ [ minimal off-shell supergravity JHEP JHEP JHEP [ JHEP [ J. Phys. JHEP [1] V. Pestun et al., [8] J.T. Liu, L.A. Pando Zayas and D. Reichmann, [9] B. Jia and E. Sharpe, [6] T.T. Dumitrescu, G. Festuccia and N. Seiberg, [7] T.T. Dumitrescu and G. Festuccia, [4] D. 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