5D SYM on 3D deformed spheres

Transcription

1 Available online at ScienceDirect Nuclear Physics B D SYM on 3D deformed spheres Teruhiko Kawano a Nariaki Matsumiya b a Department of Physics University of Tokyo Hongo Tokyo 3-33 Japan b Sumitomo Heavy Industries Ltd. 9 Natsushima-cho Yokosuka-shi Kanagawa Japan Received 8 June 5; accepted 6 July 5 Available online 6 July 5 Editor: Stephan Stieberger Abstract We reconsider the relation of superconformal indices of superconformal field theories of class S with five-dimensional N = supersymmetric Yang Mills theory compactified on the product space of a round three-sphere and a Riemann surface. We formulate the five-dimensional theory in supersymmetric backgrounds preserving N = and N = supersymmetries and discuss a subtle point in the previous paper concerned with the partial twisting on the Riemann surface. We further compute the partition function by localization of the five-dimensional theory on a squashed three-sphere in N = and N = supersymmetric backgrounds and on an ellipsoid three-sphere in an N = supersymmetric background. 5 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license Funded by SCOAP 3.. Introduction In the previous papers we have attempted to give a physical proof for the conjecture of 3. The conjecture states that the Schur limit of the superconformal index 4 of a fourdimensional N = superconformal theory of class S 56 can be computed by two-dimensional q-deformed Yang Mills theory 7. The N = superconformal theory of class S is defined in 65 as the infrared limit of M5-branes wrapped on a Riemann surface and according to the conjecture one may compute the Schur index of the theory by making use of the q-deformed Yang Mill theory on in the zero area limit. * Corresponding author. address: kawano@hep-th.phys.s.u-tokyo.ac.jp T. Kawano. The Riemann surface is commonly denoted by C in the recent literature and is often referred to as a UV curve / 5 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license Funded by SCOAP 3.

2 T. Kawano N. Matsumiya / Nuclear Physics B The superconformal index may be captured by the partition function of the four-dimensional theory compactified on S S 3. See 8 for the extension to with nonzero area. Since the infrared limit of M5-branes gives rise to the putative six-dimensional N = superconformal theory it is conceivable to obtain the index by computing the partition function of the N = theory compactified on S S 3 with a partial twisting on. The idea has been argued in 9 as a top-down approach to uncover the relation of a generic superconformal index of the theories of class S with a topological field theory. For a review of the superconformal indices of theories of class S see 5. In the previous papers we put the idea into practice by exchanging the order of the compactifications; regarding M5-branes wrapped on a circle as D4-branes we compactified five-dimensional N = supersymmetric Yang Mills theory on with a partial twisting and further on the round S 3 in a somewhat ad hoc way. Computing the partition function of the compactified theory by localization we have found that the fixed points give the fields of the q-deformed Yang Mills theory and that the one-loop contributions and the classical action at the fixed points yield the measure of the partition integral of it; namely the partition function of the five-dimensional compactified theory is reduced to that of the two-dimensional q-deformed Yang Mill theory. However there was a confusion about the partial twisting in. The supersymmetric background used in especially the partial twisting preserved only N = supersymmetry in four dimensions. Since the conjecture in 3 is concerned with the four-dimensional N = superconformal theories the results in seem to have nothing to do with the conjecture. The construction of the N = superconformal theories by the twisted compactification of the N = theory on has been generalized for N = supersymmetric theories in four dimensions 6 9 which we will refer to as N = theories of class S. We can see that the twisting used in is identical to what is called 3 the N = twist in 6. Superconformal indices of the N = superconformal theories of class S have been calculated in four dimensions in. A simple comparison shows that the result in is in good agreement with the Schur limit of the mixed Schur index in as we will see later. The questions we raise are two-fold; first when the N = theory compactified on with the N = twisting so that N = supersymmetry remains unbroken in four dimensions whether will we obtain the q-deformed Yang Mills theory on via localization? This was the original motivation in the previous paper. Second when replacing the round S 3 by a deformation of the S 3 such as a squashed S 3 and an ellipsoid S 3 as discussed in whether will we obtain a deformation of the Schur index for the round S 3 like the mixed Schur index in? We will make an attempt to answer both of the questions in this paper which is organized as follows: in Sections and 3 we will begin with the construction of the five-dimensional supersymmetric Yang Mills theory on a curved space based on the idea of 3 that the fields of an off-shell supergravity multiplet are utilized as background fields to preserve supersymmetries of the field theory on a curved space. In fact through the dimensional reduction of the six-dimensional N = conformal supergravity in 5 on-shell supersymmetry transforma- Similar constructions have been used in 6 to explore four-dimensional N = superconformal field theories of class S. See also for N = supersymmetric theories. See 3 also 4 for related works on three-dimensional Chern Simons theory from M5-branes. 3 See Appendix D of 6 about the embedding of the spin connection on to the R-symmetry group. The N = twist corresponds to the case of l = l in their Calabi Yau construction of 9.

3 458 T. Kawano N. Matsumiya / Nuclear Physics B tions and an on-shell action of the five-dimensional theory compactified on a curved space have been derived in 4 following the idea 3. Therefore Sections 3 and 4 are essentially devoted to a review of 4 up to a few points that we perform the dimensional reduction in the time direction of the six-dimensional theory instead of the spatial direction as in 4. And we obtain off-shell supersymmetry transformations and an off-shell action of the five-dimensional theory on a curved space in Section 6 which are necessary to carry out localization. In Section 5 we will discuss the partial twistings mentioned above the N = twisting and the N = twisting in more details in the language of the background gauge field of the R-symmetry group and we will describe the supersymmetric background on a round S 3 in in terms of supergravity background fields for the N = twisting in Subsection 5. and give a supersymmetric background on the round S 3 for the N = twisting in Subsection 5.. We will proceed to consider two supersymmetric backgrounds on a squashed S 3 the analog of the background in and of the one in in Subsections 5.3 and 5.4 respectively. Especially for the former we will give supersymmetry backgrounds for both of the twistings. In Subsection 5.5 we will discuss a supersymmetric background for the N = twisting on an ellipsoid S 3 in an analogous way to. After the discussions about the off-shell formulation of the five-dimensional theory in Section 6 as mentioned above we will explain our localization method in depth in Section 7. We will compute the partition functions by localization on the round and squashed S 3 s in Section 8 for the background in Section 5.3 and that on the ellipsoid S 3 in Section 9 for the background in Section 5.5. However the computation of the partition function on the squashed S 3 for the background in Section 5.4 somewhat doesn t seem straightforward to be done by localization and we will leave it as an open question. Finally Section is devoted to the summary and discussions of this paper. Appendix A is a simple collection of our conventions about the anti-symmetrization of various indices and about differential forms used in this paper and the gamma matrices of the Lorentz groups in five and six dimensions are shown in our representation in Appendix B. The R-symmetry group of the six- and five-dimensional theories are commonly Spin5 R Sp R and the associating gamma matrices in our representation are given in Appendix C. The spinors in the theories are symplectic Majorana Weyl spinors and in Appendix D our convections about those spinors are explained. After the dimensional reduction of the conformal supergravity supersymmetry transforms of the fermionic fields in the supergravity multiplet the Weyl multiplet yield supersymmetry conditions on the background fields to preserve supersymmetries on the curved background. Besides the supersymmetry condition derived from the gravitino field there is another supersymmetry condition from the fermionic auxiliary field in the Weyl multiplet and it is too long to write down explicitly in the text. Therefore the explicit form of the supersymmetry condition is written in Appendix E. In Appendix F a few formulas which we think are useful to verify the invariance of the actions in Sections 3 and 4 under the supersymmetry transformations are given. In Appendix G Killing spinors and metrics are discussed on the round squashed and ellipsoid S 3 following. Appendix H explains the difference among the notations used in 5 in 4 and in this paper and further the difference between the notations used here and in the previous paper.

4 T. Kawano N. Matsumiya / Nuclear Physics B Euclidean 5D N = SYM in SUGRA backgrounds In this section the dimensional reduction along the time direction will be performed for the six-dimensional N = conformal supergravity derived in 5. This section Sections 3 and 4 are essentially a review of 4 but the spatial dimensional reduction was carried out there. In Subsection. we will recapitulate the main results of 5 which we will need in this paper about the supergravity multiplet called the Weyl multiplet in the conformal tensor calculus. In Subsection. we will discuss the dimensional reduction of the Weyl multiplet which play roles of supersymmetric background fields to retain supersymmetries of the five-dimensional Yang Mills theory on a curved space. Subsection.3 is just a small digression about the relation of Killing spinors with Killing vectors... Weyl multiplet in 6D N = conformal supergravity In this paper following 4 we will carry out dimensional reduction of the six-dimensional N = supergravity in 5 to obtain a five-dimensional Euclidean maximally supersymmetric Yang Mills theory in supergravity backgrounds. It has been discussed in 3 that the supergravity backgrounds provide a systematic method for supersymmetric compactifications of supersymmetric field theories. The construction of the supergravity in 5 is based on the conformal tensor calculus. See the textbook 6 for the conformal tensor calculus and references therein. In this approach one starts with a gauge field theory by gauging the six-dimensional N = superconformal symmetry group OSp 6 4 whose bosonic part consists of the conformal group SO 6 and the R-symmetry group Spin5. The symmetry group OSp 6 4 is generated by P a : translation D: translation M ab : Lorentz K a : special conformal R IJ : R-symmetry Q α : supersymmetry S α : conformal supersymmetry whose corresponding gauge fields are shown in Table. Let us list the notations of the various indices on the generators and the gauge fields: a b = 5; the Lorentz indices μ ν = 5; the coordinate frame indices I J = 5; the vector indices of the Spin5 R symmetry α β = 4; the spinor indices of the Spin5 R symmetry. The fermionic fields ψ μ α and φ μ α are the gauge fields of the supersymmetry and the conformal supersymmetry respectively. They are symplectic Majorana Weyl spinors of positive and negative chirality respectively. See Appendix D for our conventions about symplectic Majorana Weyl spinors. A straightforward manner of gauging translations doesn t lead to general coordinate transformations which is indispensable to a theory of gravity. To gain general coordinate transformations from translations in the conformal tensor calculus approach auxiliary fields 4 in Table are intro- 4 They are referred to as matter fields in 54. It is not always necessary to introduce auxiliary fields for the deformation and it depends on the numbers of supersymmetries and the dimensions of spacetime i.e. superconformal algebras. See 6 for more details.

5 46 T. Kawano N. Matsumiya / Nuclear Physics B Table The gauge fields of the 6D N = superconformal symmetry. Gauge fields Transformations Restrictions Spin5 R Weight Boson E a μ P a : translations sechsbein b μ D: dilatation V IJ μ R IJ : R-symmetry V IJ μ = V JI μ Fermion ψ α μ Q α : supersymmetry gravitini 4 / Ɣ 7 ψ α μ = ψ α μ Dependent gauge fields Boson ab μ M ab : local Lorentz spin connection f a μ K a : special conformal + Fermion φ α μ S α : conformal supersymmetry Ɣ 7 φ α μ = φ α μ 4 +/ Table The auxiliary fields for the deformation of the superconformal symmetry. Auxiliary fields Symmetries Spin5 R Weight Bosonic fields T αβ abc T αβ abc = 3! ε abc def T αβ def 5 T αβ abc = T βα abc αβ T αβ abc =. M αβ γδ M αβγ δ = M γδαβ = M βαγδ = M αβδγ 4 αβ M αβγ δ = γδ M αβγ δ = αγ βδ M αβγ δ =. Fermionic field χ αβ γ Ɣ 7 χ αβ γ = χ αβ γ χ αβ γ = χ βα γ αβ χ αβ γ = χ γβ γ = 6 3/ χ αβ γ Ɣ = χ α β γ T C α α β β γ γ. duced and the transformation laws of the gauge fields are deformed by imposing some constraints on the gauge field strengths and the auxiliary fields such that the resulting transformation laws give a closed algebra as explained in 6. Furthermore one requires the invertibility of the gauge field E a μ of translations to solve the constraints which allows us to regard it as the sechsbein. Solving the constraints makes the gauge fields μ ab f a μ and φ α μ dependent fields given in terms of the other gauge fields and the auxiliary fields. In fact they are given by μ ab = ω μ ab + E μ a b b E μ b b a + φ α μ = f a μ = 8 R μ a 8 E μ a R c c + 3 T αβ μcd T αβ acd + where the ellipses denote the contributions from the fermionic fields. One can see that the spin connection μ ab is a generalization of the Levi-Civita spin connection ω μ ab satisfying de a + ω a b E b = E a = E μ a dx μ

6 T. Kawano N. Matsumiya / Nuclear Physics B and R μ a is the Ricci tensor R μ a = ν b R νμ ba of the curvature tensor of the spin connection a b R a b = R cd a b E c E d = d a b + a c c b where ν b denotes the inverse of the sechsbein E a μ i.e. coframe. After the deformation one finds a closed algebra with the covariant general coordinate transformations. The remaining independent gauge fields and auxiliary fields form a multiplet called the Weyl multiplet including the graviton the gravitini and the others. We show the resulting bosonic transformations of the independent gauge fields except for the covariant general coordinate transformations δe μ a = D E μ a + a b E μ b δψ α μ = D ψ α μ + 4 R IJ ρ IJ α β ψ β μ + 4 ab Ɣ ab ψ α μ δv μ IJ = R IJ + R I K V μ KJ + R J L V μ IL δb μ = μ D Ka E μ a where D ab Ka and R IJ are the parameters of dilatation the Lorentz special conformal and R-symmetry transformations respectively and under the first four transformations the auxiliary fields transform as δt αβ μνρ = D T αβ μνρ δm αβ γδ = D M αβ γδ δχ αβ γ = 3 D χ αβ γ + 4 ab Ɣ ab χ αβ γ. Under the R-symmetry transformations they transform in the representations shown in Table respectively. The resulting supersymmetry Q- transformations and superconformal S- transformations on the gauge fields and the auxiliary fields are given by δe a μ = i αβ ɛ α T CƔ a ψ β μ δψ α μ = D μ ɛ α + 4! T αβ abc Ɣ abc Ɣ μ ɛ β + Ɣ μ η α δb μ = αβ ɛ α T Cφ β μ αβ η α T Cψ β μ δv IJ μ = ρ IJ ɛ α T αβ C φ β μ + η α T C ψ β μ ρ IJ α β γδ e γ T Ɣμ χ δβ α 5 δt αβ abc = ɛ α T CƔ de Ɣ 6 abc R β deq ɛ γ T CƔabc χ αβ γ trace 5 δm αβ γδ = ɛ α T CƔ μ D μ χ β γδ + η α T Cχ β γδ trace δχ αβ γ = 5 3 Ɣabc Ɣ μ ɛ γ D μ T αβ abc Ɣμν ɛ α R μν γ β 4 Dαβ γδɛ δ Ɣabc η γ T αβ abc trace 3

7 46 T. Kawano N. Matsumiya / Nuclear Physics B with trace denoting necessary terms to give the same irreducible representations of the R-symmetry group as the fields on the left hand sides. The parameter ɛ α of a supersymmetry transformation and η α of a superconformal transformation are symplectic Majorana Weyl spinors of positive and negative chirality respectively; Ɣ 7 ɛ α = ɛ α Ɣ 7 η α = η α. The operation T denotes transpose and so ɛ α T and η α T are the transposes of ɛ α and η α respectively. The curvature R α abq is the field strength of the supersymmetry gauge field gravitini ψ α μ whose exact form can be seen in 5 but it will not be necessary in this paper. Here the covariant derivatives of ɛ α and T αβ abc are given by D μ ɛ α = μ ɛ α + b μ ɛα + 4 μ abɣab ɛ α 4 V μ IJ ρij αβ ɛ β D μ T αβ abc = μ T αβ abc + μ a d T αβ bcd b μ T αβ abc + 4 V μ IJ ρ IJ αγ T βγ abc. Here the field strength of the R-symmetry gauge field V μ IJ is given by R μν α β = R μν IJ ρ IJ αβ = μ V ν IJ ν V μ IJ V μ I K V ν KJ + V ν I K V μ KJ ρij αβ... Temporally dimensional reduction of the Weyl multiplet In this subsection the dimensional reduction of the Weyl multiplet along the time direction will be considered in the same way as the dimensional reduction along one spatial direction was performed in 4 where the strategy in 7 was followed. For the usual ansatz for the metric where ds6 = 5 α dt + C + ds5 = E E + ds5 = E a E a E = dt + C α ds 5 5 = E a E a a= as a gauge-fixing condition the six-dimensional coframe E a μ μ = t 5; a = 5 can be taken by a local Lorentz transformation to be E a e μ = t e μ α α e a t e a = C μ μ e a μ where μ = 5; a = 5. Here α is a scalar field a.k.a. dilaton which is sometimes denoted by exp ϕ but we will follow 74 to denote it by α. Therefore one can see that E = α dt + C Ea = e a = e a μdx μ. a=

8 T. Kawano N. Matsumiya / Nuclear Physics B For the gauge field C = C μ dx μ we define the field strength G = dc = G μνdx μ dx ν. Since the six-dimensional coframe μ a is inverse to the sechsbein E a μ it takes the form μ θ t a = θ t a α Ca θ μ θ μ = a θ μ a under the gauge-fixing condition with C a = θ ν ac ν where the funfbein e a μ and the fivedimensional coframe θ μ b satisfy e a μ θ μ b = δ a b θ μ ae a ν = δ μ ν. One then finds the Levi-Civita spin connection ω ab = ω c ab e c satisfying de a + ω a b e b = ω a = α θ μ a μ α ω ab = ω a b = α G ab ωa bc = ωa bc with the five-dimensional Levi-Civita spin connection ω ab = ω c ab e c satisfying de a + ω a b e b = and G ab = θ μ aθ ν bg μν. As in 74 we will continue the partial gauge-fixing by using the conformal supersymmetry transformation S α to set ψ α = ; the special conformal transformations K to set b = and K a to b μ = α μα μ = 5. The latter condition makes the dilaton field α covariant constant 7; D μ α = μ α b μ α = which will be convenient for the calculations below. The partial gauge fixing conditions are summarized as E a t = ψ α = b = b μ = α μ α μ= 5. 4 We will use b μ as shorthand for α μ α and b a = θ a μ b μ. Therefore under the gauge fixing condition one has the dependent gauge field μ ab in a t = t ab = α G a ab μ = α G μa μ ab = ω μ ab + α C μg ab + e aμ θ ν b e bμ θ ν a α να. Among them after the dimensional reduction the component c μ ab = ω c ab + δ ac b b δ bc b a θ μ c μ ab ab = μ c often appears in the covariant derivatives and we refer to it as c ab. The auxiliary fields V a IJ T αβ abc are decomposed into five-dimensional fields S IJ V a IJ t I ab by

9 464 T. Kawano N. Matsumiya / Nuclear Physics B V a IJ = { V IJ S IJ V a IJ A a IJ T αβ T αβ ab t I ab ρ abc = I αβ T αβ abc = /ε de abc T αβ de with ε 345 = ε 345 =. Note that the gauge field A μ IJ is given by A μ IJ = e a μa a IJ = e a μ ν av ν IJ = V μ IJ C μ V t IJ = V μ IJ α C μ S IJ. Let us remove the underline from M αβ γδ to denote its reduced one as M αβ γδ. It is sometimes convenient to replace the spinor indices α β of M αβ γδ by the vector indices I J as M αβ γδ = M IJ ρ I αβ ρ J γδ. The field M IJ is in the representation 4 of the Spin5 R group and enjoys the symmetry properties M IJ = M JI δ IJ M IJ =. The time component of the gravitini is set to zero by the gauge fixing condition 4; ψ α t = and we will denote the remaining components ψ α μ μ = 5 simply as ψ α ψ α μ = μ since it is of positive chirality and our convention of the chirality is found in Appendix D. Since the auxiliary spinor χ αβ γ is also of positive chirality we will take χ αβ γ = 5 6 χ αβ γ with the convenient coefficient 5/6 in 4. The parameters ɛ α and η α of supersymmetry and conformal supersymmetry transformations are of positive and negative chirality respectively and we will take ɛ α ɛ α = η α = η α. The gauge fixing condition 4 is changed under the supersymmetry Q- transformation 3. In particular the zeroth component of the gravitino transforms under the supersymmetry Q and the conformal supersymmetry S as δψ α = 8α G abɣ ab ɛ α 4 S IJ ρ IJ α β ɛ β + 4 tαβ ab Ɣ ab ɛ β + Ɣ η α 5 under the gauge fixing condition 4. However combining the supersymmetry Q- and the conformal supersymmetry S- transformations one can find that one linear combination of them leaves the condition ψ α = unchanged. For any ɛ α one can see that the conformal supersymmetry transformation with the parameter η α = 8α G abγ ab ɛ α 4 S IJ ρ IJ α β ɛ β + 4 tαβ ab γ ab ɛ β 6 compensates for the deviation 5 from the gauge fixing condition on the gravitini.

10 T. Kawano N. Matsumiya / Nuclear Physics B Among the other gauge fixing conditions in 4 the condition E a t = remains unchanged under the supersymmetry Q- and the conformal supersymmetry S- transformations. But the remaining gauge fixing conditions b = and b μ = α μ α are changed under those transformations. However the deviations can be canceled by the special conformal K- transformations with appropriate parameters Ka. Note here that E a t and ψ α are left invariant under the special conformal K- transformations. Thus one may define a supersymmetry transformation in the reduced five-dimensional theory as the linear combination of supersymmetry Q- conformal supersymmetry S- and special conformal K- transformations. Following the ideas in 3 we are seeking for supersymmetric backgrounds of the reduced theory to obtain supersymmetric compactifications of the N = supersymmetric Yang Mills theory in five dimensions. Since we would like to consider bosonic backgrounds we will turn off background spinor fields and we will find the supersymmetric bosonic backgrounds leaving the spinor fields ψ α μ χ αβ γ unchanged under some of supersymmetry transformations in the reduced theory. From the supersymmetry transformation of the gravitini δ ɛ ψ α a = θ μ a δψ α μ = D a ɛ α 4 SIJ ρ IJ α βγ a ɛ β + α G abγ b ɛ α + 8α G bcγ a bc ɛ α ti bcρ I α βγ a bc ɛ β with the covariant derivative of the supersymmetry parameter D a ɛ α = θ μ ad μ ɛ α = θ μ a μ ɛ α + b aɛ α + 4 a bc γ bc ɛ α 4 A a IJ ρ IJ α βɛ β one can see that the supersymmetric bosonic backgrounds should obey D a ɛ α = 4 SIJ ρ IJ α βγ a ɛ β α G abγ b ɛ α 8α G bcγ a bc ɛ α + ti bcρ I α βγ a bc ɛ β. Under a supersymmetry transformation the auxiliary spinor χ αβ γ transforms as 7 δ ɛ χ αβ γ = 4! εabcde D f t αβ faγ bcde ɛ γ + 4! εabcde D a S γ α γ bcde ɛ β t αβ abt γδcd γ abcd ɛ δ εabcde D a t αβ bcγ de ɛ γ + α G a c t αβ bcγ ab ɛ γ α Gab S γ α γ ab ɛ β Sα δt βδ abγ ab ɛ γ t αβ abs γδ γ ab ɛ δ + 4t αβ act γδb c γ ab ɛ δ F ab γ α γ ab ɛ β 4 5 Mαβ γδɛ δ α Gab t αβ abɛ γ + t αβ abt γδ ab ɛ δ + 8 with t αβ ab = t I abρ I αβ and S α β = /S IJ ρ IJ α β where the ellipse denotes the necessary terms 5 to leave the right hand side in the representation 6 of the Spin5 R symmetry since χ αβ γ is in the representation 6. Here the two covariant derivatives are given by 5 In 45 they are denoted as trace.

11 466 T. Kawano N. Matsumiya / Nuclear Physics B D μ t αβ ab = μ t αβ ab + μ a c t αβ cb + μ b c t αβ ac b μ t αβ ab A μ α γ t γβ ab A μ β γ t αγ ab D μ S αβ = μ S αβ b μ S αβ A μ α γ S γβ A μ β γ S αγ with A μ α β = /A μ IJ ρ IJ α β whose curvature tensor F μν α β = /F μν IJ ρ IJ α β is defined by F μν I J = μ A ν I J ν A μ I J A μ I K A ν K J + A ν I K A μ K J. Therefore the other condition for the supersymmetric backgrounds is that the right hand side of 8 should vanish. The explicit form 9 of the supersymmetry condition is given in Appendix E because the equation is very lengthy to write it here. Thus 7 gives the Killing spinor equation and supersymmetric backgrounds have to allow the existence of the solutions the Killing spinors to the equation. One may interpret that 9 determines the background field M αβ γδ which will appear in the mass term of the scalar fields in the five-dimensional N = supersymmetric theory as will be seen below..3. The Killing vectors and the Killing spinors The Killing spinors ɛ α η α obeying the equation 7 form the bilinear ξ a = η α T Cγ a ɛ β αβ η γ a ɛ and its covariant derivative D μ ξ a μ ξ a + b μ ξ a + μ a b ξ b = = S IJ + ε μ abcd η ρ IJ γ a μ ɛ Dμ η α T Cγ a ɛ β + η α T Cγ a D μ ɛ β αβ α G μ a η ɛ 4α G bc η γ d ɛ t I bc η ρ I γ d ɛ satisfies D a ξ b + D b ξ a =. See Appendix D for the notations for the bilinears η ρi I n γ a a m ɛ. The vector field ξ a obeys the conformal Killing vector equation a ξ b + b ξ a = 5 η ab c ξ c 9 with the covariant derivative μ ξ a μ ξ a + ω μ a b ξ b which is related to the previous covariant derivative as D a ξ b = a ξ b + η ab bc ξ c + b a ξ b. In fact the equation D a ξ b + D b ξ a = leads to a ξ b + b ξ a = η ab bc ξ c which gives the conformal Killing vector equation 9.

12 T. Kawano N. Matsumiya / Nuclear Physics B Table 3 The tensor multiplet in the six-dimensional supergravity. Tensor multiplet Symmetries Spin5 R Weight Bosonic fields B μν B μν = B νμ φ αβ φ αβ = φ βα αβ φ αβ = 5 Fermionic field χ α Ɣ 7 χ α = χ α 4 5/ 3. Tensor multiplet in the supergravity theory To the conformal supergravity tensor multiplets can be added as matters and after the dimensional reduction they give rise to N = gauge multiplets in five dimensions. It therefore yields a five-dimensional N = supersymmetric Abelian theory in the supergravity background. It is the topic of this section. A tensor multiplet B μν φ αβ χ α of the N = supergravity is listed in Table 3 and the field strength of the two-form B is given by H = 3! H abc E a E b E c = db. The transformation rules and the equations of motion of the tensor multiplet were derived in 5. Under a fermionic transformation supersymmetry+ conformal supersymmetry the tensor multiplet transforms as δb μν = i ɛ α Ɣ Ɣ μν χ α δφ αβ = i ɛ α Ɣ χ β + i ɛ β Ɣ χ α i αβ ɛ γ Ɣ χ γ δχ α = 8 3! H + μνρ Ɣ μνρ ɛ α + 4 D μφ αβ Ɣ μ ɛ β φ αβ η β where H ± = / H ± H. See the definition of the Hodge dual in Appendix A. The covariant derivative of the scalar field φ αβ is D μ φ αβ = μ φ αβ b μ φ αβ 4 V μ IJ ρ IJ α γ φ γβ 4 V μ IJ ρ IJ β γ φ αγ. The equations of motion of the tensor multiplet are given by H φ αβ T αβ = D a D a φ αβ 5 M αβ γδ φ γδ + 3 H + abct αβ abc = Ɣ a D a χ α T αβ abc Ɣ abc χ β = 3 with the covariant derivatives

13 468 T. Kawano N. Matsumiya / Nuclear Physics B D a D a φ αβ = μ a μ 3b μ D a φ αβ + a ab D b φ αβ 4 V a IJ ρ IJ α γ D a φ γβ 4 V a IJ ρ IJ β γ D a φ αγ 5 R φ αβ D μ χ α = μ 5 b μ + ab μ Ɣ ab χ α 4 4 V μ IJ ρ IJ α βχ β. 3.. Dimensional reduction of the tensor multiplet From the six-dimensional Minkowski space to the five-dimensional Euclidean space the dimensional reduction of the tensor multiplet gives rise to the five-dimensional abelian gauge multiplet A μ φ I χ α B ab B a αa a = αθ μ aa μ a = 5 φ αβ αφ αβ = αφ I ρ I αβ χ α α 4 χ α. The remaining components B ab are described by A μ and φ I through the equation of motion of H which is reduced to H ab = φ αβ T αβ ab αφ I t I ab. Since the components H ab reduce to the field strength F μν of A μ H μνt = μ B νt + ν B tμ + t B μν μ A ν ν A μ = F μν H ab αθ μ aθ ν bf μν = αf ab one can see that the components H abc are reduced as H abc = H + abc + H abc = ε abc de H + de H de = ε abc de H de H de α ε abc de F de 4 φ i t i de. We have previously seen that a six-dimensional supersymmetry transformation with a transformation parameter ɛ α combined with the superconformal transformation with η α in 6 is reduced to a five-dimensional supersymmetry transformation. Substituting the parameter η α in 6 into the fermionic transformation rules in of the tensor multiplet one can see that their reduction gives the supersymmetry transformation of the abelian gauge multiplet δ ɛ A μ = i 4 αβ ɛ α T Cγμ χ α δ ɛ φ I = i ρ I αβ ɛ α T Cχ β 4 δ ɛ χ α = F abγ ab ɛ α γ μ D μ φ I ρ I α βɛ β + α G abφ I ρ I α βγ ab ɛ β + S I J φ J ρ I α βɛ β + ε IJKLMS IJ φ M ρ KL α β ɛ β + t I abφ J ρ IJ α βγ ab ɛ β 4

14 T. Kawano N. Matsumiya / Nuclear Physics B with the covariant derivative of φ I D μ φ I = μ φ I b μ φ I A μ I J φ J. The reduction of the external derivative of the equation d H + φ αβ T αβ = yields the equation of motion of the gauge field A μ where with d α F 4 φ I t I + F G = 5 F = F μνdx μ dx ν t I = ti μνdx μ dx ν G= G μνdx μ dx ν. The equations 3 of motion are reduced into γ μ D μ χ α 8α G abγ ab χ α 4 S IJρ IJ α βχ β + ti abρ I α βγ ab χ β = D a D a φ I S I J S J K φ K 5 R φi 4 5 MI J φ J α G abg ab φ I + 4 t I ab t J ab φ J t I ab F ab = 6 D a D a φ I = θ μa μ b μ Da φ I + μa b D b φ I 4 A μ I J D a φ J where the covariant derivative of χ α D μ χ α = μ χ α 3 b μχ α + 4 μ bc γ bc χ α 4 A μ IJ ρ IJ α βχ β with the spin connection μ ab = ω μ ab + e a μθ νb e b μθ νa b ν and the scalar curvature R of μ ab is defined by R = θ μ aθ ν b μ ν ab + μ ae νe b which comes from R = R + 4α G abg ab. From the equations of motion 5 6 one obtains the bosonic part of the action of the abelian gauge multiplet with L B = α + M BIJ = 5 δ IJ and the fermionic part F 4φ I t I F 4φ J t J + C F F dx 5 gα D a φ I D a φ I + M BIJ φ I φ J 7 R + 4α G abg ab M IJ + 4t ab I t Jab S K I S JK

15 47 T. Kawano N. Matsumiya / Nuclear Physics B L F = i 8 dx 5 gα χ α T C γ a D a χ β αβ 8α G abγ ab χ β αβ 4 S IJχ β ρ IJ αβ + ti abγ ab χ β ρ I αβ. 8 One can verify that the total action L = L F + L B is left invariant under the supersymmetry transformation 4. However it is a lengthy calculation to verify the supersymmetry invariance of the action L. Although we do not intend to pause for a detailed demonstration of it we will discuss a supersymmetry transformation of the mass term of the scalar fields φ I in the action in Appendix F which we think is one of the keys to verify the supersymmetry invariance of the action. 4. The generalization for a non-abelian gauge group The reduced theory of the six-dimensional tensor multiplet gives rise to the abelian gauge theory in five dimensions. We will extend the abelian gauge multiplet A μ φ I χ α to the adjoint representation of a non-abelian gauge group G and replace the partial derivatives by covariant ones: μ φ I μ φ I + ig A μ φ I μ χ α μ χ α + ig A μ χ α. We will henceforth denote the covariant derivatives as D μ φ I = μ φ I b μ φ I I A μ J φ J + ig A μ φ I D μ χ α = μ χ α 3 b μχ α + 4 μ bc γ bc χ α 4 A μ IJ ρ IJ α βχ β + ig A μ χ α. For the non-abelian extension of the supersymmetry transformations 4 and the equations of motion 5 6 there are two conditions to be satisfied. In the flat limit where all the backgrounds go to zero they should be reduced to the ones in the N = supersymmetric Yang Mills theory on a flat space and in the abelian limit g the extension has to go back to Our ansatz for the non-abelian extension of the supersymmetry transformations is δ ɛ A μ = i 4 αβ ɛ α T Cγμ χ α δ ɛ φ I = i ρ I αβ ɛ α T Cχ β 4 δ ɛ χ α = F abγ ab ɛ α γ μ D μ φ I ρ I α βɛ β + α G abφ I ρ I α βγ ab ɛ β + S I J φ J ρ I α βɛ β + ε IJKLMS IJ φ M ρ KL α β ɛ β + t I abφ J ρ IJ α βγ ab ɛ β + i g φ I φ J ρ IJ α βɛ β 9 with the field strength of the non-abelian gauge field A μ F ab = θ μ aθ ν bf μν = θ μ aθ ν b μ A ν ν A μ + ig A μ A ν. In the abelian gauge theory the algebra of the supersymmetry transformations 4 is closed on-shell and in the flat limit of the non-abelian gauge theory it is also closed on-shell. Therefore

16 T. Kawano N. Matsumiya / Nuclear Physics B in order to see the closure of the algebra of the supersymmetry transformations 9 we make an ansatz for the equation of motion of the spinor χ α γ μ D μ χ α + ig ρ I α β φ I χ β 8α G abγ ab χ α 4 S IJρ IJ α βχ β + ti abρ I α βγ ab χ β =. The supersymmetry transforms of D μ φ I and F ab φ I t I ab may be useful to see that the algebra of the supersymmetry transformations is closed on-shell; δ ɛ D μ φ I = i ɛ ρ I D μ χ + 4 8α G bc ɛ ρ I γ bc μ χ α G μb ɛ ρ I γ b χ 4 S KL ɛ ρ KL ρ I γ μ χ tj bc ɛ ρ J ρ I γ bc μ χ δ ɛ F ab φ I t I ab = i ɛ γ a D b χ 4 4α G cd ɛ γ cd ab 3δ c aγ d b 4δ c aδ d b χ + S IJ ɛ ρ IJ γ ab χ+ t I cd ɛ ρ I γ cd ab δ c aγ d b δ c aδ d b χ. Using the equation of motion and the Killing spinor equation 7 one can verify that the algebra of the supersymmetry transformations 9 is closed on-shell. δɛ δ η Aμ = i F μν ξ ν + D μ φ I η ρ I ɛ = i ξ ν ν A μ + μ ξ ν A ν D μ G δɛ δ η φ I = i ξ μ μ φ I ξ a b a φ I + ig G φ I I J φ J ξ μ μ χ α 3 ξ a b a χ α + ig G χ α δɛ δ η χ α = i + 4 ab γ ab χ α 4 IJ ρ IJ α β χ β with the Killing vector ξ a = η γ a ɛ where the parameters are given by IJ = i A aij ξ a + S IJ η ɛ ε IJKLM S KL η ρ M ɛ + α G ab η ρ IJ γ ab ɛ t K ab η ρ IJK γ ab ɛ ab = i D a ξ b + ξ c c ab G = i ξ a A a + φ I η ρ I ɛ see Appendix D for the abbreviation η ρ I I n γ a a m ɛ and the covariant derivative of φ I η ρ I ɛ is D μ φ I η ρ I ɛ = μ φ I η ρ I ɛ + ig A μ φ I η ρ I ɛ.

17 47 T. Kawano N. Matsumiya / Nuclear Physics B Since we have seen that the supersymmetry transformation 9 gives an on-shell closed algebra with the equation of motion we will proceed with 9 and to obtain the non-abelian extension of the action 7 8. A simple calculation shows that the equation of motion may be derived from the fermionic part of the non-abelian action S F = i dx 5 gαtr χ γ a D a χ 8 8α G ab χ γ ab χ 4 S IJ χ ρ IJ χ + ti ab χ ρ I γ ab χ + ig χ ρ I φ I χ where the symbol tr denotes a trace in the adjoint representation of the gauge group G. In the abelian limit g the non-abelian action should go to 7 more precisely the abelian action of the G abelian gauge multiplets with G denoting the dimension of the adjoint representation of G and in the flat limit we must regain the familiar non-abelian action in the N = supersymmetric Yang Mills theory. It therefore seems natural to take the ansatz where S B = + M BIJ = 5 δ IJ tr α F 4φ I t I F 4φ J t J + C F F dx 5 gαtr D a φ I D a φ I + M BIJ φ I φ J ig φ I φ J φ I φ J 3 R + 4α G abg ab M IJ + 4t ab I t Jab S K I S JK. 4 In order to examine the supersymmetry invariance of the sum S F + S B one needs to perform a similar calculation to what is done for the abelian action L. The calculation may be painful especially in the mass term of the scalar fields φ i of which the details is shown in Appendix F. However it turns out that the variation of the sum S F + S B under the supersymmetry transformation 9 doesn t vanish at the order Og. Therefore in order to obtain a supersymmetric action as discussed in 4 one needs the additional term S B = dx 5 gαtr ig ε IJKLM S IJ φ K φ L φ M 5 6 to cancel the supersymmetry variation of S B + S F. Thus one may see that S = S B + S F = + S F yields a supersymmetric non-abelian action. S B + S B 5. Supersymmetric backgrounds In this section we will discuss the supersymmetric solutions to the Killing spinor equation 7 and the condition 9 from the spinor variation δχ αβ γ which gives rise to supersymmetric backgrounds for the five-dimensional supersymmetric Yang Mills theory. In this paper we will make an assumption b μ = t i ab = S i5 = S 5i = A μ i5 = A μ 5i = i = 4 6

18 T. Kawano N. Matsumiya / Nuclear Physics B which is satisfied by the background in the previous papers as will be seen below. In we have considered the product space of a round S 3 and a Riemann surface. In this paper we are especially interested in supersymmetric backgrounds for deformed 3-spheres a squashed and an ellipsoid S 3. We will find supersymmetric backgrounds on the product spaces of those 3-spheres and which turn out to satisfy the assumption 6. It is convenient under the assumption 6 to decompose the supersymmetry parameter ɛ α as ρ 5 ɛ α = ɛ α ɛ α ɛ α = ρ 5 ɛ α = ɛ α ɛ α = in the representation with ρ 5 = diag.+. While the Killing spinor equation 7 in a generic background gives a differential equation of ɛ α and ε α coupled to each other the assumption 6 splits them into D μ ɛ α = 4 S ij σ ij α β γ μɛ β α G μνγ ν ɛ α 8α G bcγ bc μ ɛ α + t bc γ bc μ ɛ α 7 D μ ε α = 4 S ij σ ij α β γ με β α G μνγ ν ε α 8α G bcγ bc μ ε α t bc γ bc μ ε α 8 with t ab t 5 ab where the covariant derivatives are defined by D μ ɛ α μ ɛ α + b μɛ α + 4 μ bc γ bc ɛ α 4 A μ ij σ ij α β ɛ β D μ ε α μ ε α + b με α + 4 μ bc γ bc ε α 4 A μ ij σ ij α β ε β. We will further make an ansatz for the Killing spinors ε α ε α= = ɛ ζ + ε α= = C 3 ɛ ζ ; ɛ α= = ɛ ζ ± ɛ α= = C 3 ɛ ζ 9 with two-dimensional spinors ɛ ɛ on the S 3 and constant two-dimensional spinors ζ ± = ±i on obeying that τ ζ ± =±ζ ± with the Pauli matrix τ. Note that they satisfy γ 45 ε α = iτ 3 α β ε β γ 45 ɛ α = iτ 3 α β ɛ β. 3 For later convenience let us consider the commutation relation of the covariant derivatives acting on ε α which by definition gives D a D b ε α = 4 R ab cd γ cd ε α 4 F ab ij σ ij α β ε β and acting γ ab on this one obtains γ ab D a D b ε α = γ ab D a D b ε α = 4 R ε α 8 F ab ij σ ij α β γ ab ε β. 3 On the other hand using the Killing spinor equation 8 twice for γ ab D a D b ε α and equating it and the right hand side of 3 one finds that

19 474 T. Kawano N. Matsumiya / Nuclear Physics B R ε α 8 F ab ij σ ij α β γ ab ε β = σ ij α β D as ij γ a ε β D a t bc γ abc ε α 3 D b t ba γ a ε α 5 4α Db G ba γ a ε α S ij S kl σ ij σ kl α β ε β + 4 G abg ab 3 t ab + 4α 4α G ab t ab + 4α Gab ε α σ ij α 9 σ ij α β t ab t cd 5 4α G abs ij γ ab ε β 3 4α β t abs ij γ ab ε β 8 4α G a c t bc γ ab ε α G abg cd γ abcd ε α. 3 Decomposing the fields χ α φ I in the representation with ρ 5 = diag.+ as χ α ψ α φ I φ i= 4 φ 5 = σ λ α one can see that the supersymmetry transformation under the assumption 6 becomes δ ɛ A μ = i 4 ε α β ɛ α T Cγμ ψ β i 4 ε α β ε α T Cγμ λ β ɛ α T Cψ β i 4 ε α β ε α T Cλ β δ ɛ σ = i 4 ε α β δ ɛ φ i = i εσ i ɛ α T Cλ β + i ε σ i ε α T Cψ β 4 α β 4 α β δ ɛ ψ α = F abγ ab + γ a D a σ α G abσγ ab ɛ α S ij σ i g φ i φ j α σij β ɛ β δ ɛ λ α = γ a D a φ i α G abφ i γ ab S ij + ε ij kl S kl φ j t ab φ i γ ab ig σ φ i σ i α β ε β α G abσγ ab F abγ ab γ a D a σ + S ij σ + i g φ i φ j α σij β ε β ε α γ a D a φ i + α G abφ i γ ab S ij ε ij kl S kl φ j + t ab φ i γ ab + ig σ φ i σ i α β ɛ β. 33 The equations of motion of the spinors ψ α λ α under the assumption 6 give γ μ D μ ψ α + ig σ ψ α + ig σ i α β φ i λ β = 8α G abγ ab ψ α + 4 S ij σ ij α β ψ β t abγ ab ψ α

20 γ μ D μ λ α ig T. Kawano N. Matsumiya / Nuclear Physics B σ λ α + ig σ i α β = 8α G abγ ab λ α + 4 S ij with the covariant derivatives φ i ψ β σ ij α β λ β + t abγ ab λ α D μ ψ α = μ ψ α 3 b μψ α + 4 μ ab γ ab ψ α 4 A μ ij σ ij α β ψ β + ig D μ λ α = μ λ α 3 b μλ α + 4 μ ab γ ab λ α 4 A μ ij σ ij α β λ β + ig 5.. The N = SUSY background in the previous paper A ψ α μ A λ α μ. We start with the background in the previous paper where the compactification on the product space of a unit round S 3 and a Riemann surface S 3 was considered and we will reinterpret it as a supersymmetric background in terms of A μ i j S ij G ab t ab t 5 ab. See Appendix H. for the differences of the old notations used in from the ones in this paper. The background in can be read in the notations of this paper as t 45 = 4r S = S 34 = r 4α G 45 = 4r 34 in the Lorentz frame t ab G ab where we have replaced the unit radius of the S 3 by r. On the Riemann surface with local coordinates x 4 x 5 the twisting is required to preserve supersymmetries by turning on the background gauge field A i j as A = A 34 = ω45 35 with the spin connection ω 45 on the surface. This together with S = S 34 break the Spin5 R R-symmetry group to SU l U r SU l SU r when regarding the subgroup Spin4 of the Spin5 R as SU l SU r. We refer to it as the N = twisting following 6. The supersymmetry condition 9 determines the background M IJ 4 5 M 55 = 5 r R 4 5 M ij = r R where the scalar curvature R is derived from the spin connection ω 45 R e4 e 5 = dω 45 and substituting these into 4 gives 6 M B 55 = r M Bij = 4 R + 4 r δ ij i j = 4. δ ij i j = 4 The Killing spinor equation 8 in the background 34 is identical to the one in D μ ε α = r γ μ 45 ε α with the ansatz 9. 6 In the previous paper v3 on the arxiv the scalar curvature R was dropped from the mass terms M Bij of the N = hypermultiplet scalars.

21 476 T. Kawano N. Matsumiya / Nuclear Physics B The scalar curvature R on the S 3 is given by R = RS 3 + R = 6 r + R for the round S 3 of radius r. Since the gauge field A i j is minus the half of the spin connection ω 45 on the surface the field strength of A ij results in F 45 = F = 4 R. The equation 3 identically holds for the curvatures and the background fields and it is consistent with the existence of the Killing spinor ε α. In fact as explained in and in Appendix G the Killing spinor is given by ε α= = ɛ ζ + ε α= = C 3 ɛ ζ with ɛ a constant spinor on the S 3 which is consistent with our ansatz 9. For the other supersymmetry parameter ɛ α the Killing spinor equation 7 in the same background gives D a ɛ α = α G abγ b ɛ α + r γ a 45 ɛ α. Note that S = S 34 obeys S ij σ ij =. With A = A 34 we have A ij σ ij = and the twisting of the background A ij have no effects inside the covariant derivative D a ɛ α. In a generic Riemann surface we don t have a solution to the above Killing spinor equation. In fact the calculation of γ ab D a D b ɛ α shows that the scalar curvature R is an obstacle to the existence of a Killing spinor for ɛ α. We can see from that the background breaks the Spin5 R group of the R-symmetry into SU l U r which is a subgroup of SU l SU r Spin4 R Spin5 R. The symmetry breaking is caused by the twisting A = A 34 and also S = S 34. As we have seen just above the twisting only retains the half of the supersymmetries. Therefore it is consistent with the fact that the SU l symmetry doesn t give rise to the SU R R-symmetry in fourdimensional N = supersymmetric theories 66. The background 34 is not a unique solution 7 to yield an N = supersymmetric background on the round S 3. Even under the ansatz S = S 34 = S with only non-zero components G 45 and t 45 there exists a Killing spinor for ε α if S + 4α G 45 = S + 4α G 45 + t 45 = r which can be read from the Killing spinor equation 8. They may therefore be parametrized by S; 4α G 45 = 4 S t 45 = 4 S + r. 7 It has been pointed out in 8 in the context of five-dimensional N = supersymmetric theories.

22 T. Kawano N. Matsumiya / Nuclear Physics B The other supersymmetric condition 9 gives one more constraint the backgrounds are constant on D 4 S = D 5 S = and determines the remaining background M IJ 4 5 M 55 = 5 R + 4 5r 3 5 S 4 5 M ij = R 5r + 3 S δ ij for i j = 4. The scalar mass parameters M BIJ are given by M B 55 = r r S M B = =M B 44 = 4 R + r. When S = /r it certainly retains the mass term of the scalar σ in the previous papers. 5.. N = SUSY backgrounds on the round S 3 While the background in preserves half of the supersymmetries we will find a new supersymmetric background preserving both of ε α and ɛ α on the S 3. Taking the breaking of the R-symmetry group Spin5 R into account we will turn on A and S = S only and it would break the Spin5 R group down to U R SU R. We could instead turn on A 34 or S 34 only but it is just a matter of convention. We refer to this partial twisting as the N = twisting. Since we have the covariant derivatives with the ansatz 9 D μ ɛ α = μ ɛ α i A 45 μ ± ω μ τ 3 α β ɛ β D μ ε α = μ ε α i A 45 μ + ω μ τ 3 α β ε β in order to cancel the spin connection ω 45 by A in both of the covariant derivatives the chirality of ɛ α on the surface should be the same as the one of ε α ; iγ 45 ε α = τ 3 α β ɛ β. Therefore the twisting A = ω 45 works for both of ε α and ɛ α. When we turn on the components G 45 and t 45 only the Killing spinor equations 7 8 become D a ɛ α = i S τ 3 α β γ aɛ β α G abγ b ɛ α 4α G 45 t 45 γ 45 a ɛ α D a ε α = i S τ 3 α β γ aε β α G abγ b ε α 4α G 45 + t 45 γ 45 a ε α. For a = 4 5 the Killing spinor equation is satisfied with ɛ α and ε α constant on if S + 4 4α G 45 =. With the ansatz 9 the Killing spinors on the round S 3 see Appendix G. are lifted to D a ε α = r γ a 45 ε α D a ɛ α = r γ a 45 ɛ α

23 478 T. Kawano N. Matsumiya / Nuclear Physics B and the comparison of this with the above Killing spinor equations for a = 3 leads to S + 4α G 45 + t 45 = r S + 4α G 45 t 45 =± r. Depending upon the sign there are two solutions: and S = 4 4α G 45 = r 4α G 45 = r t 45 = S = 4α G 45 = t 45 = r. We will call the former background type B and the latter type A respectively. Let us begin with the type A background: S ij = 4α G ab = t 45 = r. In the background since the Killing spinor equation 7 is reduced into D μ ɛ α = r γ μ 45 ɛ α D μ ε α = r γ μ 45 ε α one obtains the solution to them ɛ α= ɛ α= = U ɛ ζ + C 3 U T ɛ ζ ε α= ε α= = ɛ ζ + C 3 ɛ ζ with ɛ and ɛ constant spinors and U the mapping of the 3-sphere to the SU group given in Appendix G and with C 3 the three-dimensional charge conjugation matrix explained in Appendix B. The supersymmetry condition 9 determines the background M IJ : 4 5 M 55 = 4 5 r 5 R 4 5 M = 4 5 M = 5 r + 3 R 4 5 M 33 = 4 5 M 44 = 5 r 5 R which gives rise to the masses M BIJ of the scalar fields φ I M B 55 = 4 r M B = M B = r + R M B 33 = M B 44 = r. Turning on the field t 45 = t I=5 45 breaks the Spin5 R symmetry group into Spin4 R and with the twisting by A = ω 45 into U U. Thus the background doesn t respect the R-symmetry of the four-dimensional N = conformal algebra but it retains the N = supersymmetry. Let us move on to the type B background: S = r 4α G 45 = r t 45 =.

24 T. Kawano N. Matsumiya / Nuclear Physics B It gives rise to the Killing spinor equation D μ ɛ α = r γ μ 45 ɛ α D μ ε α = r γ μ 45 ε α and one gives the same constant solution for the both ɛ α and ε α : ɛ α= ɛ α= = ɛ ζ + ɛ ζ ε α= ε α= = ɛ ζ + C 3 ɛ ζ with ɛ and ɛ constant spinors as above. The supersymmetric condition 9 is obeyed by the background if the background fields M IJ satisfy 4 5 M 55 = 4 5 M 33 = 4 5 M 44 = 8 5 r 5 R 4 5 M = 4 5 M = 5 r + 3 R which surely respects the R-symmetry group U R SU R. The scalars σ φ 3 φ 4 remain massless while the remaining φ φ are lifted by a half of the scalar curvature R : M B 55 = M B 33 = M B 44 = M B = M B = R. Thus they respect the remaining R-symmetry group U R SU R. Turning on either of A or A 34 without t 45 breaks the R-symmetry group SO5 R into SO SO3 U R SU R which can be identified with the R-symmetry group of the N = superconformal group if the theory flows into an infrared fixed point. On the other hand as in the previous papers turning on both A and A 34 such that A = A 34 the SO5 R group is broken to SU l U r which is the subgroup of SU l SU r SO4 SO5 R. The subgroup SU l cannot be identified to the R-symmetry group SU R because the above results shows that such a background preserves only half of the supersymmetries. 8 This is consistent with the result in A squashed 3-sphere with constant Killing spinors A squashed 3-sphere is a deformation of a round S 3 and regarding it as a circle fibration over a round -sphere i.e. the Hopf fibration the radius of the fiber differs from the radius of the base. See Appendix G for more details. In three-dimensional supersymmetric field theories on the squashed 3-sphere has been discussed and we will make use of their construction for the five-dimensional theory. The constant solution ε α on the round S 3 : ε ε = ɛ = C 3 ɛ = ɛ ζ + C 3 ɛ ζ with γ 3 ε α = ε α 36 solves the differential equation d + 4 ωab γ ab ε α ĩ r r r τ 3 α β e 3ε β = r r ea γ 45 a ε α 37 8 We thank Yuji Tachikawa for clarification on this point.

25 48 T. Kawano N. Matsumiya / Nuclear Physics B where ω ab is the spin connection of the squashed S 3 with the fiber radius r and the base radius r. See Appendix G for the squashed S 3. We will begin with the N = twisting by turning on A only. A comparison of 37 with the Killing spinor equation 8 suggests that A = r r r e 3 ω 45 4α G 45 = r r S = r r. 38 For the other supersymmetry parameter ɛ α it is easy to find a Killing spinor on the squashed S 3 if we make the same ansatz as for ε α ; it is a constant spinor ɛ ɛ = ɛ ζ + C 3 ɛ ζ obeying ɛ = T. One then see that it obeys the same differential equation 37 and thus the background 38 preserves the both Killing spinors ε α ɛ α. The other supersymmetry condition 9 determines the background fields M IJ 4 5 M = 4 5 M = 3 R + 5 r 4 5 M 33 = 4 5 M 44 = 4 5 M 55 = 5 R 8 5 r and plugging them into 4 one obtains the scalar masses M BIJ M B = M B = R + 4 r r r M B 33 = M B 44 = M B 55 =. Let us proceed to the N = twisting so that we will turn on the gauge field A ij of only one SU subgroup of the Spin5 R group by requiring that A = A 34 and then a comparison with the Killing spinor equation 7 identifies the background R-symmetry gauge field A = A 34 = r r r e 3 ω45 and for the other background fields taking account of 9 one finds that S = S 34 = S 4α G 45 = 4 S t 45 = 4 S + r r 4 5 M 55 = 4 r 5 r 4 5 R 3 5 S 8 5 r r r 4 5 M ij = r 5 r 4 + R + 3 S + 5 r r r δ ij 39 for i j = 4. In the limit r r one regains the N = supersymmetric background on the round S 3 in the previous subsection. It follows from 4 that M B 55 = r r r r S M B = M B = M B 33 = M B 44 = r R + 4 r 4 + r r r.