PreprinttypesetinJHEPstyle-HYPERVERSION Self-Duality of Green-Schwarz Sigma-Models 1 1 0 2 n a J 1 Amit Dekel and Yaron Oz 1 Raymond and Beverly Sackler School of Physics and Astronomy ] Tel-Aviv University, Ramat-Aviv 69978, Israel h t [email protected], [email protected] - p e h [ Abstract: We study fermionic T-duality symmetries of integrable Green-Schwarz 2 sigma-models on Anti-de-Sitter backgrounds with Ramond-Ramond fluxes, con- v 0 structed as Z4 supercosets of superconformal algebras. We find three algebraic con- 0 ditions that guarantee self-duality of the backgrounds under fermionic T-duality, we 4 0 classify those that satisfy them and construct the map of the monodromy matrix. . 1 We introduce new T-duality directions, where some of them contain no bosonic di- 0 rections, along which the backgrounds are self-dual. We find that the only self-dual 1 1 backgrounds are AdS ×Sn for n = 2,3,5. In addition we find that the backgrounds : n v AdS ×S1 for n = 2,3,5, AdS ×S2 and AdS ×S4 are self-dual at the level of the i n 4 2 X classical action, but have a non-trivial transformation of the dilaton. r a Keywords: Duality in Gauge Field Theories, String Duality. Contents 1. Introduction 2 2. Properties of Superconformal Algebras 3 2.1 The conformal basis and Z-gradation 3 2.2 Z automorphism 5 4 2.3 More on Z-gradations 5 2.3.1 Z-gradation of type I SCA’s 6 2.3.2 Z-gradation of type II SCA’s 8 3. Green-Schwarz Sigma-models on Semi-Symmetric backgrounds 10 3.1 The action 10 3.2 Integrability 10 4. T-duality of the GS Sigma-Models 11 4.1 Assumptions 11 4.2 T-duality of the Green-Schwarz sigma-model 12 4.3 Flat-connection transformation under T-self-duality 15 4.4 Quantum consistency of the T-Self-Duality transformation 16 5. Classification of the backgrounds 17 5.1 Type I 17 5.2 Type II 18 5.3 AdS ×M semi-symmetric spaces 20 3 5.4 AdS semi-symmetric space 20 2 5.5 When does an AdS semi-symmetric space satisfies Ω(U) = −U? 21 6. Discussion 21 A. Notations 24 B. Superalgebras 25 B.1 OSP(2N|2) 25 B.2 OSP(2N|4) 26 B.3 F(4;2) 26 B.4 F(4;0) 27 B.5 SU(1,1|N), N 6= 2 27 B.6 SU(2,2|N), N 6= 4 28 B.7 PSU(N,N|2N), N = 1,2 28 – 1 – B.8 D(2,1;ζ) 29 B.9 OSP(4∗|2N) 29 B.10 OSP(8∗|2N) 30 C. Rank of kappa symmetry 31 C.1 Type-U2 32 C.2 Type-U4 33 C.3 Type-O1 33 C.4 Type-O2 34 C.5 Type-Tu 35 C.6 Type-To 36 1. Introduction Self-duality of the Green-Schwarz sigma model (GSSM) on AdS × S5 background 5 is used to explain the existence of the dual-superconformal symmetry of scattering amplitudes in N = 4 SYM, and their connection to Wilson-loops [1][2]. The su- perconformal symmetry together with the dual one generate a Yangian symmetry algebra, which is related to the integrability properties of the theory. It is well known that GSSM’s on semi-symmetric spaces (Z supercoset spaces) 4 exhibit an infinite set of conserved charges [3] which satisfy the Yangian algebra [4][5]. It is thus natural to ask whether GSSM’s on other (than AdS × S5) semi- 5 symmetric backgrounds are self-dual under T-duality. In previous papers [6][7][8], some backgrounds were checked to be self-dual, while other were found not to be self-dual. In those papers, the background’s self-duality was checked on a case by case basis. A general argument for self-duality is still lacking. In the present paper we will take a rather general approach and formulate criteria for semi-symmetric backgrounds to be self-dual. We present three sufficient algebraic conditions for self-duality, and explain the lack of self-duality of backgrounds that do not satisfy them. We denote the superconformal algebras (SCA’s) by g, with the Z decomposition 2 g = g ⊕ g to its even and parts respectively. We further decompose the SCA’s ¯0 ¯1 according to a Z-gradation with gradings ±1,0 only, where the charges are assigned by a generator U. The T-duality is performed along all the directions associated with the grading 1 generators, which form an abelian subalgebra. We will prove that a background is self-dual if : 1. Ω(U) = −U, where Ω is the Z automorphism map. 4 – 2 – 2. Rank(κ-symmetry) ≥ dim(g )/4. ¯1 3. The SCA’s Killing-form vanishes. The first condition ensures a non-singular coupling of the fermionic coordinates. Thesecondconditionallowsaparticularrepresentationofthesupergroupthatisused in the T-duality procedure. The third condition guarantees the quantum consistency of the transformation, that is a non-trivial dilaton is not generated. We find that the only self-dual GSSM’s are the AdS × Sn for n = 2,3,5. All n of them were found previously to be self-dual [1][2][6]. We find there are also back- grounds that areself-dual at theclassical level, but at thequantum level their dilaton shifts, these are the AdS ×S1 for n = 2,3,5 (the case of n = 5 was discussed in [8]), n AdS ×S4, and AdS ×S2. In addition to the usual self-duality along the flat AdS 2 4 directions followed by some odd directions, namely the directions associated with span{P,Q}, we find other abelian subalgebras along which the GSSM is self-dual (one of them was discussed in [1]). Some of these directions involve only fermionic directions. We give the general transformation of the action and the flat-connection for any such abelian subalgebra. The transformation, as in the AdS × S5 case, is 5 a spectral parameter dependent automorphism, which is a composition of the Z - 4 automorphism map and an automorphism induced by the Z-gradation. The paper is organized as follows. In section 2 we briefly review some properties of the SCA’s, including a discussion of their Z-gradation structure. In section 3 we briefly discuss the GSSM and their basic integrability properties. In section 4 we prove T-self-duality of the GSSM’s using the three algebraic conditions stated above. In section 5 we classify the SCA’s according to the conditions for T-self- duality. In section 6 we discuss the results and various open questions. In appendix A we summarize our notations. In appendix B we provide technical computations concerning the SCA’s and their classification according to the first condition. In appendix C we compute the kappa-symmetry needed for the second condition. 2. Properties of Superconformal Algebras 2.1 The conformal basis and Z-gradation The generators of the SCA in d-dimensions are g = span{P,K,D,L} - the so(2,d− C 1) conformal subalgebra generators, span{R} - the R-symmetry subalgebra genera- tors, andspan{Q}andspan{S}-the(odd)supercharges andsuperconformal charges respectively. Altogether we have g = span{P,K,L,D;R;Q,S}. The SCA’s super- SC commutation relations are given by the commutation relations of g and span{R} C together with [P,Q] = 0, [K,S] = 0, [P,S] ∼ Q, [K,Q] ∼ S, (2.1) – 3 – [R,Q] ∼ Q, [R,S] ∼ S, {Q,Q} ∼ P, {S,S} ∼ K, {Q,S} ∼ D+L+R. These commutation relations can summarized using the charge of the generators under the dilatation generator D, see figure 1. This charge assignment is an example K S D,L,R Q P - D −2 −1 0 1 2 Figure 1: The charge of the SCA’s generators under D. of Z-gradation of the SCA, which is a decomposition of the algebra such that g = g and [g ,g ] ⊂ g . In this case i = −2,−1,0,1,2. Besides this Z-gradation, i∈Z i i j i+j the SCA may have others. L The superalgebras are classified according to their type, I or II [9]. The ter- minology type I and type II refers to the representation of the even part of the superalgebra on the odd part. If the representation is irreducible the superalgebra is called type II and if it is a direct sum of two irreducible representations the super- algebra is called type I. In the case of type I, the odd part decomposes according to another Z-gradation, which is called the distinguished gradation [9]. This gradation is associated with the generator B (which we call the hypercharge) which is in the algebra for A(m,n 6= m) and C(n + 1) and not for A(m,m). The generators are decomposed as g = g ⊕g ⊕g = {Q,Sˆ}⊕{P,K,D,L;R}⊕{Qˆ,S}. (2.2) I 1 0 −1 For type II SCA’s we do not have such a gradation, but when the number of space- time supersymmetries is even, N ∈ 2N , we do have another Z-gradation associated 1 ˇ with a generator of the R-symmetry subalgebra which we call λ. This decomposition, which further decomposes the R-symmetry generators to R,λ,Rˆ, is given by g = g ⊕g ⊕g ⊕g ⊕g = {R}⊕{Q,Sˆ}⊕{P,K,D,L;λ}⊕{Qˆ,S}⊕{Rˆ}. (2.3) II 2 1 0 −1 −2 In order to present the SCA’s, one has to work with real-forms of the SCA, since we have to take complex combinations of the odd generators and the R-symmetry generators. We summarize some relevant properties of the SCA’s in table 1. Fur- ther decomposition of the commutation relations (2.1) should be obvious from the gradations introduced above. Another characteristic of the SCA’s is whether the Killing-form is degenerate or not, see table 1. The Killing-form is defined as the supertrace of every two generators – 4 – Table 1: Some properties of SCA’s d SCA R-symmetry dim(g¯1) N type Killing-form 1 osp(N|2) so(N) 2N N IforN =2,elseII ND,exceptforN =4 1 su(1,1|N 6=2) u(N) 4N 2N I ND 1 psu(1,1|2) su(2) 8 4 I Zero 1 osp(4∗|2N) su(2)×usp(2N) 8N 4N II ND 1 G(3) g2 14 7 II ND 1 F(4;0) so(7) 16 8 II ND 1 D(2,1;α) so(4) 8 4 II Zero 3 osp(N|4) so(N) 4N N IforN =2,elseII ND,exceptforN =6 4 su(2,2|N 6=4) u(N) 8N N I ND 4 psu(2,2|4) su(4) 32 4 I Zero 5 F(4;2) su(2) 16 2 II ND 6 osp(8∗|N) usp(N) 8N N II ND,exceptforN =6 The table gives the SCA’s as classified in [10]. The spinor representations for d = 3,4,5,6 are su(2),su(2)×su(2),sp(4),su(4) respectively. ND- stands for non-degenerate. N is the number of space-timesupersymmetries. in theadjoint representation [11][9], K = Str(LadjLadj). The SCA’s with degenerate ab a b Killing-formareknowntohavespecial propertiesinthecontextoftheGreen-Schwarz sigma-models, e.g [12][13][14], and as we shall see they are also special with respect to the self-duality properties of the sigma-models. 2.2 Z automorphism 4 Every SCA has at least one Z automorphism [15]. A SCA is decomposed under this 4 automorphism into four sets g = H ⊕H ⊕H ⊕H , (2.4) 0 1 2 3 such that [H ,H } ⊂ H , B(H ,H ) 6= 0 only if i + j = 0 mod 4, and i j i+j mod 4 i j Ω(H ) = ikH , where B represents the Cartan-Killing bilinear-form and Ω(·) is the k k automorphism map. Using the Z automorphism property we can define a semi-symmetric space by 4 taking the quotient with respect to the invariant locus H (so the bosonic part is a 0 symmetric-space). A SCA may have several different Z automorphisms and so one 4 can identify a semi-symmetric space with respect to each automorphism. Some of the semi-symmetric spaces will have a bosonic AdS sub-space in which we are mainly interested in the present paper, although we will also consider non-AdS spaces. 2.3 More on Z-gradations Aswehaveshownabove, anySCAhasaZ-gradation,g = g suchthat[g ,g ] ⊂ i∈Z i i j g . When i takes a finite number of values, say i ≤ i ≤ i , the set imax g i+j min Lmax i=imax/2 i defines an abelian subalgebra if i > 0, and similarly for the set imin/2 g if max Li=imin i i < 0. min L – 5 – In the present paper we will be interested in Z-gradations with |i| ≤ 1, that is g = A ⊕ B ⊕ A , with A abelian subalgebras1 and B a subalgebra, so these 1 0 −1 ±1 0 are the Z-gradations we will consider from now on. Any such decomposition can be induced by introducing a U(1) generator U, with respect to g (which may or 0 may not be part of the SCA), satisfying ad (L ) = [U,L ] = aL , ∀ L ∈ g, where U a a a a a = ±1,0 is the charge of the generator. The decomposition under ad induces a U one-parameter dependent automorphism σ (L ) = λUL λ−U = λaL , λ ∈ C. (2.5) λ a a a ThevariousZ automorphisms[15]oftheSCA’smayhavedifferentrelationswith 4 the U(1) generator inducing the Z-gradation. We are interested in those satisfying Ω(U) = −U. (2.6) Ω is defined to act on a commutator as Ω([L ,L ]) = [Ω(L ),Ω(L )], thus a b a b σ (Ω(L )) = λ−a(Ω(L )), (2.7) λ a a and the non-trivial bilinear-form is of the form B(L Ω(L )) with a = b. For back- a b grounds satisfying (2.6) and some other conditions (to be discussed later) we will be able to prove T-self-duality. Next we consider four classes of Z-gradations that will be of interest in the study of T-self-duality of GS-sigma-models. We study the type I and type II SCA’s separately. Comment: when talking about the grading, one usually use integer labeling, while the charges with respect to the U(1) generators we will use (D,B,Rˇ,λˇ) are integer fortheevengeneratorsandhalfintegerfortheoddgenerator(e.gthereshould really be a factor of 2 in figure 1, and later in figure 2). In the next sections when we will write the charges with respect to the generators we will use integer numbers although they should be understood to be divided by 2, so at the end of the day when we will discuss the gradations with charges ±1,0 only, these will really be the charges under the combination of the U(1) generators. 2.3.1 Z-gradation of type I SCA’s The type I SCA’s include the su(1,1|N 6= 2), su(2,2|N 6= 4), psu(1,1|2), psu(2,2|4), osp(2|2) ≃ su(1,1|1), osp(2|4). Generally, the bosonic part of these SCA’s is g = ¯0 so(2,d)⊕su(N)⊕u(1). We will refer to the last u(1) as the hypercharge, which in the case of the ’psu’ SCA’s decouples from the SCA. The two ’osp’ SCA’s are missing the su(N) subalgebra. 1The Z-gradationconsidered should not necessarily be consistent, namely g±1 and g0 may con- tain even and odd generators respectively [11][9]. – 6 – We will consider three u(1)’s, generating consistent Z-gradations of the SCA’s, and their combinations which generates Z-gradations with |i| ≤ 1. We already considered D, which induces the following consistent Z-gradation g = span{P}, g = span{Q,Qˆ}, (2.8) 2 1 g = span{D,L}⊕u(N), 0 g = span{S,Sˆ}, g = span{K}, −1 −2 and B which induces the distinguished Z-gradation g = span{Q,Sˆ}, g = su(M,M)⊕u(N), g = span{Qˆ,S}. (2.9) 1 0 −1 Lastly, we decompose the su(N) generators, Rk with k,l = 1,...,N and Rk = 0, l k k under su(N) → s(u(P)×u(N−P)). This divides the R-symmetry indices to k,l,.. = P 1,..,P and k′,l′,... = P +1,...,N, so the u(P) generators are Rk and the u(N −P) l generators are Rk′, with the relation Rk+ Rk′ = 0. We define the generator, l′ k k k′ k′ Rˇ = Rk − Rk′, inducing the Z-gradation k k k′ k′ P P P P g = span{Rk′}, g = span{Qk,Qˆ ,S ,Sˆk}, (2.10) 2 l 1 l′ l′ g = s(u(P)×u(N −P))⊕su(M,M), 0 g = span{Qk′,Qˆ ,S ,Sˆk′}, g = span{Rk}. −1 l l −2 l′ Next we consider the combinations of D,B and Rˇ that give the Z-gradations with |i| ≤ 1 (up no normalization). First we have the well known U = D +B [2][1] which gives the non-consistent Z-grading decomposition g = (P,Q) ⊕(L,D,Qˆ,Sˆ,R) ⊕(K,S) . (2.11) 1 0 −1 This decomposition holds for all the type I SCA’s. In this case the invariant sub- algebras are (s)u(M|N) for (p)su(2M|N), and u(1|2) for osp(2|4). Similarly for U = D − B we get the same gradation with the hatted and unhatted generators interchanged. Next, we consider U = D +Rˇ which generates the decomposition g = (P,Qk,Qˆ ,R l) ⊕(L,D,Qk′,Qˆ ,S ,Sˆk,R l,R l′) ⊕(K,S ,Sˆk′,R l′) k′ k′ 1 k k′ k k′ 0 k k −1 (2.12) which was mentioned in [1] for psu(2,2|4). This decomposition cannot be applied for the ’osp’ SCA’s. Forthis decomposition the invariant subalgebras are (p)s(u(M|P)× u(M|N −P)) for (p)su(2M|N). Next, weconsiderU = 2B whichgeneratestheconsistentdistinguished-gradation g = (Q,Sˆ) ⊕(P,K,D,L;R) ⊕(Qˆ,S) . (2.13) 1 0 −1 – 7 – Thisdecompositionwasnotconsidered beforeinthecontext ofT-duality, andimplies the model may be self-dual under T-duality along fermionic directions only. Lastly, we consider U = B +Rˇ which generates the inconsistent-gradation g = (Qk,Sˆk,R l) ⊕(P,K,L,D,Qk′,Sˆk′,S ,Qˆ ,R l,R l′) ⊕(S ,Qˆ ,R l′) . k′ 1 k′ k′ k k′ 0 k k k −1 (2.14) This decomposition also was not considered before in the context of T-duality. It is similar to the decomposition (2.12), interchanging the roles of the AdS and the sphere. Wesummarize theabove decompositions in figure2(a-c). The results forAdS × 5 S5 background are also given in table 3. Note that in cases where the SCA is ’psu’, the hypercharge is not a part of the SCA and so the Z-gradation automorphisms are outer. The rest of the automorphisms are inner. Decompositions with respect to combination, different from the ones presented by changing the relative signs of the generatorsareobviousandaddressedinthefigure. Onecanfindmoredecompositions of the SCA’s, which we find less interesting with respect to the AdS backgrounds2. 2.3.2 Z-gradation of type II SCA’s The type II SCA’s include the osp(N 6= 2|2), osp(N 6= 2|4), osp(4∗|2), osp(8∗|N), D(2,1;α), F(4) and G(3), but we’ll consider only osp(2N 6= 2|2), osp(2N 6= 2|4), osp(4∗|2), osp(8∗|2N), D(2,1;α) and F(4) which can be decomposed according to (2.3) (the ones with even number of space-time supersymmetries). As mentioned above the type II SCA’s decompose under the charge assignment of D, g = span{P}, g = span{Q,Qˆ}, (2.15) 2 1 g = span{D,L}⊕span{R,λ,Rˆ}, 0 g = span{S,Sˆ}, g = span{K}. −1 −2 and according to the gradation (2.3), where the R-symmetry decomposes to R ⊕ 2 λ ⊕ Rˆ (the subscript indicates the gradation). The R-symmetry decomposition 0 −2 for all type II SCA’s with such decomposition is given in table 2. The Z-gradation ˇ of (2.3) is induced by the generator λ given in the table. ˇ Combining the two, U = D +λ, we find that all type II SCA’s have the incon- sistent Z-gradation g = (P,Q,R) ⊕(L,D,Qˆ,Sˆ,λ) ⊕(K,S,Rˆ) . (2.16) 1 0 −1 2For example, in the notation of AdS5×S5 we have (Pαα˙,Kαα˙,Lαα,Lα˙α˙,Qiα,Qˆαi˙′,Siα′,Sˆiα˙,Ri′i)⊕(Pα¯α˙,Pαα¯˙,Kα¯α˙,Kαα¯˙,Lαα¯,Lα˙α˙,Lα˙α¯˙,Qiα¯,Qi′α,Qˆαi˙,Qˆαi¯˙′, Siα,Siα¯′,Sˆi′α˙,Sˆiα¯˙,Rii,Ri′i′)⊕(Kα¯α¯˙,Pα¯α¯˙,Lα¯α¯,Lα¯˙α¯˙,Siα¯,Sˆi′α¯˙,Qi′α¯,Qˆαi¯˙,Rii′) where(α,α˙)=(1,1),(1,2),(2,1),or(2,2),(α¯,α¯˙)takedifferentvaluethen(α,α˙),andi=1,..,n≤4, i′ =n+1,..,4. The abeliansubalgebrainvolvesunphysicaldirections, L. The invariantsubalgebra is ps(u(2|n)×u(2|4−n)). – 8 – D R 2 P 2 Rii’ 1 Q^ Q 1 Q^i, Si Qi’,S^ i’ L,D P,K,L,D 0 R 0 Rii, R i’i’ -1 S S^ -1 Q^i’, Si’ Qi,S^ i -2 K -2 Ri’i -1 0 1 B -1 0 1 B (a) (b) R λ 2 Rii’ 2 R 1 Si,S^i’ Qi,’ Q^i 1 S^ Q L,D 0 K Rii, R i’i’ P 0 K Lλ,D P -1 Si’,S^ i Qi, Q^i’ -1 S Q^ -2 Ri’i -2 R^ -2 -1 0 1 2 D -2 -1 0 1 2 D (c) (d) Figure 2: Z-gradation of type-I SCA’s with 4N-odd generators and R-symmetry SU(2M) × U(1). The abelian subalgebras are circled. (a) Decomposition under B and D. In this case the relevant U(1)’s are (±)(D±B) (circled with solid blue contours) where the abelian subalgebra contains d-bosonic and N-fermionic generators, and ±2B (circled with dashed red contours) where the abelian subalgebra contains 2N-fermionic genera- tors. (b) Decomposition under B and Rˇ. In this case the relevant U(1)’s are (±)(Rˇ ±B) (circled with solid blue contours) where the abelian subalgebra contains M2-bosonic and 2M-fermionic generators, and ±2B (circled with dashed red contours) where the abelian subalgebra contains 2N-fermionic generators. (c) Decomposition under D and Rˇ. In this case the relevant U(1)’s are (±)(Rˇ ±D) where the abelian subalgebra contains d+M2- bosonic and N-fermionic generators. (d) Z-gradation of type-II SCA’s with 4N-odd gen- erators and R-symmetry R ⊕λ ⊕R . The abelian subalgebras are circled. We have the 1 0 −1 decomposition under λˇ and D. In this case the relevant U(1)’s are (±)(D±λˇ) where the abelian subalgebra contains d+dim(R )-bosonic and N-fermionic generators. 1 – 9 –