Generalised Permutation Branes on a product of cosets $G_{k_1}Htimes G_{k_2}H$

a r X i v :h e p -t h /0601061v 1 10 J a n 2006Generalised Permutation Branes on a product ofcosets G k 1/H ×G k 2/HGor Sarkissian∗Dipartimento di FisicaUniversita di Roma ”Tor Vergata”,I.N.F.N sezione di Roma ”Tor Vergata”via della Ricerca Scientifica 1,00133Roma,ItalyAbstractWe study the modifications of the generalized permutation branes de-fined in hep-th/0509153,which are required to give rise to the non-factorizable branes on a product of cosets G k 1/H ×G k 2/H .We find that for k 1=k 2there exists big variety of branes,which reduce to the usual permutation branes,when k 1=k 2and the permutation symmetry is restored.1Introduction and summaryIn the recent paper[1]were suggested the generalized permutation branes on aproduct of the WZW models G k1×G k2with the not necessarily equal levels k1and k2.Geometrically the branes wrap the following submanifolds:(g1,g2)={(h1fh−12)k′2,(h2fh−11)k′1|h1,h2∈G}(1)where k′i=k i/k and k=gcd(k1,k2).Obviously for k1=k2(1)reduces to the usual permutation branes[2–8](g1,g2)={h1fh−12,h2fh−11}(2)It is well known that for a submanifold to serve as D-brane in the WZW model the boundary two-formωC should exist trivializing the Wess-Zumino three-form on the brane[9–11]:ωW ZW|brane=dωC.(3)It was found in[1]that the restriction of the WZW form k1ωW ZW(g1)+k2ωW ZW(g2) to the submanifold(1)indeed satisfies to this condition withω(f)Cgiven by the equation:ω(f) C (h1,h2)=k1k2κ,whereλis an integral weight of G Lie algebra,andκ=lcm(k1,k2).Comparing the formulae(1)and(2),describing generalized and usual permu-tation branes respectively one can deduce that the generalized branes preserve less symmetry.The usual permutation branes preserve two different twisted ad-joint actions:(g1,g2)→(mg1,g2m−1),(g1,g2)→(g1m−1,mg2),(5) while the generalized branes preserve only the diagonal subgroup:(g1,g2)→(mg1m−1,mg2m−1)(6)Using the form(4)boundary equations of motion have been analyzed in[1]and indeed found to beJ1+J2=¯J1+¯J2.(7)Motivated by the recently established connection between the permutation D-branes in the minimal N=2supersymmetric models and the Landau-Ginzburg superpotential factorisation[12–14]it was suggested in[1]that the generalizedpermutation branes should exist also for cosets of the form G k1/H×G k2/H forthe different levels k1and k2.To produce such D-brane from the generalized D-branes(1)one should modify them in a way to preserve adjoint action of product H×H:(g1,g2)→(m1g1m−11,m2g2m−12)(8)where m i∈H.This problem is remained unsolved in[1].Aim of this paper is to study the modifications of the ansatz(1)giving rise to the required symmetry.In the next section we show that one can propose two kinds of boundary conditions possessing with the necessary symmetries.I.(g1,g2)={(h1fh−12)k′2ptp−1,L−1(h2fh−11)k′1nrn−1L}(9)where L,p,n∈H and run all the H subgroup,r and t arefixed quantized elements of H.In other words we multiply both elements of the ansatz(1)by the(quantized)conjugacy classes of the subgroup H,and then smear the derived object along the adjoint action of H.The chain of the transformations:h1→m1h1,h2→m1h2,L→m1Lm−12,p→m1p,n→m1n(10)reproduces(8).II.(g1,g2)={(h1fh−12)k′2(s1ls−12)k′2,L−1(h2fh−11)k′1(s2ls−11)k′1L}(11) where L,s1,s2∈H run all the subgroup H and l isfixed quantized element.By words,we multiply the generalized permutation brane of G k1×G k2by thegeneralized permutation brane of H k3×H k4(k3=x e k1,k4=x e k2,x e is theembedding index of H in G[15])and then again smear derived object along the adjont action of H.We show in the next section that for k1=k2(9)and(11) are equivalent.Using the Polyakov-Wiegmann identityωW ZW(gh)=ωW ZW(g)+ωW ZW(h)−d(tr(g−1dgdhh−1))(12)one obtains that the branes(9)satisfy to the condition(3)with the following boundary two-form:Ω(h1,h2,L,p,n)=ω(f)C(h1,h2)+k2ω(2)(C2C4,L)+k1ωt(p)−k1(tr(C−11dC1dC3C−13))−k2(tr(C−12dC2dC4C−14))+k2ωr(n)(13) where we denotedC1=(h1fh−12)k′2,C2=(h2fh−11)k′1,C3=ptp−1,C4=nrn−1,(14)ω(f)Cis the two-form given by the formula(4),ωt(p)is the two-form found in[10,11]ωt(p)=tr(p−1dptp−1dpt−1)(15)andω(2)is the following useful two-argument two-formω(2)(g,U)=tr(dUU−1(gdUU−1g−1+g−1dg+dgg−1))(16)which we frequently encounter in this paper.For an abelian subgroup H=U(1),and equal levels k1=k2the brane (9)was studied in[16].(In that paper this brane was written in the form(h1fh−12L−1,h2fh−11L),but after the redefinition L−1h′2=h2we derive(9).)It was checked for this case in[16],that the full Lagrangian with boundary term (13)enjoys with the symmetry(8).It was also shown in[16]that the geometry ofthis brane coincides with the shape corresponding to the permutation boundarystate of the parafermions product.This serves for us as the hint that we foundthe correct solution.One can also check that the branes(11)as well satisfy(3)with the two-formΩ(h1,h2,s1,s2,L)=ω(f)C (h1,h2)+ω(l)C(s1,s2)+k2ω(2)(C2C6,L)−k1(tr(C−11dC1dC5C−15))−k2(tr(C−12dC2dC6C−16))(17)whereC 5=(s 1ls −12)k ′2,C 6=(s 2ls −11)k ′1.(18)The rest of the paper is organized as follows.In the section 2we obtain boundary conditions (9)and (11)by analyzing the action of the gauged WZW model on a world-sheet with boundary.In the section 3we consider these branes for product SU (2)k 1/U (1)×SU (2)k 2/U (1).In the appendix A we deliver some algebraical calculations proving gauge invariance of action with boundary term given by (13).In the appendix B we review different coordinates systems on S 3sphere used in the section 3.2Lagrangian and symmetriesIn this section we obtain the boundary conditions (9)and (11)by considering gauged WZW model action on a world-sheet with boundary along the way worked out in [17]and [18].First of all we remind the bulk action of the gauged WZW model [19–23]:S G/H (g,A )=S G +k G4πΣd 2zL kin +BωW ZW (g )(20)is bulk WZW action,L kin =tr(∂z g∂¯z g −1),ωW ZW=1is manifestly gauge invariant under the transformation:g→mgm−1,U→mU,˜U→m˜U(23) For the case under consideration the bulk action is:S G/Hbulk(˜g1,˜g2,˜h1,˜h2)=S G(˜g1)+S G(˜g2)−S H(˜h1)−S H(˜h2)(24) where˜g1=U−11g1˜U1˜g2=U−12g2˜U2˜h1=U−11˜U1˜h2=U−12˜U2(25) The levels k3and k4of the third and forth terms in(24),as explained above,are related to the levels k1and k2of thefirst and second terms respectively through the embedding index x e of H in Gk3=x e k1k4=x e k2.(26) Consider the action(24)in the presence of boundary.We should specify boundary conditions.Let us choose for the sum of thefirst two actions the generalized permutation boundary conditions(1)and impose the arguments of the third and forth terms to take their values in the quantized conjugacy classes:˜g1=((U−11h1)f(U−11h2)−1)k′2˜h1=(U−11p)t−1(U−11p)−1˜g2=((U−11h2)f(U−11h1)−1)k′1˜h2=(U−11n)r−1(U−11n)−1(27) It is easy to see that(27)brings to us conditions(9)for g1and g2withL=U1U−12(28) Using the forms(4)and(15)one can write the full action with the boundary term:S=S G/Hbulk (˜g1,˜g2,˜h1,˜h2)−1where∂B=Σ+D.The action(29)is manifestly gauge invariant under the gauge transformation:h1→m1h1,h2→m1h2,U1→m1U1,U2→m2U2,p→m1p,n→m1n(30) implying,recalling the definition(28),L→m1Lm−12.(31)We derived the transformation rules(10).Using the Polyakov-Wiegmann identi-ties the action(29)can be written as:1S=S G/H(g1,A1)+S G/H(g2,A2)−The full action with boundary term is:S =S G/H bulk (˜g 1,˜g 2,˜h1,˜h 2)−14πDΩ(39)whereΩ=ω(f )C (U −11h 1,U −11h 2)+ω(l )C (U −11s 1,U −11s 2)+k 1ω(g 1,U 1,˜U1)+k 2ω(g 2,U 2,˜U 2)(40)Repeating the same steps as outlined in the appendix A one can show that(40)coincides with (17).Some comments:Let us consider the branes (9)and (11)for equal levels k 1=k 2,when permu-tation symmetry is restored:(g 1,g 2)=(C 1C 3,L −1C 2C 4L )=(h 1fh −12ptp −1,L −1h 2fh −11nrn −1L )(41)(g 1,g 2)=(C 1C 5,L −1C 2C 6L )=(h 1fh −12s 1ls −12,L −1h 2fh −11s 2ls −11L )(42)By the redefinitionh −12C 5=h ′2,L ′=C −15L(43)one can write the brane (42)in the form (C 1,L −1C 2C 6C 5L ).Taking into accountthat C 6C 5in this case is the usual conjugacy class,we see that at the point of the restored symmetry the family of the branes (42)coincides with the family (41).Performing the same kind of redefinition in (41)h −12C 3=h ′2,L ′=C −13L(44)one can write all the branes(41)in the form(g1,g2)=(C1,L−1C2C4C3L)(45)Presumably we can multiply both elements in(9)by the chain of conjugacy classes,as in[8],but for the case of k1=k2,when permutation symmetry is restored,we see,that one can cover all the family already multiplying just oneof them.The conclusion is,that generically when k1=k2we have two familiesof branes(9)and(11),which reduce to the branes of the form(45),when the permutation symmetry is restored.3Generalized permutation branes on SU(2)k1/U(1)×SU(2)k2/U(1)In this section we consider permutation branes on product of SU(2)k1/U(1)×SU(2)k2/U(1)cosets.At the beginning we consider usual permutation brane fork1=k2and show that the geometrical description given above coincide with the permutation boundary state[3]overlap with the graviton wave packet.Actuallythis calculation was performed in[16],but for the completeness and the reader’s convenience we repeat it(slightly generalized and with corrected typos)here. Then we elaborate the geometry of the simplest generalized permutation brane.With the U(1)subgroup generated byσ3the brane(9)for k1=k2=k takesthe form:(g1,g2) brane=(h1fh−12,e iασ32e iπM2).(46)where f=e iˆψσ3k,j=0,...,k kσ32g2e iασ3kσ3cosˆ˜θ=cos˜θ1cos˜θ2−sin˜θ1sin˜θ2cos(χ2+ϕ1),(48)e iˆ˜φ=e iχ1+ϕ2cosˆ˜θ2cos˜θ22−sin˜θ12e−iχ2+ϕ12g2and using(48)and(49)we can rewrite(47)ascos ˜Θ2k)=cosˆψ,(50)wherecos˜Θ=cos˜θ1cos˜θ2−sin˜θ1sin˜θ2cosγ,(51) and we have introduced new labelsγ=χ2+ϕ1−αandξ/2=˜Φ−χ1+ϕ22=12cos˜θ12e iγ2sin˜θ22 .(52)Let us recall that the vectorial gauging of U(1)symmetry is corresponding to the translation ofφand the resulting target space of the SU(2)k/U(1)model, derived after the gaugefixingφ=0and integrating out of the gaugefield,is the two-dimensional disc,parameterized byθand˜φ.In the case of product the target space is parameterized byθ1,θ2,˜φ1,˜φ2.Hence the brane consists of those points for which equation(50)admits a solution forγ.Θandξare considered here as the complicated functions of˜θ1,˜θ2andγgiven by(51)and(52)respectively. Forˆψ=0there are additional constraints,which imply that in this case the brane is two dimensional and given by the equations˜θ1=−˜θ2,˜φ1=−˜φ2+πMS0je iπMm/k N1,N2|j,m,N1 1⊗|j,m,N2 1(54)where S Lj is matrix of the modular transformation of SU(2)kS Lj= k+2sin (2L+1)(2j+1)π2j+1D j nm(g( θ))=S0j ∼(k+2)k+2,one obtains that in the large-k limit the overlap reduces to θ1, θ2|L,M ∼ j m sin[(2j+1)ˆψ]e iπMm/k D j mm(g1( θ1))D j mm(g2( θ2)).(59) It is known[25]that d j nm are satisfying the relation(note that there is no summation assumed for the repeated indices)d j mm(cos˜θ1)d j mm(cos˜θ2)=1k )d jmm(cos˜Θ)dγ(61)Now using that m D j mm(g)=sin(2j+1)ψsinˆψdγ,(62)wherecosψ=cos ˜Θ2k)(63)From this equation it follows that the brane consist of all those points for whichthe expression in the argument of theδfunction has a root forγ.This is the same condition as the one coming from equation(50),obtained in the Langrangian approach.Let us turn now to the generalized permutation brane on a product SU(2)k1/U(1)×SU(2)k2/U(1).The branes(9)and(11)for abelian case take formsH={(h1fh−12)k′2e iπM12,e iασ32e iπM22}(64)H={(h1fh−12)k′2e iβk′2σ3k1σ32(h2fh−11)k′1e−iβk′1σ32e iπM22}(65) We see that for k1=k2we have much bigger variety of branes,which all degen-erate to(46)when k1=k2.Now we describe geometry of the generalized permutation branes(64)and (65)for the simplest case f=e.For this case the branes are:(g1,g2)=(g k′2e iπM12,e iασ32e iπM22)(66) (g1,g2)=(g k′2e iβk′2σ3k1σ32g−k′1e−iασ32e iπM22)(67)To elaborate the geometry of(66)and(67)wefirst recall the useful fact mentionedin[1]that the element g n in the coordinates(88)has the same anglesξandηasg butψhas to be replaced by nψ.Then we need the formulae of transformation from the coordinates(88)to the Euler coordinates:tan˜φ=tanψcosξsinθ=sinψsinξ(68) Takingfinally into account that the adjoint action of U(1)does not changeθand˜φangles of an element one can describe the geometry of(66)and(67)by the following equations of embedding respectively:tan ˜φ1−πM12k2 =−tan k′1ψcosξsinθ2=−sin k′1ψsinξ(69)tan ˜φ1−k′2β−πM12k2 =−tan k′1ψcosξsinθ2=−sin k′1ψsinξ(70) The brane(66)is two-dimensional with the world-volume coordinates(ψ,ξ), whereas the brane(67)is three-dimensional with the world-volume coordinates (ψ,ξ,β).It is easy to check that when k1=k2,and k′1=k′2=1,(69)and(70) reduce to(53).Tofind various other properties of the branes like mass,spectrum et.c.,is left for the future work.A Details of calculationsTo show that(33)coincides with(13)at the beginning we solve(27):˜U1=C−13U1,(71)˜U2=U2U−11C−14U1,(72)g1=C1C3(73)g2=L−1C2C4L=(U1U−12)−1C2C4(U1U−12)(74) where C1,C2,C3,C4are defined in(14).Inserting(71)and(73)in(34)for i=1one can show thatω(g1,U1,˜U1)=−tr(C−11dC1dC3C−13)−ω(2)(C1,U1)−ω(2)(C3,U1)(75) Inserting(72)and(74)in(34)for i=2one obtainsω(2)(g2,U2,˜U2)=−ω(2)(C2,U1)−ω(2)(C4,U1)+ω(2)(C2C4,U1U−12)−tr(C−12dC2dC4C−14)(76)To deal with the second and third terms in(33)we recall the following useful identity derived in[17]:ω(t)(U−11p)=ω(t)(p)+ω(2)(C3,U1)(77)It was shown in[17]that(77)guarantees that the full WZW Lagrangian on a world-sheet with boundary,with boundary conditions specified by the conjugacy class C3,enjoys with the full diagonal subalgebra:g(z,¯z)→k L(z)g(z,¯z)k−1R(¯z),k L|boundary=k R|boundary(78) The last ingredient which we need is the formula giving transformation prop-erties ofωC:ω(f) C (U−11h1,U−11h2)=ω(f)C(h1,h2)+k1ω(2)(C1,U1)+k2ω(2)(C2,U1)(79)It is possible to derive this formula using the definition(4).But this formula is nothing else as the global form of the equation(7),reflecting symmetry properties of the generalized permutation brane.It is straightforward to check that(79) guarantees that the WZW Lagrangian on a world-sheet with boundary with theboundary conditions given by the generalized permutation brane(1)enjoys with the symmetryg1(z,¯z)→k L(z)g1(z,¯z)k−1R (¯z),g2(z,¯z)→h L(z)g2(z,¯z)h−1R(¯z),k L|boundary=k R|boundary=h L|boundary=h R|boundary(80) Inserting(75),(76),(77)and(79)and in(33)one ends up with(13).B Various coordinate systems for the sphere andrelations between themA three-sphere S3is a group manifold of the SU(2)group.A generic element in this group can be written asg=X0σ0+i(X1σ1+X2σ2+X3σ3)= X0+iX3X2+iX1−(X2−iX1)X0−iX3 (81) subject to condition that the determinant is equal to oneX20+X21+X22+X23=1.(82) The metric on S3can be written in the following three ways,which will be used in the main text.Firstly,using the Euler parametrization of the group element we haveg=e iχσ32e iϕσ34 (dχ+cos˜θdϕ)2+d˜θ2+sin2˜θdϕ2 .(84) The ranges of coordinates are0≤˜θ≤π,0≤ϕ≤2πand0≤χ≤4π.Secondly,we can use coordinates that are analogue to the global coordinate for AdS3X0+iX3=cosθe i˜φ,X2+iX1=sinθe iφ(85)d s2=dθ2+cos2θd˜φ2+sin2θdφ2.(86) The relation between the metrics(83)and(85)is given byχ=˜φ+φ,ϕ=˜φ−φ,θ=˜θThe ranges of coordinates are−π≤˜φ,φ≤πand0≤θ≤π2,d s2=dψ2+sin2ψ(dξ2+sin2ξdη2)(88) X0+iX3=cosψ+i sinψcosξ,X2+iX1=sinψsinξe iη.(89)The ranges of the coordinates are0≤ψ,ξ≤πand0≤η≤2π.References[1]S.Fredenhagen and T.Quella,“Generalised permutation branes,”JHEP0511(2005)004[arXiv:hep-th/0509153].[2]J.M.Figueroa-O’Farrill and S.Stanciu,“D-branes in AdS(3)x S(3)x S(3)x S(1),”JHEP0004(2000)005[arXiv:hep-th/0001199].[3]A.Recknagel,“Permutation branes,”JHEP0304(2003)041[arXiv:hep-th/0208119].[4]M.R.Gaberdiel and S.Schafer-Nameki,“D-branes in an asymmetric orb-ifold,”Nucl.Phys.B654(2003)177[arXiv:hep-th/0210137].[5]J.Fuchs,I.Runkel and C.Schweigert,“Boundaries,defects and Frobeniusalgebras,”Fortsch.Phys.51(2003)850[Annales Henri Poincare4(2003) S175][arXiv:hep-th/0302200].[6]J.Fuchs and C.Schweigert,“Symmetry breaking boundaries.II:More struc-tures,examples,”Nucl.Phys.B568(2000)543[arXiv:hep-th/9908025].[7]J.Fuchs and C.Schweigert,“Symmetry breaking boundaries.I:Generaltheory,”Nucl.Phys.B558(1999)419[arXiv:hep-th/9902132].[8]T.Quella,“On the hierarchy of symmetry breaking D-branes in group man-ifolds,”JHEP0212(2002)009[arXiv:hep-th/0209157].[9]C.Klimcik and P.Severa,“Open strings and D-branes in WZNW models,”Nucl.Phys.B488(1997)653[arXiv:hep-th/9609112].[10]A.Y.Alekseev and V.Schomerus,“D-branes in the WZW model,”Phys.Rev.D60(1999)061901[arXiv:hep-th/9812193].[11]K.Gawedzki,“Conformalfield theory:A case study,”arXiv:hep-th/9904145.[12]I.Brunner and M.R.Gaberdiel,“Matrix factorisations and permutationbranes,”JHEP0507(2005)012[arXiv:hep-th/0503207].[13]H.Enger,A.Recknagel and D.Roggenkamp,“Permutation branes and linearmatrix factorisations,”arXiv:hep-th/0508053.[14]C.Caviezel,S.Fredenhagen and M.R.Gaberdiel,“The RR charges of A-type Gepner models,”arXiv:hep-th/0511078.[15]P.Di Francesco,P.Mathieu and D.Senechal,“Conformalfield theory”.Springer-Verlag,1997,New York.[16]G.Sarkissian and M.Zamaklar,“Symmetry breaking,permutation D-braneson group manifolds:Boundary states and geometric description,”Nucl.Phys.B696(2004)66[arXiv:hep-th/0312215].[17]S.Elitzur and G.Sarkissian,“D-branes on a gauged WZW model,”Nucl.Phys.B625(2002)166[arXiv:hep-th/0108142].[18]T.Kubota,J.Rasmussen,M.A.Walton and J.G.Zhou,“Maximally sym-metric D-branes in gauged WZW models,”Phys.Lett.B544(2002)192 [arXiv:hep-th/0112078].[19]K.Bardakci,E.Rabinovici and B.Saering,“String Models With C≤1Components,”Nucl.Phys.B299(1988)151.[20]K.Gawedzki and A.Kupiainen,“G/H Conformal Field Theory FromGauged WZW Model,”Phys.Lett.B215(1988)119.[21]K.Gawedzki and A.Kupiainen,“Coset Construction From Functional Inte-grals,”Nucl.Phys.B320(1989)625.[22]D.Karabali,Q.H.Park,H.J.Schnitzer and Z.Yang,“A Gko ConstructionBased On A Path Integral Formulation Of Gauged Wess-Zumino-Witten Actions,”Phys.Lett.B216(1989)307.[23]D.Karabali and H.J.Schnitzer,“Brst Quantization Of The Gauged WZWAction And Coset Conformal Field Theories,”Nucl.Phys.B329(1990) 649.[24]J.M.Maldacena,G.W.Moore and N.Seiberg,“Geometrical interpre-tation of D-branes in gauged WZW models,”JHEP0107(2001)046 [arXiv:hep-th/0105038].[25]N.Vilenkin,”Special functions and the theory of group representations”.Providence,R.I.,American Mathematical Society(AMS),(1968)613p.。