We consider the maximal central extension of the supertranslation algebra in
d=4 and 3, which includes tensor central charges associated to topological
defects such as domain walls (membranes) and strings. We show that for all
N-extended superalgebras these charges are related to nontrivial configurations
on the scalar moduli space. For N=2 theories obtained from compactification on
Calabi-Yau threefolds, we give an explicit realization of the moduli-dependent
charges in terms of wrapped branes.Comment: 11 pages. Sections 2 and 4 expanded, misprints corrected, references
  adde

Ferrara, S.

Porrati, M.

Physics Letters B

English

arXiv

We consider the maximal central extension of the supertranslation algebra in d=4 and 3, which includes tensor central charges associated to topological defects such as domain walls (membranes) and strings. We show that for all N-extended superalgebras these charges are related to nontrivial configurations on the scalar moduli space. For N=2 theories obtained from compactification on Calabi-Yau threefolds, we give an explicit realization of the moduli-dependent charges in terms of wrapped branes.We consider the maximal central extension of the supertranslation algebra in d=4 and 3, which includes tensor central charges associated to topological defects such as domain walls (membranes) and strings. We show that for all N-extended superalgebras these charges are related to nontrivial configurations on the scalar moduli space. For N=2 theories obtained from compactification on Calabi-Yau threefolds, we give an explicit realization of the moduli-dependent charges in terms of wrapped branes.We consider the maximal central extension of the supertranslation algebra in d =4 and 3, which includes tensor central charges associated to topological defects such as domain walls (membranes) and strings. We show that for all N -extended super-algebras these charges are related to nontrivial configurations on the scalar moduli space. For N =2 theories obtained from compactification on Calabi–Yau threefolds, we give an explicit realization of the moduli-dependent charges in terms of wrapped branes

Ferrara, Sergio

Porrati, Massimo

CERN Document Server

hep-th/9711116, NYU-TH-97/11/01CERN-TH/97-318, UCLA/97/TEP/24Central Extensions of Supersymmetry in Four and ThreeDimensionsSergio Ferraraa;b 1 and Massimo Porratic;d 2a Theory Division, CERN, CH-1211 Geneva 23, Switzerlandb Dept. of Physics UCLA, 405 Hilgard Av. Los Angeles, CA 90024, USAc Department of Physics, NYU, 4 Washington Pl., New York NY 10003, USA 3d Rockefeller University, New York NY 10021-6399, USAABSTRACTWe consider the maximal central extension of the supertranslation algebra in d=4and 3, which includes tensor central charges associated to topological defects suchas domain walls (membranes) and strings. We show that for all N-extended super-algebras these charges are related to nontrivial congurations on the scalar modulispace. For N = 2 theories obtained from compactication on Calabi-Yau threefolds,we give an explicit realization of the moduli-dependent charges in terms of wrappedbranes.November 19971e-mail: ferraras@vxcern.cern.ch2e-mail: massimo.porrati@nyu.edu3Permanent address.1 IntroductionCentral extensions of extended supersymmetry algebras in any dimensions have been consideredin recent years in connection to the dynamics of BPS saturates p-branes. Generically, for anN-extended supersymmetry algebra in d dimensions, the structure of the central extension ofthe supertranslation algebra has the form:fQA ; QB g =XPZAB1::p(γ1::pC); A;B = 1; ::; N (1)where, for p = 1, ZAB must contain a singlet under the automorphism group of the supercharges(the R-symmetry). This singlet is a linear combination of the translation generator P and asinglet vector charge Z. This central charge can be absorbed in a redenition of P, but onlywhen spacetime is uncompactied [1]. Upon compactication, its spectrum diers, in general,from the Kaluza Klein spectrum of P. The total number of central charges, including P,correctly adds up to the dimension of the symmetric product of N spinorial representations:(N dim s)(N dim s+ 1)=2, where dim s is the dimension of an irreducible spinor representationin space-time dimension d. The existence of tensor charges in the presence of p-branes has beenderived in ref. [2]; for N = 8, the algebra in eq. (1) has been found in [3].The central extension of Poincare supersymmetry limits the central extension to Lorentzscalars, i.e. p = 0. On the other hand, already some years ago it has been realized that p-branes, i.e. extended objects present in consistent extensions of supergravity theories, can giverise to nonzero tensor central charges [2].Tensorial central charges can be divided in two categories: charges that are sources of of dy-namical gauge elds, and charges that have no associated dynamical gauge eld, and correspondto topological defects of the theory.We will study charges in both categories. Particularly: a) vector charges in 4 dimensions,associated to rank-2 antisymmetric tensors, which are dual to 4-d scalars; b) rank-2 tensorcharges, associated to rank 3 antisymmetric tensors, which are non-dynamical in 4-d. Thelatter case has received quite some attention in recent times in the context of supersymmetricYang-Mills theories [4], since the associated extended object is a BPS saturated membrane.In this paper, we rst consider the central extension of the supersymmetry algebra in dimen-sions 4 and we consider particular physical situations in which these new central charges arenon-vanishing. For the particular case of type II Calabi-Yau compactications, we show that theextended objects source of the tensorial charges lead to new phenomena in the hypermultipletmoduli space. These phenomena can be described in a general setting valid for all N-extendedtheories. We also discuss central extensions of the supertranslation algebra in 3 dimensions, andits relation to three-dimensional U-duality.12 Central Extension of Supersymmetry Algebras in d=4The Haag-Lopuszanski-Sohnius central extension of the super-Poincare algebra for N-extendedsupersymmetry is [5]:fQA ; Q _Bg =  _PAB;fQA ; QB g = Z[AB]: (2)By counting the spinor components in the left-hand side of these equations, it is evident thatsome terms have been omitted in the right-hand side. These additional terms are central chargesof the supertranslation algebra, rather than of the super-Poincare algebra. The complete algebrahas additional terms, as in equation (1), namelyfQA ; Q _Bg =  _PAB +  _ZAB; (ZAA = 0);fQA ; QB g = Z[AB] + Z(AB) : (3)These new terms correspond to string charges in the adjoint of SU(N), and membrane chargesin the twofold symmetric of SU(N).One can easily see that the total number of bosonic generators in the rhs of eqs. (3) addsup to 8N2 + 2N . This is the same dimension as the lhs, which is a symmetric 4N  4N matrix.The tensor charges in eq. (3) can be easily related to geometrical quantities that dependon the asymptotic value of massless scalar elds (moduli). To see this, we need only recallthat in any theory with local supersymmetry, the supersymmetry charge can be expressed in aGauss-like form which involves only the gravitino eld ( A,  A _):QA =ZSdx ^ dx _  A _; QA _ =ZSdx ^ dx _ A : (4)S is a large surface inside a 3-dimensional space-like hypersurface. Quite generally, one maywrite the change of the gravitino under a supersymmetry transformation as:  A _ = DA_ + VAB B_ +12 _MABB + :::: (5)An analogous law holds for  A. Here, D is the standard Lorentz-covariant derivative, thescalar MAB is the gravitino mass matrix, while V AB  is a composite gauge vector, function of thescalar elds in the theory. Notice that it can be decomposed into a traceless part, in the adjointof the R-symmetry SU(N), plus a singlet. The commutators of two supersymmetry charges canbe found most easily by varying eq. (4) with respect to a supersymmetry transformation, andusing eq. (5). The result isfQA ; QB _g =ZSdx ^ dx _VAB  + :::fQA ; QB g =ZSdx ^ dx MAB + ::: (6)2Here the ellipsis stand for the standard terms in the supersymmetry algebra (P and the scalarcentral charges Z [AB]). These equations give the explicit relation between tensor charges andscalar moduli.Eqs. (6) shows that for generic N , the central charge of the domain wall is given in termsof the gravitino mass matrix, MAB, which belongs to the twofold symmetric representation ofthe R-symmetry SU(N), while the central charge of the strings is generated by the compositegauge connection of the R-symmetry.It is easy to show that in the presence of a string there can exist a nonzero vector chargeZ. Explicitly, in the presence of an innite string (either fundamental or solitonic), one maychoose as surface of integration S a cylinder coaxial with the string. For an innite string inuncompactied space, the total vector charge diverges, but the charge density per unit length,dZAB=dl, is nite. At any point along the string it reads:dZAiB=dl = l^iIdxjV Aj B; i; j = 1; 2; 3: (7)Here l^ is a unit vector tangent to the string and the integral is taken around any loop orthogonalto l^ and encircling the string once.Notice that eq. (7) gives not only the vector charge in the adjoint of SU(N), but also thesinglet vector charge. As we said before, even though this charge is not algebraically independentfrom P, and it is indistinguishable from it in uncompactied space, it becomes an independentcharge upon compactication on non-simply connected manifolds.Similarly, using eqs. (6), we can nd that the charge Z(AB) is nonzero in the presence ofa domain wall. Consider for simplicity a flat domain wall located at z = const in Cartesiancoordinates. Then, the second of eqs. (6) gives us the following expression for the tensor-chargedensity (i.e. the charge per unit area of the membrane):Z(AB)12 = [MAB(z = +1)−MAB(z = −1)]; Z(AB)ij = 0 otherwise: (8)In N = 1 (rigid) supersymmetry, there exist examples of both BPS saturated strings (theN = 1 Abelian Higgs model) and BPS saturated membranes [4]. In local supersymmetry, thetensor charges associated to these objects can be expressed in a model-independent way by usingthe superconformal tensor calculus (see for instance [6]), which allows us to write the o-shellgravitino transformation law as:   _ = D _ +34iA _ −34i(; )   _ −  _(; ): (9)In this universal equation, ,  are the fermionic generators of the superconformal group; A and are, respectively, the gauge eld and the generator of the local R-symmetry. The parameters and  are not independent, rather, in any specic model, their dependence on  and the elds3of the theory, , is xed by choosing a conformal gauge in which the Einstein action and thegravitino kinetic term are canonically normalized [7]. The gauge eld A is non-dynamical; bysolving its algebraic equations of motion it can be expressed as a function of the  and theircovariant derivatives. Eqs. (4,9) give a universal formula for the charge density of strings anddomain walls. For domain walls the formula reads (cfr. eq. (8)):Z12 =12[(z = +1)− (z = −1)]; Zij = 0 otherwise: (10)This formula applies in particular to the the case in which the domain wall is due to strong-coupling phenomena or fermion condensates [4]. It holds also when the supergravity Lagrangiancontains higher curvature terms, as the ones due to string or M-theory corrections.In N = 2 supersymmetry, the membrane charges and the algebraically independent stringcharges form a triplet of SU(2). The string charges in the triplet of SU(2) are given by theWilson loopHdxj [V Aj B− (1=2)ABVCj C ]. They do not depend at all on the vector multiplets, sinceonly hypermultiplets contribute to the composite gauge connection of the SU(2) R-symmetry [8].The string charge singlet of SU(2) instead, is a function of vector multiplets only, since hyper-multiplets do not appear in the N = 2 formula for V CC [8].The membrane central charge depends on both hypermultiplet and vector-multiplet moduli,since the formula for the gravitino mass matrix reads [8, 9, 10]MAB = LPAB ; (11)and L is a covariantly holomorphic section of the U(1) bundle associated to the vector moduli.This is in agreement with (in fact, it is required by) the centrally extended algebra in eqs. (3).Indeed, since Z(AB) enters in the fQA ; QB g commutators, it must carry the same U(1) weightas the scalar charge. This is indeed the case, since [10]Z [AB] = ABZ; Z = Lq − Fp: (12)The string charge ZAB, instead, is neutral under U(1), since it carries the same U(1) as thetranslations P.An explicit realization of extended objects with moduli-dependent charge densities can beobtained in type II superstrings compactied on Calabi-Yau threefolds. The study of thesecompactications also provides a check of the properties discovered above, namely that: a) thecharge density of triplet strings only depends on hypermultiplets, b) the charge density of singletstrings only depends on vector multiplets, c) the charge density of domain walls depends onboth vector and hypermultiplet moduli.Let us introduce the following notation: 1 is the string charge density, 2 is the domainwall charge density; J is the Ka¨hler form of the CY space, Ω is the CY holomorphic 3-form,4and ak are the homology k-cycles of the CY space. The string dilaton, , is a hypermultipletboth in type IIA and type IIB theory.We consider rst type IIB superstrings. In this case, the Ka¨hler form is a function of thehypermultiplet moduli only, while Ω depends only on the vector moduli. The 4-D strings presentin this theory are:1. The fundamental 10-d string. In the 4-d Einstein frame its charge density is:1 = exp(2) exp(KJ); KJ  − log(Za6J ^ J ^ J): (13)2. The D-string. Its charge density is: 1 = exp() exp(KJ)3. The D3 brane wrapped on a 2-cycle. Here:1 = exp() exp(KJ)Za2J: (14)4. The D5 brane wrapped on a 4-cycle:1 = exp() exp(KJ)Za4J ^ J: (15)5. The D7 brane wrapped on the CY threefold: 1 = exp().6. The NS5 brane wrapped on a 4-cycle.1 = exp(KJ)Za4J ^ J: (16)All the charge densities listed above are functions of hypermultiplets only; thus all the stringsof type IIa on CY are triplets.To get a domain wall in 4-d, on the other hand, one must wrap a D5 or NS5 brane on a3-cycle. The charge densities now read2 = exp() exp(3KJ=2)V3; for NS5; 2 = exp(2) exp(3KJ=2)V3; for D5: (17)The volume of the minimal-volume 3-cycle is [11]V3 = exp(KΩ=2−KJ)jZa3Ωj; KΩ = − log(iZΩ ^ Ω): (18)Since V3 depends on vector moduli, 2 is a function of both vectors and hypermultiplets, aspredicted by the general SUSY algebra in eq. (3).The type IIA string provides us with another subtle check of our formulae. In type IIAsuperstrings, indeed, J is a function of the vector moduli, while Ω depends on the hypermultipletmoduli only. The 4-d strings in this case come from:51. D4 branes wrapped on 3-cycles, with charge density:1 = exp() exp(KJ)V3 = exp(KΩ=2)jZa3Ωj: (19)2. NS5 branes wrapped on 4-cycles:1 = exp(KJ)Za4J ^ J: (20)Notice that the string obtained from the D4 brane is a triplet of SU(2), since its charge densityeq. (19) depends only on hypermultiplets, but the string obtained from the NS5 brane is asinglet, since its 1 depends only on vector moduli.Finally, the 4-d domain walls of type IIA are:1. D2 branes. Their charge density is: 2 = exp(2) exp(3KJ=2)2. D4 branes wrapped on 2-cycles: 2 = exp(2) exp(3KJ=2)Ra2J .3. D6 branes wrapped on 4-cycles: 2 = exp(2) exp(3KJ=2)Ra4J ^ J .4. D8 branes wrapped on the CY threefold: 2 = exp(2) exp(KJ=2).5. NS5 branes wrapped on 3-cycles: 2 is given by the rst equation in formula (17).As expected, they depend on both hypermultiplets and vector multiplets.3 Central Extension in d=3The analysis done above for four-dimensional extended supersymmetry can be repeated in threedimensions. In d=3, the R-symmetry of theN extended supertranslation algebra isO(N), ratherthan SU(N), and its maximal central extension readsfQA ; QB g = (γC)PAB + (γC)Z(AB) + CZ[AB]; (Z(AB)AB = 0): (21)We see again that by adding to the translations a vector charge, traceless and symmetric underO(N), together with a scalar charge, in the adjoint of O(N), we nd N(2N+1) bosonic charges.This is the dimension of the lhs of eq. (21).The adjoint of O(2N) decomposes as follows under U(N)  O(2N):adj O(2N) = adj U(N) + [2] + [2]; (22)where [2] denotes the twofold antisymmetric of U(N). This property shows that the scalarcentral charges in 3 dimensions come from the scalar central charges in four dimensions plusthe extra scalar charges coming from the dimensional reduction of ZAB and P.6The simplest model with a nonzero central charge Z [AB] is the N = 2 Abelian Higgs model.It is obtained by dimensional reduction of the 4-d Abelian Higgs model, which contains a BPSsaturated string. Upon dimensional reduction, the associated string charge gives rise to a scalarcentral charge Z [AB] = [AB]Z. Physically, this is the charge of the 3-d vortex generated by thedimensionally reduced BPS string.In 3 dimensions, the U-duality group of the N = 16 theory is E8(8). This strongly suggeststhat the 120 scalar charges {in the adjoint of O(16){ together with the charges of the 128\elementary" massless elds {in the left spinor representation of O(16){ should complete the248-dimensional fundamental representation of E8(8).4 ConclusionsIn this paper, we found an explicit universal formula relating the tensor central charges to thegeometry of scalar moduli elds. These formulae are easily generalized to take into account thepossible existence of non-vanishing fermion condensates. We presented this generalization ex-plicitly for the (o-shell) N = 1 superalgebra. Condensates are expected in strongly interactingtheories. In rigid supersymmetry, indeed, it has been shown explicitly that they give rise tomembrane central charges [4].AcknowledgmentsS.F. is supported in part by DOE under grant DE-FGO3-91ER40662, Task C, and by EECScience Program SC1*-CI92-0789. M.P. is supported in part by NSF under grant PHY-9722083.References[1] P.K. Townsend, hep-th/9507048, published in the proceedings of PASCOS ’95 (J.Bagger Ed., World Scientic, Singapore, 1996).[2] J.A. de Azcarraga, J.P. Gauntlett, J.M. Izquierdo and P.K. Townsend, Phys. Rev.Lett. 63 (1989) 2443.[3] I. Bars, Phys. Rev. D54 (1996) 5203, hep-th/9604139.[4] G. Dvali and M. Shifman, Phys. Lett. B396 (1997) 64, ERRATUM-ibid. B407 (1997)452, hep-th/9612128; A. Kovner, M. Shifman and A. Smilga, hep-th/9706089; B.Chibisov and M. Shifman, hep-th/9706141; E. Witten, hep-th/9706109.[5] R. Haag, J.T. Lopuszanski and M. Sohnius, Nucl. Phys. B88 (1975) 257.7[6] P. van Nieuwenhuizen, Phys. Rep. 68 (1981) 189.[7] T. Kugo and S. Uehara, Nucl. Phys. B222 (1983) 125.[8] B de Wit, P.G. Lauwers and A. Van Proeyen, Nucl. Phys. B255 (1985) 569.[9] R. D’ Auria, S. Ferrara and P. Fre, Nucl. Phys. B359 (1991) 705.[10] A. Ceresole, R. D’ Auria, S. Ferrara and A. Van Proeyen, Nucl. Phys. B444 (1995)92, hep-th/9502072.[11] K. Becker, M. Becker and A. Strominger, Nucl. Phys. B456 (1995) 130, hep-th/9507158.8

Central Extensions of Supersymmetry in Four and Three Dimensions

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