Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors

We review our construction of a bifundamental version of the fuzzy 2-sphere and its relation to fuzzy Killing spinors, first obtained in the context of the ABJM membrane model. This is shown to be completely equivalent to the usual (adjoint) fuzzy sphere. We discuss the mathematical details of the b...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2010
Автори:	Nastase, H., Papageorgakis, C.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
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Цитувати:	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors / H. Nastase, C. Papageorgakis // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 45 назв. — англ.

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spelling	Nastase, H. Papageorgakis, C. 2019-02-09T09:39:46Z 2019-02-09T09:39:46Z 2010 Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors / H. Nastase, C. Papageorgakis // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 45 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T75; 81T30 DOI:10.3842/SIGMA.2010.058 https://nasplib.isofts.kiev.ua/handle/123456789/146360 We review our construction of a bifundamental version of the fuzzy 2-sphere and its relation to fuzzy Killing spinors, first obtained in the context of the ABJM membrane model. This is shown to be completely equivalent to the usual (adjoint) fuzzy sphere. We discuss the mathematical details of the bifundamental fuzzy sphere and its field theory expansion in a model-independent way. We also examine how this new formulation affects the twisting of the fields, when comparing the field theory on the fuzzy sphere background with the compactification of the 'deconstructed' (higher dimensional) field theory. This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html. It is a pleasure to thank Sanjaye Ramgoolam for many comments, discussions and collaboration in [9]. CP is supported by the STFC grant ST/G000395/1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors
spellingShingle	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors Nastase, H. Papageorgakis, C.
title_short	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors
title_full	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors
title_fullStr	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors
title_full_unstemmed	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors
title_sort	bifundamental fuzzy 2-sphere and fuzzy killing spinors
author	Nastase, H. Papageorgakis, C.
author_facet	Nastase, H. Papageorgakis, C.
publishDate	2010
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We review our construction of a bifundamental version of the fuzzy 2-sphere and its relation to fuzzy Killing spinors, first obtained in the context of the ABJM membrane model. This is shown to be completely equivalent to the usual (adjoint) fuzzy sphere. We discuss the mathematical details of the bifundamental fuzzy sphere and its field theory expansion in a model-independent way. We also examine how this new formulation affects the twisting of the fields, when comparing the field theory on the fuzzy sphere background with the compactification of the 'deconstructed' (higher dimensional) field theory.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146360
citation_txt	Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors / H. Nastase, C. Papageorgakis // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 45 назв. — англ.
work_keys_str_mv	AT nastaseh bifundamentalfuzzy2sphereandfuzzykillingspinors AT papageorgakisc bifundamentalfuzzy2sphereandfuzzykillingspinors
first_indexed	2025-11-25T21:08:27Z
last_indexed	2025-11-25T21:08:27Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 058, 28 pages Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors? Horatiu NASTASE † and Constantinos PAPAGEORGAKIS ‡ † Instituto de F́ısica Teórica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil E-mail: nastase@ift.unesp.br ‡ Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, UK E-mail: costis.papageorgakis@kcl.ac.uk Received March 26, 2010, in final form July 09, 2010; Published online July 20, 2010 doi:10.3842/SIGMA.2010.058 Abstract. We review our construction of a bifundamental version of the fuzzy 2-sphere and its relation to fuzzy Killing spinors, first obtained in the context of the ABJM membrane model. This is shown to be completely equivalent to the usual (adjoint) fuzzy sphere. We discuss the mathematical details of the bifundamental fuzzy sphere and its field theory expansion in a model-independent way. We also examine how this new formulation affects the twisting of the fields, when comparing the field theory on the fuzzy sphere background with the compactification of the ‘deconstructed’ (higher dimensional) field theory. Key words: noncommutative geometry; fuzzy sphere; field theory 2010 Mathematics Subject Classification: 81T75; 81T30 1 Introduction and Motivation Noncommutative geometry is a tool that finds numerous applications in the description of a wide range of physical systems. A celebrated example appearing in String Theory is in terms of the polarisation phenomenon discovered by Myers, in which N Dp-branes in the presence of transverse Ramond–Ramond flux distribute themselves onto the surface of a higher-dimensional sphere [1]. The physics of the simplest case are captured by a U(N) theory, with the solution involving fuzzy 2-spheres [2, 3, 4]. These are related to families of Hermitian matrices obeying the SU(2) algebra [Xi, Xj ] = 2iεijkXk. (1.1) The Xi enter the physics as ground state solutions to the equations of motion via (1.1). Then their commutator action on the space of all N × N matrices organises the matrices into rep- resentations of SU(2) ' SO(3). An important aspect of the geometry of the fuzzy 2-sphere involves the construction of fuzzy (matrix) spherical harmonics in SU(2) representations, which approach the space of all classical S2 spherical harmonics in the limit of large matrices [3]. This construction of fuzzy spherical harmonics allows the analysis of fluctuations in a non-Abelian theory of Dp-branes to be expressed at large N in terms of an Abelian higher dimensional theory. This describes a D(p + 2) brane wrapping the sphere, with N units of worldvolume magnetic flux. At finite N the higher dimensional theory becomes a noncommutative U(1) with a UV cutoff [5, 6, 7, 8]. ?This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html mailto:nastase@ift.unesp.br mailto:costis.papageorgakis@kcl.ac.uk http://dx.doi.org/10.3842/SIGMA.2010.058 http://www.emis.de/journals/SIGMA/noncommutative.html 2 H. Nastase and C. Papageorgakis In this article we review a novel realisation of the fuzzy 2-sphere involving bifundamental matrices. The objects that crucially enter the construction are discrete versions of Killing spinors on the sphere [9, 10]1. The motivation is similar to the above and comes from the study of the model recently discovered by Aharony, Bergman, Jafferis and Maldacena (ABJM) describing the dynamics of multiple parallel M2-branes on a Zk M-theory orbifold [11], which followed the initial investigations of Bagger–Lambert and Gustavsson (BLG) [12, 13, 14, 15]. The ABJM theory is an N = 6 superconformal Chern–Simons-matter theory with SO(6) R-symmetry and gauge group U(N) × U(N̄). The two Chern–Simons (CS) terms have equal but opposite levels (k,−k) and the matter fields transform in the bifundamental representation. One can use the inverse CS level 1/k as a coupling constant to perform perturbative calculations. At k = 1 the theory is strongly coupled and describes membranes in flat space. For k = 1, 2 the supersymmetry and R-symmetry are nonperturbatively enhanced to N = 8 and SO(8) respectively [11, 16, 17]. It is then possible to use this action to investigate aspects of the AdS4/CFT3 duality, with the role of the ’t Hooft coupling played by λ = N k . The action of the Zk orbifold on the C4 space transverse to the M2’s is such that taking k → ∞ corresponds to shrinking the radius of the M-theory circle and entering a IIA string theory regime. Of particular interest are the ground-state solutions of the maximally supersymmetric mas- sive deformation of ABJM found by Gomis, Rodŕıguez-Gómez, Van Raamsdonk and Verlinde (GRVV) [18]2. The theory still has a U(N) × U(N̄) gauge group and N = 6 supersymmetry but conformal invariance is lost and the R-symmetry is broken down to SU(2)× SU(2)×U(1). Its vacua are expected to describe a configuration of M2-branes blowing up into spherical M5- branes in the presence of transverse flux through a generalisation of the Myers effect. At k = 1 these solutions should have a dual description in terms of the 1 2 -BPS M-theory geometries with flux found in [20, 21]. Interestingly, the matrix part of the above ground-state equation is given by the following simple relation, which we will refer to as the GRVV algebra3: Gα = GαG† βG β −GβG† βG α, (1.2) where Gα are N × N̄ and G† α are N̄ × N matrices respectively. Given that the Myers effect for the M2-M5 system should employ a 3-dimensional surface, one might initially expect this to represent the defining relation for a fuzzy 3-sphere. Moreover, the explicit irreducible solutions of (1.2) satisfy GαG† α = 1, which seems to suggest the desired fuzzy 3-sphere structure. However, we will see that the requisite SO(4) R-symmetry, that would be needed for the existing fuzzy S3 construction of Guralnik and Ramgoolam (GR) [24, 25, 26], is absent in this case. As was also shown in [9], the GR fuzzy S3 construction implies the following algebra εmnpqX+ n X − p X + q = 2 ( (r + 1)(r + 3) + 1 r + 2 ) X+ m, εmnpqX− n X + p X − q = 2 ( (r + 1)(r + 3) + 1 r + 2 ) X− m, (1.3) which must be supplemented with the sphere condition XmXm = X+ mX − m +X− mX + m = (r + 1)(r + 3) 2 ≡ N and the constraints X+ mX + n = X− mX − n = 0. 1The work in [9] was carried out in collaboration with S. Ramgoolam. 2The mass-deformed theory was also presented in [19]. 3The same defining matrix equation appears while looking for BPS funnel solutions in the undeformed ABJM theory and first appeared as such in [22]. Its relation to the M2–M5 system was also investigated in [23]. Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 3 Here r defines a representation of SO(4) ' SU(2)×SU(2) by R+ r and R− r , with labels ( r+1 2 , r−1 2 ) and ( r−1 2 , r+1 2 ) respectively for the two groups, and the X± m are constructed from gamma mat- rices. Even though the algebra (1.3) looks similar to the GRVV algebra (1.2), they coincide only in the ‘fuzziest’ case with r = 1, i.e. the BLG A4-algebra, which in the Van Raamsdonk SU(2)× SU(2) reformulation [27] is R2Xm = −ikεmnpqXnX†pXq. This fact suggests that equation (1.2) does not describe a fuzzy S3. Furthermore, the perturba- tive calculations that lead to the above equation are valid at large k, where the ABJM theory is describing IIA String Theory instead of M-theory and as a result a D2–D4 bound state in some nontrivial background. In the following, we will review how solutions to equation (1.2) actually correspond to a fuzzy 2-sphere, albeit in a realisation involving bifundamental instead of the usual adjoint matrices, by constructing the full spectrum of spherical harmonics. This is equivalent to the usual construc- tion in terms of the SU(2) algebra (1.1). In fact there is a one-to-one correspondence between the representations of the SU(2) algebra Xi and the representations in terms of bifundamental matrices. We will also show how the matrices Gα, which are solutions of the GRVV algebra up to gauge transformations, correspond to fuzzy Killing spinors on the sphere, recovering the usual Killing spinors in the large N limit. The purpose of this article is to present the mathematical aspects of the above construc- tion in a completely model-independent way and highlight some of its features simply starting from (1.2). The reader who is interested in the full background and calculations in the con- text of the ABJM model is referred to [9, 10], where an analysis of small fluctuations around the ground-states at large N , k showed that they can be organised in terms of a U(1) theory on R2,1 × S2, consistent with an interpretation as a D4-brane in Type IIA. The full 3-sphere expected from M-theory then appeared as the large N , k = 1 limit of a fuzzy Hopf fibration, S1/Zk ↪→ S3 F /Zk π→ S2 F , in which the M-theory circle S1/Zk is fibred over the noncommutative sphere base, S2 F . We also discuss how this bifundamental formulation affects the twisting of the fields when ‘deconstructing’ a higher dimensional field theory. This is achieved by studying the field theory around a fuzzy sphere background, where the twisting is necessary in order to preserve supersym- metry. Even though the twisting is usually described in the context of compactifying the higher dimensional ‘deconstructed’ theory, we show how this naturally arises from the bifundamental fuzzy sphere field theory point of view. The rest of this paper is organised as follows. In Section 2 we give the harmonic decomposition of the GRVV matrices and relate them to the fuzzy supersphere. In Section 3 we present a one- to-one map between the adjoint and bifundamental fuzzy sphere constructions, while in Section 4 we establish that connection in terms of the fuzzy Hopf fibration and define the fuzzy version of Killing spinors on S2. We then discuss the resulting ‘deconstruction’ of higher dimensional field theories on the 2-sphere, specifically the issue of twisting of the fields in order to preserve supersymmetry. In Section 5 we review the process and discuss the differences between the adjoint and bifundamental cases, while in Section 6 we briefly discuss a particular application by summarising the results of [9, 10]. We conclude with some closing remarks in Section 7. 2 Constructing the fluctuation expansion Notation. In this section, we will denote by k, l, m, n the matrix indices/indices of states in a vector space, while keeping i, j = 1, . . . , 3 as vector indices on the fuzzy S2. We will also use j for the SU(2) spin and Ylm for S2 spherical harmonics, following the standard notation. The distinction should be clear by the context. 4 H. Nastase and C. Papageorgakis 2.1 Ground-state matrices and symmetries We begin by writing the ground-state solutions to (1.2), found in [18] and given by( G1 ) m,n = √ m− 1δm,n,( G2 ) m,n = √ (N −m)δm+1,n,( G† 1 ) m,n = √ m− 1δm,n,( G† 2 ) m,n = √ (N − n)δn+1,m. (2.1) Using the decomposition of the above complex into real coordinates G1 = X1 + iX2, G2 = X3 + iX4, (2.2) one easily sees that these satisfy 4∑ p=1 XpX p ≡ GαG† α = N − 1, which at first glance would seem to indicate a fuzzy S3 structure. However, note that in the above G1 = G† 1 for the ground-state solution. With the help of (2.2) this results in X2 = 0, which is instead indicative of a fuzzy S2. As usual in the case of fuzzy sphere constructions, the matrices Gα will be used to construct both the symmetry operators (as bilinears in G, G†, also acting on Gα themselves) as well as fuzzy coordinates, used to expand in terms of spherical harmonics on the fuzzy sphere. 2.1.1 GG† relations As a first step towards uncovering the S2 structure we calculate the GG† bilinears( G1G† 1 ) m,n = (m− 1)δmn,( G2G† 2 ) mn = (N −m)δmn,( G1G† 2 ) mn = √ (m− 1)(N −m+ 1)δm,n+1,( G2G† 1 ) mn = √ (N −m)mδm+1,n,( GαG† α ) mn = (N − 1)δmn. Defining Jα β = GαG† β we get the following commutation relation [Jα β , J µ ν ] = δµ βJ α ν − δα ν J µ β . These are commutation relations of the generators of U(2). Then the Ji = (σ̃i)α βJ β α are the generators of SU(2) that result in the usual formulation of the fuzzy4 S2, in terms of the algebra [Ji, Jj ] = 2iεijkJk. (2.3) The trace J ≡ Jα α = N−1 is a trivial U(1) ' U(2)/SU(2) generator, commuting with everything else. 4Note that more correctly, we should have written Jα β = GαG† β and Ji = (σ̃i) α βJβ α = (σi)β αJβ α, but in the following we will stick to the notation Jα β . The kind of matrix multiplication that one has will be made clear from the context. Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 5 2.1.2 G†G relations Next, we calculate the G†G combinations( G† 1G 1 ) mn = (m− 1)δmn,( G† 2G 2 ) mn = (N −m+ 1)δmn −Nδm1δn1,( G† 1G 2 ) mn = √ (m− 1)(N −m)δm+1,n,( G† 2G 1 ) mn = √ (m− 2)(N −m+ 1)δm,n+1,( G† αG α ) mn = Nδmn −Nδm1δn1 and define J̄β α = G† αGβ. The commutation relations for the above then form another copy of U(2) [J̄α β , J̄ µ ν ] = δµ β J̄ α ν − δα ν J̄ µ β and similarly, J̄i = (σ̃i)α β J̄ β α once again satisfy the usual SU(2) algebra5, for another fuzzy S2 [J̄i, J̄j ] = 2iεijkJ̄k. The trace (J̄)mn = (J̄α α )mn = Nδmn −Nδm1δn1, (2.4) which is a U(1) ' U(2)/SU(2) generator, commutes with the SU(2) generators J̄i, though as a matrix does not commute with the generators J1 2 and J2 1 of the first set of SU(2) generators. At this point, it seems that we have two SU(2)’s, i.e. SO(4) ' SU(2) × SU(2) as expected for a 3-sphere, even though we have not yet shown that these are proper space symmetries: We have only found that the J, J̄ satisfy a certain symmetry algebra. In fact, we will next see that these are not independent but rather combine into a single SU(2). 2.1.3 Symmetry acting on bifundamental (N, N̄) matrices All the (N, N̄) bifundamental scalar matrices are of the type G, GG†G, GG†GG†G, . . .. The simplest such terms are the Gα matrices themselves, the action of the symmetry generators on which we will next investigate. It is easy to check that the matrices Gα satisfy G1G† 2G 2 −G2G† 2G 1 = G1, G2G† 1G 1 −G1G† 1G 2 = G2. Using the definitions of Ji and J̄i, we find JiG α −GαJ̄i = (σ̃i)α βG β. (2.5) The G1, G2 transform like the (1, 0) and (0, 1) column vectors of the spin-1 2 representation with the J ’s and J̄ ’s matrices in the u(N)× u(N̄) Lie algebra. 5Again, note that we should have written J̄α β = G† αGβ which emphasises that for J̄ , the lower index is the first matrix index, and J̄i = (σ̃i) α β J̄α β = (σi)β αJ̄α β , which emphasises that as matrices, the J̄i are defined with the Pauli matrices, whereas Ji was defined with their transpose. However, we will again keep the notation J̄β α . 6 H. Nastase and C. Papageorgakis By taking Hermitian conjugates in (2.5), we find that the antibifundamental fields, G† α, transform as G† αJi − J̄iG † α = G† β(σ̃i)β α. (2.6) Therefore the Gα, G† α form a representation when acted by both Ji and J̄i, but neither symmetry by itself gives a representation for Gα, G† α. This means that the geometry we will be constructing from bifundamental fluctuation modes has a single SU(2) symmetry, as opposed to two. Equations (2.5) and (2.6) imply relations giving transformations between Ji and J̄i, thus showing they represent the same symmetry G† γJiG γ = (N + 1)J̄i, Gγ J̄iG † γ = (N − 2)Ji. Writing the action of the full SU(2)×U(1) on the Gα, including the U(1) trace J̄ , we obtain Jα βG γ −Gγ J̄α β = δγ βG α − δα βG γ , (2.7) while taking Hermitian conjugates of (2.7) we obtain the U(2) transformation of G† α, J̄α βG † γ −G† γJ α β = −δα γG † β + δα βG † γ . The consequence of the above equations is that Gα has charge 1 under the U(1) generator J̄ . Thus a global U(1) symmetry action on Gα does not leave the solution invariant, and we need to combine with the action of J̄ from the gauge group to obtain an invariance. We next turn to the construction of fuzzy spherical harmonics out of Gα. 2.2 Fuzzy S2 harmonics from U(N) × U(N̄) with bifundamentals All bifundamental matrices of U(N)×U(N̄), are maps between two different vector spaces. On the other hand, products of the bilinears GG† and G†G are adjoint matrices mapping back to the same vector space. Thus, the basis of ‘fuzzy spherical harmonics’ on our fuzzy sphere will be constructed out of all possible combinations: U(N) adjoints like GG†, GG†GG†, . . ., U(N̄) adjoints like G†G, G†GG†G, . . ., and bifundamentals like G, GG†G, . . . and G†, G†GG†, . . .. 2.2.1 The adjoint of U(N) Matrices like GG† act on an N dimensional vector space that we call V+. Thus the space of linear maps from V+ back to itself, End(V+), is the adjoint of the U(N) factor in the U(N) × U(N̄) gauge group and GG† are examples of matrices belonging to it. The space V+ forms an irreducible representation of SU(2) of spin j = N−1 2 , denoted by VN V+ = VN . The set of all operators of the form GG†, GG†GG†, . . . belong in End(V+) and can be expanded in a basis of ‘fuzzy spherical harmonics’ defined using the SU(2) structure. Through the SU(2) generators Ji we can form the fuzzy spherical harmonics as Y 0 = 1, Y 1 i = Ji, Y 2 ((i1i2)) = J((i1Ji2)), Y l ((i1···il)) = J((i1 · · ·Jil)). In the above, the brackets ((i1 · · · il)) denote traceless symmetrisation. The complete space of N × N matrices can be expanded in the fuzzy spherical harmonics with 0 ≤ l ≤ 2j = N − 1. One indeed checks that N2 = 2j∑ l=0 (2l + 1). Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 7 Then, a general matrix in the adjoint of U(N) can be expanded as A = N−1∑ l=0 l∑ m=−l almYlm(Ji), where Ylm(Ji) = ∑ i f ((i1···il)) lm Ji1 · · ·Jil . The Ylm(Ji) become the usual spherical harmonics in the ‘classical’ limit, when N →∞ and the cut-off in the angular momentum is removed. In conclusion, all the matrices of U(N) can be organised into irreps of SU(2) constructed out of Ji, which form the fuzzy spherical harmonics Ylm(Ji). 2.2.2 The adjoint of U(N̄) In a fashion similar to the U(N) case, the matrices G†G, G†GG†G, . . ., are linear endomorphisms of V−. These matrices are in the adjoint of the U(N̄) factor of the U(N)×U(N̄) gauge group, and will be organised into irreps of the SU(2) constructed out of J̄i. However, we now have a new operator: We have already noticed in (2.4) that the U(1) generator J̄ is nontrivial. We can express it as J̄ = G† αG α = N −NĒ11. This means that End(V−) contains in addition to the identity matrix, the matrix Ē11 which is invariant under SU(2). If we label the basis states in V− as \|e−k 〉 with k = 1, . . . , N , then Ē11 = \|e−1 〉〈e − 1 \|. This in turn means that V− is a reducible representation V− = V − N−1 ⊕ V − 1 . The first direct summand is the irrep of SU(2) with dimension N − 1 while the second is a one- dimensional irrep. Indeed, one checks that the J̄i’s annihilate the state \|e−1 〉, which is necessary for the identification with the one-dimensional irrep to make sense. As a result, the space End(V−) decomposes as follows End(V−) = End(V − N−1)⊕ End(V − 1 )⊕Hom(V − N−1, V − 1 )⊕Hom(V − 1 , V − N−1), that is, the matrices split as Mµν = (Mij ,M11,M1i,Mi1). The first summand has a decomposi- tion in terms of another set of fuzzy spherical harmonics Ylm(J̄i) = ∑ i f ((i1···il)) lm J̄i1 · · · J̄il , for l going from 0 to N − 2, since (N − 1)2 = N−2∑ l=0 (2l + 1). This gives only matrices in the (N − 1) block, i.e. the End(V − N−1). The second summand is just one matrix transforming in the trivial irrep, Ē11. The remaining two N − 1 dimensional spaces of matrices cannot be expressed as products of J̄i. They are spanned by Ē1k = \|e−1 〉〈e − k \| ≡ g−−1k , Ēk1 = \|e−k 〉〈e − 1 \| ≡ g−−k1 , 8 H. Nastase and C. Papageorgakis which are like spherical harmonics for Hom(V − N−1, V − 1 )⊕Hom(V − 1 , V − N−1). They transform in the N−1 dimensional irrep of SU(2) under the adjoint action of J̄i and are zero mode eigenfunctions of the U(1) symmetry operator J̄ . Therefore, one can expand a general matrix in the adjoint of U(N̄) as Ā = ā0Ē11 + N−2∑ l=0 l∑ m=−l ālmYlm(J̄i) + N∑ k=2 bkg −− 1k + N∑ k=2 b̄kg −− k1 (2.8) and note that we could have replaced Ē11 with the U(1) generator J̄ by redefining ā0 and ā00. In the large N limit the Ylm(J̄i) become the ordinary spherical harmonics of S2, just like Ylm(Ji). There are order N2 of these modes, which is appropriate as the fuzzy S2 can roughly be thought of as a 2-dimensional space with each dimension discretised in N units. The mode ā0, bk and b̄k can be neglected at large N , as they have much less than N2 degrees of freedom. 2.2.3 SU(2) harmonic decomposition of bifundamental matrices As in the case of the U(N̄) matrices, the bifundamental matrices of the form G, GG†G, . . . giving physical fluctuating fields, are not enough to completely fill Hom(V−,V+). Given the decomposition V− = V − N−1 ⊕ V − 1 , we decompose Hom(V−,V+) as Hom(V−,V+) = Hom(V − N−1, V + N )⊕Hom(V − 1 , V + N ), i.e. the matrices Mµν as (Miν ,M1ν). The first summand has dimension N(N − 1), while the second has dimension N and forms an irreducible representation of SU(2). Since the V − N−1 and V + N are irreps of SU(2) we can label the states with the eigenvalue of J̄3, J3 respectively. Given our normalisation of the SU(2) generators in (2.3), the usual spin is Jmax 3 2 . The matrices in Hom(V − N−1, V + N ) are of the form \|e+m〉〈e−n \|, where m = −N+1 2 , −N+3 2 , . . . , N−1 2 , n = −N+2 2 , −N+4 2 , . . . , N−2 2 denote the eigenvalues of J3 2 . These are spanned by matrices of the form G(J̄i1)(J̄i2) · · · (J̄il), i.e. the matrix G times matrices in End(V − N−1). The operators in Hom(V − N−1, V + N ) transform in representations of spin l+ 1 2 for l = 0, . . . , N−2. The dimensions of these representations correctly add up to N−2∑ l=0 (2l + 2) = N(N − 1). This then gives the SU(2) decomposition of Hom(V − N−1, V + N ) as Hom(V − N−1, V + N ) = N−2⊕ l=0 Vl+1/2. On the other hand, matrices \|e+k 〉〈e − 1 \| ≡ Êk1 ∈ Hom(V − 1 , V + N ) cannot be written in terms of the G’s and G†’s alone, because Gα acting on \|e−1 〉 gives zero. The index k runs over the N states in V+. Here Êk1 are eigenfunctions of the operator Ē11 with unit charge, Êk1Ē11 = Êk1. Combining all of the above, the bifundamental fluctuations rα can be expanded as follows rα = rα βG β + N∑ k=1 tαk Êk1, Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 9 with rα β = N−2∑ l=0 l∑ m=−l (rlm)α βYlm(Ji). We further decompose rα β into a trace and a traceless part and define sα β = rα β − 1 2 δα β r γ γ , r = rγ γ , Tα = tαk Êk1. Thus the complete expansion of rα is given simply in terms of rα = rGα + sα βG β + Tα. (2.9) We could equivalently have written rα = N−2∑ l=0 l∑ m=−l (rlm)α βG βYlm(J̄i) + N∑ k=1 tαk Êk1 using the spherical harmonics in J̄ in (2.8). In the following, we will choose, without loss of generality, to work with (2.9). Until now we have focused on matrices in Hom(V−,V+) but the case of Hom(V+,V−) is similar. The matrices G†, G†GG†, . . . will also form a representation of SU(2) given by J̄ ∼ G†G, times a G† matrix. Once again one needs to add an extra T †α = (tαk )∗F̂1k fluctuation in order to express the matrices F̂1k ≡ \|e−1 〉〈e + k \| ∈ Hom(V + N , V − 1 ). In fact, the result for the complete fluctuating field can be obtained by taking a Hermitian conjugate of (2.9), yielding r†α = G† αr +G† βs β α + T †α. 2.3 Fuzzy superalgebra The matricesGα and Ji can be neatly packaged into supermatrices which form a representation of the orthosymplectic Lie superalgebra OSp(1\|2). The supermatrix is nothing but the embedding of the N × N̄ matrices into U(2N). The adjoint fields live in the ‘even subspace’, while the bifundamentals in the ‘odd subspace’. For a generic supermatrix M = ( A B C D ) the superadjoint operation is M ‡ = ( A† C† −B† D† ) . For Hermitian supermatrices this is X = ( A B −B† D ) , with A = A† and D = D† [28]. This gives the definition of the supermatrices Ji = ( Ji 0 0 J̄i ) and Jα = ( 0 √ NGα − √ NG† α 0 ) , 10 H. Nastase and C. Papageorgakis where we raise and lower indices as Gα = εαβG β, with ε = iσ̃2 = −iσ2. Then the SU(2) algebra together with the relation (2.5) and the definition of Ji, J̄i result in the following (anti)commutation relations [Ji,Jj ] = 2iεijkJk, [Ji,Jα] = (σ̃i)αβJ β, {Jα,Jβ} = −(σ̃i)αβJi = −(iσ̃2σ̃i)αβJi, which is the defining superalgebra OSp(1\|2) for the fuzzy supersphere of [29]. It is known that the only irreducible representations of OSp(1\|2) split into the spin-j plus the spin-(j− 1 2) representations of SU(2), which correspond precisely to the irreducible representation for the Ji (spin j) and J̄i (spin j − 1/2) that we are considering here6. As a result, the most general representations of the fuzzy superalgebra, including Gα be- sides Ji, J̄i, coincide with the most general representations of the two copies of SU(2). This points to the fact that perhaps the representations in terms of Gα are equivalent to the repre- sentations of SU(2). Next we will see that this is indeed the case. 3 Equivalence of fuzzy sphere constructions We now prove that our new definition of the fuzzy 2-sphere in terms of bifundamentals is equivalent to the usual definition in terms of adjoint representations of the SU(2) algebra. The ABJM bifundamental scalars are interpreted as Matrix Theory (N × N) versions of Euclidean coordinates. Accordingly, for our fuzzy space solution in the large N -limit one writes Gα → √ Ngα, with gα some commuting classical objects, to be identified and better under- stood in due course. In that limit, and similarly writing Ji → Nxi, J̄i → Nx̄i, one has from Sections 2.1.1 and 2.1.2 that the coordinates xi = (σ̃i)α βg βg∗α, x̄i = (σ̃i)α βg ∗ αg β (3.1) are two versions of the same Euclidean coordinate on the 2-sphere, xi ' x̄i. In the above construction the 2-sphere coordinates xi, x̄i are invariant under multiplication of the classical objects gα by a U(1) phase, thus we can define objects g̃α modulo such a phase, i.e. gα = eiα(~x)g̃α. The GRVV matrices (2.1), that from now on we will denote by G̃α instead of Gα, are fuzzy versions of representatives of g̃α, chosen such that g̃1 = g̃†1 (one could of course have chosen a different representative for g̃α such that g̃2 = g̃†2 instead). In terms of the gα, equation (3.1) is the usual Hopf map from the 3-sphere gαg†α = 1 onto the 2-sphere xixi = 1, as we will further discuss in the next section. In this picture, the phase is simply the coordinate on the U(1) fibre of the Hopf fibration, while the g̃α’s are coordinates on the S2 base. While gα are complex coordinates acted upon by SU(2), the g̃α are real objects acted upon by the spinor representation of SO(2), so they can be thought of as Lorentz spinors in two dimensions, i.e. spinors on the 2-sphere. The fuzzy version of the full Hopf map, Ji = (σ̃i)α βG βG† α, can be given either using Gα = UG̃α or Gα = ˜̂ GαÛ . The U and Û are unitary matrices that can themselves be expanded in terms of fuzzy spherical harmonics U = ∑ lm UlmYlm(Ji), with UU † = Û Û † = 1, implying that in the large-N limit (U, Û) → eiα(~x). 6See for instance Appendix C of [28]. The general spin-j is the Ji representation constructed from the GRVV matrices, while the general spin j − 1 2 is the J̄i representation constructed from the GRVV matrices. Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 11 That means that by extracting a unitary matrix from the left or the right of Gα, i.e. modulo a unitary matrix, the resulting algebra for G̃α −G̃α = G̃βG̃† βG̃ α − G̃αG̃† βG̃ β (3.2) should then be exactly equivalent to the usual SU(2) algebra that appears in the adjoint con- struction: Both should give the same description of the fuzzy 2-sphere. We would next like to prove this equivalence for all possible representations. 3.1 Representations We first note that the irreducible representations of the algebra (3.2), given by the matrices (2.1), indeed give the most general irreducible representations of SU(2). Defining J± = J1 ± iJ2, J̄± = J̄1 ± iJ̄2, we obtain from (2.1) that (J+)m,m−1 = 2 √ (m− 1)(N −m+ 1) = 2αN−1 2 ,m−N+1 2 , (J−)n−1,n = 2 √ (n− 1)(N − n+ 1) = 2αN−1 2 ,n−N+1 2 , (J3)mn = 2 ( m− N + 1 2 ) δmn and (J̄+)m,m−1 = 2 √ (m− 2)(N −m+ 1) = 2αN−2 2 ,m−N+2 2 , (J̄−)n−1,n = 2 √ (n− 2)(N − n+ 1) = 2αN−2 2 ,n−N+2 2 , (J̄3)mn = 2 ( m− N + 2 2 ) δmn +Nδm1δn1, whereas the general spin-j representation of SU(2) is (J+)m,m−1 = αj,m, (J−)n−1,n = αj,n, (J3)mn = mδmn (and the rest zero), where αjm ≡ √ (j +m)(j −m+ 1) and m ∈ −j, . . . ,+j takes 2j + 1 values. Thus the representation for Ji is indeed the most general N = 2j + 1 dimensional representation, and since (J̄+)11 = (J̄−)11 = (J̄3)11 = 0, the representation for J̄i is also the most general (N − 1) = 2(j− 1 2)+1 dimensional representation. We still have the U(1) generators completing the U(2) symmetry, which in the case of the irreducible GRVV matrices G̃α are diagonal and give the fuzzy sphere constraint G̃αG̃† α ∝ 1l, G̃† αG̃α ∝ 1l, J = J1 1 + J2 2 = (N − 1)δmn, J̄ = J̄1 1 + J̄2 2 = Nδmn −Nδm1δn1, where again (J̄)11 = 0, since J̄i is in the N −1×N −1 dimensional representation: The element E11 = δm1δn1 is a special operator, so the first element of the vector space on which it acts is also special, i.e. V− = V − N−1 ⊕ V − 1 . Moving to reducible representations of SU(2), the Casimir operator ~J2 = JiJi giving the fuzzy sphere constraint is diagonal, with blocks proportional to the identity. The analogous object that gives the fuzzy sphere constraint in our construction is the operator J = GαG† α. 12 H. Nastase and C. Papageorgakis Indeed, in the case of reducible matrices modulo unitary transformations, G̃α, we find (in the same way as for ~J2 = JiJi for the SU(2) algebra) J = diag((N1 − 1) 1lN1×N1 , (N2 − 1) 1lN2×N2 , . . . ) (3.3) and similarly for J̄ = G† αGα J̄ = diag ( N1 ( 1− E (1) 11 ) 1lN1×N1 , N2 ( 1− E (2) 11 ) 1lN2×N2 , . . . ) . (3.4) 3.2 GRVV algebra → SU(2) algebra For this direction of the implementation one does not need to consider the particular represen- tations of the algebra; the matrices G̃α will be kept as arbitrary solutions. We define as before, but now for an arbitrary solution Gα, GαG† β ≡ Jα β ≡ Ji(σ̃i)α β + Jδα β 2 . (3.5) Using the GRVV algebra it is straightforward to verify that GαG† α ≡ J commutes with Jk. Multiplying (3.2) from the right by (σ̃k)γ αG † γ , one obtains −Jk = GβG† βJk − Jα βJ β γ(σ̃k)γ α. Using the definition for the Jα β factors in (3.5) and the relation [J, Jk] = 0, one arrives at −Jk = i 2 εijkJiJj , which is just the usual SU(2) algebra. It is also possible to define G† αG β ≡ J̄α β ≡ J̄i(σ̃i)β α + J̄δβ α 2 and similarly obtain [J̄ , J̄k] = 0. By multiplying (3.2) from the left by (σ̃k)γ αG † γ , we get in a similar way −J̄k = i 2 εijkJ̄iJ̄j . Thus the general SU(2) algebras for Ji and J̄i indeed follow immediately from (3.2) without restricting to the irreducible GRVV matrices. 3.3 SU(2) algebra → GRVV algebra This direction of the implementation is a priori more problematic since, as we have already seen, the representations of Ji and J̄i are not independent. For the irreducible case in particular, V + N is replaced by the representation V − N−1 ⊕ V − 1 , so we need to generalise this identification to reducible representations in order to prove our result. As we will obtain this relation at the end of this section and it should have been the starting point of the proof, we will close with some comments summarising the complete logic. We will first try to understand the classical limit. The Hopf fibration (3.1) can be rewritten, together with the normalisation condition, as gαg∗β = 1 2 [ xi(σ̃i)α β + δα β ] . Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 13 By extracting a phase out of gα, we should obtain the variables g̃α on S2 instead of S3. Indeed, the above equations can be solved for gα by gα = ( g1 g2 ) = eiφ√ 2(1 + x3) ( 1 + x3 x1 − ix2 ) = eiφg̃α, (3.6) where eiφ is an arbitrary phase. In the fuzzy case Gα and G† β do not commute, and there are two different kinds of equations corresponding to Ji and J̄i, GαG† β ≡ 1 2 [ Ji(σ̃i)α β + δα βJ ] , G† βG α ≡ 1 2 [ J̄i(σ̃i)α β + δα β J̄ ] . (3.7) We also impose that [J, Jk] = 0, [J̄ , J̄k] = 0, so that J and J̄ are diagonal and proportional to the identity in the irreducible components of Ji. We solve the first set of equations in (3.7) by writing G1G† 1 = 1 2(J + J3), for which the most general solution is G1 = TU , with T a Hermitian and U a unitary matrix. Since J + J3 is real and diagonal, by defining T = 1√ 2 (J + J3)1/2 we obtain Gα = ( G1 G2 ) = ( J + J3 J1 − iJ2 ) T−1 2 UN×N = G̃αUN×N . (3.8) Thus G̃α is also completely determined by Ji, J . Similarly, the second set of equations in (3.7) can be solved by considering G† 1G 1 = 1 2(J̄+ J̄3), for which the most general solution is G1 = Û T̃ , where as before T̃ = 1√ 2 ( J̄ + J̄3 )1/2 , to obtain Gα = ( G1 G2 ) = ÛN̄×N̄ T̃−1 2 ( J̄ + J̄3 J̄1 − iJ̄2 ) = Û ˜̂ Gα. (3.9) Thus ˜̂ Gα is completely determined by J̄i, J̄ . Comparing the two formulae for Gα we see that they are compatible if and only if Û = TUT̃−1 and J̄1 − iJ̄2 = T̃ 2U−1T−1(J1 − iJ2)T−1U, (3.10) where U is an arbitrary unitary matrix. These equations define an identification between the two representations of SU(2), in terms of Ji and J̄i, needed in order to establish the equivalence with the GRVV matrices. We now analyse the equivalence for specific representations. For the irreducible represen- tations of SU(2), we define J̄i from Ji as before (V + N → V − N−1 ⊕ V − 1 ) and J = (N − 1) 1lN×N , J̄ = N(1 − E11) 1lN×N . For reducible representations of SU(2), Ji can be split such that J3 is block-diagonal, with various irreps added on the diagonal. One must then take J and J̄ of the form in (3.3) and (3.4). The condition (3.10) is solved by U = 1 and J1, J2 block diagonal, with the blocks being the irreps of dimensions N1, N2, N3, . . . , and the J̄1, J̄2 being also block diagonal, but where each Nk × Nk irrep block is replaced with the (Nk − 1) × (Nk − 1) irrep block, plus an E(k) 11 , just as for the GRVV matrices. We can hence summarise the proof a posteriori in the following steps: 14 H. Nastase and C. Papageorgakis 1. Start with Ji (i = 1, 2, 3) in the reducible representation of SU(2), i.e. block diagonal with the blocks being irreps of dimensions N1, N2, N3, . . . . 2. Take J = GαG† α and J̄ = G† αGα as in (3.3) and (3.4) since these are necessary conditions for the Gα to satisfy the GRVV algebra. The condition [J, Jk] = 0 is used here. 3. The J̄i are completely determined (up to conventions) from Ji, J and J̄ by (3.10) and the condition [J̄ , J̄k] = 0. 4. The G̃α are then uniquely determined by (3.8), while the ˜̂ Gα by (3.9). 5. The G̃α and ˜̂ Gα defined as above indeed satisfy the GRVV algebra. 4 Fuzzy Hopf fibration and fuzzy Killing spinors Having established the equivalence between the adjoint (usual) and the bifundamental (in terms of G̃α) formulations of the fuzzy S2 we turn towards ascribing an interpretation to the matri- ces G̃α themselves. 4.1 Hopf fibration interpretation One such interpretation was alluded to already in (2.2), where the fuzzy (matrix) coordinates Gα were treated as complex spacetime coordinates. The irreducible GRVV matrices satisfy G̃1G̃† 1 + G̃2G̃† 2 = N − 1 and G̃1 = G̃† 1. The first relation suggests a fuzzy 3-sphere, but the second is an extra constraint which reduces the geometry to a 2d one. This is in agreement with the fuzzy S2 equivalence that we already established in the previous section. The matrices G̃α are viewed as representatives when modding out the U(N) symmetry, and the condition G̃1 = G̃† 1 amounts to a choice of representative of the equivalence class. The construction of the fuzzy S2 in usual (Euclidean) coordinates was obtained by Ji = (σ̃i)α βG βG† α, xi = Ji√ N2 − 1 ⇒  x1 = J1√ N2 − 1 = 1√ N2 − 1 ( G1G† 2 +G2G† 1 ) , x2 = J2√ N2 − 1 = i√ N2 − 1 ( G1G† 2 −G2G† 1 ) , x3 = J3√ N2 − 1 = 1√ N2 − 1 ( G1G† 1 −G2G† 2 ) , Gα √ N → gα and we already stated that the relation between gα and xi is the classical Hopf map S3 π→ S2, (3.1). Indeed, the description of the Hopf map in classical geometry is given as follows: One starts with Cartesian coordinates X1, X2, X3, X4 on the unit S3 with X2 1 +X2 2 +X2 3 +X2 4 = 1 and then goes to complex variables Z1 = X1 + iX2, Z2 = X3 + iX4, satisfying ZαZ∗ α = 1. The Hopf map defines Cartesian coordinates on the unit S2 base of the fibration by xi = (σ̃i)α βZ βZ∗ α, (4.1) Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 15 which is invariant under an S1 fibre defined by multiplication of Zα by a phase. The xi are Euclidean coordinates on an S2 since xixi = (σ̃i)α β(σ̃i)µ νZ βZ∗ αZ νZ∗ µ = 1 and this identifies Zα ≡ gα from above. Let us now work in the opposite direction, starting from the classical limit and discretising the geometry by demoting the Hopf map (4.1) from classical coordinates to finite matrices. We need matrices for Zα which we call Gα. The coordinates xi transform in the spin-1 representation of SU(2). If we want to build them from bilinears of the form G†G we need G, G† to transform in the spin-1 2 representation. We also want a gauge symmetry to extend the U(1) invariance of Zα (the S1 fiber of the Hopf map), and for N -dimensional matrices U(N) is the desired complex gauge invariance that plays that role. In the usual fuzzy 2-sphere, the xi are operators mapping an irreducible N -dimensional SU(2) representation VN to itself. It is possible to do this in an SU(2)-covariant fashion because the tensor product of spin-1 with VN contains VN . Since Gα are spin-1 2 , and 1 2 ⊗VN = VN+1⊕VN−1 does not contain VN , we need to work with reducible representations in order to have Gα map the representation back to itself. The simplest thing to do would be to consider the representation VN ⊕ VN−1. The next simplest thing is to work with VN ⊕ (VN−1 ⊕ V1) and this possibility is chosen by the GRVV matrices [18] and allows a gauge group U(N) × U(N̄) which has a Z2 symmetry of exchange needed to preserve parity. So the unusual property of the GRVV matrices G̃, the difference between V+ = VN and V− = VN−1 ⊕ V1 follows from requiring a matrix realisation of the fuzzy S2 base of the Hopf fibration. These in turn lead to the SU(2) decompositions of End(V+), End(V−), Hom(V+,V−), Hom(V−,V+), for the fluctuation matrices that we saw in Section 2.2.7 The xi, G, G† are operators in V+⊕V− which is isomorphic, as a vector space, to VN ⊗V2. The endomorphisms of VN correspond to the fuzzy sphere. The N states of VN generalise the notion of points on S2 to noncommutative geometry. The 2-dimensional space V2 is invariant under the SU(2). It is acted on by G, G† which have charge +1, −1 under the U(1) (corre- sponding to (J, J̄)) acting on the fibre of the Hopf fibration, so we also have two points on top of our fuzzy S2. Since in this subsection we looked at a fibration of S3, we need to emphasise that the fluctuation analysis does not have enough modes to describe the full space of functions on S3, even if we drop the requirement of SO(4) covariance and allow for the possibility of an SU(2)× U(1) description. As we explained above, the only remnant of the circle in the matrix con- struction is the multiplicity associated with having states \|+〉, \|−〉 in V+ and V−. A classical description of the S3 metric as a Hopf fibration contains a coordinate y transverse to the S2. Instead, the matrix fluctuations of our solution are mapped to functions on S2 and hence lead to a field theory on S2. 4.2 Killing spinor interpretation We will close this circle of arguments by interpreting the classical objects g̃α, obtained in the large-N limit of G̃α, as Killing spinors and fuzzy Killing spinors on the 2-sphere respectively. We have seen that in the classical limit the relation between Ji and Gα becomes the first Hopf map (3.1), and hence can be thought of as its fuzzy version. However, the above Hopf relation is invariant under multiplication by an arbitrary phase corresponding to shifts on the S1 fibre, 7The usual fuzzy S2 has also been discussed in terms of the Hopf fibration, where the realisation of the SU(2) generators in terms of bilinears in Heisenberg algebra oscillators yields an infinite dimensional space which admits various projections to finite N constructions [30]. In that case the xi are not bilinears in finite matrices. 16 H. Nastase and C. Papageorgakis so the objects g̃α obtained by extracting that phase in (3.6), i.e. g̃α = 1√ 2(1 + x3) ( 1 + x3 x1 − ix2 ) , (4.2) are instead defined on the classical S2. In the Hopf fibration, the index of gα is a spinor index of the global SO(3) symmetry for the 2-sphere. By extracting the S1 phase one obtains a real (or rather, subject to a reality condition) g̃α and the α can be thought of as describing a (Majorana) spinor of the SO(2) local Lorentz invariance on the 2-sphere. We will argue that the latter is related to a Killing spinor. Note that this type of index identification easily extends to all even spheres. In the fuzzy version of (4.2), the G̃α obtained from Gα by extracting a unitary matrix, are real objects defined on the fuzzy S2. They equal the GRVV matrices in the case of irreducible representations, or G̃ = ( J + J3 J1 − iJ2 ) T−1 2 in general. The standard interpretation, inherited from the examples of the SU(2) fuzzy 2-sphere and other spaces, is that the matrix indices give rise to the dependence on the sphere coordinates and the index α is a global symmetry index. However, we have just seen that already in the classical picture one can identify the global symmetry spinor index with the local Lorentz spinor index. Therefore we argue that the correct interpretation of the classical limit for G̃α is as a spinor with both global and local Lorentz indices, i.e. the Killing spinors on the sphere ηαI . In the following we will use the index α interchangeably for the two. In order to facilitate the comparison with the Killing spinors, we express the classical limit of the Ji–G̃α relation as xi ' x̄i = (σi)α β g̃†β g̃ α. (4.3) Killing spinors on Sn We now review some of the key facts about Killing spinors that we will need for our discussion. For more details, we refer the interested reader to e.g. [31, 32, 33, 34, 35]. On a general sphere Sn, one has Killing spinors satisfying Dµη(x) = ± i 2 mγµη(x). There are two kinds of Killing spinors, η+ and η−, which in even dimensions are related by the chirality matrix, i.e. γn+1, through η+ = γn+1η −, as can be easily checked. The Killing spinors on Sn satisfy orthogonality, completeness and a reality condition. The latter depends on the application, sometimes taken to be the modified Majorana condition, which mixes (or identifies) the local Lorentz spinor index with the global symmetry spinor index of Sn. For instance, on S4 the orthogonality and completeness are respectively8, η̄IηJ = ΩIJ and ηα J η̄ J β = −δα β , 8The charge conjugation matrix in n dimensions satisfies in general CT = κC, γT µ = λCγµC−1, where κ = ±, λ = ± and it is used to raise/lower indices. The Majorana condition is then given by η̄ = ηT C. Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 17 where the index I is an index in a spinorial representation of the SO(n+1)G invariance group of the sphere and the index α is an index in a spinorial representation of the SO(n)L local Lorentz group on the sphere. The indices are then identified by the modified Majorana spinor condition as follows9 η̄I ≡ ( ηI )T C (n) − = − ( ηJ )† γn+1ΩIJ , where ΩIJ = iσ2 ⊗ 1ln 2× n 2 is the invariant tensor of Sp(n 2 ), satisfying ΩIJΩJK = δI K . The Euclidean coordinates of Sn are bilinear in the Killing spinors xi = (Γi)IJ η̄ Iγn+1η J , (4.4) where η are of a single kind (+ or −), or equivalently η̄I +η J −. In the above the Γ are in SO(n+1)G, while the γ in SO(n)L. Starting from Killing spinors on Sn, one can construct all the higher spherical harmonics. As seen in equation (4.4), Euclidean coordinates on the sphere are spinor bilinears. In turn, symmetric traceless products of the xi’s construct the scalar spherical harmonics Y k(xi).10 One can also construct the set of spinorial spherical harmonics by acting with an appropriate operator on Y kηI Ξk,+ = [(k + n− 1 + iD/)Y k]η+, Ξk,− = [(k + n− 1 + iD/)Y k]η− = [(k + 1 + iD/)Y k+1]η+. Note that in the above the derivatives act only on the scalar harmonics Y k. Any spinor on the sphere can be expanded in terms of spinorial spherical harmonics, Ψ =∑ k ψkΞk,±. Consistency imposes that the Ξk,± can only be commuting spinors. The Killing spinors are then themselves commuting spinors, as they are used to construct the spinorial spherical harmonics. For higher harmonics the construction extends in a similar way but the formulae are more complicated and, as we will not need them for our discussion, we will not present them here. The interested reader can consult e.g. [37]. Killing spinors on S2 and relation between spinors For the particular case of the S2, γi = Γi = σi for both the SO(2)L and the SO(3)G Clifford algebras. Then the two C-matrices can be chosen to be: C+ = −σ1, giving κ = λ = +, and C− = iσ2 = ε, giving κ = λ = −. Note that with these conventions one has C−γ3 = iσ2σ3 = −σ1 = C+. In the following we will choose the Majorana condition to be defined with respect to C−. Equation (4.4) then gives for n = 2 η̄I = (ηT )IC− ⇒ xi = (σi)IJ(ηT )IC−γ3η J . (4.5) The orthonormality and completeness conditions for the Killing spinors on S2 are η̄IηJ = εIJ and ηα J η̄ J β = −δα β , while the modified Majorana condition is (ηJ)† = εIJ η̄ I ≡ εIJ(ηI)TC−. 9For more details on Majorana spinors and charge conjugation matrices see [31, 36] and the Appendix of [35]. 10These are the higher dimensional extensions of the usual spherical harmonics Y lm(xi) for S2. 18 H. Nastase and C. Papageorgakis Since C− = ε, by making both indices explicit and by renaming the index I as α̇ for later use, one also has (ηαα̇)† = ηαα̇ ≡ εαβεα̇β̇η ββ̇ . (4.6) Finally, the spinorial spherical harmonics on S2 are Ξ±lm = [(l + 1 + iD/ )Ylm]η± and thus the spherical harmonic expansion of an S2-fermion is (writing explicitly the sphere fermionic index α) ψα = ∑ lm,± ψlm,±Ξ±,α lm = ∑ lm,± [ψlm,±(l + 1 + iD/ )Ylm]αβη β ±. To construct explicitly the Killing spinor, we must first define a matrix S, that can be used to relate between the two different kinds of spinors on S2, spherical and Euclidean. On the 2-sphere, one defines the Killing vectors Ka i such that the adjoint action of the SU(2) generators on the fuzzy sphere fields becomes a derivation in the large-N limit11 [Ji, ·] → 2iKa i ∂a = 2iεijkxj∂k. One can then explicitly check that Ka i produces a Lorentz transformation on the gamma matri- ces12 Ka i (σ̃i)α β = −eam ( SσmS−1 ) β α ≡ − ( SγaS−1 ) β α , where eam is the vielbein on the sphere and S is a unitary matrix defining the transformation (\|a\| = 1) S = a − sin θ 2 e iφ/2 −i cos θ 2 eiφ/2 cos θ 2 e−iφ/2 −i sin θ 2 e−iφ/2  . Imposing the (symplectic) reality condition on S εαβ ( S−1 )β γ εγδ = ( ST ) α δ = Sδ α, (4.7) we fix a = √ i ∗ and obtain the relations( SσiS −1 ) α β = ( SσiS −1 )β α , ( Sγ3S −1 )α β = −xi(σ̃i)α β ,( SγaS −1 )α β = −habK b i (σ̃i)α β. (4.8) If one has real spinors obeying (χαα̇)† = χαα̇ ≡ εαβεα̇β̇χββ̇, which was identified in (4.6) as the modified Majorana spinor condition, it follows from (4.7) that rotation by the matrix S preserves this relation, i.e. ((χα̇S)α)† = ( S−1χα̇ )α ≡ −εα̇β̇ ( S−1 )αβ χββ̇ = εα̇β̇εαβ(χβ̇S)β . (4.9) 11Precise expressions for the Killing vectors as well as a set of useful identities can be found in Appendix A of [10]. 12A Lorentz transformation on the spinors acts as Λµ νγν = SγµS−1, with S unitary. Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 19 We can now define the explicit form of the Killing spinor ηIα = ( S−1 )α β ηIβ 0 = 1√ 2 ( S−1 )α β εβI = 1√ 2 SI Jε αJ , where in the last equality we used the (symplectic) reality condition (4.7) on S. From (4.9) it is clear that the ηIα obey the modified Majorana condition. It is then possible to use (4.8) to prove that xi = (σi)IJ η̄ Iγ3η J , hence verifying that the ηIα are indeed Killing spinors. One can also explicitly check that Da (( S−1 )α β εβI ) = + i 2 (γa)α β ( S−1 )β γ εγI , which in turn means that 1√ 2 ( S−1 )α β εβI = ηαI + . Identif ication with Killing spinor Using (4.6), we rewrite (4.5) as xi = (σi)I J ( ηI )† γ3η J = (σ̃i)I J (√ 2P+η I )†(√2P+η J ) , (4.10) where P± = 1 2(1 ± γ3). Now comparing (4.10) with (4.3) one is led to the following natural large-N relation, G̃α → √ 2NP+η I , provided the spinor indices α and I get identified, i.e. G̃α √ N ≡ g̃α ↔ g̃I ≡ √ 2P+η I = (P+)α β(S−1)β γε γI = (P+)α βS I Jε βJ = SI J(P−)J Kε αK . Thus, the Weyl projection can be thought of as ‘removing’ either α or I, since only one of the two spinor components is non-zero. In order to further check this proposed identification at large-N we now calculate ∂a (√ 2P+η I ) = − i 2 ( SγaS −1 )I J (√ 2P+η J ) + T̃a (√ 2P+η I ) , (4.11) where T̃θ = 0 and T̃φ = i 2 cos θ and (∂aS)S−1 = − i 2 SγaS −1 + STaS −1 by explicitly evaluation, with Tθ = 0 and Tφ = − i 2 cos θγ3. This needs to be compared with the analogous result given in equation (4.48) of [9] from the classical limit of the adjoint action of Ji on G̃α, i.e. from [Ji, G̃ α], ∂ag̃ α = i 2 ĥabK b i (σ̃i)a β g̃ β = − i 2 ( SγaS −1 )α β g̃β . (4.12) In [9] it was also verified that the above could reproduce the correct answer for ∂axi, which can be rewritten as ∂axi = − i 2 g̃†α [ (σ̃i)α β ( SγaS −1 )β γ − ( SγaS −1 )α β (σ̃i)β γ ] g̃γ . 20 H. Nastase and C. Papageorgakis Note that even though there is a difference between (4.11) and (4.12), given by the purely imaginary term T̃a that is proportional to the identity, the two answers for ∂axi exactly agree, since in that case the extra contribution cancels. This extra term is a reflection of a double ambiguity: First, the extra index α on ηI can be acted upon by matrices, even though it is Weyl- projected, in effect multiplying the Weyl-projected ηI by a complex number; if the complex number is a phase, it will not change any expressions where the extra index is contracted, thus we have an ambiguity against multiplication by a phase. Second, g̃α is just a representative of the reduction of gα by an arbitrary phase, so it is itself only defined up to a phase. The net effect is that the identification of the objects in (4.11) and (4.12) is only up to a phase. Indeed, locally, near φ ' 0, one could write g̃αe i 2 φ cos θ ↔ √ 2P+η I but it is not possible to get an explicit expression for the phase over the whole sphere. 4.3 Generalisations On a general S2n some elements of the above analysis of fuzzy Killing spinors carry through. That is because even though it is possible to write for every S2n xA = η̄I(ΓA)IJγ2n+1η J , where ηI are the Killing spinors, one only has possible fuzzy versions of the quaternionic and octonionic Hopf maps to match it against, i.e. for 2n = 4, 8. We will next find and interpret the latter in terms of Killing spinors on the corresponding spheres. S4 The second Hopf map, S7 π→ S4, is related to the quaternionic algebra. Expressing the S7 in terms of complex coordinates gα, now with α = 1, . . . , 4, the sphere constraint becomes gαg†α = 1 (gαg†α = 1 ⇒ xAxA = 1; A = 1, . . . , 5). The map in this case is (see for instance [38]) xA = gβ(ΓA)α βg † α, with (ΓA)α β the 4× 4 SO(5) gamma matrices13. Here we have identified the spinor index I of SO(5) with the Lorentz spinor index α of SO(4). Initially, the gα’s are complex coordinates acted upon by SU(4), but projecting down to the base of the Hopf fibration we replace gα in the above formula with real g̃α’s, instead acted upon by the spinorial representation of SO(4), i.e. by spinors on the 4-sphere. This process is analogous to what we saw for the case of the 2-sphere. Once again, it is possible to identify g̃α with the Killing spinors, this time on S4. This suggest that one should also be able to write a spinorial version of the fuzzy 4-sphere for some bifundamental matrices G̃α, satisfying JA = G̃β(ΓA)α βG̃ † α, J̄A = G̃† α(ΓA)α βG̃ β, where JA, J̄A generate an SO(5) spinor rotation on G̃α by JAG̃ α − G̃αJ̄A = (ΓA)α βG̃ β. This in turn implies that the fuzzy sphere should be described by the same GRVV algebra as for the S2 case G̃α = G̃αG̃† βG̃ β − G̃βG̃† βG̃ α 13These are constructed as: σ1 and σ3 where 1 is replaced by 1l2×2 and σ2 where i is replaced by iσ1, iσ2, iσ3. Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 21 but now with G̃α being 4 complex matrices that describe a fuzzy 4-sphere, which poses an interesting possibility that we will however not further investigate here. S8 The third Hopf map, S15 π→ S8, is related to the octonionic algebra. The S15 is expressed now by the real objects gT αg α = 1, α = 1, . . . , 16 that can be split into two groups (1, . . . , 8 and 9, . . . , 16). The Hopf map is expressed by [39] (gT αgα = 1 ⇒ xAxA = 1) xA = gT α (ΓA)αβgβ , where (ΓA)αβ are the SO(9) gamma-matrices14. Similarly for the case of the S4 above, even though gα’s are initially 16-dimensional variables acted by the spinor representation of SO(9), one can project down to the base of the Hopf fibration and replace the gα’s with real 8-dimensional objects on the 8-sphere g̃α. Then the g̃α’s are identified with the Killing spinors of S8. This once again suggests that one should be able to write a spinorial version of the fuzzy 8-sphere for some bifundamental matrices G̃α satisfying JA = G̃α(ΓA)αβG̃T β , J̄A = G̃T α(ΓA)αβG̃β, where JA, J̄A generate an SO(9) spinor rotation on G̃α by JAG̃α − G̃αJ̄A = (ΓA)α βG̃β and implies the same GRVV algebra, but with the G̃α’s now being 16 dimensional real matrices that describe the fuzzy 8-sphere. 5 Deconstruction vs. twisted compactification We now describe certain changes which occur when ‘deconstructing’ a supersymmetric field theory on the bifundamental fuzzy S2, in contrast to the usual S2, and comparing with the compactified higher-dimensional theory. The term ‘deconstruction’ was first coined in [40] for a specific four-dimensional model but more generally extends to creating higher dimensional theories through field theories with matrix degrees of freedom of high rank. In our particular case, the fuzzy S2 background arises as a solution in a d-dimensional field theory and fluctuations around this background ‘deconstruct’ a d + 2-dimensional field theory. We will focus on the case where the d + 2-dimensional field theory compactified on S2 is supersymmetric. 5.1 Adjoint fuzzy S2 This construction is familiar in the context of D-branes, though any field theory with a fuzzy S2 background will also do. For instance, the example we will follow is [41], where an N = 1 supersymmetric massive SU(N) gauge theory around a fuzzy S2 background solution, coming 14The gamma-matrices are constructed similarly to the S4 case as follows: Γi = ( 0 λi −λi 0 ) , Γ8 =( 0 1l8×8 1l8×8 0 ) , Γ9 = ( 1l8×8 0 0 − 1l8×8 ) , i.e. from σ2 with λi replacing i, and from σ1 and σ3 with 1 replaced by 1l8×8. The λi satisfy {λi, λi} = −2δij (similarly to the iσi in the case of S4) and are constructed from the structure constants of the algebra of the octonions [39]. An explicit inversion of the Hopf map is given by gα = [(1 + x9)/2]1/2uα for α = 1, . . . , 8 and gα = [2(1 + x9)] −1/2(x8 − xiλi)uα−8 for α = 9, . . . , 16, with uα a real 8-component SO(8) spinor satisfying uαuα = 1 thus parametrising the S7 fibre. 22 H. Nastase and C. Papageorgakis from the low energy theory on a stack of D3-branes in some nontrivial background, was identified with the Maldacena–Núñez theory of IIB 5-branes with twisted compactification on S2 [42]. This construction was known to give an N = 1 massive theory after dimensional reduction that can be identified with the starting point, thus the D3-brane theory around the fuzzy sphere deconstructs the 5-brane theory. The twisting of the 5-brane fields can be understood both in the compactification as well as the deconstruction pictures. In compactification, and for the [41] model, it is known from [43] that in order to preserve supersymmetry on D-branes with curved worldvolumes one needs to twist the various D-brane fields. Specifically, that means embedding the S2 spin connection, taking values in SO(2) ' U(1), into the R-symmetry. As a result, the maximal supersymmetry one can obtain after compactification is N = 1 (corresponding to U(1)R). On the other hand, in deconstruction, the need for twisting will instead appear by analysing the kinetic operators of the deconstructed fields. The brane intuition, though useful, is not necessary, and in the following we will understand the twisting as arising generally from requiring supersymmetry of the dimensionally reduced compactified theory. This will be matched by looking at the kinetic term diagonalisation of the deconstructed theory. Compactif ication On a 2-sphere, scalar fields are decomposed in the usual spherical harmonics Ylm(xi) = Ylm(θ, φ) and can thus give massless fields after compactification (specifically, the l = 0 modes). However, that is no longer true for spinors and gauge fields. In that case, the harmonic decomposition in terms of Ylm(xi) must be redefined in order to make explicit the Lorentz properties of spinors and vectors on the 2-sphere, i.e. to make them eigenvectors of their corresponding operators. Spinors on the sphere are eigenvectors of the total angular momentum J2 i . These are of two types: Eigenvectors Ω of the orbital angular momentum L2 i (Cartesian spherical spinors) and eigenvectors Υ of the Dirac operator on the sphere −i∇̂S2 = −iĥabema σm∇b (spherical basis spinors), whose square is R2(−i∇̂S2)2 = J2 i + 1 4 . The two are related by a transformation with a sphere-dependent matrix S, already described in Section 4.2. The former are decomposed in the spinorial spherical harmonics Ωα̂ jlm = ∑ µ=±1 2 C(l, 1 2 , j;m− µ, µ,m)Yl,m−µ(θ, φ)χα̂ µ, where j = q± = l ± 1 2 and α̂ = 1, 2, as ψα̂ = ∑ lm ψ (+) lm Ωα̂ l+ 1 2 ,lm + ψ (−) lm Ωα̂ l− 1 2 ,lm . Both have a minimum mass of 1 2R , since the Dirac operator squares to J2 i + 1 4 = j(j + 1) + 1 4 . Similarly, the vector fields do not simply decompose in Ylm’s, but rather in the vector spherical harmonics 1 R Tjm = 1√ j(j + 1) [ sin θ∂θYjmφ̂− csc θ∂φYjmθ̂ ] , 1 R Sjm = 1√ j(j + 1) [ ∂θYjmθ̂ + ∂φYjmφ̂ ] , with j ≥ 1. It is more enlightening to show the decomposition of the field strength on the 2-sphere 1 R csc θFθφ = R2 ∑ lm Flm 1√ l(l + 1) ∆S2Ylm, Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 23 with l = 1, 2, . . . . Thus again only massive and no massless modes are obtained after dimensional reduction [41]. Note that as we can see, the expansion in spinorial or vector spherical harmonics corresponds to redefining the expansion in terms of Ylm (rearranging its coefficients). Therefore in the absence of twisting supersymmetry will be lost after dimensional reduction, since all S2-fermions will be massive but some massless S2-scalars will still remain. Twisting, however, allows for the presence of fermionic twisted-scalars (T-scalars), i.e. fermions that are scalars of the twisted SO(2)T Lorentz invariance group (with charge T ), which will stay massless. In this way the number of supersymmetries in the dimensionally reduced theory equals the number of fermionic T-scalars. One chooses the twisted Lorentz invariance of the sphere as QT = Qxy + QA, where Qxy is the charge under the original Lorentz invariance of the sphere SO(2)xy, and QA is the charge under the U(1) subgroup of R-symmetry. This is necessary because one needs to identify the U(1) spin connection (‘gauge field of Lorentz invariance’) with a corresponding connection in the R-symmetry subgroup, i.e. a gauge field from the transverse manifold. An example of an action for twisted fields is provided by the result of [43], for a bosonic T-spinor Ξ, fermionic T-scalars Λ and T-vectors ga∫ ddxd2σ √ h [ − i 2 µΛ̄γµ∂µΛ− i 2 µḡaγ µ∂µg a+µωabḠabΛ−2∂µΞ†∂µΞ−8Ξ† ( −i∇̂S2 )2Ξ], (5.1) where µ is the mass parameter, Gab = ∂agb−∂bga is the field strength of the fermionic T-vector, and as usual ωab = 1√ g ε ab is the symplectic form on the sphere. We note that the kinetic terms in the flat directions (µ, ν) are given by their bosonic or fermionic nature, while the type of kinetic terms in the sphere directions (a, b) are dictated by their T-spin and the number of derivatives on it are again dictated by their statistics (bosons have two derivatives, fermions only one). These fields are decomposed in spherical harmonics corresponding to their T-charge. Then e.g. the fermionic T-scalar can have a massless (l = 0) mode, which after dimensional reduction will still be a fermion and give N = 1 supersymmetry. Deconstruction To have a fuzzy sphere background of the usual type, we need in the worldvolume theory at least 3 scalar modes φi to satisfy [φi, φj ] = 2iεijkφk, but usually there are more. Then the need for e.g. bosonic T-spinors is uncovered by diagonalising the kinetic term for all the scalar fluctuations around the fuzzy sphere background. For instance in [41], there are 6 scalar modes forming 3 complex scalars Φi, with fluctuations δΦi = ai + ibi and kinetic term∫ ddxd2σ √ hδΦ† i [( 1 + J2 ) δij − iεijkJk ] δΦj . The (complete set of) eigenvectors of this kinetic operator are given by the vector spherical har- monics JiYlm and the spinorial spherical harmonics Ωα̂ jlm. This kinetic operator is then diagona- lised by defining T-vectors na coming from the vector spherical harmonics and T-spinors ξα̂ coming from the spinor spherical harmonics. When completing this program, the deconstructed action is the same as the compactified one, e.g. for [41] one again obtains the twisted action (5.1). At finite N , the matrices are expanded in the fuzzy spherical harmonics Ylm(Ji), becoming the Ylm(xi) of classical S2, but the above diagonalisation corresponds in the classical limit to re- organising the expansion (this includes a nontrivial action on the coefficients of the expansion) to form the spinorial, vector, etc. spherical harmonics. Thus for the adjoint construction all the fields on the classical S2 appear as limits of functions expanded in the scalar fuzzy spherical harmonics, Ylm(Ji), and the various tensor structures of S2 fields were made manifest by diagonalising the various kinetic operators. 24 H. Nastase and C. Papageorgakis 5.2 Bifundamental fuzzy S2 The case of the bifundamental fuzzy S2 is richer. One wants to once again compare with the same compactification picture. However, the particulars of the deconstruction will be different. Deconstruction Here we need a fuzzy sphere background of GRVV type, hence at least 2 complex scalar modesRα in the worldvolume theory giving the fuzzy sphere background in terms of Rα = fGα, with Gα satisfying (1.2). The fluctuation of this field will be called rα. Performing the deconstruction follows a set of steps similar to the adjoint fuzzy S2, namely one wants to expand in the fuzzy spherical harmonics and in the classical limit reorganise the expansion (acting nontrivially on the coefficients of the expansion) to construct the spinor, vector, etc. spherical harmonics. However now there are some subtle points that one needs to take into account. We have two kinds of fuzzy spherical harmonics, Ylm(Ji) and Ylm(J̄i), both giving the same Ylm(xi) in the classical limit. Adjoint fields, e.g. the gauge fields, will be decomposed in terms of one or the other according to their respective gauge groups. On the other hand for bifundamental fields one must first ‘extract’ a bifundamental GRVV matrix, G̃α or G̃† a, before one is left with adjoints that can be decomposed in the same way. We detailed this procedure for rα in Section 2.2.3. The expansion in Ylm(xi) must be then reorganised as in the usual fuzzy S2 in order to diagonalise the kinetic operator, thus producing the spinor, vector, etc. spherical harmonics. The most important difference is that G̃α has a spinor index on S2; in particular we saw in Section 4.2 that in the classical limit g̃α is identified with a Killing spinor. That means that the operation of ‘extracting’ G̃a corresponds to automatically twisting the fields! Let us make this concrete by considering a specific example. In the mass-deformed ABJM theory, one has besides the Rα field a doublet of scalar fields Qα̇ with fluctuation qα̇, where α̇ is an SU(2) index transverse to the sphere. Thus the qα̇ start off life as scalars. However, due to their bifundamental nature, one must first ‘extract’ G̃α → √ Ng̃α, by writing qα̇ = Qα̇ αG̃ α. In order to diagonalise the kinetic operator, we perform an S-transformation and construct Ξα α̇ = i(P+S −1Qα̇)α + ( P−S −1Qα̇ )α , (5.2) after which the kinetic term becomes the twisted action N2 ∫ d3xd2σ √ ĥ [ 1 2 Ξ̄α̇(−i2µ∇̂S2)2Ξα̇ − 1 2 ∂µΞ̄α̇∂µΞα̇ − 3µ2Ξ̄α̇Ξα̇ ] . (5.3) More generally, the functions on the sphere are actually sections of the appropriate bundle: Either ordinary functions, sections of the spinor or the line bundle. Specifically, anything without an α index is a T-scalar, one α index implies a T-spinor and two α indices a T-scalar plus a T-vector in a (1 ⊕ 3) decomposition. That is, the U(1)T invariance is identified with the SO(2)L ' U(1)L Lorentz invariance of the sphere, described by the index α. In addition to this, an interesting new alternative to the above construction also arises. We can choose to keep G̃α in the spherical harmonic expansion (by considering it as part of the spherical harmonic in the classical limit). The derivative of the spherical harmonic expansion then includes the derivative of g̃α given in (4.12) and one obtains a fuzzy version of the classical derivative operator q† β̇ Ji − J̄iq † β̇ → 2iKa i ∂aq † β̇ + q† β̇ xi. Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors 25 This operator acts on all bifundamental fields, including the ABJM fermions ψ†α. In this new kind of expansion, we recover the usual Lorentz covariant kinetic term. For instance for the scalar fields qα̇ of ABJM we obtain (after a rescaling of the fields) 1 g2 Y M ∫ d3xd2σ √ h [ −∂Aq†α̇∂Aq α̇ ] , where A = µ, a is a total (worldvolume + fuzzy sphere) index. The price one pays for this simplicity (compared to (5.3)) is however that the classical N → ∞ limit of the supersym- metry transformation is very subtle, since a naive application will relate fields with different finite N gauge structures (bifundamentals with adjoints), naively implying a gauge-dependent supersymmetry parameter. But at least formally, by keeping G̃α inside the spherical harmonic expansion, we obtain an un-twisted, fully supersymmetric version of the action on the whole worldvolume plus the fuzzy sphere. 6 Supersymmetric D4-brane action on fuzzy S2 from ABJM As a concrete application of the whole discussion thus far, we present the final results for the Lagrangian obtained by studying fluctuations around the fuzzy S2 ground-state of the mass- deformed ABJM model. The fluctuating fields are the rα scalars forming the fuzzy sphere background, transverse scalars qα̇, gauge fields Aµ and Âµ, fermions ψα and χα̇. The spherical harmonic expansion on the fuzzy sphere is for each of the above rα = rG̃α + sα βG̃ β = [ (r)lmδ α β + (sα β)lm ] Ylm(Ji)G̃β, qα̇ = Qα̇ αG̃ α = (Qα̇ α)lmYlm(Ji)g̃α, ψα = ψ̃G̃α + Uα βG̃β = [ (ψ̃)lmδ β α + (Uα β)lm ] Ylm(Ji)G̃β, χα̇ = χα̇αG̃ α = (χα̇α)lmYlm(Ji)G̃α, Aµ = Alm µ Ylm(Ji), Âµ = Âlm µ Ylm(J̄i), becoming in the classical limit rα = rg̃α + sα β g̃ β = [ (r)lmδ α β + (sα β)lm ] Ylm(xi)g̃β, qα̇ = Qα̇ αg̃ α = (Qα̇ α)lmYlm(xi)g̃α, ψα = ψ̃g̃α + Uα β g̃β = [ (ψ̃)lmδ β α + (Uα β)lm ] Ylm(xi)g̃β , χα̇ = χα̇αg̃ α = (χα̇α)lmYlm(xi)g̃α, Aµ = Alm µ Ylm(xi), Âµ = Âlm µ Ylm(xi). These can be further redefined as sα β(σ̃i)β α = Ka i Aa + xiφ, Υα α̇ = (P−S−1χα̇)α, with Aa becoming the sphere component of the gauge field and Φ = 2r + φ becoming a scalar, while 2r−φ is ‘eaten’ by the gauge field in a Higgs mechanism that takes us from nonpropagating CS gauge field to propagating YM field in 3d [44]. The final supersymmetric version of the action is then Sphys = 1 g2 YM ∫ d3xd2σ √ h [ −1 4 FABF AB − 1 2 ∂AΦ∂AΦ− µ2 2 Φ2 − ∂Aq†α̇∂Aq α̇ + µ 2 ωabFabΦ 26 H. Nastase and C. Papageorgakis + ( 1 2 Ῡα̇D̃5Υα̇ + i 2 µῩα̇Υα̇ + h.c. ) − (ψS)D̃5 ( S−1ψ† ) + i 2 µ(ψS) ( S−1ψ† )] . The twisting of the fields that have a G̃α in their spherical harmonic expansion is done as follows: First, we twist by expressing qα̇ as Qα̇ α and ψα as ψ̃, Uα β. We then redefine the twisted fields in order to diagonalise their kinetic operator by further writing Qα̇ α according to (5.2) and Uα β = 1 2 Ui(σ̃i)α β , Ūα β = 1 2 Ui(σ̃i)α β, Ui = Ka i ga + ψ̂xi, Ūi = Ka i ḡa + ¯̂ ψxi. The final twisted action is Sphys = 1 g2 YM ∫ d3xd2σ √ h [ − 1 4 FABF AB − 1 2 ∂AΦ∂AΦ− µ2 2 Φ2 + µ 2 ωabFabΦ + ( 1 2 Ῡα̇D5Υα̇ + i 2 µῩα̇Υα̇ + h.c. ) + 1 4 Ξ̄α̇ ( −2i µ ∇S2 )2 Ξα̇ − ∂µΞ̄α̇∂µΞα̇ − 3 2 µ2Ξ̄α̇Ξα̇ + 1 4 Λ̄/∂Λ + 1 4 ḡa/∂g a + i 4 ωabḠabΛ + i 2 µΛ̄Λ ] . 7 Conclusions In this paper we reviewed our fuzzy S2 construction in terms of bifundamental matrices, origi- nally obtained in the context of the ABJM model in [9, 10], focusing on its model-independent mathematical aspects. We found that this is completely equivalent to the usual adjoint SU(2) construction, but that it involves fuzzy versions of Killing spinors on the 2-sphere, which we defined. We described the qualitative differences that appear when using the bifundamental S2 to ‘deconstruct’ higher dimensional field theories. The expansion of the fields involving fuzzy Killing spinors result in an automatic twisting of the former on the sphere. 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Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors

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