Braz J Phys (2015) 45:481–492 DOI 10.1007/s13538-015-0336-9 PARTICLES AND FIELDS On Abelianizations of the ABJMModel and Applications to Condensed Matter Jeff Murugan1 ·Horatiu Nastase2 Received: 1 April 2015 / Published online: 13 June 2015 © Sociedade Brasileira de Fı́sica 2015 Abstract In applications of AdS/CFT to condensed matter systems in 2+1 dimensions, the ABJM model is often used; however, the condensed matter models are usually abelian and contain charged fields. We show that a naive reduction of the ABJM model to N = 1 does not have the desired features, but we can find an abelian reduction that has most features, and we can also add fundamental fields to the ABJM model to obtain other models with similar properties. Keywords AdS/CMT · Abelian reduction · ABJM mode 1 Introduction The last few years have seen an overwhelming amount of work devoted to applying the AdS/CFT correspondence to condensed matter systems (see [1, 2] for a review). The usual approach to this problem is somewhat phenomenolog- ical since there does not exist any good way to derive an explicit AdS/CFT duality that gives a condensed matter sys- tem of interest. One obstacle for this is the necessity to use a large N gauge theory in order to have a pure gravity dual � Jeff Murugan jeff@nassp.uct.ac.za Horatiu Nastase nastase@ift.unesp.br 1 The Laboratory for Quantum Gravity & Strings, Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag, Rondebosch, 7700, South Africa 2 Instituto de Fı́sica Teórica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, São Paulo 01140-070, SP, Brazil (as opposed to a full string theory dual), whereas condensed matter models of interest are usually abelian. Another is that it appears to be very difficult to obtain relevant condensed matter models from a full string theory construction. In two previous papers (together with A.Mohammed) [3, 4], we have begun to address these questions. We have shown, in particular, that a bosonic abelian reduction of the ABJM model (which is considered a primer for 2+1 dimen- sional condensed matter systems in the same way that N = 4 SYM in 3+1 dimensions is considered a primer for QCD) reproduces a relativistic Landau-Ginzburg model which has been used for describing the quantum critical phase near a superconductor-insulator transition [5, 6]. We have also shown that this is a nontrivial abelian truncation, still allow- ing for a gravity dual description, and that the abelianization procedure can be mimicked in a simple condensed matter model, at least as far as its general features. In this paper, we continue this program and ask if it is possible to construct abelianizations of the ABJM model that result in actions with the same general features as con- densed matter models with scalars, fermions and gauge fields, and both global and local charges. We will investi- gate the N = 1 reduction of the ABJM model, as well as a generalization of the abelianization in [3, 4] that includes fermions, and the introduction of fundamental fields in the ABJM model.1 We will find that simply setting N = 1 does 1After the first version of this paper appeared on the arXiv, the paper [7] appeared, where however a different ansatz for the fermions is con- sidered (with an extra εαβ ), which leads to an N = 4 supersymmetric model (an additional N = 2 supersymmetric model uses two sets of matrices). It is also stated there that the gravity dual correspond- ing to the abelianization, viewed as a perturbation around a vacuum, is strongly coupled and cannot be trusted. However, in our case, it is impossible to view a general solution of the abelian reduction as a small pertubation, as it can be easily checked; hence, this is not a problem for our construction. mailto:jeff@nassp.uct.ac.za mailto: 482 Braz J Phys (2015) 45:481–492 not serve the desired purpose, but the other two procedures have features that are similar to the condensed matter model. The paper is organized as follows. In Section 2, we describe the features of the condensed matter models that we are after followed by an analysis of the N = 1 case in Section 3. Section 4 concerns itself with the general- ization of the abelianization in [3, 4] to include fermions. In Section 5, we add fundamental degrees of freedom to the ABJM model and compare to existing literature, and in Section 6, we conclude with some thoughts on future directions. 2 Condensed Matter Models In order to facilitate the utility of the gauge/gravity cor- respondence to (planar) condensed matter physics, various models have recently been proposed that bear at least some resemblance to what is quickly becoming the canonical example of an AdS4/CFT3 relation, the ABJM model [8]. These models were then thought to be well described by the gravity dual to ABJM, namely type IIA superstring the- ory (or, at least, its supergravity limit) on AdS4 × CP 3. One characteristic they all share is that they are abelian and, depending on the condensed matter application in question, contain an emergent (non-electromagnetic) U(1) gauge field. One such model, proposed in [9], which we will take as a relevant example, was used to describe compressible Fermi surfaces. It is this that we will take as a template to work with since it exhibits several of the general fea- tures observed in most condensed matter systems modeled by the ABJM action. Defined through the nonrelativistic action S = ∫ d3x ⎡ ⎢⎣f † + ⎛ ⎜⎝(∂τ − iAτ ) − ( �∇ − i �A )2 2mf − μ ⎞ ⎟⎠ f+ +f † − ⎛ ⎜⎝(∂τ + iAτ ) − ( �∇ + i �A )2 2mf − μ ⎞ ⎟⎠ f− +b † + ⎛ ⎜⎝(∂τ − iAτ ) − ( �∇ − i �A )2 2mb + ε1 − μ ⎞ ⎟⎠ b+ +b † − ⎛ ⎜⎝(∂τ + iAτ ) − ( �∇ + i �A )2 2mb + ε1 − μ ⎞ ⎟⎠ b− +u 2 ( b † +b+ + b † −b− )2+vb † +b † −b−b+−g1 ( b † +b † −f−f+ + h.c. ) +c† ⎛ ⎜⎝∂τ − ( �∇ )2 2mc + ε2 − μ ⎞ ⎟⎠ c−g2(c †(f+b−+f−b+)+h.c.) ⎤ ⎥⎦ , (2.1) it has an emergent gauge field (i.e., not the electromagnetic gauge field), a U(1) global symmetry current corresponding to electric charge, Q = f † +f+ + f † −f− + b † +b+ + b † −b− + 2c†c, (2.2) and both fundamental charged bosons b± and fermions f̃± coupled to the gauge field as well as a neutral fermion c. When writing a relativistic version of this, we should replace (∂τ − iqAτ ) − ( �∇ − iq �A )2 2m − μ → (∂μ − iqAμ)2 − m2, (2.3) for bosons, and for fermions, by the square root of the Klein- Gordon operator above. We can now see the general features that we need for a good model: it needs to have bosons and fermions, cou- pled to an emergent abelian gauge field with both positive and negative charges, and a global charge (corresponding to the electric charge) with values for the charges of the fields independent of the local charge. Moreover, we would like to find a model that has all the fields with positive global charges, but both positive and negative local charges. The interaction terms will have up to fourth-order couplings in the bosons, and two bose-two fermi interactions. We will try to obtain models that have the same general properties using abelianization procedures for the ABJM model and later by adding fundamental fields. 3 ABJM U(N) × U(N) Model for N = 1 The naive first thing to do in order to obtain an abelian field theory from the U(N)×U(N) ABJM model would be to set N = 1. One might well be concerned about the sensibility of such an attempt but since it is the U(N) × U(N) version of the ABJM model that is related to a brane construction and not the SU(N) × SU(N) one, it is a consistent thing to try. Nevertheless, we demonstrate below that it is too naive. The action was written down in [10, 11], as SN=1 ABJM = ∫ d3x [ k 4π εμνλ ( A(1) μ ∂νA (1) λ − A(2) μ ∂νA (2) λ ) −iψA†γ μDμψA − DμC † ADμCA ] , (3.1) where the gauge covariant derivative DμCA = ∂μCA + i ( A(1) μ − A(2) μ ) CA . (3.2) In order to understand the physics of this truncation, we will need to Higgs the model. We know from general con- siderations [12, 13] that Higgsing the ABJM model means that the Chern-Simons gauge field will eat a scalar degree Braz J Phys (2015) 45:481–492 483 of freedom and become physical, i.e., of the Maxwell type. It is this behavior that we want to reproduce here. The Higgsing procedure in a rather general case, for both SU(N) × SU(N) and U(N) × U(N) models was set out in [13]. To specialize to our case, we will follow this quite closely and simply set N = 1. We begin by defining A± μ = 1 2 ( A(1) μ ± A(2) μ ) , F± μν = ∂μAν − ∂νAμ. (3.3) With this, we can easily check that the Chern-Simons part of the action becomes∫ d3x k 2π εμνλA− μF+ νλ. (3.4) In order to Higgs the model, it is necessary to write CA = XA + iXA+4 √ 2 + vδA4, (3.5) where the XA and XA+4 are real, and we consider the scalar field VEV v to be large. Then, the covariant derivative of the scalars becomes DμCA = ∂μ ( XA + iXA+4 √ 2 ) + 2iA− μ ( XA + iXA+4 √ 2 + vδA4 ) , (3.6) while the kinetic term for the scalars reduces to ∣∣∣DμCA ∣∣∣2 = ( ∂μXA − A− μXA+4 )2 2 + ( ∂μXA+4 + 2A− μXA + 2 √ 2vA− μδA4 )2 2 = ( ∂μXA )2 + ( ∂μXA+4 + 2 √ 2vA− μδA4 )2 2 , (3.7) where we have dropped terms subleading in v. On the other hand, the term k 2π εμνλ 1 2 √ 2v ∂μX8F+ νλ (3.8) vanishes by partial integration and use of the Bianchi identity, so it can be added to the action with impunity. Consequently, the bosonic part of the action, Sbose = ∫ d3x [ k 2π εμνλ ( A− μ + 1 2 √ 2v ∂μX8 ) F+ νλ − (∂μXA)2 2 − ( ∂μXA+4 + 2 √ 2vA− μδA4 )2 2 ⎤ ⎥⎦ . To clarify this, we now make the shift A− μ → A− μ − 1 2 √ 2v ∂μX8, after which the bosonic action becomes Sbose = ∫ d3x ⎡ ⎢⎣ k 2π εμνλA− μF+ νλ− ( ∂μXI ′)2 8 − 4v2 ( A− μ )2 ⎤ ⎥⎦ , (3.9) where I ′ = 1, ..., 7 and again, to leading order in v. At this point, we recognize that the field A− μ is, in fact, auxilliary and can be integrated out to give A− μ = k 16πv2 εμνλF +νλ. (3.10) Substituting this expression back in the bosonic action, we finally find for the Higgsed action S = ∫ d3x ⎡ ⎢⎣− k2 32π2v2 F+μνF+ μν − ( ∂μXI ′)2 2 − iψ̄†Aγ μDμψA ] . (3.11) We see that we must define k2 32π2v2 = 1 4g2 , (3.12) and therefore, we must also have k → ∞, such that the ratio k/v remains fixed. This in turn means that, in the covariant derivative DμψA → ( ∂μ + 2i ( A− μ − 1 2 √ 2v ∂μX8 )) ψA = ( ∂μ + 2i ( 1 4g2 2π k εμνλF +νλ )) ψA, all terms additional to the partial derivative ∂μψA are sub- leading in v and can be dropped, giving the final form of the Higgsed action, SHiggs = ∫ d3x ⎡ ⎢⎣− 1 4g2 F+μνF+ μν − ( ∂μXI ′)2 2 − iψ̄†Aγ μ∂μψA ⎤ ⎥⎦ . (3.13) 484 Braz J Phys (2015) 45:481–492 In this form, it is clear that, while we retain the the global SO(7) symmetry (with A a spinor index and I ′ a fundamental index), the local symmetry does not act on the matter fields. In other words, setting N = 1 results in nothing more than a free Maxwell supermultiplet, with the scalars and fermions not coupled to the gauge field. This “abelianization” therefore does not produce the desired fea- tures for applying to the condensed matter phenomena of interest to us. 4 Nontrivial Abelianization with Fermions In a series of recent papers [3, 4], we introduced a nontrivial abelianization procedure which, when further restricted, led to a relativistic Landau-Ginzburg model relevant for condensed matter. While promising, the focus there was strictly on the bosonic sector of the ABJM model. We now ask whether it is possible to extend this procedure to include fermions and, moreover, whether the resulting effective theory (with scalars, gauge fields and fermions) exhibits any of the desired features of (2.1). In keeping with our previous notation, we split the scalars in the ABJM supermultiplet as CA = ( Qα, Rα̇ ) , with the abelianization ansatz Qα = φαGα; Rα̇ = χα̇Gα̇ (4.1) with no sum over α, α̇ = 1, 2, and the Gα matrices as intro- duced in [14] (see also [15]). Now, we do the same for the fermions, splitting ψA = ( ψα, ψ̃α̇ ) . In analogy with the scalars, the abelianization ansatz for the fermions is ψα = ηαGα; ψ̃α̇ = η̃α̇Gα̇, (4.2) again with no sum over α, α̇. Then, by virtue of the fact that the bifundamental abelianization ansatz is the same as for the scalars, we can immediately show that the ABJM covariant derivative on fermions reduces to the same abelian covariant derivative that acts on the abelian scalars, Dμηi =( ∂μ − ia (i) μ ) ηi , Dμη̃i = ( ∂μ − ia (i) μ ) η̃i . It follows then that the kinetic term for the fermions is the standard Dirac one. It remains only to calculate the interaction terms between the scalars and the fermions. That is easily done, using the defining relation for the GRVV matrices Gα = GαG † βGα −GβG † βGα . After some algebra, we find that the only non-vanishing term contributes, Tr [ C † ACAψB†ψB − ψB†CAC † AψB ] = N(N − 1) 2 [( |φ1|2 + |χ1|2 ) ( η † 2η2 + η̃ † 2η̃2 ) + ( |φ2|2 + |χ2|2 ) ( η † 1η1 + η̃ † 1η̃1 )] (4.3) to the interaction terms 2πi k N(N − 1) 2 [( |φ1|2 + |χ1|2 ) ( η † 2η2 + η̃ † 2η̃2 ) + ( |φ2|2 + |χ2|2 ) ( η † 1η1 + η̃ † 1η̃1 )] . (4.4) The fermion kinetic terms, on the other hand, contribute −iTr [ ψ†A(γ μDμ + μ)ψA ] = −i N(N − 1) 2∑ i=1,2 [ η † i ( /D + μ)ηi + η̃ † i ( /D + μ)η̃i ] , (4.5) so that the total effective abelian action is then S = −N(N − 1) 2 ∫ d3x { k 4π εμνλ ( a(2) μ f (1) νλ + a(1) μ f (2) νλ ) +|Dμφi |2 + |Dμχi |2 + ∑ i=1,2 [ η † i ( /D + μ)ηi + η̃ † i ( /D + μ)η̃i ] − 2πi k [( |φ1|2 + |χ1|2 ) ( η † 2η2 + η̃ † 2 η̃2 ) + ( |φ2|2 + |χ2|2 ) × ( η † 1η1 + η̃ † 1 η̃1 )] + ( 2π k )2 [( |φ1|2 + |χ1|2 ) ( |χ2|2−|φ2|2−c2 )2 + ( |φ2|2 + |χ2|2 ) ( |χ1|2 − |φ1|2 − c2 )2 + 4|φ1|2|φ2|2( |χ1|2 + |χ2|2 ) + 4|χ1|2|χ2|2 ( |φ1|2 + |φ2|2 )]} (4.6) Having started off with an N = 6 supersymmetric, the- ory in (2 + 1)−dimensions, at this point it is worth asking how much (if any) of the supersymmetry is preserved by our abelianization procedure. 4.1 Supersymmetry The susy rules for the massless ABJM model are [13, 14] δCA = iε̄ABψB, δψB = γ μεABDμCA + 2π k 2CCC † BCDεCD −2π k ( CAC † CCC − CCC † CCA ) εAB, δAμ = −2π k ( ε̄ABγμCBψ†A − ε̄ABγμψAC † B ) , (4.7) δÂμ = −2π k ( ε̄ABγμψ†ACB − ε̄ABγμC † BψA ) , Braz J Phys (2015) 45:481–492 485 where indices are raised and lowered with the SU(4) invari- ant metric δB A , and the susy parameters εAB are antisymmet- ric in AB and satisfy εAB = 1 2 εABCDεCD, (4.8) which means they live in the 6 representation of SU(4). The mass deformation introduces an extra term in the fermion supersymmetry transformation rules [16] δ(μ)ψA = 1 2 εDF CF ⎛ ⎜⎜⎝ μ 0 0 0 0 μ 0 0 0 0 −μ 0 0 0 0 −μ ⎞ ⎟⎟⎠ D A , (4.9) where the normalization is such that the scalar mass term in the Lagrangian is −μ2Tr [ C̄ACA ] . The reality condition for the supersymmetry parameter means (in α, α̇ components) that the independent compo- nents of εAB are ( ε12, ε11̇, ε12̇ ) , which are complex, i.e., N = 6 real supersymmetries. The other components are related by the reality condition to these ones as ε1̇2̇ = ε12; ε22̇ = −ε11̇; ε1̇2 = −ε12̇, (4.10) and as before, the ε’s are antisymmetric, so for example, ε1̇1 = −ε11̇, etc. We now split the susy laws in α, α̇ compo Qα = φαGα, Rα̇ = χα̇Gα̇, Aμ = a(2) μ G1G † 1 + a(1) μ G2G † 2, Âμ = a(2) μ G † 1G 1 + a(1) μ G † 2G2, (4.11) ψα = ηαGα, ψ̃α̇ = η̃α̇Gα̇, To understand what will happen, we write out explicitly one component, namely Q1, for which we obtain (δφ1)G1 = iε̄12 ( η2G 2 ) + iε̄11̇ ( η̃1̇G 1̇ ) + iε̄12̇ ( η̃2̇G 2̇ ) , (4.12) For this to make sense, i.e., for this susy transformation law to remain a symmetry after the abelianization, the right- hand side needs to be proportional to G1 also. This restricts us to ε12 = ε12̇ = 0, and only ε11̇ �= 0. Repeating the argument for the other scalar components, we find that (δφ1)G 1 = ( iε̄11̇η̃1̇ ) G1̇, (δφ2)G 2 = ( iε̄22̇η̃2̇ ) G2̇, (4.13) (δχ1̇)G 1̇ = ( iε̄1̇1η1 ) G1, (δχ2̇)G 2̇ = ( iε̄2̇2η2 ) G2, or using the relations between the epsilon components and peeling off the G matrices, δφ1 = iε̄11̇η̃1̇, δφ2 = −iε̄11̇η̃2̇, (4.14) δχ1̇ = −iε̄11̇η1, δχ2̇ = iε̄11̇η2. Clearly then, for the scalars only, the ε11̇ independent component is nonzero. For the gauge fields, substituting the abelianization ansatz in the transformation law for Aμ gives ( δa(2) μ ) G1G † 1 + ( δa(1) μ ) G2G † 2 = − 2π k [ ε̄11̇γμ ( Q1ψ̃†1̇ − R1̇ψ†1 ) +ε̄22̇γμ ( Q2ψ̃†2̇ − R2̇ψ†2 ) −ε̄11̇γμ ( ψ1R † 1̇ − ψ̃1̇Q † 1 ) −ε̄22̇γμ ( ψ2R † 2̇ − ψ̃2̇Q † 2 )] , (4.15) where we have already kept only the terms proportional to G1G † 1 or G2G † 2 on the right-hand side and set to zero the rest. Again, only the independent ε11̇ component survives. Identifying the coefficients of G1G † 1 and G2G † 2 on the left- hand side and the right-hand side, we find that δa(1) μ = 2π k ε̄11̇γμ [ φ2η̃ ∗̇ 2 + φ∗ 2 η̃2̇ − χ2̇η ∗ 2 − χ ∗̇ 2 η2 ] , (4.16) δa(2) μ = −2π k ε̄11̇γμ [ φ1η̃ ∗̇ 1 + φ∗ 1 η̃1̇ − χ1̇η ∗ 1 − χ ∗̇ 1 η1 ] . Next, we move on to the fermion rules. Here, again we keep only the independent ε11̇ component (having checked that the rest give a different matrix dependence on the right-hand side from the left-hand side), and find for ψ1, that (δη1)G 1 = γ με11̇DμR1̇ −2π k [ R1̇ ( Q † 2Q 2 + R † 2̇ R2̇ ) − ( Q † 2Q 2 + R † 2̇ R2̇ ) R1̇ ] ε1̇1 + 1 2 με11̇R 1̇. (4.17) Using the relations G1 = G1G † 2G 2−G2G † 2G 1; G2 = G2G † 1G 1−G1G † 1G 2, (4.18) peeling off G1 and using the relations between epsilons, we get δη1 = γ με11̇Dμχ1̇ + 2π k ε11̇χ1̇ ( |φ2|2 + |χ2̇|2 ) + μ 2 ε11̇χ1̇ (4.19) 486 Braz J Phys (2015) 45:481–492 Repeating for ψ2, ψ1̇, ψ2̇, we get δη2 = −γ με11̇Dμχ2̇ − 2π k ε11̇χ2̇ ( |φ1|2 + |χ2̇|2 ) − μ 2 ε11̇χ2̇, δη̃1̇ = −γ με11̇Dμφ1 − 2π k ε11̇φ1 ( |φ2|2 + |χ2̇|2 ) + μ 2 ε11̇φ1, δη̃2̇ = γ με11̇Dμφ2 + 2π k ε11̇φ2 ( |φ1|2 + |χ1̇|2 ) − μ 2 ε11̇φ2. (4.20) Finally then, for ease of notation, renaming ε11̇ as simply ε, we write down the susy transformation rules δφ1 = iε̄η̃1̇, δφ2 = −iε̄η̃2̇, δχ1̇ = −iε̄η1, δχ2̇ = iε̄η2, δa(1) μ = 2π k ε̄γμ [ φ2η̃ ∗̇ 2 + φ∗ 2 η̃2̇ − χ2̇η ∗ 2 − χ ∗̇ 2 η2 ] , δa(2) μ = −2π k ε̄γμ [ φ1η̃ ∗̇ 1 + φ∗ 1 η̃1̇ − χ1̇η ∗ 1 − χ ∗̇ 1 η1 ] , (4.21) δη1 = γ μDμχ1̇ + 2π k εχ1̇ ( |φ2|2 + |χ2̇|2 ) + μ 2 εχ1̇, δη2 = −γ μεDμχ2̇ − 2π k εχ2̇ ( |φ1|2 + |χ2̇|2 ) − μ 2 εχ2̇, δη̃1̇ = −γ μεDμφ1 − 2π k εφ1(|φ2|2 + |χ2̇|2) + μ 2 εφ1, δη̃2̇ = γ μεDμφ2 + 2π k εφ2 ( |φ1|2 + |χ1̇|2 ) − μ 2 εφ2. Since the parameter ε is complex, we have an SO(2) = U(1) R-symmetry and this is therefore N = 2 susy in three dimensions. An immediate question that arises is whether we can supersymmetrize the further truncation to the Landau- Ginzburg system in [3, 4]. From the above rules, we see that setting φ1 = φ2 = 0 and χ1̇ = b requires that η̃1̇ = η̃2̇ = 0 for consistency. However, because of the rela- tion δχ1̇ = −iε̄η1, this would seem to also require η1 = 0. With this in place, most things are consistent, except the relation for δη1, whose right-hand side gives the consistency condition b [ +iγ μa(1) μ ε − 2π k |χ2̇|2ε + μ 2 ε ] = 0 (4.22) which is a constraint on fields and so is not satisfied in gen- eral. Apparently then, the Landau-Ginzburg system cannot be supersymmetrized in general. 4.2 Global Charge Analysis and Applicability to Condensed Matter To summarize our findings of the previous section, the reduced effective action in (4.6) has N = 2 supersym- metry. It has a local U(1) × U(1) invariance, where the fields φ1, χ1, η1, η̃1 are charged with charge +1 under the first U(1), and φ2, χ2, η2, η̃2 all carry charge +1 under the second U(1). With respect to global symmetries, the original ABJM model had SU(4)×U(1) R-symmetry before the addition of the mass term. Adding the mass term breaks this to SU(2)× SU(2) × U(1)A × U(1)B × Z2, where the SU(2)’s act on Q and R, respectively. Specifically, U(1)A acts on Q with charge +1 and on R with charge −1, and Z2 interchanges Q and R. We note now that the action (4.6) has an overall U(1)8 × Z2 global invariance. Each of the U(1)’s acts on just one of the eight fields φ1, φ2, χ1, χ2, η1, η2, η̃1, η̃2 and not on any of the rest. Out of these global symmetries, two linear com- binations defined above are promoted to local invariances, leaving a total of six factors of U(1) as just global invari- ances. In addition, the Z2 acts by interchanging indices 1 and 2, i.e., φ1 ↔ φ2, χ1 ↔ χ2, η1 ↔ η2, η̃1 ↔ η̃2, a(1) μ ↔ a(2) μ . (4.23) Compared to the action in (2.1), here the gauge charge is either +1 or zero for both groups (for half of the fields, it is +1 and the other half, 0), which is already different. We could, of course, choose the equivalent of the global electric charge of (2.1) to be, completely diagonal, Q ∼ φ † i φi + χ † i χi + η † i ηi + η̃ † i η̃i (4.24) but this would be just the sum of the two gauge charges (for the two U(1) gauge groups), and so would not constitute an independent charge. We could also have considered that the field φ, for example, carries charge +1 and φ†, −1, but that would only mean that there are positive and negative charges, and we would still have a global charge that is the sum of the two local charges. As an alternative, one could choose as the electric charge some other combination of the eight global U(1)’s like, for example, the charge under which the φi, ηi have charge +1, and χi, η̃i have charge −1. Then, considering that the local charges are +1 for the fields and −1 for their conjugates, the global charge is different from any lin- ear combination of the local charges. However, this global charge has both positive and negative carriers, unlike the expression in (2.2). In conclusion, while our abelianization produces a model that exhibits some of the more general features of the condensed matter model (2.1), it differs in some of the finer details and since, as they say, this is where the devil is, it is worth exploring other avenues as well. Braz J Phys (2015) 45:481–492 487 5 Adding Fundamentals to the ABJM Model To this end, we now consider the addition of fundamental fields to the ABJM model, with the ultimate goal of special- izing to N = 1 at the end and thus obtaining the required charged fields even in the abelian case. In the case of Nc coincident Dp-branes, with an SU(Nc) gauge theory liv- ing on them, adding fundamentals can be done in two ways. The first is via the addition of a probe D(p + 4)-brane (or D(p + 2)-brane) to the gravitational background set up by the Nc Dp-branes with the role of the fundamental scalars in the SU(Nc) gauge theory played by open strings stretched between the stack of Dp-branes and the probe brane. In this case, however, a Dp −D(p + 4) system with the D(p + 4)- brane wrapped on a compact space is not consistent since the flux has nowhere to go on a compact space2. To cancel the flux, one needs to add negative charge. Normally, this would require the addition of anti-branes, but if one wants to preserve supersymmetry, this is no longer an option. In this case, the only consistent solution is to add an orientifold plane, for a total system of Dp − D(p + 4) − O(p + 4). A further complication in our case is that the ABJM model has a product gauge group, SU(N) × SU(N), with bifundamental scalars and fermions under both gauge groups, so it is not obvious at all how to add the fundamen- tals. Fortunately, all is not lost since we do know that giving a VEV to one of the scalars, say 〈X1〉 = v, turns the ABJM model into a D2-brane gauge theory with SU(N) gauge group, via a 3D Higgs mechanism [12, 13]. Thereafter, the construction should reduce to the general one. 5.1 A Review of the D3-D7-O7 Case To better understand how to proceed, let us begin by reviewing the case of D3-branes. In this case, consistently adding fundamental fields while preserving supersymmetry is achieved by adding D7-branes and an orientifold O7- plane, as in the construction of [17]. The D3-branes by themselves, of course, provide an SU(N) gauge group, and the gravity dual is the all too familiar AdS5 × S5. Adding D7-branes to this background means that they need to be parallel to AdS5 and wrapping a three-cycle inside the S5. That means that for consistency of the 7-brane flux, we need to add an orientifold plane, which acts as a sink for the flux. This orientifolding means that the gauge group is now USp(2N) and we have an N = 2 superconformal field the- ory. Since the orientifold plane carries charge −4, one needs to add four D7-branes together with the O7-plane, resulting in a global SO(2Nf ) = SO(8) group. 2As a simple intuitive picture, a positive electric charge on a circle has flux lines going away from the charge on both sides, so it needs a corresponding negative charge somewhere else to sink the flux lines incoming from both sides. 5.1.1 The Gravity Picture For the gravity dual, the spatial Z2 orbifold part of the orien- tifold acts on AdS5×S5 as follows. The metric on AdS5×S5 is written as ds2 = R2 ( − cosh2 ρdt2 + dρ2 + sinh2 ρd�2 3 + cos2 θdψ2 + dψ2 + sin2 θd�̃2 3 ) d�̃2 3 = cos2 θ ′dψ ′2 + dθ ′2 + sin2 θ ′dφ2, (5.1) and then, the Z2 acts by sending ψ ′ → ψ ′+π . The effect of the orientifolding is to have θ ′ ∈ (0, π/2) instead of (0, π), and the invariant plane (the orientifold O(7)-plane) is situ- ated at θ ′ = π/2 and carries −4 units of D7-brane charge. It is this that is canceled by the addition of the four D7- branes. At this point, it is worth noticing two things: First, the plane is actually an S3 inside S5 and second, that we could also have chosen the plane at θ = π/2 instead of the one at θ ′ = π/2 as the O7-plane. Since the decoupling limit leaves the D7-branes unchanged, there are the four D7-branes on top of the O7- plane wrapping the S3. This results in a (7+1)-dimensional Super Yang-Mills theory at the location of the O7-plane in the gravity dual. The KK modes of the eight- dimensional SYM reduced on S3 are fields in a representation of the SO(4) group, and charged under the global SO(2Nf ) = SO(8) group. 5.1.2 The Field Theory Picture In order to establish the line of argument as well as our notation, let us briefly review the analysis presented in [17, 18]. The N = 4 SYM multiplet on N D3-branes is com- posed of ( Aa μ, ψaI α , Za[IJ ]), where a is an index in the adjoint of U(N), which can also be written as a = ij̄ , with i, j̄ = 1, ..., N in the (anti)fundamental of U(N). The Z are complex scalars, with I = 1, ..., 4 an index in the fun- damental of SU(4) (or equivalently the spinor of SO(6)), meaning that [IJ ] is in the antisymmetric 6 representation of SU(4). These fields satisfy the reality condition Za[IJ ] = εIJKL ( Z† )a KL , (5.2) and correspond to the six real transverse coordinates of the D3-brane. Orientifolding corresponds to identifying some of the fields. Under this identification, the gauge group changes from SU(N) to USp(2N). Importantly, adjoint fields remain in the adjoint, except that instead of an unrestricted ij , the adjoint of USp(2N) is the symmetric (ij), or 2N(2N + 1)/2 representation. The scalars in the adjoint 488 Braz J Phys (2015) 45:481–492 representation now are simply singlets under the remain- ing SU(2) × SU(2) global symmetry and carry unit charge under the U(1). Adding the D7-branes means that we also need an anti- symmetric field YAA′ [ij ] in the 2N(2N − 1)/2 representation as well as four fundamentals, qAm i , of the gauge group USp(2N). This breaks supersymmetry down to N = 2, which in turn means that the R-symmetry splits as SO(6) → SO(4) × SO(2), or equivalently SU(4) → SU(2)×SU(2)×U(1). Of this, only the SU(2)R ×U(1) is still an R-symmetry and the other, SU(2)L, factor becomes a simple global symmetry. The index I splits into AA′, with A, A′ = 1, 2 corresponding to the two SU(2) factors. While YAA′ [ij ] respects the SU(4) symmetry, qAm i does not since it is charged only under SU(2)R , which is the only R-symmetry of the final model. The index m = 1, ..., 8 belongs to the flavor group SO(8). In summary, • Z is a singlet of SU(2)L, SU(2)R , and SO(8), with unit charge under U(1); • Y is a doublet of SU(2)L and a doublet of SU(2)R and singlet under U(1) and • q is a singlet under SU(2)L, a doublet under SU(2)R , has no charge under U(1), and is in the fundamental of SO(8). The fields Y and q satisfy reality conditions qA i = εABJij ( q† )j B (5.3) YAA′ [ij ] = εABεA′B ′ JikJjl ( Y † )[kl] BB ′ where Jij is the antisymmetric invariant matrix of USp(2N), Jij = ( 0 1 −1 0 ) (5.4) Because of the reality conditions, qA are two real scalars, corresponding to the two overall transverse coor- dinates, i.e., transverse to both the D3-branes and the D7-branes (as do the two real components of the com- plex adjoint Z scalars), and YAA′ are four real scalars, corresponding the four relative transverse coordinates, namely parallel to the D7-branes, but transverse to the D3-branes. In N = 1 language, where we can write the scalars qAm i as belonging to two chiral superfields qm and q̃m, and the scalars YAA′ [ij ] as belonging to two chiral superfields YA′ and Y ′A′ , the superpotential can be written as W ∼ ( Zijq imq̃jm + ZijJ jkYA′ kk′J k′lY ′A′ ll′ J l′i ) , (5.5) with the resulting F-terms Fq̃ = Zq, Fq = Zq̃, FY ′ il = ZijJ jkYkl − (i ↔ l), (5.6) FY il = ZijJ jkY ′ kl − (i ↔ l), FWil = Yij J jkY ′ kl + (i ↔ l) + (qq̃)(il). For the component action, the scalar potential is given by the squares of the F-terms plus the square of the D-terms. To close this discussion of the field theory side of the orientifolding, we point out that • there are operators with SO(4)R = SU(2) × SU(2) indices as well as flavor SO(8) indices, like for instance OABmn = q̄AmqBn (with an implicit sum over the gauge indices i), which couple to the fields coming from the S3 reduction of 7 + 1-dimensional SYM in the gravity dual and • there are also the usual gauge invariant operators with- out flavor indices, only with SO(6)R = SU(4)R indices, which couple to the fields coming from the S5 reduction of supergravity fields in the gravity dual. 5.2 The Gravity Dual Returning to our case at hand, we consider a construc- tion which should reduce to a D2-D6-O6 system when we reduce the ABJM model down to type IIA string theory, i.e., a T-dual construction to the one above. The background that we work with is AdS4 × CP 3 instead of AdS5 × S5 but, as before, we look for a Z2 symmetry leading to an O6-plane inside the gravity dual. The Fubini-Study metric on CP 3 is ds2 = dξ2 + cos2 ξ 4 ( dθ2 1 + sin2 θ1dφ2 1 ) + sin2 ξ 4 ( dθ2 2 + sin2 θ2dφ2 2 ) + cos2 ξ sin2 ξ(dψ + A1 − A2), (5.7) Ai = 1 2 cos θidφi. Just as in the D3 − D7 case, we will choose the O6-plane parallel to AdS4 and wrapping the codimension-3 cycle in CP 3, θ1 = θ2 = π 2 ; ψ = π, (5.8) which is a fixed plane for the Z2 action φ1 → φ1 + π; φ2 → φ2 + π; ψ → ψ + π. (5.9) The metric on this fixed plane is ds2 = dξ2 + cos2 ξ 4 dφ2 1 + sin2 ξ 4 dφ2 2 , (5.10) Braz J Phys (2015) 45:481–492 489 with coordinate ranges 0 < ξ < π/2 and 0 ≤ φi ≤ 2π and the action (5.9), which is an S3/Z2 ∈ CP 3. The same cycle has been used in [19], where more details of the con- struction are given, so we will wait until we compare to that paper to comment further. We note also that in [19], it was observed that this construction makes an orbifold but it was not required that an orientifold O6-plane live there, as we do. 5.3 The Field Theory We start by trying to understand the system when thought of as a D2-D6-O6 system, i.e., when one of the scalars has aquires VEV and the Higgs mechanism on the ABJM model leads to N D2-branes. That analysis is T-dual to the D3-D7- O7 case studied above, so it will be quite similar. 5.3.1 D2-brane Analysis The field theory living on the stack of N D2-branes is an N = 8 Super Yang-Mills, with field content( Aa μ, ψaA α , XaI ′) , where a = ij is in the adjoint of SU(N), I ′ = 1, ...7 is in the fundamental of SO(7) and A = 1, ..., 4 is in the fundamental of SU(4) or spinor of SO(6), which can be also understood as the spinor of SO(7). We have an overall SO(8) R-symmetry, of which only SO(7)R is manifest. When we do the orientifold projection, the gauge group changes from SU(N) to USp(2N), and the fields are still in the adjoint, though the adjoint of USp(2N) is now a sym- metric representation, a = (ij) ∈ 2N(2N +1)/2. However, the scalar fields X are now restricted to be in a three- dimensional (adjoint) representation of the SU(2) × SU(2) relative transverse global symmetry. The addition of the D6-branes again breaks SO(7) to SO(4) × SO(3), or SU(2) × SU(2) × SU(2), and splits the index A into MM ′ under the two SU(2) factors. We will also have an SO(8) flavor group, with fundamental index m = 1, ..., 8 because, again, we need to add four D6-branes to cancel the −4 charge of the O6-plane. As before, we also need to add antisymmetric tensor fields ZMM ′ [ij ] in the 2N(2N − 1)/2 representation of USp(2N), which satisfy the same reality condition as before, ZMM ′ [ij ] = εMNεM ′N ′ JikJjl ( Z† )[kl] NN ′ , (5.11) meaning that again, we have four scalars corresponding to the four relative transverse directions, parallel to the D6- branes and transverse to the D2-branes. Finally, we need to add fundamental scalars qMM ′′m i com- ing from the strings stretching between the D2-branes and the D6-branes, and corresponding to four overall transverse coordinates (transverse to both D2-branes and D6-branes). Note that naively, there should be three coordinates, but because of supersymmetry, the chiral multiplet has to have four scalars. This fits quite nicely with our interpretion of the following model as coming from eleven dimensions where it will be clear that we need four scalars. So, unlike the D3 − D7 case now the number of X’s (3) differs from the number of q’s (4), pointing to the fact that there is a better interpretation in eleven dimensions. Here, the index M is in a two-dimensional representa- tion of one of the relative transverse SU(2)’s, while M ′′ is in a two-dimensional representation of the overall trans- verse SU(2). Since the scalars need to be four, they satisfy a reality condition, qMM ′′m i = Jij ε MNεM ′′N ′′ ( q† )mj NN ′′ . (5.12) We will write the superpotential in the ABJM case only since it is easier to understand, and this case can be obtained by Higgsing. 5.3.2 Lifting to the ABJM Model The ABJM model is an N = 6 supersymmetric model with gauge group U(N) × U(N) and gauge fields A a(1) μ , A a′(2) μ and bifundamental fields ZAii′ and ψAii′ α , where a and a′ are in the adjoint of U(N), while i, i′ = 1, ..., N are in the fundamental of U(N), and A = 1, ..., 4 is in the fundamen- tal of SU(4)R . The R-symmetry group is SU(4) × U(1) = SO(6) × SO(2). The orientifold projection changes the adjoints a, a′ into symmetric tensor adjoints a = (ij) and a′ = ( i′j ′) of USp(2N). However, it now also requires that we identify the two gauge group factors. To see how this works, it will be useful to review how the ABJM model is constructed. We start with N D2-branes in type IIA wrapping a compact direction and broken in two places by an NS5-brane and a NS5′-brane. Then, one adds k D6-branes to one of the NS5-branes to turn it into a (1, k) 5-brane and rotates. The rest of the procedure is not important for our discussion. Bifundamental fields arise from strings stretching between one half of the D2-branes to the other half, through the 5-brane. Orientifolding cor- responds to adding an O6-plane at a certain point in the compact direction. This can only be the location of one of the 5-branes because of symmetry (since otherwise we would get a set-up which is asymmetric between the two gauge groups). That means that the orientifold projection will identify the two half-branes, i.e., the two gauge groups. The identification means that now we can decompose (ii′) in irreducible representations of the unique gauge group, i.e., ZAii′ = ZA(ii′) + ZA[ii′] (5.13) 490 Braz J Phys (2015) 45:481–492 where the symmetric tensor is the adjoint of USp(2N), and [ii′] is the antisymmetric representation in the decomposi- tion. The orientifold projection will also impose the reality conditions ZMM ′[ii′] = εMNεM ′N ′ Jij Ji′j ′ ( Z† )[jj ′] NN ′ , (5.14) ZMM ′(ii′) = εMNεM ′N ′ Jij Ji′j ′ ( Z† )(jj ′) NN ′ , which means that the ZA[ii′] now describe the four scalars corresponding to the relative transverse directions (paral- lel to the D6-branes, but transverse to the D2-branes), whereas ZA(ii′) describe as usual the four scalars corre- sponding to the overall transverse directions (transverse to both D2-branes and D6-branes). Adding D6-branes breaks the R-symmetry from SU(4)× U(1) to SU(2)× SU(2)×U(1), while adding fundamental fields coming from the strings stretching between the D6- branes and the two halves of the D2-branes. Naively, this leads to fields corresponding to four coordinates, qMm̃ i and q̃Mm̃′ i′ , which are therefore in a two-dimensional represen- tation of the SU(2) × SU(2) group and are charged under the U(1). However, since the orientifold projection identi- fies the gauge groups, as well as the D2 − D6 fields, so too do the q and q̃ combine. At this point, we can either describe this as taking m̃, m̃′ = 1, ..., 4 and combining them into m = 1, ..., 8, or we can consider that both indices take eight values, but then q is identified with q̃. Either way, the result is that the fields qMm i , q̃Mm i , satisfy the reality condition qMm i = Jij ε MN ( q† )mj N ; q̃Mm i = Jij ε MN ( q̃† )mj N . (5.15) In summary then, the field content of our flavored ABJM model ends up being ZA(ii′), ZA[ii′], qMm i , q̃Mm i . To write the superpotential for the theory, we first recall that the superpotential in the usual ABJM model (without orien- tifolding) can be written in terms of bifundamental fields Bi, Ai and auxiliary adjoint fields φi , as WABJM = k 8π Tr [ φ2 1 − φ2 2 ] + Tr[Biφ1Ai] + Tr[Aiφ2Bi], (5.16) or eliminating the auxiliary fields, as the quartic form WABJM = 2π k Tr(AiBiAjBj − BiAiBjAj ) = 2π k Tr(A1B1A2B2 − B1A1B2A2). (5.17) Then, the superpotential for our model is W ∼ εM ′N ′ ( Z(ii′) + Z[ii′] )MM ′ J i′j ( Z(jj ′) + Z[jj ′] )MN ′ J j ′k × ×εP ′R′ ( Z(kk′) + Z[kk′] )NP ′ J k′l (Z(ll′) + Z[ll′] )NR′ J l′i + qiNm [ εM ′N ′ (Z(ii′) + Z[ii′])MM ′ J i′j ′ ( Z(j ′j) + Z[j ′j ] )MN ′ ] q̃jNm. (5.18) As a check, we verify what happens under Higgsing. This corresponds to decomposing the ZA(ii′) into the seven scalars X(ii′)I ′ and an eigth scalar that gets eaten by the gauge fields to become dynamical, with all other fields unchanged. The resulting model matches the D2-brane analysis above as it should, providing a consistency check of the construction. 5.4 Comparison with Previous Constructions Another construction for adding fundamentals to ABJM was found in [19]. It corresponds to basically adding a probe D6-brane without considering the issues of the flux on a compact space or of exact conformality of the system (which is required in order to have a gravity dual with an AdS4 factor). In four dimensions, one can do the same by considering a probe D7-brane in AdS5 × S5, wrapping a codimension-2 cycle in S5 (see, e.g., [20]). But this con- truction is not without subtleties. First, among these is that any such system will be afflicted with the aforementioned problem with the flux, in that we need a negative sink of flux on a compact space; otherwise, the flux lines will meet at a singular point away from the D7-brane.3 This would manifest itself in general by uncancelled tadpoles in the field theory. A second problem is that the theory cannot be exactly conformal because the uncancelled flux will set a scale. It would only be so in a limited energy range which, in the probe approximation, can be considered large enough; hence, the gravity dual cannot be purely AdS5 times another factor. These problems are solved by the construction in [17]. which introduces an O7-plane of charge −4 and four D7-branes to compensate at the same fixed point. Conse- quently, there is no uncancelled flux on the compact space, and the field theory is exactly conformal. The same conclusion applies to our case. The construc- tion of [19] is only valid in the probe approximation and can be considered to be obtained by separating the O6-planes and the other D6-branes, and moving them far away on the 3Having a D-brane on a collapsable cycle as opposed to a point charge does not help from the point of view of charge, though it avoids tad- poles due to its instability [20]: consider a D1-brane wrapping an S1 cycle inside S2. The D1-brane can shrink until the cycle wraps is very small and it looks almost pointlike, say around the South Pole of the S2. But then we have the same problem with the electric charge: the flux lines will meet again at the North Pole, where therefore there should be a sink of negative charge of equal absolute value. Braz J Phys (2015) 45:481–492 491 compact space from the D6-brane we retain. The construc- tion corresponds to a D6-brane wrapping a codimension-3 cycle in CP 3, which is in fact the same S3/Z2 defined above, but without any orientifolding. We have, instead, considered this cycle to be the fixed plane of an O6 orien- tifold plane and have added four D6-branes there. In [19], the cycle was initially defined in a different way, but one can easily show it is the same cycle. It was also proven that it corresponds to a supersymmetric brane configuration, at it should. The superpotential for the model in [19] is similar to ours. In N = 1 language, the ABJM model has bifundamental fields (A1, A2) in the (N, N̄) representation and (B1, B2) in the (N̄, N) representation, and auxiliary adjoints φ1, φ2. The fundamental fields we add are q1, q2 in the (N, 1) and (1, N) representation, and q̃1, q̃2 in the (N̄, 1) and (1, N̄) representation. The ABJM model superpotential is WABJM = k 8π Tr [ φ2 1 − φ2 2 ] + Tr[Biφ1Ai] + Tr[Aiφ2Bi], (5.19) and the flavor deformation is Wflavor = Tr [ q̃1φ1q1 ] + Tr [ q̃2φ2q2 ] . (5.20) Clearly then, after eliminating the auxiliary fields, the con- struction is similar to ours. To conclude this section, let us recall the symmetries of the model. The R-symmetry is an SU(2)R , acting on( Ai, B̄i ) , with an internal SU(2), acting on the doublets (A1, A2) and (B1, B2). We can also, of course, introduce several flavors Nf , to produce a global flavor symmetry group SO ( Nf ) . 5.5 Applicability to Condensed Matter To find possible applications of these flavored models to condensed matter physics, we need to understand the physics of N = 1 truncations. To this end, the first point to note is that the orientifold model is still nonabelian for for N = 1 since we have now a USp(2) gauge theory. There appears nothing to be done about this, as it is just the result of the orientifold procedure. We can however choose a global U(1) charge inside the SO(8) carried by the q’s, and in this way get fundamental fields q, q̃ and their conjugates, coupling to the local gauge group with charges +1 and −1, and contributing +1 to the global charge (the analog of elec- tric charge for the condensed matter model), as we wanted. As an added bonus, in this case, the theory is conformal. For the probe model in [19], we can now consider N = 1, and the q, q̃ fields are in the fundamental of the resulting U(1) gauge group, with positive and negative charges. If we choose several flavors, with a SO(Nf ) symmetry group, we can again choose a U(1) subset that corresponds to the global charge with +1 charge contributions, with the important caveat of the potential problems discussed above. 6 Conclusions In this article, we have analyzed several ways in which one can obtain an abelian theory out of the ABJM model, for the purpose of simulating condensed matter models of interest. In particular, we have analyzed features of a model used for, among other things, the description of compressible Fermi surfaces in [9]. We have seen that simply setting N = 1 in the ABJM model does not work since we obtain a free abelian theory, with scalars which are not charged under the U(1) Maxwell gauge group (after using the Higgsing procedure in (2 + 1)-dimensions to go from CS to Maxwell gauge fields). Instead, one possibility that we found is to generalize the nontrivial abelian reduction in [3, 4] to include fermions. In this way, we obtained an abelian theory with N = 2 super- symmetry and six global U(1) charges, a combination of which can be taken to be somewhat similar to the global electric charge in condensed matter models, in that the posi- tive and negative charges of various fields are different from the positive and negative local charges. Another possibility that we found was to add fundamental fields to the ABJM model. 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JHEP 0206, 043 (2002). arXiv:hep-th/0205236[hep-th] http://arxiv.org/abs/1205.5833 http://arxiv.org/abs/1010.0443 http://arxiv.org/abs/1108.1197 http://arxiv.org/abs/1301.0518 http://arxiv.org/abs/0806.1218 http://arxiv.org/abs/1104.5022 http://arxiv.org/abs/0912.0903 http://arxiv.org/abs/1010.3808 http://arxiv.org/abs/0803.3218 http://arxiv.org/abs/1012.5969 http://arxiv.org/abs/0807.1074 http://arxiv.org/abs/0807.0197 http://arxiv.org/abs/1203.3546 http://arxiv.org/abs/hep-th/9806159 http://arxiv.org/abs/hep-th/0203249 http://arxiv.org/abs/0903.2194 http://arxiv.org/abs/hep-th/0205236 On Abelianizations of the ABJM Model and Applications to Condensed Matter Abstract Introduction Condensed Matter Models ABJM U(N)U(N) Model for N=1 Nontrivial Abelianization with Fermions Supersymmetry Global Charge Analysis and Applicability to Condensed Matter Adding Fundamentals to the ABJM Model A Review of the D3-D7-O7 Case The Gravity Picture The Field Theory Picture The Gravity Dual The Field Theory D2-brane Analysis Lifting to the ABJM Model Comparison with Previous Constructions Applicability to Condensed Matter Conclusions Acknowledgments References