Twisted 3D N=4 Supersymmetric YM on deformed A ∗ 3 Lattice
aa r X i v : . [ h e p - t h ] J u l LPHE-MS rabat
Twisted N = A ∗ Lattice
El Hassan Saidi ∗
1. LPHE-Modeling and Simulations, Faculty Of Sciences, Rabat, Morocco2. Centre of Physics and Mathematics, CPM- Morocco
August 30, 2018
Abstract
We study a class of twisted N = 4 supersymmetric Yang-Mills (SYM)theory on particular 3- dimensional lattice denoted as L su × u D and given by nontrivial fibration L u D × L su D with base L su D = A ∗ , the weight lattice of SU (3).We first, develop the twisted N = 4 SYM in continuum by using superspacemethod where the scalar supercharge Q is manifestly exhibited. Then, we showhow to engineer the 3D lattice L su × u D that host this theory. After that we buildthe lattice action S latt invariant under the 3 following: ( i ) U ( N ) gauge invariance,( ii ) BRST symmetry, ( iii ) the hidden SU (3) × U (1) symmetry of L su × u D . Otherfeatures such as reduction to twisted supersymmetry with supercharges livingon L su × u D , the extension to twisted maximal SYM with supercharges onlattice L su × u D as well as the relation with known results are also given. Keywords : Reduction of chiral N = (1 ,
0) SYM, BRST symmetry and Scalarsupersymmetry, Twisted SYM on lattice, Root and weight lattices of SU ( k ). Contents ∗ [email protected] ma Twisted SYM with 8 supercharges 6
Classification of so ( t, s ) spinors . . . . . . . . . . . . . . . . . . . 72.1.2 Chiral supersymmetric YM in 6D and 10D . . . . . . . . . . . . 112.2 Twisted N = 4 SYM in from N = 1 SYM . . . . . . . . . . . . . . . 142.2.1 the 6D N = 1 vector multiplet . . . . . . . . . . . . . . . . . . . 142.2.2 Reduction from 6D to 3D and twisting . . . . . . . . . . . . . . . 16 N = 4 algebra and superfields 19 N = 4 supersymmetry in . . . . . . . . . . . . . . . . . . . . 193.1.1 anticommutators . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.2 twisted superspace and superderivatives . . . . . . . . . . . . . . . 203.2 Superfields of twisted N = 4 SYM . . . . . . . . . . . . . . . . . . . 213.2.1 Gauge covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2
Supersymmetric transformations . . . . . . . . . . . . . . . . . . . 26 N = 4 SYM theory on lattice 31 L su × u D . . . . . . . . . . . . . . . . . . . . . . . . 325.1.1 Discretizing continuum . . . . . . . . . . . . . . . . . . . . . . . . 325.1.2
First result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Complex extension and orientation . . . . . . . . . . . . . . . . . . . . . 385.2.1 twisted supersymmetric YM on A ∗ . . . . . . . . . . . . . . . . . 415.2.2 action of twisted supersymmetric YM on A ∗ . . . . . . . . . . . . 45 A ∗ . . . . . . . . . . . . . . . . . . . . . . . 476.2 Building the lattice L su × u D . . . . . . . . . . . . . . . . . . . . . . . . . 506.2.1 Construction of the lattice A ∗ . . . . . . . . . . . . . . . . . . . . 506.2.2 The lattice L su × u D as a twist of A ∗ . . . . . . . . . . . . . . . . . 526.3 Lattice interpretation of BRST symmetry . . . . . . . . . . . . . . . . . 546.3.1 Closest neighbors in L su × u D . . . . . . . . . . . . . . . . . . . . . 556.3.2 BRST symmetry on lattice . . . . . . . . . . . . . . . . . . . . . . 582
Action of 3D N = 4 on Lattice L su × u D L su × u D : a dictionary . . . . . . . . . . . . . . . . . . . . . . . . 597.2 Useful identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.3 The action on L su × u D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 the set of superfields . . . . . . . . . . . . . . . . . . . . . . . . . 729.2.2 Deriving eqs(9.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Following [1]-[9] and refs therein, the lattice version of maximal euclidian four di-mensional N = 4 supersymmetric Yang Mills theory (SYM) with U ( N ) gauge invari-ance may be approached by twisting supersymmetry and requiring invariance underthe scalar supercharge Q of the resulting twisted gauge theory. In this method, the16 = 2 supersymmetric charges ( Q iα , Q ˙ αi ), transforming in the spinorial representationof SO E (4) × SO R (6), are thought of in terms of 2 × matrix Q × that can be ex-panded on products of 4 × γ µ matrices; for a general review see [1, 10] and[11]-[14] for related works. The expansion of Q × leads, on one hand, to the integralspin decomposition Q × = IQ + γ µ Q µ + γ [ µν ] Q µν + γ µ γ ˜ Q µ + γ ˜ Q (1.1)where the supercharges are split as 16 = 1 + 4 + 6 + 4 + 1; and, on the other hand,to a remarkable packaging of the field spectrum of the twisted 4D N = 4 SYM theoryinto SU (5) × U (1) representations likebosons : 10 → ⊕ ¯5fermions : 16 → ⊕ ¯5 ⊕
10 (1.2)Because of the algebraic property Q = 0, the scalar supercharge behaves as a topolog-ical object [15, 16]; a feature that allows to: ( i ) put the fields of the twisted N = 4 Euclidian QFTs are generally thought of in terms of a Wick rotation of corresponding LorentzianQFTs. However this analytic continuation is not a soft operation especially for spinors. This issue isnot directly addressed in this paper; but results of the Osterwalder-Schrader (OS) method are used.For more details on this issue, including the OS method and other approaches to overcome difficultiesinduced by analytic continuation, see [30] and refs therein; see also eq(2.22) to fix the ideas. A ∗ with a hidden SU (5) symmetry [8, 9];and ( ii ) write down a U ( N ) gauge invariant lattice field action S latt having, in additionto the SU (5) symmetry of A ∗ , a BRST symmetry generated by Q governing its quantumproperties [9].In this paper, we borrow this idea to study the lattice version of the class of twisted supersymmetric Yang-Mills theories with supercharges having an SU (3) × U (1)symmetry. This twisted supersymmetric YM theory follows from the reduction ofchiral N = (1 ,
0) SYM in and living on a particular lattice to be built in thepresent work (see section 6). Our interest into this class of twisted YM theories has beenmotivated by the two following: extend the approach of [1, 3] to the class of lattice supersymmetric YM models basedon twisting SYM theories with supercharges. It turns out that the twisted lattice gauge theory is very suggestive; it lives on a particular crystal denoted hereas L su × u D having a hidden SU (3) × U (1) symmetry; and given by a non trivial fibration L su D × L u D with 2-dimensional base sublattice L su D = A ∗ and fiber L u D isomorphicto Z , the set of integers. This kind of fibration, encoded by eq(5.8), allows also toget more insight into literature results; especially in the case of twisted maximalsupersymmetry living on the lattice A ∗ with SU (5) symmetry.To approach the case of twisted SYM with supercharges, and in a subsequentstep the class with supercharges as done in section 8, we develop a method ofengineering ( k + 1)-dimensional crystals with SU ( k ) × U (1) symmetry; and useresults on the breaking mode of the real SO (2 k ) euclidian symmetries down to thecomplex SU ( k ) × U (1) , k odd integerto get the packaging of the twisted fields into representations of SU ( k ) × U (1);and also to determine their interpretation on lattice L su k × u kD in terms of links andplaquettes.Recall that N = 4 supersymmetric Yang Mills with SO E (4) × SO R (6) sym-metry is a maximal supersymmetric YM theory that has the same number of con-served supercharges as N = (1 ,
0) SYM in euclidian 10-dimensions with isotropy4ymmetry SO E (10) , k = 5Similarly, the twisted N = 4 YM theory we are interested in here can beobtained in a quite analogous manner; but by dimensional reduction of the chiral N = (1 ,
0) SYM in 6-dimensions with euclidian symmetry SO E (6) , k = 3 explore the role of the extra U (1) symmetry that appears in twisted supersymmetricYM theories; in particular in the case of supercharges with SU (3) × U (1) sym-metry; and also in twisted maximal supersymmetry with an SU (5) × U (1). A wayto exhibit this global abelian invariance is through the breaking of SO E (2 k ) downto SU ( k ) × U (1) , which for k = 5 and k = 3 , read respectively as follows SO E (10) → SU (5) × U (1) SO E (6) → SU (3) × U (1)Under these symmetry breaking modes, real vector and spinorial k − repre-sentations of SO E (10) and SO E (6) decompose as sums of representations withrespect to the complex symmetries. For SO E (10), the decomposition of the 10 v and the 16 s are given by SO E (10) → SU (5) × U (1)10 v : 5 +2 q + ¯5 − q s : 1 − q + ¯5 +3 q + 10 − q with q a unit U (1) charge; and for the case of SO E (6), their analogue read like SO E (6) → SU (3) × U (1)6 v : 3 +2 q + ¯3 − q s : 1 − q + 3 + q These breakings show that in twisted supersymmetric YM theories, the twistedfields and the twisted supersymmetric operators, in particular the BRST charge Q ( − kq ) , are in general sections of a U (1) bundle. This property teaches us in turnsthat on lattice side SU (3) scalars carrying non trivial U(1) charges have also a nontrivial interpretation; they are associated with links along the 1- dimensional fiberand, in some sense, constitute a refining of results of [1, 3] since a similar conclusionis also valid for the case of twisted maximal supersymmetry on the lattice L su × u D .5n what follows, we focus on the study of the lattice version of twisted N = 4 SYM;and, to exhibit the role played by U (1) subsymmetry, we distinguish the two cases: thegeneric q = 0 and the singular q = 0. A similar analysis can be performed for the caseof twisted maximal supersymmetric YM in as reported in the section conclusion andcomments.The organization is as follows: In sections 2 and 3, we first review some useful toolson SO ( t, s ) spinors in diverse dimensions D = t + s . Then, we study the twisted N = 4 supersymmetric YM theory in continuum. In section 4, we build the action insuperspace and derive its component field expression. In section 5, we study the twisted N = 4 supersymmetric YM on the base sublattice A ∗ having SU (3) symmetry andcorresponding to the singular limit q = 0. In section 6, we study twisted N = 4SYM on the crystal L su × u D with SU (3) × U (1) symmetry and q = 0. In section 7,we build the action of the twisted field on L su × u D . In section 8, we give a conclusionand make two comments; one on the reduction down to twisted N = 4 and thesecond concerns the extension to N = 4 on the lattice L su × u D containing A ∗ as abase sublattice. In section 9, we give an appendix where we give explicit computationsand technical details on the construction of gauge covariant superfields. After recalling useful tools on SO ( t, s ) spinors in diverse dimensions and briefly describ-ing the reduction of chiral N = (1 ,
0) SYM in Lorentzian down to D , we studytwisted N = 4 SYM in continuum. More on continuum and the lattice version ofthis SYM theory will be developed in next sections. Here we collect some results on SO ( t, s ) spinors living on the flat space R ( t,s ) with spacetime dimension D = s + t and signature s − t where s and t stand respectively for thenumbers of space like and time like directions. A particular interest will be given to theLorentzian ( t = 1) and euclidian ( t = 0) signatures that we are interested in this work.6 .1.1 Classification of so ( t, s ) spinors Generally speaking, spinors Ψ A living on R ( t,s ) with metric η MN = diag ( − .. − , + .. +) havecomplex 2[ D ] components transforming under the space isotropy symmetry SO ( t, s ) asΨ A → Ψ ′ A = S BA Ψ B (2.1)with matrix transformation given by S = e i ω MN Σ [ MN ] Σ [ MN ] = Γ M Γ N − Γ N Γ M (2.2)In these relations, the 2[ D ] × D ] matrices Γ M are the usual gamma matrices generatingthe Clifford algebra Cl ( t, s ) defined byΓ M Γ N + Γ N Γ M = 2 η MN (2.3)The matrices Γ M are generally realized in terms of particular monomials of tensor prod-ucts of the usual 2 × σ , σ , σ that we take as σ = ! , σ = − ii ! = iε, σ = − ! (2.4)obeying amongst others the property σ T = σ , σ T = σ but σ T = − σ . For the exampleof the euclidian (0 ,
6) signature, where the metric η MN coincides with the Kroneckersymbol δ MN , the Γ M ’s are 8 × = σ ⊗ I ⊗ I Γ = σ ⊗ I ⊗ I Γ = σ ⊗ σ ⊗ I Γ = σ ⊗ σ ⊗ I Γ = σ ⊗ σ ⊗ σ Γ = σ ⊗ σ ⊗ σ Γ = σ ⊗ σ ⊗ σ (2.5)with the remarkable properties (Γ i ) † = +Γ i (Γ k +1 ) T = − Γ k +1 (Γ k ) T = +Γ k (2.6)7or the cases of the Lorentzian (1 ,
5) and (2 ,
4) signatures, the realization of the corre-sponding Γ M ’s is obtained from the euclidian representation by using Wick like rotationsas follows signature ( , ) signature ( , ) Υ = i Γ Υ = i Γ , Υ = i Γ Υ m = Γ m +1 , m > m = Γ m +1 , m > A , to which we refer to as so ( t, s ) Dirac spinors, exhibitsseveral features whose useful ones are summarized below:( i ) adjoint spinors Along with the complex Ψ, one has three cousin spinors namely Ψ T , Ψ ∗ and Ψ † = Ψ ∗ T respectively associated with the Clifford algebras generated byΓ TM , Γ ∗ M , Γ † M (2.8)and which are related to Γ M by similarity transformations as given belowΓ † M = ( − ) t A Γ M A − Γ TM = − η C Γ M C − Γ ∗ M = − η ( − ) t B Γ M B − (2.9)with A = Γ .. Γ t C T = − ε CB T = CA − (2.10)and A = η t ( − ) t ( t +1)2 CAC − B ∗ B = − εη t ( − ) t ( t +1)2 (2.11)and where ε = ± η = ±
1. We also have( C Γ M ) T = εη C Γ M ( C Γ .. Γ m ) T = − εη m ( − ) m ( m +1)2 ( C Γ .. Γ m ) (2.12)8otice that for odd dimensions, there is one solution for the matrix C ; but for evendimensions, we distinguish two kinds of possible C matrices as illutrated below on theexample of D = 6 : C + = σ ⊗ σ ⊗ σ , for η = +1 , ε = − C − = σ ⊗ σ ⊗ σ , for η = − ε = +1 (2.13)satisfying ( C ± ) T = ±C ± and ( C ± ) = I id . Notice moreover that for 6D, both of thematrices C + and C − have as product εη = −
1; and so( C Γ M ) T = −C Γ M (2.14)which is an undesirable property that requires the use of SU (2) symplectic spinors Q iA = ( Q A , Q A ) in order to recover the symmetric feature of the anticommutation relationbetween the supercharges of supersymmetric YM theory in 6D namely Q iA Q jB + Q jB Q iA = ε ij (cid:0) C Γ M (cid:1) AB P M (2.15)where ε ij is the usual 2 × N antisymmetric matrix obeying ( ε ij ) ∗ ε jl = − δ il .( ii ) Weyl spinors
In odd dimensions, the Dirac fermion Ψ satisfying (2.1-2.2) is an irreducible spinor; butin even dimensions, say D = 2 k , it can be decomposed into two irreducible Weyl spinorsΨ L and Ψ R having each 2 k − complex components.2 k = 2 k − ⊕ k − (2.16)The two chiral spinors Ψ L and Ψ R are related to the Dirac Ψ through the followingprojections Ψ L = ( I + Γ D +1 ) ΨΨ R = ( I − Γ D +1 ) Ψ (2.17)with chirality operator Γ D +1 = ( − i ) k + t Γ .. Γ k which, by using the realization (2.5), readsas Γ D +1 = ( − i ) D Γ .. Γ D = σ ⊗ σ ⊗ .. ⊗ σ | {z } k (2.18)This matrix obeys Γ D +1 Γ D +1 = I and is independent of the space time signature. Noticethat in even dimensions, the anticommutation relations between two Weyl superchargessay the left one Q L = ( I + Γ D +1 ) Q (2.19)9ead as follows { Q L , Q L } = 1 − ( − ) D I + Γ D +1 ) C Γ M P M (2.20)Therefore, non vanishing anticommutators limits the possible dimensions where { Q L , Q L } 6 =0 since the non vanishing condition requires that D has to be odd; i.e: D = 4 l + 2 , l = 1 , , ... (2.21)The leading dimensions where this is possible are D=2 , D=6 and
D=10 .( iii ) reality conditions Complex Dirac spinors might be also subject to reality conditions such as Majorana orMajorana Weyl conditions. These conditions are not usually possible since reality con-dition depends both on the space time dimension D and the signature s − t . The generalresult on the possibility of putting a reality condition on spinors in diverse Lorentzianand euclidian dimensions is collected in the following table [17]-[20], dimension D Lorentzian R ,D − Euclidian R D M M M W M − M SM M + SM W SM SM SM W M + SM M M − M W M M M W M − M SM (2.22)with M standing for Majorana and M W for Majorana Weyl spinors. We also have M ± referring to Majorana spinors with charge conjugation C ± . Notice that in the case wherethere is no Majorana spinor, one can have symplectic Majorana spinors or symplecticMajorana Weyl spinors referred in the table respectively by the symbols SM and SMW .10y symplectic Majorana spinor, we mean a set of 2N Dirac spinors Ψ A , .., Ψ NA constrainedas follows (Ψ iA ) ∗ = Ω ij B BA Ψ jB (2.23)where Ω ij is the usual 2 N × N antisymmetric symplectic matrix obeying (Ω ij ) ∗ Ω jl = − δ il and where (cid:0) B BA (cid:1) ∗ B BC = − δ AC . We also have SMW whenever B and Γ D +1 commuteknowing that (Γ D +1 ) ∗ = ( − ) t + D B Γ D +1 B − (2.24)Since (Γ D +1 ) ∗ = Γ D +1 due to (2.18), it follows that B and Γ D +1 commute for t + D = 0mod 2. From the classification table (2.22), we learn a set of interesting features inparticular: • there is no Majorana spinor in euclidian ; and nor in the Lorentzian and . Therefore, when studying the euclidian N = 4 SYM and euclidian N = 4 SYM, one is constrained to use symplectic Majorana spinors. • the Majorana and Majorana-Weyl conditions are not preserved by analytic con-tinuation from Lorentzian to euclidian signature. This is a well known problemthat has been considered from various view points [30] and refs therein; in partic-ular from the approach of Osterwalder-Schrader where the hermiticity in euclidianspace is abandoned [31]. Chiral supersymmetric YM in 6D and 10D
Like for the well known case of Lorentzian maximal SYM theories with real super-charges, supersymmetric QFTs with real supercharges can be formulated in diversedimensions. These are the N = 8 , N = (4 , , N = 4, N = 2 and N = 2 theories; they may be obtained by reduction of the chiral , N = (1 , . This Lorentzian SYM theory can be then viewed asthe mother of supersymmetric theories with supercharges. From this point of view,chiral N = (1 ,
0) SYM in shares a kind of maternity property with N = (1 ,
0) SYMin which is the mother theory of supersymmetric QFTs’ having 16 supercharges. Notice that for those dimensions D where there is no Mjaorana spinor like in D = 1 + 4 or euclidian and , we shall also use the standard conventional notations N = 4 ( N = 4) to refer tothe 16 real ( 8 real) supercharges although strictly speaking this convention is not rigorous. wisted 3D N = 4 SYM
In twisted N = 4 supersymmetric YM theory, the 8 real supersymmetric chargesare represented by a complex 2 × Q × that can be expanded in terms of the Pauli matrices σ µ as follows Q × = QI + Q µ σ µ (2.25)Similarly as for the case of eq(1.2) of twisted maximal supersymmetric YM theory, thefield content of the spectrum of the twisted N = 4 SYM theory can be packaged into SU (3) × U (1) representations likebosons : 6 → ⊕ ¯3fermions : 4 → ⊕ Correspondence 3D N = 4 and N = 4Pushing forward the similarity between twisted SYM with supercharges and twistedmaximal supersymmetric YM ( see footnote 2 ), one finds the following correspondenceto be established throughout this study: • twisted SYM with 16 supercharges lattice hidden symmetry N = 4 L su × u D SU (5) × U (1) N = 4 L su D = A ∗ SU (5) (2.27)with 5- dimensional lattice L su × u D given by the fibration L u D → L su × u D ↓L su D (2.28)The base sublattice L su D is given by the 4- dimensional lattice [1, 3] L su D = A ∗ (2.29)12enerated by the 4 fundamental weight vectors ~ Ω , ~ Ω , ~ Ω , ~ Ω (2.30)of the SU (5) symmetry group. These weight vectors are the dual of the 4 simpleroots of the Lie algebra of SU (5); ~a , ~a , ~a , ~a (2.31)generating the 4-dimensional root lattice A of SU (5). So the crystal A ∗ is thedual of A ; see later on for further details and [21]-[29] for related constructions. • twisted SYM with 8 supercharges lattice hidden symmetry N = 4 L su × u D SU (3) × U (1) N = 4 L su D SU (3) (2.32)with L u D → L su × u D ↓L su D (2.33)and base sublattice given by the 2- dimensional lattice [33]-[34] L D = A ∗ (2.34)generated by the 2 fundamental weight vectors ~ω , ~ω (2.35)of the SU (3) symmetry. These weight vectors are the dual of the simple roots ~α , ~α of SU (3); then A ∗ is the dual of the 2-dimensional root lattice A of SU(3)which may be thought of as the honeycomb, see fig. 1 for illustration.In what follows, we study the twisted N = 4 SYM in continuum. First, we describesome special features on SYM in ; then we derive the SU (3) × U (1) covariant spectrum13igure 1: the 2D lattice A ∗ generated by ~ω , ~ω ; the 2 fundamental weight vectors ofSU(3). Each (green) node in A ∗ has 3 + 3 first nearest neighbors forming respectively atriplet (red sites) and an anti-triplet (blue sites) of SU (3) . of twisted N = 4 SYM in Next, we give the twisted N = 4 superalgebra having an SU (3) × U (1) isotropy symmetry with supersymmetric generators as Q (+3 q ) , Q ( − q ) a (2.36)transforming respectively as a complex scalar and a complex triplet of SU (3). After, weuse superspace method to realize the scalar supersymmetric charge Q (+3 q ) which may bealso thought of as a BRST charge operator. N = 4 SYM in from N = 1 SYM In Lorentzian one distinguishes 3 kinds of supersymmetric YM theories: two of themhave real conserved supercharges and the third one has real supercharges [35]-[37]. The field theories having supercharges are given by the well known non chiral N = (1 ,
1) and the chiral N = (2 , supercharges isgiven by the chiral N = (1 ,
0) SYM or equivalently N = (0 , the 6D N = 1 vector multiplet First, recall that in Lorentzian there are only Weyl Ψ W D and Dirac fermions Ψ Dirac D = (cid:0) Ψ W L D , Ψ W R D (cid:1) ; so that the smallest supermultiplet contains a left Ψ W L D fermion, transform-14ng in the SO (1 ,
5) spinor representation 4 + , or right Ψ W R D Weyl fermion transformingin 4 − . Recall also that in the language of fermions, a Weyl spinor in , say the leftone, Ψ W L D ∼ + having 4 complex (8 real ) degrees of freedom, is made of a dotted and an undotted Weyl spinors as follows Ψ
W L D = (cid:16) ξ α + , ¯ λ − ˙ α (cid:17) , α = 1 , (cid:0) Ψ W L D (cid:1) c = (cid:16) λ α + , ¯ ξ − ˙ α (cid:17) ∼ c + (2.38)and has the same 6D chirality as 4 + . The ± charges carried by the fields refer to quantumnumbers of SO R (2) ∼ U R (1) resulting from the reduction of the SO (1 ,
5) Lorentzsymmetry down to SO (1 , × SO R (2). The corresponding 6D Weyl right fermionΨ W R D ∼ − with negative chirality is given byΨ W R D = (cid:16) ξ α − , ¯ λ +˙ α (cid:17)(cid:0) Ψ W R D (cid:1) c = (cid:16) λ α − , ¯ ξ +˙ α (cid:17) (2.39)The chiral N = (1 ,
0) supersymmetric YM theory has two types of supermultipletsdescribing matter and gauge fields with on shell degrees of freedom as follows:( a )
6D hypermultiplets
These supermultiplets describe matter; they have 2 complex (4 real) scalars and a 6DWeyl spinor H D : (cid:0) , (cid:1) D (2.40)they may belong to any representation of the gauge symmetry including complex ones;see [38]-[17] for other properties.( b )
6D vector multiplets V D These multiplets have a gauge field and a Weyl spinor V N =(1 , D : (cid:0) , (cid:1) D (2.41)15hey transform in the adjoint representation of the gauge symmetry. Below, we refer tothese supermultiplets like V N =16 D = (cid:0) A M , ψ A (cid:1) D (2.42)where the field A M is the 6D hermitian gauge field and ψ A the complex 4- dimensionWeyl spinor of SO (1 , ≃ SU ∗ (4) (2.43)the fields A M and ψ A are valued in the Lie algebra of the U( N ) gauge symmetry. Reduction from 6D to 3D and twisting
We give two approaches to build the twisted field spectrum of N = 4 supersymmetricYM that follows from the reduction of (2.41-2.42). We also comment on the link betweenthe two methods.(1) first approach This approach is a rephrasing of eq(2.25); it involves two steps: ( i ) dimension reductionfrom to ; ( ii ) twisting the symmetries resulting from the breaking of SO (1 , N = (1 ,
0) supersymmetric YM theory down tothe space, the SO (1 ,
5) breaks down to SO (1 , × SO R (3)and so the 6 local coordinates X M of R , decompose like( x µ , y m )with x µ ∈ R , and y m ∈ R R with respective isotropy symmetries SO (1 ,
2) and SO R (3).Similarly, the on shell 4 + 4 real degrees of freedom of the chiral N = (1 ,
0) gaugemultiplet (2.41), decomposes into a gauge field A µ , three real scalars φ m and 4 Majo-rana spinors ψ α , ..., ψ α . In the euclidian version of this theory, the SO (1 , × SO R (3)isotropy gets mapped to the compact SO E (3) × SO R (3) and the 4 Majorana spinors ψ αI into 2 complex Dirac spinors ξ α , ξ α like (cid:18) , Dirac , (cid:19) D = (cid:0) A µ , ξ αi , φ m (cid:1) D (2.44)with: • the field A µ being a real gauge field transforming as (3 ,
1) under SO E (3) × SO R (3), 16 the fields ξ αi are complex fermions of transforming (2 ,
2) spinors of SO E (3) × SO R (3) ≃ SU E (2) × SU R (2), • the fields φ m are 3 real scalars transforming as (1 ,
3) under SO E (3) × SO R (3).Notice that in practice these fields should be taken as functions depending only on the x coordinates; A (0) µ = A µ ( x ), ξ αi (0) = ξ αi ( x ) , ... (2.45)but generally speaking they are functions of both coordinates ( x, y ) ; A µ = A µ ( x, y ), ξ αi = ξ αi ( x, y ) , ... (2.46)By taking y m as the coordinates of a real 3-torus T with large volume1(2 πl ) Z T d y = 1 , vol (cid:0) T (cid:1) = (2 πl ) (2.47)one may expand these fields into infinite harmonic series like F ( x, y ) = X n ,n ,n e in m ym πl F ( n ,n ,n ) ( x ) (2.48)where eqs(2.45) appear as the zero mode of the expansions and the extra others asmassive modes that break gauge symmetry in the restricted real 3D.Under twisting, the quantum numbers of SO E (3) and SO R (3) groups are identified andthe SO E (3) × SO R (3) symmetry is reduced down to the diagonal SO (3) = SO E (3) × SO R (3) SO ′ (3) (2.49)As a consequence of the twisting, the fields of the chiral N = (1 ,
0) gauge multiplet(2.41) are mapped to the twisted onesfields : twisted fields SO (3) repres A µ A µ φ m B µ ξ α ± ξ, ξ µ ⊕ two gauge fields A µ , B µ that we combine into a complex gauge field and its adjointlike G µ = A µ + iB µ ¯ G µ = A a − iB µ (2.51) • four complex fermionic fields ξ, ξ µ transforming respectively as a singlet and tripletof SO (3).(2) second approach This approach involves one step; and, in some sense, is a direct method. The idea ofthis way of doing relies on the fact that since the fields G µ and ξ, ξ µ are complex fields,one may be tempted to use complex groups to deal with them; this extension can beimplemented by considering other breaking modes of the SO E (6) isotropy symmetry ofthe euclidian space time R (following from the Wick rotation of R , ); in particular SO E (6) −→ SU (3) × U (1)6 v ∼ +2 q + ¯3 − q s ∼ + q + 1 − q (2.52)where q is a unit charge of the abelian U (1) factor that can be fixed to a number q .But here we will keep it free for later use when considering the singular limit q = 0.Under the breaking mode (2.52), the euclidian space R , parameterized by the localcoordinates X M = ( x µ , y µ ) with y µ = x µ +3 , get mapped to the complex C with localcoordinates z a = x a + iy a (2.53)where ( x a ) the coordinates of the real space R and ( y a ) the coordinates of the internalspace R int . Moreover, the fields of the multiplet (2.41) decompose as follows SO E (6) −→ SU (3) × U (1) A M : G a ( − q ) , ¯ G (+2 q ) a Ψ A : ψ a (+ q ) , ψ ( − q ) (2.54)18nd may be treated in general as functions of ( z, ¯ z ); this property will be manifested onthe lattice side through orientations of the links; complex p-tensors and their duals areassociated with p-plaquettes with opposite orientations.Moreover, comparing eqs(2.50-2.51) with eq(2.52-2.54), we end with the following re-sults:( a ) the spectrum of twisted fields of the two approaches are quasi the same; the maindifference is that (2.52) depend on the extra charge q and transform in SU (3)representations rather than SO (3).( b ) Eqs(2.50-2.51) are recovered from eqs(2.52-2.54) by taking the limit q → SU (3) × U (1) symmetry down to the real SO (3) whichmay be thought of as its ”real part”. In practice, this corresponds to dropping outthe y- dependence into the fields and using (2.47) to integrate it out in the fieldaction. N = 4 algebra and superfields We first give the basic anticommutators defining this superalgebra; then we describe thegeneral structure of twisted superspace and superfields. N = 4 supersymmetry in For a generic charge q , the twisted N = 4 supersymmetric algebra is generated by Q (+3 q ) , Q ( − q ) a , P (+2 q ) a (3.1)and P a ( − q ) (3.2)having no supersymmetric partner; a property that makes asymmetric the formulationof twisted supersymmetric YM. 19 .1.1 anticommutators These operators transform under U (1) × SU (3) as in (2.52); and satisfy the followingbasic anticommutation relations, n Q (+3 q ) , Q ( − q ) a o = 2 P (+2 q ) a n Q ( − q ) a , Q ( − q ) b o = 0 (3.3)together with the topological one (cid:8) Q (+3 q ) , Q (+3 q ) (cid:9) = 0 (3.4)and h Q (+3 q ) , P (+2 q ) a i = h Q ( − q ) a , P (+2 q ) b i = 0 (cid:2) Q (+3 q ) , P a ( − q ) (cid:3) = h Q ( − q ) a , P b ( − q ) i = 0 (3.5)These graded commutation relations preserve the U(1) charge and are covariant under SU (3) symmetry. twisted superspace and superderivatives The twisted N = 4 superalgebra may be realized in superspace by using complexbosonic and fermionic coordinates z a ( − q ) , z (+2 q ) a , θ ( − q ) , ϑ a (+ q ) (3.6)with (cid:0) z (+2 q ) a (cid:1) † = z a ( − q ) (3.7)Using the usual supersymmetric covariant derivatives D (+3 q ) and D ( − q ) a , instead of thesupercharges Q (+3 q ) and Q ( − q ) a , a suitable superspace representation of the twisted su-peralgebra (3.3) is given by D (+3 q ) = ∂∂θ ( − q ) D ( − q ) a = ∂∂ϑ a (+ q ) + 2 θ ( − q ) ∂ (+2 q ) a (3.8)with P a ( − q ) = ∂ a ( − q ) = ∂∂z (+2 q ) a ¯ P (+2 q ) a = ∂ (+2 q ) a = ∂∂z a ( − q ) (3.9)20o implement gauge interactions, these superspace derivatives are covariantized by in-troducing gauge connexions as follows D (+3 q ) = D (+3 q ) + ig Y M Υ (+3 q ) D ( − q ) a = D ( − q ) a + ig Y M Υ ( − q ) a L (+2 q ) a = ∂ (+2 q ) a + ig Y M V (+2 q ) a L a ( − q ) = ∂ a ( − q ) + ig Y M U a ( − q ) (3.10)These extended superderivatives are needed for building the gauge covariant superfieldsΦ ( q i ) i of the twisted YM theory to be considered later. N = 4 SYM
To build the field action S twisted of the twisted N = 4 supersymmetric YM theory,we require invariance under the three following symmetries:( a ) the gauge symmetry which we take as U ( N ),( b ) the scalar supersymmetric charge Q (+3 q ) or equivalently D (+3 q ) ; and,( c ) the U (1) × SU (3) space isotropy symmetryFirst, observe that the gauge invariant action S twisted = 1 l Z L twist (3.11)with the scalar supercharge Q (+3 q ) manifestly exhibited reads in superspace as follows L twist = (cid:18)Z dθ ( − q ) L ( − q ) (cid:19) ϑ a (+ q ) =0 (3.12)The scale factor l is as in eq(2.47). The superspace density L ( − q ) transforms in the U (1) × SU (3) representation 1 − ; that is having − U (1), and hasthe form L ( − q ) = T r (cid:16) L ( − q ) twist (cid:17) (3.13)with the N × N superfield matrix L ( − q ) twist = L ( − q ) twist (Φ) (3.14)The L ( − q ) twist depends on a set of superfieldsΦ ( q i ) i = Φ ( q i ) i ( z, θ, ϑ ) (3.15)21hat describe the off shell degrees of freedom of twisted N = 4 supersymmetricYM theory. Below, we describe this set of superfields; for more details on the explicitderivation see the analysis given in the appendix. Gauge covariance
The U ( N ) gauge symmetry of the action (3.12) acts on the superfield matrix density L ( − twist like L ( − q ) twist → g L ( − q ) twist g − (3.16)since T r (cid:16) L ( − q ) twist (cid:17) = T r (cid:16) g L ( − q ) twist g − (cid:17) (3.17)where, for convenience as described in the appendix, the matrix element g is chosen asfollows g = g ( z, ¯ z, ϑ a (+ q ) ) , D (+3 q ) g = g depend on z, ¯ z, ϑ a (+ q ) but has no θ ( − q ) The property (3.17) is ensured by requiring the superfields Φ ( q i ) i to be also gauge covari-ant; this means that under a generic gauge symmetry transformation g , we haveΦ ( q i ) i → g Φ ( q i ) i g − (3.19)General results on covariant formulation of supersymmetric YM theories in superspace[43] applied to our present study lead to the following set of gauge covariant superfields. Fermionic sector : Ψ ( − q ) Ψ a (+ q ) Φ (+ q ) ab SU (3) × U (1) : 1 − q + q + q scale mass dim 1 1 1 Bosonic sector : J (0) , E ab ( − q ) F (+4 q ) ab SU (3) × U (1) : 1 ¯3 − q +4 q scale mass dim
32 32 32 (3.20)22uilt out of commutators of the gauge covariant superderivatives Ψ ( − q ) = ig Y M h D ( − q ) a , L a ( − q ) i Ψ a (+ q ) = ig Y M (cid:2) D (+3 q ) , L a ( − q ) (cid:3) Φ (+ q ) ab = ig Y M h D ( − q ) a , L (+2 q ) b i (3.21)and J (0) = ig Y M h L (+2 q ) a , L a ( − q ) i E ab ( − q ) = ig Y M (cid:2) L a ( − q ) , L b ( − q ) (cid:3) F (+4 q ) ab = ig Y M h L (+2 q ) a , L (+2 q ) b i (3.22)with gauge coupling constant g Y M scaling like ( mass ) . Notice that as far as super-fields with scaling dimension as ( mass ) are concerned, eqs(3.20) may also contain thefermionic superfield Ψ (+5 q ) a = ig Y M h D (+3 q ) , L (+2 q ) a i (3.23)it is constrained to be equal to zero in our construction. The above fermionic and bosonicgauge covariant superfields obey as well constraint relations; in particular D (+3 q ) Ψ ( − q ) = 2 J (0) − D ( − q ) a Ψ a (+ q ) D (+3 q ) E ab ( − q ) = L a ( − q ) Ψ b (+ q ) − L b ( − q ) Ψ a (+ q ) D (+3 q ) Φ (+ q ) ab = F (+4 q ) ab (3.24)and remarkably D (+3 q ) Ψ a (+ q ) = 0 D (+3 q ) F (+4 q ) ab = 0 L (+2 q ) b Ψ ( − q ) = L a ( − q ) Φ (+ q ) ab (3.25)23o deal with these constraint relations, it is helpful to use the following θ - expansionsΨ a (+ q ) = ψ a (+ q ) + θ ( − q ) f a (+4 q ) Ψ ( − q ) = ψ ( − q ) + θ ( − q ) F (0) Φ (+ q ) ab = φ (+ q ) ab + θ ( − q ) F (+4 q ) ab J (0) = F (0) + θ ( − q ) ∇ (+2 q ) a ψ a (+ q ) E ab ( − q ) = F ab ( − q ) + θ ( − q ) h ∇ a ( − q ) ψ b (+ q ) − ∇ b ( − q ) ψ a (+ q ) i F (+4 q ) ab = F (+4 q ) ab + θ ( − q ) κ (+7 q ) ab (3.26)with the fields ψ ( − q ) , ψ a (+ q ) (3.27)being the twisted fermionic fields of eqs(2.54); the bosonic fields f a (+4 q ) and F (0) scalingas ( mas ) are auxiliary fields; and finally F (0) , F ab ( − q ) , F (+4 q ) ab (3.28)are as follows F (+4 q ) ab = ig Y M h ∇ (+2 q ) a , ∇ (+2 q ) b i F ab ( − q ) = ig Y M h ∇ a ( − q ) , ∇ b ( − q ) i F (0) = ig Y M h ∇ (+2 q ) a , ∇ a ( − q ) i (3.29)with the gauge covariant ∇ a ( − q ) , ∇ (+2 q ) a derivatives given by ∇ a ( − q ) = ∂ a ( − q ) + ig Y M G a ( − q ) ∇ (+2 q ) a = ∂ (+2 q ) a + ig Y M G (+2 q ) a (3.30)with G (+2 q ) a , G a ( − as in (2.54).We also have the relations ∇ (+2 q ) a ψ a (+ q ) = ∂ (+2 q ) a ψ a (+ q ) + ig Y M h G (+2 q ) a , ψ a (+ q ) i ∇ a ( − q ) ψ b (+ q ) = ∂ a ( − q ) ψ b (+ q ) + ig Y M h G a ( − q ) , ψ b (+ q ) i (3.31)24otice that the constraint relation D (+3 q ) Ψ a (+ q ) = 0, which by using the gauge fixingdescribed in the appendix reads also like D (+3 q ) Ψ a (+ q ) = 0 (3.32)is solved as follows Ψ a (+ q ) = ψ a (+ q ) , f a (+4 q ) = 0 (3.33)This solution shows that ψ a (+ q ) is a supersymmetric invariant field in agreement withthe θ - expansion of the the gauge superfield V a ( − q ) involved in (3.10), U a ( − q ) = G a ( − q ) + θ ( − q ) ψ a (+ q ) (3.34)Similarly, we have for the constraint D (+3 q ) F (+ q ) ab = 0, the following F (+4 q ) ab = F (+4 q ) ab , κ (+7 q ) ab = 0 (3.35)showing that F (+ q ) ab is a supersymmetric invariant field in agreement with the θ - expansionof the the gauge superfield Υ ( − q ) a involved in eqs(3.10) namelyΥ ( − q ) a = γ ( − q ) a + θ ( − q ) G (+2 q ) a (3.36)From this relation, we also learn that G (+2 q ) a is supersymmetric invariant and so thesuperfield V (+2 q ) a appearing in (3.10) has no θ ( − q ) dependence and then should be as V (+2 q ) a = G (+2 q ) a (3.37)Regarding the constraint relation L (+2 q ) b Ψ ( − q ) = L a ( − q ) Φ (+ q ) ab , we use the θ - expansionsof the superfields to end, on one hand, with L (+2 q ) b Ψ ( − q ) = ∂ (+2 q ) b Ψ ( − q ) + ig Y M h G (+2 q ) a , Ψ ( − q ) i = ∇ (+2 q ) a ψ ( − q ) + ig Y M θ ( − q ) ∇ (+2 q ) a F (0) (3.38)and, on the other hand, with L a ( − q ) Φ (+ q ) ab = ∂ a ( − q ) Φ (+ q ) ab + ig Y M h U a ( − q ) , Φ (+ q ) ab i = ∇ a ( − q ) φ (+ q ) ab + ig Y M θ ( − q ) (cid:16)h G a ( − q ) , F (+4 q ) ab i + n ψ a (+ q ) , φ (+ q ) ab o(cid:17)
25y equating, we obtain ∇ (+2 q ) b ψ ( − q ) = ∇ a ( − q ) φ (+ q ) ab h G (+2 q ) b , F (0) i = h G a ( − q ) , F (+4 q ) ab i + n ψ a (+ q ) , φ (+ q ) ab o (3.39)Under gauge transformations with matrix element g chosen, for simplicity, as g = g (cid:16) z, ¯ z, ϑ a (+ q ) (cid:17) , D (+3 q ) g = ( − q ) → g Ψ ( − q ) g − Φ (+ q ) ab → g Φ (+ q ) ab g − (3.41)and E ab ( − q ) → g E ab ( − q ) g − J (0) → g J (0) g − (3.42) Supersymmetric transformations
First, we give the supersymmetric transformations of the on shell degrees of freedom ofeq(2.54); then we consider the transformations of a particular set of off shell ones. • On shell multiplet
Using the equations of motion of the on shell twisted fields; in particular ∇ (+2 q ) a ψ a (+ q ) =0 and ∇ (+2 q ) a ψ ( − q ) = 0, we can write down the supersymmetric transformationsgenerated by the scalar operator Q (+3 q ) ; they are given by Q (+3 q ) G a ( − q ) = ψ a (+ q ) Q (+3 q ) ψ a (+ q ) = 0 Q (+3 q ) G (+2 q ) a = 0 Q (+3 q ) ψ ( − q ) = F (0) Q (+3 q ) F (0) = ∇ (+2 q ) a ψ a (+ q ) = 0 (3.43)26here we used F (0) = ∂ (+2 q ) a G a ( − q ) − ∂ a ( − q ) G (+2 q ) a + ig Y M (cid:2) G (+2 q ) a , G a ( − q ) (cid:3) (3.44) • Off shell case
A set of off shell degrees of freedom is as in eqs(3.20-3.26); the supersymmetrictransformations of the fields are therefore given by Q (+3 q ) ψ ( − q ) = F (0) Q (+3 q ) F (0) = 0 Q (+3 q ) φ (+ q ) ab = F (+4 q ) ab Q (+3 q ) F (+4 q ) ab = 0 (3.45)where F (0) is an auxiliary field; and Q (+3 q ) F ab ( − q ) = ψ ab ( − q ) Q (+3 q ) ψ ab ( − q ) = 0 Q (+3 q ) F (0) = ∇ (+2 q ) a ψ a (+ q ) Q (+3 q ) ∇ (+2 q ) a ψ a (+ q ) = 0 (3.46)where we have set ψ ab ( − q ) = ∇ a ( − q ) ψ b (+ q ) − ∇ b ( − q ) ψ a (+ q ) (3.47) The action of twisted fields of chiral N = 4 supersymmetry, exhibiting manifestlythe scalar supercharge Q (+3 q ) , reads in superspace like L twist = (cid:18)Z dθ ( − q ) L ( − q ) (cid:19) ϑ =0 , (4.1)with lagrangian superdensity L ( − q ) depending on the Grassman variable θ ( − q ) ; but alsoon ϑ a (+ q ) , (4.2)Because of the role played by the supersymmetric generator D ( − q ) a in our construction;eg (3.21-3.24) and appendix, the dependence into the ϑ a (+ q ) is implicit; and is killed atthe end after performing integration with respect to θ ( − q ) .27 .1 Lagrangian superdensity The general form of the fermionic superdensity L ( − q ) scaling as ( mass ) one may con-struct out of the set of gauge covariant superfields (3.20) is as follows L ( − q ) = α T r (cid:2) Ψ ( − q ) D (+3 q ) Ψ ( − q ) (cid:3) + α T r (cid:2) Ψ ( − q ) J (0) (cid:3) + α T r (cid:2) ε abc Ψ a (+ q ) E bc ( − q ) (cid:3) + α T r h Φ (+ q ) ab E ab ( − q ) i + ν F I
T r (cid:2) Ψ ( − q ) (cid:3) (4.3)where the α i ’s are normalization numbers and the coupling scaling as ( mass ) ν F I (4.4)is a Fayet-Iliopoulous coupling constant. This term breaks scalar supersymmetry; it willbe dropped out below.Notice that the integration of with respect to the Grassamn variable of ν F I Z dθ ( − q ) T r (cid:2) Ψ ( − q ) (cid:3) (4.5)leads in general to ν F I
T r (cid:0) F (0) (cid:1) = ν F I dim U ( N ) X A =1 F (0) A T r (cid:0) T A (cid:1) (4.6)which does’nt vanish due to the abelian gauge subsymmetry U (1) = U ( N ) SU ( N ) (4.7)In the above relation, the matrices T A stand for the generators of U ( N ).28 .2 Component field action Using the θ - expansions (3.26-3.35) and integrating with respect to the Grassman variable θ ( − q ) ; we obtain L twist = α T r (cid:2) F (0) F (0) (cid:3) + α T r (cid:2) F (0) F (0) (cid:3) − α T r h ψ ( − q ) ∇ (+2 q ) a ψ a (+ q ) i +2 α T r h ε abc ψ a (+ q ) ∇ b ( − q ) ψ c (+ q ) i + α T r h F (+4 q ) ab F ab ( − q ) i − α T r h φ (+ q ) ab h ∇ a ( − q ) ψ b (+ q ) − ∇ b ( − q ) ψ a (+ q ) ii (4.8)Notice that the terms with coefficients α , α , α are manifestly invariant with respectto the scalar supersymmetric transformations. However the variation of the term T r h ε abc ψ a (+ q ) ∇ b ( − q ) ψ c (+ q ) i (4.9)leads to T r h ε abc n ψ a (+ q ) , n ψ b (+ q ) , ψ c (+ q ) ooi (4.10)or equivalently13 ε abc ψ a (+ q ) A (cid:16) ∇ b ( − q ) ψ c (+ q ) (cid:17) B ψ c (+ q ) C T r (cid:0)(cid:2) T A , (cid:2) T B , T C (cid:3)(cid:3) + cyclic perm (cid:1) which vanishes identically due to Jacobi-Identity.Notice that the component field action (4.8) can be rewritten in a more convenient formby eliminating φ (+ q ) ab through the constraint relation ∇ (+2 q ) b ψ ( − q ) = ∇ a ( − q ) φ (+ q ) ab (4.11)following from the superfield constraint eqs(3.25-3.39) namely L (+2 q ) b Ψ ( − q ) = L a ( − q ) Φ (+ q ) ab (4.12)29ndeed, starting from T r (cid:16) φ (+ q ) ab ∇ a ( − q ) ψ b (+ q ) (cid:17) = T r (cid:16) φ (+ q ) ab ∂ a ( − q ) ψ b (+ q ) (cid:17) ig Y M
T r (cid:16) φ (+ q ) ab h G a ( − q ) , ψ b (+ q ) i(cid:17) (4.13)and integrating by part, we get up to a total divergence, T r (cid:16) φ (+ q ) ab ∇ a ( − q ) ψ b (+ q ) (cid:17) = − T r h(cid:16) ∇ a ( − q ) φ (+ q ) ab (cid:17) ψ b (+ q ) i = − T r h ψ b (+ q ) ∇ (+2 q ) b ψ ( − q ) i (4.14)By substituting back into (4.8), we end with the lagrangian density L twist = α T r (cid:2) F (0) F (0) (cid:3) + α T r (cid:2) F (0) F (0) (cid:3) + α T r h F (+4 q ) ab F ab ( − q ) i + ( α + 2 α ) T r h ψ a (+ q ) ∇ (+2 q ) a ψ ( − q ) i +2 α T r h ε abc ψ a (+ q ) ∇ b ( − q ) ψ c (+ q ) i (4.15)Eliminating the auxiliary field F (0) through its equation of motion F (0) = − α α F (0) (4.16)and putting back into the lagrangian density, we end with L twist = α T r h F (+4 q ) ab F ab ( − q ) i − ( α ) α T r (cid:2) F (0) F (0) (cid:3) + ( α + 2 α ) T r h ψ a (+ q ) ∇ (+2 q ) a ψ ( − q ) i +2 α T r h ε abc ψ a (+ q ) ∇ b ( − q ) ψ c (+ q ) i (4.17)30ith scaling mass dimension ( mass ) . Notice that the YM coupling constant g Y M iswithin the gauge covariant derivatives ∇ (+2 q ) a , ∇ a ( − q ) and the field strengths F (+4 q ) ab , F ab ( − q ) as shown on eqs(3.29-3.30). N = 4 SYM theory on lattice
In all what follows, we focuss on the study of twisted N = 4 supersymmetric YM onparticular 3- dimensional lattice L su × u D . This crystal is given by the fibration L u D → L su × u D ↓L su D = A ∗ (5.1)with the two following components:( i ) the base sublattice L su D given by A ∗ , the dual of the 2-dimensional root lattice A that is associated with the SU (3) symmetry [33, 34]; and( ii ) the fiber L u D associated with the U (1) factor of the symmetry SU (3) × U (1), it isa 1-dimensional lattice with direction normal to A ∗ .To fix the ideas, L su × u D will be realized as a twist of the 3- dimensional lattice A ∗ ; thedual to the 3D root lattice A generated by the 3 simple roots of SU (4); that is: L su × u D ∼ twist of A ∗ (5.2)Notice that the A ∗ crystal is generated by the 3 fundamental weight of SO (6) ≃ SU (4);and the twist of A ∗ we are looking for is the one induced by the breaking mode SU (4) → SU (3) × U (1) (5.3)For the explicit engineering of L su × u D ; see next section; in due time let us focus onbuilding the lattice analogue of the twist field of continuum.31 .1 Tensor fields on lattice L su × u D In this subsection, we study the discretization of the twisted N = 4 SYM theory tothe lattice L su × u D of (5.1) by first focussing on the projection of this gauge theory onthe base sublattice L su D = A ∗ The implementation of the effect of the fiber L u D will be considered later on. Discretizing continuum
We begin by considering real scalar fields and then real tensor ones living on 3D space.After that, we give the extension to complex space and complex fields appearing in theformulation 3D N = 4 SYM theory given in previous sections.1) Discretizing space
In the discretization of the real 3D continuum space, generic points P with local coor-dinates ( x µ ) = ( x, y, z ) get mapped to lattice nodes N = N ( n , n , n )described by 3- dimensional integral position vectors ~R n In the example of a simple cubic lattice with spacing parameter L , the nodes N arerepresented by ~R n = x n ~e + y n ~e + z n ~e (5.4)with ~e i the usual canonical basis obeying ~e i .~e j = δ ij Each site ~R n in this simple lattice has 6 first nearest neighbors located at ~R n ± L~e , ~R n ± L~e , ~R n ± L~e (5.5)In the case of the lattice L su × u D with an SU (3) × U (1) symmetry, the site positions ~R n are given by ~R n = n ~L + n ~L + n ~L n = ( n , n , n ) (5.6)32ith the basis ~L i satisfying a non trivial 3 × J su × u ij = ~L i .~L j ~R n . ~R m = n i J su × u ij m j (5.7)capturing the shape of the crystal L su × u D .The intersection matrix J su × u ij has a set of features; in particular the 2 following usefulones.( a ) the matrix J su × u ij is exactly given by J su × u ij = + q + 2 q q + 2 q + 4 q q q q q (5.8)It depends on the number q that encodes the charges of the twisted supersymmetricYM fields under the abelian U (1) symmetry of (5.3) and moreover defines thefibration (2.33).( b ) In the particular and remarkable case q = 0, the intersection matrix J su × u ij reducesto the singular matrix (cid:0) J su × u ij (cid:1) q =0 = J su ij
00 0 ! (5.9)with J su ij = 13 ! (5.10)This singular case corresponds exactly to the projection of sites ~R ( n ,n ,n ) (5.11)in the 3- dimensional L su × u D onto sites ~r ( n ,n ) (5.12)in the base sublattice A ∗ L su × u D depend on the parameter q ; and so canbe parameterized like ~R ( q )( n ,n ,n ) = ~r ( n ,n ) qZ n ! (5.13)with third component belonging to the fiber, Z n ∈ L u D , L u D ≃ q Z (5.14)The same property is valid for the ~L i basis generators; they depend on the charge q and may be decomposed as well like ~L ( q )1 = ~l q ! , ~L ( q )2 = ~l q ! , ~L ( q )3 = ~ q ! (5.15)with the 2-dimensional vectors ~r n giving the sites in the base sublattice A ∗ ~r n = n ~l + n ~l n = ( n , n ) (5.16)So the case q = 0 define a projection from L su × u D down to the base A ∗ ; we have ~R (0) n = ~r n ! (5.17)2) Discrete field variables • scalar fields Under discretization of the real continuum space into L su × u D , local scalar fieldsΦ ( x ) of the continuum get mapped to an infinite set of discrete variablesΦ ( R n ) = Φ ( q )( n ,n ,n ) ≡ Φ ( q ) n (5.18)living at the lattice nodes R n , n = ( n , n , n ) ∈ Z ( q ) n may have either an even statistics or an odd one dependingon whether Φ ( x ) is bosonic or fermionic. In the case of twisted N = 4 su-persymmetric YM, odd variables are given by the twisted fermion ψ ( − q ) and theGrassman variable θ ( − q ) . • antisymmetric p-tensors Real p- form fields T [ p ] ( x ) in continuum T [ p ] = p ! dx µ ∧ dx µ ... ∧ dx µ p T µ ...µ p (5.19)are associated with p- dimensional plaquettes in the lattice. In the case of L su × u D ,we have vectors and and their duals namely the rank 2 antisymmetric tensors; rank3 antisymmetric tensors are dual to scalars. So we havefields → p-plaquettes T sites T µ links T µν ∼ links (5.20)Let us illustrate the construction on the particular case of the gradient ∂ µ Φ ( x ) . To get the discrete expression representing ∂ µ Φ ( x ) on the lattice L su × u D , it isuseful to consider d Φ ( x ) = dx µ ∂ µ Φ ( x ) (5.21)which is a particular in 3D. This differential, which involves the operator d = dx µ ∂ µ , behaves as a scalar under SO (3) and is related to Φ ( x ) by d Φ ( ~x ) = lim d~x → ~ [Φ ( ~x + d~x ) − Φ ( ~x )] (5.22)In the standard case of a simple cubic lattice with spacing parameter L , the arbi-trary elementary variations d~x of the 3-dimensional continuum are given by the first nearest neighbors namely ± L~e , ± L~e , ± L~e (5.23)In the case of discretization of space to the L su × u D of eq(5.3), vectors ~x in con-tinuum are mapped to ~R ( q ) n and the elementary variations d~x are mapped to firstnearest neighbors of ~R ( q ) n ; that is ~x + d~x → ~R ( q ) n + ~V ( q ) I (5.24)35ith ~V ( q ) I = ~υ I qZ I ! (5.25)and the 6 non zero ~υ I ’s as ~υ i = υ i υ i ! , i = 1 , ..., A ∗ . So the 3D crystalanalogue of d Φ ( x ) is given byΦ I ( R n ) = Φ( R n + V ( q ) I ) − Φ ( R n ) (5.27)with projection on the A ∗ base sublattice corresponding to q = 0 as follows φ I ( ~r n ) = φ ( ~r n + ~υ I ) − φ ( ~r n ) (5.28)where the ~υ I ’s are as in (5.25). For later use, it is convenient to denote φ I ( ~r n ) like φ I ( ~r n ) = φ r n → ( r n + υ I ) ≡ φ n,I (5.29) First result
From eqs(5.27-5.28), we learn a set of useful features that we collect below: • the field ∂ µ Φ ( x ) is mapped to link variablesΦ ( q ) n,I (5.30)living on edges of the 3D crystal L su × u D .For q = 0, the link variables Φ ( q ) n,I are projected down to φ n,I (5.31)living on the A ∗ the edge −−−→ P r n P r n + υ I ∼ ˜ υ I (5.32) • the analogue of the gauge field A µ ( x ) on the lattice L su × u D is given by the linkvariables U I ( R ( q ) n ) = U ( q ) n,I (5.33)36ith projection on the A ∗ base sublattice as U I ( ~r n ) = U n,I (5.34)living on the links (5.32).The usual gauge transformation in continuum with G a generic gauge group element ∂ µ + ig Y M A µ ( x ) → G ( x ) [ ∂ µ + ig Y M A µ ( x )] G † ( x ) (5.35)is mapped to U I ( R ( q ) n ) → G ( R ( q ) n ) U I ( R ( q ) n ) G † ( R ( q ) n + V I ) (5.36)On the A ∗ base sublattice, these transformations reduce to U I ( ~r n ) → G ( ~r n ) U I ( ~r n ) G † ( ~r n + ~υ I ) (5.37) • the discrete analogue of the field strength F µν ( x ) is given by the plaquette variables W [ IJ ] (cid:16) R ( q ) n (cid:17) = W ( q ) n, [ IJ ] , I = J (5.38)which is dual to 1-dimensional link. On the A ∗ base sublattice, these variables areprojected to the 2- dimensional plaquette variables W [ IJ ] ( r n ) = W n, [ IJ ] , I = J (5.39)associated with −−−→ P r n P r n + υ I ∧ −−−→ P r n P r n + υ J ∼ ˜ υ I ∧ ˜ υ J (5.40)This plaquette has 4 vertices located at P r n ∼ ~r n , P r n + υ I ∼ ~r n + ~υ I P r n + υ J ∼ ~r n + ~υ J , P r n + υ I + υ J ∼ ~r n + ~υ I + ~υ J (5.41)and has an interpretation in terms of the vector surface ~s IJ = ~υ I ∧ ~υ J with com-ponents s µIJ = 12 ε µνρ υ µI υ ρJ (5.42)37 Non abelian U ( N ) gauge fieldsIn the case of YM theory with non abelian U ( N ) gauge symmetry, the fields arevalued in the adjoint representation of the Lie algebra of the gauge symmetry; sothe gradient ∂ µ Φ ( x ) and the field strength F µν involve gauge covariant derivatives D µ Φ = ∂ µ Φ + [ A µ , Φ] F µν = [ D µ , D ν ] (5.43)By discretization, D µ Φ ( x ) and F µν ( x ) are respectively mapped to discrete N × N matrix variables Φ ( q ) I ( R n ) , W ( p ) IJ ( ~r n ) (5.44)carrying moreover charges under U (1). On the base sublattice A ∗ where q = 0,these quantities becomeΦ I ( ~r n ) = U I ( ~r n ) Φ ( ~r n + ~υ I ) − Φ ( ~r n ) U I ( ~r n ) W IJ ( ~r n ) = U I ( ~r n ) U J ( ~r n + ~υ I ) − U J ( ~r n ) U I ( ~r n + ~υ J ) (5.45) In the case of the complex 3D space, on which the 3-dimensional twisted N = 4 super-symmetric YM has been formulated, one distinguishes two kinds of quantities: • complex antisymmetric tensor fields of type B ( q ) a ...a p transforming in some complexrepresentation R q of SU (3) × U (1) , • the adjoint fields B ( − q ) a ...a p transforming in the adjoint conjugate ¯ R − q .On the lattice L su × u D , these objects are interpreted in terms of oriented p-simplex. Tothat purpose, recall the objects appearing in the field action (4.8-4.17); there, we havebosons and fermions: In the bosonic sector, each object have an adjoint as shown belowObject Adjoint z a ( − q ) ¯ z (+2 q ) a dz a ( − q ) d ¯ z (+2 q ) a G a ( − q ) ¯ G (+2 q ) a ∇ a ( − q ) ∇ (+2 q ) a F ab ( − q ) ¯ F (+4 q ) ab (5.46)38nd so both orientations of bosonic link variables are involved; contrary to the chiralfermionic sector Object Adjoint θ ( − q ) - ϑ a (+ q ) - ψ ( − q ) - ψ a (+) - ∇ a ( − q ) ψ b (+) - ∇ (+2 q ) a ψ a (+) - (5.47)where we have only one orientation for fermionic lattice link variables.To study the discrete version of (4.8-4.17), we proceed in two steps: a) step 1 : we describe lattice theory living on the 2-dimensional A ∗ given by fig 2.Figure 2: the lattice A ∗ generated by the 2 basic weight vectors ~ω , ~ω of SU (3). Greennodes are associated with the lattice variable ψ ( − q ) n ; red nodes with ψ I (+ q ) n and bluenodes with the lattice gauge variables ( U I ( − q ) n , U (+2 q ) In ). More precisely ψ I (+ q ) n transformsinto the representation ; it is given by the link from Green to red nodes. Similarly, U I ( − q ) n ; it transforms in the representation and is given by links from the blue to thegreen nodes while U (+2 q ) In transforms in the adjoint ¯3 and is given by links from the greento the blue sites. b) step 2 : we extend the construction from A ∗ to the 3-dimensional lattice L su × u D byimplementing the fiber direction L u D . This corresponds to unfolding the normal directionto A ∗ as illustrated on fig 3. 39igure 3: the 3D lattice L su × u D given by a L su D × L u D fibration with 2D base sublatticegiven by A ∗ and 1D fiber L u D isomorphic to Z , the set of integers. The superpositionof the 3 sublattices G , R and B making A ∗ as given by eq(5.57) is lifted. Sheets with Q = 3 q corresponds to the sublattice G , sheets with Q = q to the sublattice R and thosewith Q = 2 q to the sublattice B . 40 .2.1 twisted supersymmetric YM on A ∗ The lattice A ∗ is generated by the 2 fundamental weight vectors ~ω , ~ω of the Lie algebraof the simple SU (3) symmetry. These are non canonical vectors obeying ~ω i .~ω i = , (cid:16) \ ~ω , ~ω (cid:17) = π (5.48)Each site ~r n in A ∗ ( say a green site of fig 2 ) has 6 first nearest neighbors located at ~r n + ~υ i and which can be organized into 2 subsets, each having 3 elements like ~υ = L q ~ω ~υ = L q ( ~ω − ~ω ) ~υ = − L q ~ω (5.49)and ~υ ′ = − ~υ ~υ ′ = − ~υ ~υ ′ = − ~υ (5.50)Notice that each triplet obeys a traceless property ~υ + ~υ + ~υ = ~ ~υ ′ + ~υ ′ + ~υ ′ = ~ G of (green) sites ~r n at the centre of the hexagons of fig 2 forma sublattice of A ∗ generated by the particular vectors ~α = 2 ~ω − ~ω ~α = ~ω − ~ω (5.52)This means that green sites are related amongst others as ~r m = X m ,m m ~α + m ~α , (5.53)and form a sublattice G ≡ { ~r m } green sites which is nothing but A , the root lattice of SU (3). So we have the isomorphism G ≃ A (5.54)41otice moreover that, because of the symmetric role played by the types of nodes(green, red, blue in fig 2), the same thing may be said about the set R of red sites andthe set B of blue ones. In other words the set R is isomorphic to a root lattice of SU(3)and similarly the set B is isomorphic as well to a root lattice of SU(3). Thus we havethe isomorphisms R ≃ A , B ≃ A (5.55)and formally R ≃ G + ~ω , B ≃ G + ~ω (5.56)From this representation, it follows that the lattice A ∗ is made by the superposition ofthe 3 sublattices G , R and B or equivalently A ∗ = G ∪ R ∪ B (5.57)For more details on the matrix describing the shape of the 2- dimensional base lattice A ∗ ; see subsection 5.Under discretization of continuum, we have the following correspondencecontinuum crystal A ∗ ( z, ¯ z ) ~r n z + dz ~r n + ~υ I ¯ z + d ¯ z ~r n − ~υ I (5.58)with n = ( n , n ) arbitrary integers; and where ~υ I stand for the 3 oriented first nearestneighbors given by (5.49). Observe that complex conjugation is captured by the changeof the orientation of ~υ I .Regarding the lattice analogue of the twist fields in continuum, we have the followingdictionary:( i ) bosonic fields : continuum crystal A ∗ G a ( z, ¯ z ) U In ¯ G a ( z, ¯ z ) U † n,I F ab ( z, ¯ z ) W IJn ¯ F ab ( z, ¯ z ) W † n,IJ F ba ( z, ¯ z ) W Jn,I (5.59)42ith U In = U I ( ~r n ) U † n,I = U † I ( ~r n ) W IJn = U I ( ~r n ) U J ( ~r n + ~υ I ) − U J ( ~r n ) U I ( ~r n + ~υ J ) W † n,IJ = U † J ( ~r n + ~υ I ) U † I ( ~r n ) − U † I ( ~r n + ~υ J ) U † J ( ~r n ) (5.60)and W In,I = U I ( ~r n ) U † I ( ~r n ) − U † I ( ~r n − ~υ I ) U I ( ~r n − ~υ I ) (5.61)( ii ) fermionic fields : continuum crystal A ∗ ψ ( z, ¯ z ) ψ n ψ a ( z, ¯ z ) ψ In ∇ a ψ b ψ IJn ¯ ∇ a ψ a ψ In,I (5.62)with ψ n = ψ ( ~r n ) ψ In = ψ I ( ~r n ) ψ IJn = ψ IJ ( ~r n ) ψ In,I = ψ II ( ~r n ) (5.63)and ψ IJ ( ~r n ) = U I ( ~r n ) ψ J ( ~r n + ~υ I ) − ψ J ( ~r n ) U I ( ~r n + ~υ J ) ψ II ( ~r n ) = ψ I ( ~r n ) U † I ( ~r n ) − U † I ( ~r n − ~υ I ) ψ I ( ~r n − ~υ I ) (5.64)The U ( N ) gauge transformations with generic unitary N × N matrix G ( ~r n ) act on the43attice fields as followsfield → gauge transform U In G ( ~r n ) U I ( ~r n ) G † ( ~r n + ~υ I ) U † n,I G ( ~r n + ~υ I ) U † I ( ~r n ) G † ( ~r n ) W IJn G ( ~r n ) W IJn G † ( ~r n + ~υ I + ~υ J ) W † n,IJ G ( ~r n + ~υ I + ~υ J ) W IJn G † ( ~r n ) ψ ( ~r n ) G ( ~r n ) ψ ( ~r n ) G † ( ~r n ) ψ I ( ~r n ) G ( ~r n ) ψ I ( ~r n ) G † ( ~r n + ~υ I ) ψ IJ ( ~r n ) G ( ~r n ) ψ IJ ( ~r n ) G † ( ~r n + ~υ I + ~υ J ) ψ II ( ~r n ) G ( ~r n ) ψ II ( ~r n ) G † ( ~r n ) (5.65)(5.66)Using these gauge transformations, one can check that the following couplings( i ) : T r (cid:2) ψ ( ~r n ) ψ II ( ~r n ) (cid:3) ( ii ) : − ε IJK
T r (cid:2) ψ K ( ~r n − ~υ K ) ψ IJ ( ~r n ) (cid:3) (5.67)are gauge invariant provided we have G † ( ~r n + ~υ I + ~υ J ) G ( ~r n − ~υ K ) = I (5.68)But this constraint equation requires ~υ I + ~υ J + ~υ K = ~ ε IJK , can be also written as ~υ + ~υ + ~υ = ~ A ∗ as shown by eq(5.51).44 .2.2 action of twisted supersymmetric YM on A ∗ Using the above dictionary giving the analogue of fields in continuum to lattice variableson L su × u D , we can work out the action of twisted supersymmetric YM on the basesublattice A ∗ . This action may decomposed in 2 parts as S lattice = S boselattice + S fermilattice (5.71)where S boselattice involving bosonic degrees of freedom and S fermilattice describing lattice fermionicvariables coupled to the gauge link variables.( α ) Bosonic term
Under discretization, the bosonic part of the field action in continuum namely S bosecont = α Z T r h F (+4 q ) ab F ab ( − q ) i − ( α ) α Z T r (cid:2) F (0) F (0) (cid:3) (5.72)gets mapped to the following gauge invariant lattice one S boselatt = α X A ∗ T r (cid:16) W IJn W † n,IJ (cid:17) − ( α ) α X A ∗ T r (cid:16) W (0) n W † (0) n (cid:17) (5.73)with W IJn W † n,IJ = P + P − P − P W (0) n W † (0) n = R + R − R − R (5.74)and P = U I ( ~r n ) U J ( ~r n + ~υ I ) U † J ( ~r n + ~υ I ) U † I ( ~r n ) P = U J ( ~r n ) U I ( ~r n + ~υ J ) U † I ( ~r n + ~υ J ) U † J ( ~r n ) P = U J ( ~r n ) U I ( ~r n + ~υ J ) U † J ( ~r n + ~υ I ) U † I ( ~r n ) P = U I ( ~r n ) U J ( ~r n + ~υ I ) U † I ( ~r n + ~υ J ) U † J ( ~r n ) (5.75)45s well as R = U I ( ~r n ) U † I ( ~r n ) U J ( ~r n ) U † J ( ~r n ) R = U † I ( ~r n − ~υ I ) U I ( ~r n − ~υ I ) U † J ( ~r n − ~υ J ) U J ( ~r n − ~υ J ) R = U I ( ~r n ) U † I ( ~r n ) U † J ( ~r n − ~υ J ) U J ( ~r n − ~υ J ) R = U † I ( ~r n − ~υ I ) U I ( ~r n − ~υ I ) U J ( ~r n ) U † J ( ~r n ) (5.76)( β ) fermionic term For the fermionic terms, the analogue of S fermicont ( α + 2 α ) Z T r h ψ a (+ q ) ∇ (+2 q ) a ψ ( − q ) i +2 α Z T r h ε abc ψ a (+ q ) ∇ b ( − q ) ψ c (+ q ) i (5.77)is given by the following gauge invariant expression S fermilatt = ( α + 2 α ) X A ∗ T r h ψ ( ~r n ) ψ I ( ~r n ) U † I ( ~r n ) i +( α + 2 α ) X A ∗ T r h ψ ( ~r n ) U † I ( ~r n − ~υ I ) ψ I ( ~r n − ~υ I ) i +2 α X A ∗ ε IJK
T r (cid:2) ψ K ( ~r n − ~υ K ) ψ IJ ( ~r n ) (cid:3) (5.78)with ψ IJ ( ~r n ) as in eq(5.64). First, we construct the lattice by giving further details on the base sublattice A ∗ ;and the fiber L u D . Then we turn to derive the gauge invariant lattice action of twistedsupersymmetric YM theory on L su × u D . 46 .1 More on the base sublattice A ∗ The base sublattice A ∗ is generated by the 2 fundamental weight vectors ~ω , ~ω of SU (3).These fundamental weight vectors, having the length and angle ( \ ω , ω ) = π , are thedual of the 2 simple roots ~α , ~α of SU (3), ~ω i .~α j = δ ij (6.1)Below, we take ~ω , ~ω and ~α , ~α in the real plane as follows ~ω = (cid:16) √ , √ (cid:17) , ~ω = (cid:16) , √ (cid:17) ~α = (cid:0) √ , (cid:1) , ~α = (cid:16) − √ , √ (cid:17) (6.2)They are related to each other like ~α = 2 ~ω − ~ω ~α = 2 ~ω − ~ω (6.3)and ~ω = (2 ~α + ~α ) ~ω = ( ~α + 2 ~α ) (6.4)The vectors ~α , ~α generate A ; the root lattice of SU (3). ~r n ∈ A ⇔ ~r n = X n,m n~α + m~α = X n,m (2 n − m ) ~ω + (2 m − n ) ~ω Using ~ω , ~ω , position vectors ~r n of sites in the wight lattice A ∗ are expanded like ~r n = r L ~ω ( n ,n ) (6.5)with ~ω ( n ,n ) = n ~ω + n ~ω (6.6)and where L stands for the spacing parameter of the lattice and n = ( n , n ) are arbitraryintegers. Using (6.4), we also have ~r n = r L (2 n + n )3 ~α + r L ( n + 2 n )3 ~α (6.7)47he architecture of the sites of the A ∗ crystal is encoded into the intersection matrix J su ij = ~ω i .~ω j (6.8)given by J su ij = 13 ! (6.9)with inverse K su ij = − − ! , K su ij = ~α i .~α j (6.10) classifying closest neighbors Each site ~r n in the base lattice A ∗ has (3 + 3) first nearest neighbors and second nearestones; they are as follows: • Up to a scaling factor L, the first nearest neighbors are given by ~λ = ~ω ~λ = ~ω − ~ω ~λ = − ~ω (6.11)and ~ζ = ~ω ~ζ = ~ω − ~ω ~ζ = − ~ω (6.12)obeying the identities ~λ + ~λ + ~λ = ~ ~ζ + ~ζ + ~ζ = ~ ~r n , its first nearestneighbors (6.11) are given by the red sites; and those of associated with (6.12) aregiven by the blue ones. • ~ω (2 , − = +2 ~ω − ~ω ~ω (1 , = + ~ω + ~ω ~ω ( − , = − ~ω + 2 ~ω ~ω ( − , = − ~ω + ~ω ~ω ( − , − = − ~ω − ~ω ~ω (1 , − = + ~ω − ~ω (6.14)and are nothing but the six roots of the SU (3) namely ± ~α , ± ~α , ± ( ~α + ~α ) (6.15)As an illustration on fig 2, each green site ~r n has 6 nearest neighbors located at ~r n ± L~α , ~r n ± L~α , ~r n ± L ( ~α + ~α ) (6.16)Observe also that a generic red site located at ~r n + L~λ i , i = 1 , , ~r n + L~λ i ± L~α , ~r n + L~λ i ± L~α , ~r n + L~λ i ± L ( ~α + ~α ) (6.18)This result may be also checked by computing the relative vector ~V ij between twonearest sites located at ~r n + L~λ i and ~r n + L~λ j . We have ~V ij = (cid:16) ~r n + L~λ i (cid:17) − (cid:16) ~r n + L~λ j (cid:17) = L (cid:16) ~λ i − ~λ j (cid:17) (6.19)which, by using eqs(6.12-6.14), the 6 vectors ~λ i − ~λ j are precisely the 6 roots of SU (3). 49 .2 Building the lattice L su × u D The lattice L su × u D is a 3-dimensional crystal that may be thought of as given by atwisting of weight lattice A ∗ of the Lie algebra of the symmetry SO (6) ≃ SU (4) (6.20)To build this 3D crystal, we begin by describing the lattice A ∗ ; after what we turn toconstruct L su × u D . Construction of the lattice A ∗ The 3- dimensional lattice A ∗ is the dual of the root lattice of SU (4); it is generated bythe 3 fundamental weight vectors of SU (4) ~ Ω , ~ Ω , ~ Ω (6.21)Using the lattice spacing parameter L su of the crystal A ∗ , we can express the positions ~R n of sites in this lattice as follows ~R n = L su r ~ Ω ( n ,n ,n ) (6.22)with ~ Ω ( n ,n ,n ) = n ~ Ω + n ~ Ω + n ~ Ω (6.23)where n i are arbitrary integers.The shape of A ∗ is encoded in the intersection matrix J su ij = ~ Ω i .~ Ω j given by J su ij =
34 12 1412 (6.24)The fundamental weight vectors ~ Ω i are the dual of the 3 simple roots ~a , ~a , ~a of SU (4)obeying ~ Ω i .~a j = δ ij (6.25)The vectors ~ Ω i are also the highest weight vectors of the complex , the adjoint conjugate ¯4 and the real 6 dimensional representations of SU (4).representation : ¯4 highest weights : ~ Ω ~ Ω ~ Ω (6.26)50he set of the weight vectors ~ Λ i defining the states of the complex 4- dimensional highestweight representations and its conjugate ¯4 are as followsweight vectors of weight vectors of ¯4 ~ Λ = ~ Ω ~ Λ ′ = ~ Ω ~ Λ = ~ Ω − ~ Ω ~ Λ ′ = ~ Ω − ~ Ω ~ Λ = ~ Ω − ~ Ω ~ Λ ′ = ~ Ω − ~ Ω ~ Λ = − ~ Ω ~ Λ ′ = − ~ Ω (6.27)These quartets obey the following traceless properties ~ Λ + ~ Λ + ~ Λ + ~ Λ = ~ ~ Λ ′ + ~ Λ ′ + ~ Λ ′ + ~ Λ ′ = ~ ~ Λ ′′ i of the realFigure 4: Each site ~R n in the lattice A ∗ has 8 = 4 + 4 first nearest neighbors associatedwith the weight vectors ~ Λ i and ~ Λ ′ i . In the figure, 4 sites of the 8 ones are representedand are related to the Ω i generators as in eqs(6.27).51- dimensional representation of SU (4) ≃ SO (6) are given by ~ Λ ′′ = ~ Ω ~ Λ ′′ = ~ Ω + ~ Ω − ~ Ω ~ Λ ′′ = ~ Ω − ~ Ω ~ Λ ′′ = ~ Ω − ~ Ω ~ Λ ′′ = ~ Ω − ~ Ω − ~ Ω ~ Λ ′′ = − ~ Ω (6.29)(6.30)satisfying also a traceless property ~ Λ ′′ + ... + ~ Λ ′′ = 0 and having an interpretation interms of second nearest neighbors. The lattice L su × u D as a twist of A ∗ Under the breaking of SU (4) down to U (1) × SU (3), the representations and ¯4 breakdown to [44] → + q ⊕ − q ¯4 → ¯3 − q ⊕ ¯1 +3 q (6.31)and the weight vectors (6.27) becomeweight vectors of + q ⊕ − q weight vectors of ¯3 − q ⊕ +3 q ~ Λ = (cid:16) ~λ , + q (cid:17) ~ Λ ′ = (cid:16) ~ , +3 q (cid:17) ~ Λ = (cid:16) ~λ , + q (cid:17) ~ Λ ′ = (cid:16) ~ζ , − q (cid:17) ~ Λ = (cid:16) ~λ , + q (cid:17) ~ Λ ′ = (cid:16) ~ζ , − q (cid:17) ~ Λ = (0 , , − q ) ~ Λ ′ = (cid:16) ~ζ , − q (cid:17) (6.32)where where ~λ i , ~ζ i are as in eqs(6.11).Similarly, we have for the 6-dimensional representation the following decomposition → − q ⊕ ¯3 +2 q (6.33)52ith weight vectors ~ Λ ′′ i as follows ~ Λ ′′ = (cid:16) ~ζ , +2 q (cid:17) ~ Λ ′′ = (cid:16) ~ζ , +2 q (cid:17) ~ Λ ′′ = (cid:16) ~ζ , +2 q (cid:17) ~ Λ ′′ = (cid:16) ~λ , − q (cid:17) ~ Λ ′′ = (cid:16) ~λ , − q (cid:17) ~ Λ ′′ = (cid:16) ~λ , − q (cid:17) (6.34)From these decompositions, we deduce the expressions of the fundamental weight vectorsof U (1) × SU (3) that we denote as ~ Γ , ~ Γ , ~ Γ . These vectors are given by ~ Γ = ( ~ω , q ) ~ Γ = ( ~ω , q ) ~ Γ = (0 , , q ) (6.35)with unit charge q to be fixed later. Computing the intersection matrix of these weights J su × u ij = ~ Γ i .~ Γ j (6.36)we find J su × u ij = + q + 2 q q + 2 q + 4 q q q q q (6.37)with det J su × u ij = 3 q (6.38)and inverse given by K su × u ij = − − − − q +19 q (6.39)By requiring the following condition on the self intersection of ~ Γ q + 19 q = 1 (6.40)it results q = 13 (6.41)53utting this value back into the above intersection matrices, we end with J su × u ij = (6.42)and K su × u ij = − − − − (6.43)From these intersection matrices, we determine the following relations ~a = 2 ~ Γ − ~ Γ ~a = 2 ~ Γ − ~ Γ − ~ Γ ~a = ~ Γ − ~ Γ (6.44)and ~ Γ = ~a + ~a + ~a ~ Γ = ~a + 2 ~a + 2 ~a ~ Γ = ~a + 2 ~a + 3 ~a (6.45)Sites ~R n in L su × u D are therefore expanded like ~R n = L ~ Γ n , L = L su r
43 (6.46)with ~ Γ n = n ~ Γ + n ~ Γ + n ~ Γ (6.47)and ~ Γ = ~ω q ! , ~ Γ = ~ω q ! , ~ Γ = ~ q ! (6.48)obeying the duality relation ~ Γ i .~a j = δ ij (6.49)We also have ~a = ~ω − ~ω ! , ~a = ~ω − ~ω ! , ~a = − ~ω q ! (6.50) Here we use the weight vectors of the SU (3) × U (1) representations carried by thefermionic and the bosonic fields of the spectrum of twisted supersymmetric YM theoryon L su × u D to give a crystal interpretation of the BRST symmetry. To that purpose, wefirst derive the closed nearest neighbors in L su × u D ; then, we give the lattice realizationof BRST symmetry. 54 .3.1 Closest neighbors in L su × u D Given a generic site ~R n in L su × u D , its closest neighbors are as follows: a) first nearest neighbors and the links ψ I (+ q ) n :Each site in L su × u D has 6 first nearest neighbors that split into 3 + 3 ones respectivelylocated at ~R n + L~ Λ I = ~R n + ~V (+ q ) I (6.51)and ~R n + L~ Λ ′ I = ~R n − ~V (+ q ) I (6.52)with ~V (+ q )1 = L~ Γ ~V (+ q )2 = L (cid:16) ~ Γ − ~ Γ (cid:17) ~V (+ q )3 = L (cid:16) ~ Γ − ~ Γ (cid:17) (6.53)satisfying ~V (+ q )1 + ~V (+ q )2 + ~V (+ q )3 = L~ Γ (6.54)Using (6.48), these relations read also like ~V (+ q )1 = L (cid:16) ~λ , q (cid:17) ~V (+ q )2 = L (cid:16) ~λ , q (cid:17) ~V (+ q )3 = L (cid:16) ~λ , q (cid:17) (6.55)with ~V (+ q )1 + ~V (+ q )2 + ~V (+ q )3 = (0 , , Lq ) (6.56)and length (cid:16) ~V (+ q ) I (cid:17) = (cid:16) − ~V (+ q ) I (cid:17) = (cid:0) + q (cid:1) L = L (6.57)since 3 q = 1.Therefore the oriented links −−−→ P R n P R n + V (+ q ) I ∼ V (+ q ) I (6.58)are associated with the fermionic lattice variables ψ I (+ q ) n = ψ I (+ q ) ( R n ) (6.59)55 ) second nearest neighbors and gauge field variables Each site R n in L su × u D has 6 second nearest neighbors that also split into 3 + 3 onesrespectively located at ~R n + ~V ( − q ) I (6.60)and at ~R n + ~V (+2 q ) I = ~R n − ~V ( − q ) I ∼ − ~V ( − q ) I (6.61)Upon the normalization L = 1, we have ~V (+2 q )1 = ~ Γ ~V (+2 q )2 = ~ Γ + ~ Γ − ~ Γ ~V (+2 q )3 = ~ Γ − ~ Γ ~V ( − q )1 = ~ Γ − ~ Γ ~V ( − q )2 = ~ Γ − ~ Γ − ~ Γ ~V ( − q )3 = − ~ Γ (6.62)or equivalently ~V (+2 q )1 = (cid:16) − ~λ , q (cid:17) ~V (+2 q )2 = (cid:16) − ~λ , q (cid:17) ~V (+2 q )3 = (cid:16) − ~λ , q (cid:17) ~V ( − q )1 = (cid:16) ~λ , − q (cid:17) ~V ( − q )2 = (cid:16) ~λ , − q (cid:17) ~V ( − q )3 = (cid:16) ~λ , − q (cid:17) (6.63)with length (cid:16) ~V (+2 q ) a (cid:17) = (cid:0) + 4 q (cid:1) L = 2 L (cid:16) ~V ( − q ) a (cid:17) = (cid:0) + 4 q (cid:1) L = 2 L (6.64)The 3 oriented links −−−→ P R n P R n + V (+2 q ) I ∼ − ~V ( − q ) I (6.65)are associated with the gauge field variables U (+2 q ) n,I = U (+2 q ) I ( R n )56hile the opposite ones −−−→ P R n P R n − V (+2 q ) I ≡ P R n + V ( − q ) I P R n ∼ V ( − q ) I (6.66)with the complex conjugate fields U I ( − q ) n = U I ( − q ) ( R n ) (6.67) c) third nearest neighbors The site ~R n of L su × u D has 2 third nearest neighbors that also split into 1 + 1 respectivelylocated at ~R n + L~ Γ = ~R n + ~V (+3 q ) ~R n − L~ Γ = ~R n + ~V ( − q ) (6.68)with ~ Γ = ~ q ! (6.69)and ~ Γ .~ Γ = 3 (6.70)The oriented links −−−→ P R n P R n + V ( − q ) ∼ V ( − q ) (6.71)are associated with the fermionic singlets ψ ( − q ) n and the Grassman variable ψ ( − q ) n = ψ ( − q ) ( R n ) , θ ( − q ) (6.72)while the opposite ones −−−→ P R n P R n − V ( − q ) ≡ P R n + V (+3 q ) P R n ∼ − V ( − q ) (6.73)with objects carrying +3 unit charges of U(1) like the discrete analogue of ∇ (+2 q ) a ψ a (+) .57 .3.2 BRST symmetry on lattice
Being a scalar objet under SU (3) but having non trivial charges of U(1), the scalarsupersymmetric Q (+3 q ) may be interpreted as a link operator living on the direction ~ Γ = (0 , , q ) (6.74)Therefore supersymmetric transformations generated by Q (+3 q ) can be interpreted asparticular shifts on the lattice L su × u D ; more precisely shifts in the L u D sublattice of thefibration L u D → L su × u D ↓ A ∗ For example, the supersymmetric transformation Q (+3 q ) U I ( − q ) n = ψ I (+ q ) n (6.75)corresponds to shifting ~V ( − q ) I as follows ~V ( − q ) I + ~V (+3 q )0 = ~V (+ q ) I (6.76)In general, the operator Q (+3 q ) acts on the various lattice variables like Q (+3 q ) U I ( − q ) n = ψ I (+ q ) n Q (+3 q ) ψ I (+ q ) n = 0 Q (+3 q ) U (+2 q ) I,n = 0 Q (+3 q ) W (+4 q ) IJ,n = 0 Q (+3 q ) ψ ( − q ) n = F (0) n Q (+3 q ) F (0) n = 0 Q (+3 q ) W IJ ( − q ) n = Υ IJ ( − q ) n Q (+3 q ) Υ IJ ( − q ) n = 0 Q (+3 q ) W (0) n = Υ (+3 q ) n Q (+3 q ) Υ (+3 q ) n = 0 (6.77)58ith Υ IJ ( − q ) n = U I ( − q ) ( R n ) ψ J (+ q ) ( R n + ~V ( − q ) I ) − ψ J (+ q ) ( R n ) U I ( − q ) ( R n + ~V ( − q ) J )Υ (0) n = ψ I (+ q ) ( R n ) U (+2 q ) I ( R n ) − U (+2 q ) I ( R n − ~V ( − q ) I ) ψ I (+ q ) ( R n − ~V ( − q ) I ) (6.78) N = 4 on Lattice L su × u D First we give the expression of the fields on the lattice L su × u D ; then we build the gaugeinvariant action on L su × u D . L su × u D : a dictionary Under discretization of the the complex 3D continuum into the crystal L su × u D , theanalogue of coordinate variables and fields are given by the following dictionary • Coordinates variables continuum : crystal L su × u D ( z a ( − q ) , ¯ z (+2 q ) a ) ~R n z a ( − q ) + dz a ( − q ) ~R n + ~V ( − q ) I ¯ z (+2 q ) a + d ¯ z (+2 q ) a ~R n + ~V (+2 q ) I = ~R n − ~V ( − q ) I (7.1)with, by setting L = 1 , ~V (+2 q ) I = (cid:16) − ~λ a , +2 q (cid:17) ~V ( − q ) I = (cid:16) + ~λ a , − q (cid:17) (7.2) • Bosonic fields continuum : crystal L su × u D G a ( − q ) ( z, ¯ z ) U I ( − q ) n G (+2 q ) a ( z, ¯ z ) U (+2 q ) n,I F ab ( − q ) ( z, ¯ z ) W IJ ( − q ) n ¯ F (+4 q ) ab ( z, ¯ z ) W (+4 q ) n,IJ F (0) ( z, ¯ z ) W (0) n (7.3)59ith U I ( − q ) n = U I ( − q ) ( R n ) U (+2 q ) n,I = U (+2 q ) I ( R n ) W IJ ( − q ) n = U I ( − q ) ( R n ) U J ( − q ) ( R n + V ( − q ) I ) − U J ( − q ) ( R n ) U I ( − q ) ( R n + V ( − q ) J ) W (+4 q ) n,IJ = U (+2 q ) J ( R n + V ( − q ) I ) U (+2 q ) I ( R n ) − U (+2 q ) I ( R n + V ( − q ) J ) U (+2 q ) J ( R n ) W (0) n = U I ( − q ) ( R n ) U (+2 q ) I ( R n ) − U (+2 q ) I ( R n − V ( − q ) I ) U I ( − q ) ( R n − V ( − q ) I ) (7.4)(7.5) • Fermionic fields continuum : crystal L su × u D ψ ( − q ) ( z, ¯ z ) ψ ( − q ) n ψ I (+ q ) ( z, ¯ z ) ψ I (+ q ) n ∇ a ( − q ) ψ b (+) ψ IJ ( − q ) n ¯ ∇ (+2 q ) a ψ a ( − ) ψ I (+) n,I (7.6)with ψ ( − q ) n = ψ ( − q ) ( R n ) ψ I (+ q ) n = ψ I (+ q ) ( R n ) ψ IJ ( − q ) n = U I ( − q ) ( R n + V ( − q ) J ) ψ J (+ q ) ( R n + V ( − q ) I ) − ψ J (+ q ) ( R n ) U I ( − q ) ( R n ) ψ I (+3 q ) n,I = ψ I (+ q ) ( R n + V ( − q ) I ) U (+2 q ) I ( R n ) − U (+2 q ) I ( R n − V ( − q ) I ) ψ I (+ q ) ( R n ) (7.7) • Gauge symmetry attice variable → gauge transform U I ( − q ) n G ( R n ) U I ( − q ) ( R n ) G † ( R n + V ( − q ) I ) U (+2 q ) n,I G ( R n + V ( − q ) I ) U (+2 q ) I ( R n ) G † ( R n ) W IJ ( − q ) n G ( R n ) W IJ ( − q ) n G † ( R n + V ( − q ) I + V ( − q ) J ) W (+4 q ) n,IJ G ( R n + V ( − q ) I + V ( − q ) J ) W IJ (+4 q ) n G † ( R n ) ψ ( − q ) n G ( R n ) ψ ( − q ) ( R n ) G † ( R n + V ( − q )0 ) ψ I (+ q ) n G (cid:16) R n + V ( − q ) I (cid:17) ψ I (+ q ) ( R n ) G † ( R n ) ψ IJ ( − q ) n G ( R n + V ( − q ) J ) ψ IJ ( − q ) ( R n ) G † ( R n + V ( − q ) I ) ψ I (+3 q ) n,I G ( R n + V ( − q ) I + V ( − q ) I ) ψ I (+3 q ) I ( R n ) G † ( R n ) (7.8) Here we collect some relations useful for checking gauge invariance of the lattice fieldaction L su × u D ~V (+ q )1 = (cid:16) ~λ , q (cid:17) ~V (+ q )2 = (cid:16) ~λ , q (cid:17) , ~V ( − q ) I = − ~V (+ q ) I ~V (+ q )3 = (cid:16) ~λ , q (cid:17) ~V (+2 q )1 = (cid:16) − ~λ , q (cid:17) ~V (+2 q )2 = (cid:16) − ~λ , q (cid:17) , ~V ( − q ) I = − ~V (+2 q ) I ~V (+2 q )3 = (cid:16) − ~λ , q (cid:17) ~V (+3 q )0 = (cid:16) ~ , q (cid:17) , ~V ( − q )0 = − ~V (+3 q )0 (7.9)61ith ~λ = − ~λ − ~λ (7.10)From these relations, we learn a set of identities; in particular V (+ q )1 + V (+ q )2 + V ( − q )3 = ~ V (+ q )1 + V ( − q )2 + V (+ q )3 = ~ V ( − q )1 + V (+ q )2 + V (+ q )3 = ~ ~V (+ q )1 + ~V (+ q )2 + ~V (+ q )3 = ~V (+3 q )0 ~V ( − q )1 + ~V ( − q )2 + ~V ( − q )3 = ~V ( − q )0 ~V (+ q ) I + ~V (+2 q ) I = ~V (+3 q )0 ~V ( − q ) I + ~V ( − q ) I = ~V ( − q )0 (7.12)These identities are important for establishing gauge symmetry of monomials like( i ) : T r h ψ ( − q ) ( R n ) ψ I (+3 q ) I ( R n ) i ( ii ) : ε IJK
T r h ψ I (+ q ) ( R n − V ( − q ) I ) ψ JK ( − q ) ( R n ) i (7.13) • checking gauge invariance of the term ( i )Under a gauge transformation G ( R n ), the lattice variables ψ ( − q ) n and the diver-gence ψ I (+3 q ) n,I transform as ψ ( − q ) n → G ( R n ) ψ ( − q ) ( R n ) G † ( R n + V ( − q )0 ) ψ I (+3 q ) n,I → G ( R n + V ( − q ) I + V ( − q ) I ) ψ I (+3 q ) I ( R n ) G † ( R n ) (7.14)Using the identity V ( − q ) I + V ( − q ) I = V ( − q )0 (7.15)the second term of (7.14) reads also like ψ I (+3 q ) n,I → G ( R n + V ( − q )0 ) ψ I (+3 q ) I ( R n ) G † ( R n ) (7.16)62herefore the gauge transformation of the term ψ ( − q ) ( R n ) ψ I (+3 q ) I ( R n ) is given by T r h G ( R n ) ψ ( − q ) ( R n ) G † ( R n + V ( − q )0 ) G ( R n + V ( − q )0 ) ψ I (+3 q ) I ( R n ) G † ( R n ) i and leads then to T r h G ( R n ) ψ ( − q ) ( R n ) ψ I (+3 q ) I ( R n ) G † ( R n ) i (7.17)which, due to the cyclic property of the trace, reduce further to T r h ψ ( − q ) ( R n ) ψ I (+3 q ) I ( R n ) i (7.18) • checking gauge invariance of the term ( ii )Starting from the gauge transformation of the lattice variables ψ I (+ q ) n and ψ IJ ( − q ) n namely ψ I (+ q ) n → G ( R n + V ( − q ) I ) ψ I (+ q ) ( R n ) G † ( R n ) ψ JK ( − q ) n → G ( R n + V ( − q ) K ) ψ JK ( − q ) ( R n ) G † ( R n + V ( − q ) J ) (7.19)the putting back into ε IJK
T r h ψ I (+ q ) ( R n + V ( − q ) K ) ψ JK ( − q ) ( R n ) i (7.20)we get G ( R n + V ( − q ) K + V ( − q ) I ) ψ I (+ q ) ( R n + V ( − q ) K ) G † ( R n + V ( − q ) K ) × G ( R n + V ( − q ) K ) ψ JK ( − q ) ( R n ) G † ( R n + V ( − q ) J ) (7.21)Invariance of the trace of this quantity requires G †† ( R n + V ( − q ) J ) G ( R n + V ( − q ) K + V ( − q ) I ) = I id (7.22)63equiring in turn the constraint relation V ( − q ) K + V ( − q ) I = V ( − q ) J , I = J = K (7.23)which is identically satisfied due to eqs(7.9). L su × u D Starting from the twisted Lagrangian density L twist (4.17) namely L twist = T r h F (+4 q ) ab F ab ( − q ) i − ( α ) α α T r (cid:2) F (0) F (0) (cid:3) + (cid:16) α α + 2 (cid:17) T r h ψ a (+ q ) ∇ (+2 q ) a ψ ( − q ) i +2 α α T r h ε abc ψ a (+ q ) ∇ b ( − q ) ψ c (+ q ) i (7.24)and using the dictionary of subsection 6.1 between fields in continuum and lattice vari-ables, we can write down the gauge invariant action on L su × u D following from the aboveone. We find: S latt = X L su × u D T r (cid:16) W IJ ( − q ) n W (+4 q ) n,IJ (cid:17) − ( α ) α α X L su × u D T r (cid:16) W (0) n W † (0) n (cid:17)(cid:16) α α (cid:17) X L su × u D T r h ψ ( − q ) ( R n ) ψ I (+ q ) ( R n + V ( − q ) I ) U (+2 q ) I ( R n ) i + (cid:16) α α (cid:17) X L su × u D T r h ψ ( − q ) ( R n ) U (+2 q ) I ( R n − V ( − q ) I ) ψ I (+ q ) ( R n ) i +2 α α X L su × u D ε IJK
T r h ψ I (+ q ) ( R n + V ( − q ) K ) ψ JK ( − q ) ( R n ) i (7.25)with W IJ ( − q ) n , W (+4 q ) n,IJ , W (0) n and ψ JK ( − q ) ( R n ) as in eqs(7.7) and where α i ’s are normal-ization numbers. 64 Conclusion and comments
In this paper, we studied twisted N = 4 supersymmetric YM on a particular 3-dimensional lattice L su × u D having an SU (3) × U (1) symmetry and realized by thefibration L su D × L u D with the 2- dimensional base L su D = A ∗ , the weight lattice of SU(3), and fiber L u D ≃ q Z .This fibration is encoded by the intersection matrix J su × u ij = + q + 2 q q + 2 q + 4 q q q q q , det J su × u ij = 3 q with q a unit charge of U (1). The SU (3) × U (1) complex symmetry appearing here is oneof the breaking modes of the SO E (6) symmetry of the chiral 6D N = 1 supersymmetricYM on R ; the usual breaking mode used in the twisting is given by the real SO E (3) × SO R (3)symmetry with SO E (3) the isotropy group of R and SO R (3) the R-symmetry. Thegroup SU (3) × U (1) may be therefore viewed as a complexification of the diagonalsymmetry of SO E (3) × SO R (3).To that purpose, we first reviewed general aspects of SO ( t, s ) spinors in diverse dimen-sions; then we built the twisted N = 4 supersymmetric algebra (3.3) generated, inaddition to the bosonic, by 4 complex fermionic generators Q (+3 q ) , Q ( − q ) a transforming respectively as a complex SU (3) singlet and a complex SU (3) triplet car-rying moreover non trivial charges under U (1), the number q is a non zero unit chargeof U (1); but its singular limit q = 0has an interpretation on lattice; it corresponds to the projection of L su × u D down to thebase sublattice L su D = A ∗ .Then extending ideas from covariant gauge formalism of supersymmetric YM theories65nd using the gauge covariant superfields (3.20), we studied the superspace formulationof the twisted gauge theory exhibiting manifestly invariance under Q (+3 q ) . This super-charge may be also interpreted as a particular BRST operator and the correspondingsupersymmetric transformation as BRST transformations. The derivation of the set ofgauge covariant superfields (3.20) is a key step in our construction since only 1 of the 4complex (8 real) supersymmetric charges are off shell; this set is explicitly derived in theappendix, eqs(9.7-9.8).After that, we studied the lattice version of twisted 3 D N = 4 supersymmetric YMliving on L D given by the fibration L u (1)1 D → L su × u D ↓L su D = A ∗ (8.1)To achieve the lattice construction, we performed the 3 following steps:( a ) developed a method of engineering the crystal L su × u D with a manifestly SU (3) × U (1) symmetry. This lattice is given by the fibration (8.1); the shape of the basesublattice A ∗ , corresponds to the projection q = 0, and is completely given by theinverse of the Cartan matrix of SU (3). The 3D lattice L su × u D is a twist of theweight lattice A ∗ of SU (4) ≃ SO (6) L su × u D ∼ twist of A ∗ ( b ) worked out the dictionary eqs(7.1-7.8) between objects O cont living in continuum andtheir analogue O lattice on the lattice L su × u D . The objects include the twisted fields,the coordinates and the supersymmetric generators.( c ) built the lattice action S lattice that is invariant under:( i ) the U ( N ) gauge symmetry,( ii ) the complex scalar supersymmetric charge Q (+3 q ) ,( iii ) the SU (3) × U (1) symmetry of L su × u D . We conclude this study by making 2 comments; one concerning the reduction to N = 4 dimensions; and the other regarding the extension of the construction to twistedmaximal supersymmetric YM in N = 4 and N = 4 dimensions.66) Reduction down to 2D
The twisted N = 4 SYM, that uses the following SU (3) packaging of the fields bosons : G a , ¯ G a , fermions : ξ , ψ a lives on the lattice A ∗ ; it follows from the N = 4 analysis by taking the limit q = 0. However, to exhibit the decomposition Q × = QI + Q µ γ µ + Q µν γ [ µν ] γ µ to decompose Q × in a similar way to thesplitting eq(1.1), we have to break the SU (3) × U (1) symmetry down to SU (2) × U (1) × U (1)As a consequence of this breaking, twisted N = 4 algebra leads to a particularclass of twisted N = 4 supersymmetry with generators as follows SU (3) × U (1) → SU (2) × U (1) diag Q (+3 q ) Q (+3 q ) Q ( − q ) a Q ( − p − q ) α , Q (+2 p − q ) P (+2 q ) a P ( − p +2 q ) α , Z (+2 p +2 q ) where U (1) diag is the diagonal subgroup of U (1) × U (1). This superalgebra hastwo complex SU (2) scalar supercharges Q (+3 q ) , Q (+2 p − q ) and an isodoublet Q ( − p − q ) α obeying amongst others the anticommutation relations n Q (+3 q ) , Q ( − p − q ) α o = 2 P ( − p +2 q ) α (cid:8) Q (+3 q ) , Q (+2 p − q ) (cid:9) = 2 Z (+2 p +2 q ) where P ( − p +2 q ) α refers to bosonic translations and where the charge Z (+2 p +2 q ) canbe taken equal to zero ( Z (+2 p +2 q ) = 0) if we want to realize both scalar supersym-metries Q (+3 q ) , Q (+2 p − q ) on lattice. 67he field spectrum describing the on shell degrees of freedom of the twisted 2D N = 4 supersymmetry, that follows from the reduction of the twisted 3D N = 4SYM is, up to some details, given by SO E (6) : SU (2) × U (1) diag A M : G α ( − q + p ) ¯ G (+2 q − p ) α φ ( − q − p ) ¯ φ (+2 p +2 q ) Ψ A : ψ α (+ q + p ) ψ ( − q ) ψ (+ q − p ) The complex bosonic fields G α ( − q + p ) , φ ( − q − p ) transform respectively in the representation 2 − q + p and 1 − q − p of SU (2) × U (1) diag .Similarly, the complex fermionic fields ψ α (+ q + p ) , ψ ( − q ) , ψ (+ q − p ) transform respectively in 2 + q + p , 1 − q and 1 + q − p .The lattice L su × u D , on which live the twisted lattice 2D N = 4 theory, follows bythe reduction of (8.1) and is given by the fibration L u (1) diag D → L su × u D ↓L su D = A ∗ where the base sublattice A ∗ is the weight lattice of SU (2).2) Extension to 5D N = 4 supersymmetry The analysis we have given in this paper extends to the case of twisted N = 4supersymmetric YM having supercharges. The generators of the underlyingtwisted N = 4 superalgebra carry charges under SU (5) × U (1) as follows fermionic generators : Q (+5 q ) Q ( − q ) a Q [ ab ](+ q ) SU (5) × U (1) : 1 +5 q ¯5 − q + q see footnote 2 N = 1 gauge multiplet (cid:0) A M , Ψ A (cid:1) in down to 5D to get (cid:0) A µ , B m , Ψ αI (cid:1) transforming into representations of SO E (5) × SO R (5)then twisting the two SO (5) factors. In doing these steps, one ends, up on com-plexification, with the complex field spectrum SO E (10) : SU (5) × U (1) A M : G a ( − q ) ¯ G (+2 q ) a Ψ A : ψ ( − q ) ψ a (+ q ) ψ ( − q )[ ab ] Applying similar techniques used in this paper, one concludes that the latticeon which live the twisted N = 4 supersymmetric YM should be given by thefibration L u (1) diag D → L su × u D ↓L su D = A ∗ with base sublattice A ∗ precisely as the one found in [1, 3]. More details and specialfeatures of this lattice will be reported in a future occasion. The aim of this appendix is to derive the set (3.20) of the gauge covariant superfieldsΦ ( q i ) i for describing twisted chiral N = 4 supersymmetric YM theory. A summary ofthis analysis has been given in subsection 3.2.69 .1 General on scalar supersymmetry in superspace First recall that the on shell degrees of freedom of the twisted chiral N = 4 super-symmetric YM are as followsFermions : ψ ( − q ) ψ a (+ q ) SU (3) × U (1) ¯1 − q + q scale mass dim 1 1Bosons : G a ( − q ) ¯ G (+2 q ) a SU (3) × U (1) 3 − q ¯3 +2 q scale mass dim
12 12 (9.1)Using the scalar Grassman variable θ ( − q ) , associated with the scalar supersymmetriccharge Q (+3 q ) , and auxiliary fields, one may a priori combine these degrees of freedominto particular superfields as followsΨ ( − q ) = ψ ( − q ) + θ ( − q ) F (0) V a ( − q ) = G a ( − q ) + θ ( − q ) ψ a (+ q ) Υ ( − q ) a = γ ( − q ) a + θ ( − q ) G (+2 q ) a (9.2)Notice that the component fields ̥ are not ordinary fields since they depend, in additionto the bosonic coordinates z, ¯z, on extra Grassman coordinates ϑ a (+ q ) associated withthe supersymmetric charges Q ( − q ) a ; that is ̥ = ̥ ( z, ϑ ) (9.3)Explicitly, we have ψ ( − q ) ( z, ϑ ) = ψ ( − q ) ( z ) + ϑ a ( − q ) ξ ( − q ) a ( z ) + ... F (0) ( z, ϑ ) = F (0) ( z ) + ϑ a ( − q ) ξ (+ q ) a ( z ) + ... G a ( − q ) ( z, ϑ ) = G a ( − q ) ( z ) + ϑ b ( − q ) ξ a ( − q ) b ( z ) + ... ψ a (+ q ) ( z, ϑ ) = ψ a (+ q ) ( z ) + ϑ b ( − q ) ∆ a (+2 q ) b ( z ) + ... γ ( − q ) a ( z, ϑ ) = γ ( − q ) a ( z ) + ϑ b ( − q ) ξ (0) ba ( z ) + ... G (+2 q ) a ( z, ϑ ) = G (+2 q ) a ( z ) + ϑ b ( − q ) ξ (+3 q ) ba ( z ) + ... (9.4)70he dependence of these component modes into ϑ a ( − q ) is eliminated at the end afterintegration with respect to θ ( − q ) by setting ϑ a ( − q ) = 0.Notice moreover that the superfields (9.2) are not good candidates for superspace for-mulation of scalar supersymmetric invariance. The point is that under gauge symmetrytransformations with generic group elements G , the bosonic gauge superfields U a ( − q ) and V (+2 q ) a do not transform covariantly since G a ( − q ) → G G a ( − q ) G − + G∂ a ( − q ) G − G (+2 q ) a → G G (+2 q ) a G − + G∂ (+2 q ) a G − (9.5)To overcome this difficulty, one needs to work with the gauge covariant superfield oper-ators D (+3 q ) = D (+3 q ) + ig Y M Υ (+3 q ) D ( − q ) a = D ( − q ) a + ig Y M Υ ( − q ) a L (+2 q ) a = ∂ (+2 q ) a + ig Y M V (+2 q ) a L a ( − q ) = ∂ a ( − q ) + ig Y M U a ( − q ) (9.6)and their graded commutators from which we learn the set of gauge covariant superfields(3.20); this set is constructed below. We first give our result regarding the set of gauge covariant superfields; then we turn toderive it explicitly. 71 .2.1 the set of superfields
Twisted chiral N = 4 supersymmetric YM exhibiting manifestly the supercharge Q (+3 q ) is described in superspace by the following superfields Fermionic sector : Ψ ( − q ) Φ (+ q ) ab Ψ a (+ q ) SU (3) × U (1) : ¯1 − q + q + q scale mass dim 1 1 1 Bosonic sector : J (0) E ab ( − q ) F (+4 q ) ab SU (3) × U (1) : 1 ¯3 − q +4 q scale mass dim
32 32 32 (9.7)obeying constraint relations to be deriver later on. Their θ - expansion are given byΨ ( − q ) = ψ ( − q ) + θ ( − q ) F (0) Φ (+ q ) ab = φ (+ q ) ab + θ ( − q ) F (+4 q ) ab Ψ a (+ q ) = ψ a (+ q ) + θ ( − q ) f a (+2 q ) J (0) = J (0) + θ ( − q ) ∇ (+2 q ) a ψ a (+ q ) E ab ( − q ) = F ab ( − q ) + θ ( − q ) h ∇ a ( − q ) ψ b (+ q ) − ∇ b ( − q ) ψ a (+ q ) i F (+4 q ) ab = F (+4 q ) ab + θ ( − q ) κ (+7 q ) ab (9.8)In these relations ψ ( − q ) , ψ a (+ q ) are the twisted fermionic fields of the on shell spectrum(9.1); and J (0) , F ab ( − q ) , F (+4 q ) ab as follows F (+4 q ) ab = ig Y M h ∇ (+2 q ) a , ∇ (+2 q ) b i E ab ( − q ) = ig Y M h ∇ a ( − q ) , ∇ b ( − q ) i J (0) = ig Y M h ∇ (+2 q ) a , ∇ a ( − q ) i (9.9)with gauge covariant derivatives as in eq(3.30) and gauge coupling constant g Y M scalinglike ( mass ) . 72 .2.2 Deriving eqs(9.7)
We begin by the superspace realization of the twisted chiral N = 4 algebra generatedby D (+3 q ) , D ( − q ) a , ∂ (+2 q ) a , ∂ a ( − q ) (9.10)obeying the anticommutation relations n D (+3 q ) , D ( − q ) a o = 2 ∂ (+2 q ) a n D ( − q ) a , D ( − q ) b o = 0 (cid:8) D (+3 q ) , D (+3 q ) (cid:9) = 0 (9.11)To implement gauge symmetry, we covariantize the supersymmetric derivatives (9.10)which become D (+3 q ) = D (+3 q ) + i Υ (+3 q ) D ( − q ) a = D ( − q ) a + i Υ ( − q ) a L (+2 q ) a = ∂ (+2 q ) a + iV (+2 q ) a L a ( − q ) = ∂ a ( − q ) + iU a ( − q ) (9.12)where Υ ( q i ) i , Υ (+3 q ) , V (+2 q ) a , U a ( − q ) are gauge connexions. These superfield operatorstransform covariantly under arbitrary gauge transformation superfield matrices G like D (+3 q ) → G D (+3 q ) G − D ( − q ) a → G D ( − q ) a G − L (+2 q ) a → G L (+2 q ) a G − L a ( − q ) → G L a ( − q ) G − (9.13)with G = G ( z, ¯ z, ϑ a (+ q ) ; θ ( − q ) ) (9.14)which, upon expanding in θ ( − q ) - series, reads also G = g + θ ( − q ) ς (+3 q ) with g = g ( z, ¯ z, ϑ a (+ q ) ) ς (+3 q ) = ς (+3 q ) ( z, ¯ z, ϑ a (+ q ) ) (9.15)73hese gauge covariant derivatives (9.12) are not independent; they obey some constraintrelations required by supersymmetry; in particular the conventional ones n D (+3 q ) , D ( − q ) a o = 2 L (+2 q ) a n D ( − q ) a , D ( − q ) b o = 0 (cid:8) D (+3 q ) , D (+3 q ) (cid:9) = 0 (9.16)and h D (+3 q ) , L (+2 q ) a i = 0 (9.17)Being the basic gauge covariant objects of the twisted SYM theory, eqs(9.12) allow tobuild gauge covariant superfields by taking graded commutators. The gauge covariantsuperfields with small scaling mass dimension are of particular interest; we have: Bosonic superfields
The bosonic gauge covariant superfields that scale as ( mass ) − are given by thecommutators of L (+2 q ) a and L a ( − q ) as follows J (0) = ig Y M h L (+2 q ) a , L a ( − q ) i E ab ( − q ) = ig Y M (cid:2) L a ( − q ) , L b ( − q ) (cid:3) F (+4 q ) ab = ig Y M h L (+2 q ) a , L (+2 q ) b i (9.18)with g Y M the gauge coupling constant scaling as ( mass ) .Because of (9.17), the superfield F (+4 q ) ab obeys the remarkable property D (+3 q ) F (+4 q ) ab = 0 (9.19)but the two others do not D (+3 q ) J (0) = 0 D (+3 q ) E ab ( − q ) = 0 (9.20)From these constraint eqs, we learn that F (+4 q ) ab should be a highest component of asuperfield while F (+4 q ) ab and E ab ( − q ) are good candidates for superspace formulationof twisted chiral supersymmetric YM.74 ) Fermionic superfields
These fermionic gauge covariant superfields we need scale as ( mass ) − ; and aregiven by Ψ ( − q ) = ig Y M h D ( − q ) a , L a ( − q ) i Ψ a (+ q ) = ig Y M (cid:2) D (+3 q ) , L a ( − q ) (cid:3) Φ (+ q ) ab = ig Y M h D ( − q ) a , L (+2 q ) b i (9.21)with Ψ a (+ q ) obeying the property D (+3 q ) Ψ a (+ q ) = 0 (9.22)but D (+3 q ) Ω ( − q ) = 0 D (+3 q ) Φ (+ q ) ab = 0 (9.23)Here also, we learn that Ψ a (+ q ) is a highest component of a superfield while Ψ ( − q ) and Φ (+ q ) ab are good candidates for the superspace formulation of twisted chiralsupersymmetric YM. relations between fermionic and bosonic superfields Using the anticommutation relations of the twisted chiral superalgebra, one findsthat the fermionic and bosonic gauge covariant superfields constructed above arenot completely independent; they are related through constraint relations; in par-ticular D (+3 q ) Ψ ( − q ) = 2 J (0) − D ( − q ) a Ψ a (+ q ) D (+3 q ) Φ (+ q ) ab = 2 F (+4 q ) ab D (+3 q ) E ab ( − q ) = L a ( − q ) Ψ b (+ q ) − L b ( − q ) Ψ a (+ q ) L (+2 q ) b Ψ ( − q ) = − L a ( − q ) Φ (+ q ) ba (9.24)Acting on the first relation by D (+3 q ) and using the identity D (+3 q ) D (+3 q ) = 0, we75et another constraint relation on the J (0) superfield D (+3 q ) J (0) = L (+2 q ) a Ψ a (+ q ) (9.25)Doing the same thing for the second relation, we end with the constraint relation(9.19).In what follows, we choose a particular frame for the gauge fields to build the θ -expansions of the superfieldsΨ ( − q ) Ψ a (+ q ) Φ (+ q ) ab J (0) E ab ( − q ) F (+4 q ) ab (9.26)solving the constraint relations (9.24-9.25).B) Gauge fixing choice
To make explicit computations in superspace, we start from the supersymmetric gaugecovariant derivatives of eqs(9.12); then make the gauge fixing choiceΥ (+3 q ) = 0 (9.27)leading to D (+3 q ) = D (+3 q ) and then D (+3 q ) = ∂∂θ ( − q ) (9.28)This particular choice also allows to expand (9.20-9.23) as in eq(9.8). To establish thisresult, notice first that eq(9.27) corresponds to reducing the set of gauge transformations(9.15) down to the subset of superfield matrices G having no dependence in θ ( − q ) , thatis D (+3 q ) G = 0 (9.29)By substituting back into (9.15), the superfield matrix G reduces to g with g = g ( z, ¯ z, ϑ a (+) ) (9.30)76nd the gauge covariant derivatives behaving generally like D (+3 q ) → g D (+3 q ) g − D ( − q ) a → g D ( − q ) a g − L (+2 q ) a → gL (+2 q ) a g − L a ( − q ) → gL a ( − q ) g − (9.31)become D (+3 q ) = ∂∂θ ( − q ) D ( − q ) a = ∂∂ϑ a (+ q ) + θ ( − q ) ∇ (+2 q ) a L (+2 q ) a = ∇ (+2 q ) a L a ( − q ) = ∇ a ( − q ) + iθ ( − q ) λ a (+ q ) (9.32)with ∇ (+2 q ) a , ∇ a ( − q ) given by (9.6). Substituting these expressions back into eqs(9.24-9.24), one ends with the following θ - expansionsΨ ( − q ) = ψ ( − q ) + θ ( − q ) F (0) J (0) = J (0) + θ ( − q ) ∇ (+2 q ) a ψ a (+ q ) E ab ( − q ) = E ab ( − q ) + θ ( − q ) (cid:16) ∇ a ( − q ) ψ b (+ q ) − ∇ b ( − q ) ψ a (+ q ) (cid:17) Φ (+ q ) ab = φ (+ q ) ab + θ ( − q ) F (+4 q ) ab (9.33)and remarkably Ψ a (+ q ) has no θ ( − q ) dependenceΨ a (+ q ) = ψ a (+ q ) (9.34)with component field modes as in eqs(9.4). We also have the constraint relations D (+3 q ) Ψ ( − q ) + D ( − q ) a Ψ a (+ q ) = 2 J (0) L (+2 q ) b Ψ ( − q ) = − L a ( − q ) Φ (+ q ) ba (9.35)77he first constraint is solved as J (0) = J (0) + θ ( − q ) ∇ (+2 q ) a ψ a (+ q ) (9.36)and the second leads to ∇ (+2 q ) b ψ ( − q ) = ∇ a ( − q ) φ (+ q ) ab (9.37) Acknowledgement 1 : I thank M. Rausch Traubenberg for helpful discussions onthe properties of Majorana spinors in diverse dimensions. This work is supported byURAC/O9/CNR.
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