A A formula of gamma matrix

# Supersymmetry Algebra in Super Yang-Mills Theories

We compute supersymmetry algebra (superalgebra) in supersymmetric Yang-Mills theories (SYM) consisting of a vector multiplet including fermionic contribution in six dimensions. We show that the contribution of fermion is given by boundary terms. From six dimensional results we determine superalgebras of five and four dimensional SYM by dimensional reduction. In five dimensional superalgebra the Kaluza-Klein momentum and the instanton particle charge are not the same but algebraically indistinguishable. We also extend this calculation including a hyper multiplet and for maximally SYM. We derive extended supersymmetry algebras in these four dimensional SYM with the holomorphic coupling constant given in hep-th/9408099.

## 1 Introduction

Supersymmetric Yang-Mills theories (SYM) in higher dimensions than four [1] have been uncovered to possess their own rich structure of supersymmetric (SUSY) quantum field theories (QFT) in spite of their nature of lack of power counting renormalizability.

In five dimensional case the structure of Coulomb branch at long distance can be determined exactly due to the fact that prepotential can be computed exactly by one-loop [2, 3]. What was interestingly found is that if the number of matter multiplets is small enough, there is no singularity of Landau pole and it becomes possible to take strong coupling limit on smooth moduli space, which leads to an ultra-violet (UV) fixed point with global symmetry enhancement depending on the matter content. This phenomenon has been further studied by using brane construction [4, 5, 6, 7, 8], a superconformal index [9, 10, 11, 12, 13] and direct state analysis [14, 15, 16].

Maximally SYM in five dimensions has also attracted a great deal of attention and studied in relation to six dimensional (2,0) superconformal field theory (SCFT) [17, 18], whose Lagrangian description is unknown. Although it was shown that UV divergence of five dimensional SYM appears at six loops [19], which indicates necessity of UV completion, BPS sector of the theory is expected to encode information of that of (2,0) SCFT due to its insensitivity to UV. It was shown that five dimensional maximally SYM contains Kaluza-Klein modes coming from the sixth direction as states with instanton-particle charge [18, 20, 21].

Search of a SUSY gauge theory enjoying a non-trivial UV fixed point has also been done in six dimensions [22]. The requirement is gauge anomaly cancellation as is the case in even dimensional QFT. It has been shown that anomaly of matter multiplets can cancel if the number is small enough for SU(2) gauge group. This was further studied in other simple gauge groups [23]. Examples of nontrivial UV fixed points are provided by compactification of string theory with strong coupling (or tensionless) limit [24, 25, 26, 27]. See [28] for other examples of six dimensional gauge theories.

In comparison to these non-trivial developments of higher dimensional SUSY gauge theories this paper performs a basic calculation for an aim to determine supersymmetry algebra (superalgebra) of six dimensional SYM. Lagrangian description allows us to compute six dimensional superalgebra explicitly and dimensional reduction for the six dimensional result enables us to compare the Kaluza-Klein momentum of the sixth direction and instanton-particle charge, which are identified in earlier study. We also recover a basic result of superalgebra of four dimensional SYM including a hyper multiplet, which leads to the formula of central charge with the holomorphic coupling constant insightfully chosen in [29].

The rest of this paper is organized as follows. In §2 we review the method to determine superalgebra by using ten dimensional SYM following [30]. In §3 we compute superalgebra of SYM in six dimensions including contribution of a hyper mutliplet (§3.2). In particular the algebra in six dimensions is determined by dimensional reduction of ten dimensional one. In §4, §5 we determine superalgebras of five and four dimensional SYM, respectively, by dimensional reduction from six or ten dimensions. §6 is devoted to conclusion and discussion. Appendix contains a formula of gamma matrix (§A) and convention in six dimensions used in this paper (§B).

## 2 Superalgebra in 10d SYM

In this section we review the supersymmetry algebra in ten dimensional supersymmetric Yang-Mills theory [30] using our convention. Results in this section are used to derive similar results of maximally SYM in other dimensions by dimensional reduction later. The fields of SYM in ten dimensions are a gauge field 1 and a Majorana-Weyl fermion (gaugino) , whose chirality we choose as positive.

 ^Γ10λ=λ,λ=C10¯λT, (2.1)

where are SO(1,9) gamma matrices, ,

 ^Γ10=Γ01⋯9,C10=−Γ03579. (2.2)

We realize the ten dimensional gamma matrices by using six dimensional ones as (3.45) in §3.3, which is useful for dimensional reduction carried out later. We employ matrix notation for spinor indices and acts only on them. The SYM Lagrangian (density) in ten dimensions is given by

 L10=1g210Tr[14FMNFMN+12¯λΓMDMλ] (2.3)

where , . The action constructed from this Lagrangian is invariant under supersymmetry transformation rule given by

 ΔAM=¯ϵΓMλ,Δλ=12FMNΓMNϵ (2.4)

where is a supersymmetry parameter of Majorana-Weyl fermion satisfying and . The supersymmetry current is obtained as

 ¯¯¯¯¯¯¯SP=1g210Tr[¯¯¯λ12FMNΓPΓMN],SP=1g210Tr[−12FMNΓMNΓPλ] (2.5)

where the SUSY current satisfies .

To compute the supersymmetry algebra of this theory, we compute variation of the SUSY current under supersymmetry transformation.

 2g210Δ¯¯¯¯¯¯¯SP= Tr[Δ¯λFMNΓPΓMN+2¯λDMΔANΓPΓMN]. (2.6)

The 1st term can be calculated as

 Tr[Δ¯λFMNΓPΓMN]= −12Tr[FQRFMN]¯ϵΓRQPMN−4Tr[FPMFMN]¯ϵΓN−Tr[FMNFMN]¯ϵΓP (2.7)

The 2nd term is calculated as follows.

 Tr[2¯λDMΔANΓPΓMN]= 2Tr[¯ϵΓNDMλ¯λΓPΓMN] = −18(Tr[¯λΓQDMλ]¯ϵΓNΓQΓPΓMN+13!Tr[¯λΓQRSDMλ]¯ϵΓNΓSRQΓPΓMN +12⋅5!Tr[¯λΓQRSTUDMλ]¯ϵΓNΓUTSRQΓPΓMN), (2.8)

in which we used ten dimensional Fierz identity

 χ¯ψ=−124(¯ψΓMχΓM+13!¯ψΓMNPχΓPNM+12⋅5!¯ψΓMNPQRχΓRQPNM)1−(−)ψ^Γ102 (2.9)

where are Weyl fermions of the same chirality and we denote the chirality of by . By using the equation of motion of gaugino and a formula

 ΓM1⋯M5ψ¯χΓM1⋯M5=0 (2.10)

where and are Weyl fermion with the same chirality, the above can be simplified as

 Tr[¯λΓM1DM2λ]¯ϵΓM2M1P+2Tr[¯λΓPDMλ]¯ϵΓM+Tr[¯λΓPM1M2DM3λ]¯ϵΓM3M2M1. (2.11)

Summing up these terms we find2

 2g210Δ¯¯¯¯¯¯¯SP= Extra open brace or missing close brace +12∂M3Tr[¯λΓPM1M2λ]¯ϵΓM3M2M1−12∂NTr[¯λΓPMNλ]¯ϵΓM (2.12)

where is the stress tensor given by

 TMP= 14g210(4Tr[FPNFNM]+ηMPTr[FMNFMN]−2Tr[¯λΓ(MDP)λ]) (2.13)

and we used and .

The supercharge is defined by

 Q=∫d9xS0. (2.14)

Under the standard convention of canonical formalism, it can be shown that

 ΔO=[−i¯ϵQ,O] (2.15)

for a gauge invariant operator and a canonical bracket. Although it is not difficult to show this computationally, it needs a little careful argument to justify this, as we shall do below. The canonical momentum of the gaugino is computed as

 Πλ=∂L∂(∂0λ)=12g210[−¯λΓ0]. (2.16)

Under the canonical commutation relation where is the unit matrix in terms of implicit space, gauge and spinor indices, one can easily show that

 Δλ=[−i¯ϵQ,λ]. (2.17)

On the other hand, the canonical momentum of the gauge field is computed as

 ΠAM=∂L∂(∂0AM)=1g210F0M, (2.18)

which has a vanishing component for time direction as ordinary Yang-Mills theory. This suggests that there is no kinetic term of the time component of the gauge field in the (off-shell) Lagrangian and the system is constrained by saddle point equation thereof, which is given by , where runs the space directions. This requires us to choose a set of dynamical (or canonical) variables to quantize the system. We naturally choose it as the gauge fields of the space directions. Then the canonical commutation relation is By using this it is not difficult to show that

 ΔAM=[−i¯ϵQ,AM]. (2.19)

We stress that the SUSY variation (2.4) is reproduced for the dynamical gauge fields () and not for the auxiliary one ().3 This argument is consistent with the fact that the SUSY variation of supercurrent derived in (2.12) is an on-shell relation. One may ask that there will be another constraint by fixing gauge symmetry which every Yang-Mills theory possesses, in which case one has to use not the canonical bracket but a Dirac one for (2.19) in order to be consistent with the gauge fixing. This should be the case though we still claim that (2.15) holds for a canonical bracket. The argument is as follows. When one fixes gauge symmetry, the initial supersymmetry transformation is not consistent with the fixed gauge in general. One can modify the SUSY transformation so as to be consistent with the gauge fixing by combining gauge transformation. Then the right-hand side of (2.19) replaced by the Dirac bracket will reproduce the modified SUSY transformation for the gauge fields. This suggests that the modified SUSY transformation for a gauge invariant operator should agree with the initial one because the modification is given by a gauge transformation. Thus one has only to use a canonical bracket and do not need to use a Dirac one in (2.15).

As a result, by using (2.12) and (2.15), algebra between the supercurrent and supercharge in SYM in ten dimensions (local form of SUSY algebra) is given by

 {Q,¯¯¯¯¯¯¯SP}= Extra open brace or missing close brace +JPM5M4M3M2M1ΓM1M2M3M4M5 (2.20)

where we define

 JPM= −i4g210∂NTr[¯λΓPMNλ], (2.21) CM1M2= −18g210∂M3Tr[¯λΓM1M2M3λ], (2.22) JPM1M2M3= i4g210∂M3Tr[¯λΓPM1M2λ], (2.23) JPM5M4M3M2M1= −i4g210Tr[FQRFMN]εPQRMNM5M4M3M2M1, (2.24)

with . Note that the contributions of fermions are total derivative terms. Especially we obtain supersymmetry algebra in ten dimensional SYM as

 {Q,¯¯¯¯Q}= −2iPMΓM+ZMΓM+ZM1M2M3ΓM3M2M1+ZM5M4M3M2M1ΓM1M2M3M4M5 (2.25)

where we used on shell, and we set

 Misplaced & (2.26)

## 3 Superalgebra in 6d SYM

In this section we compute supersymmetry algebra of six dimensional SYM with eight and sixteen supercharges. We derive results of maximally SYM in six dimensions by dimensional reduction of ten dimensional one obtained in the previous section.

### 3.1 Vector multiplet

First we consider a vector multiplet. This theory has global symmetry. The bosonic field contents are a gauge field , and the SU(2) triplet auxiliary fields which satisfy , . The super partner is a Sp(1)-Majorana Weyl fermion satisfying

 ^ΓλA=+λA,εABC6(¯¯¯¯¯¯λB)T=λA (3.1)

where , and and are a chirality matrix and a charge conjugation in six dimensions, respectively, defined by

 ^Γ=Γ012345,C6=Γ035. (3.2)

See AppendixB for more details on our convention in six dimensions. In this convention, the supersymmetric Lagrangian reads

 LV= 1g26Tr[14FMNFMN+12¯¯¯¯¯¯λAΓM[DM,λA]+12DABDBA] (3.3)

where , . The supersymmeric transformation rule is

 ΔAM=¯¯¯¯¯ϵAΓMλA,ΔλA=12FMNΓMNϵA+αDABϵB,ΔDAB=α(DM¯¯¯¯¯¯λBΓMϵA−12δABDM¯¯¯¯¯¯λCΓMϵC), (3.4)

where is also a symplectic-Majorana Weyl fermion such that and . Thus the type of SUSY is (1,0). is arbitrary parameter and thus one can set to zero as long as one considers only vector multiplet due to the fact that the auxiliary field can be integrated out to be zero. Once one introduces coupling to a hyper multiplet, which is done in the next subsection, is uniquely determined as . The supersymmetry current of this theory is computed in the same way as in ten dimensions.

 ¯¯¯¯¯¯¯SAP=1g26Tr[12¯¯¯¯¯¯λAFMNΓPΓMN],SAP=1g26Tr[−12FMNΓMNΓPλA], (3.5)

where they are determined so as to satisfy .

Let us compute the SUSY algebra of SYM consisting of a vector multiplet.

The 1st term can be calculated as

 Tr[Δ¯¯¯¯¯¯λAFMNΓPΓMN]= −12Tr[FQRFMN]¯¯¯¯¯ϵAΓRQPMN−4Tr[FPMFMN]¯¯¯¯¯ϵAΓN−Tr[FMNFMN]¯¯¯¯¯ϵAΓP −αTr[(DAB)†FMN]¯¯¯¯¯ϵB(ΓPMN+δMPΓN−δNPΓM). (3.7)

The 2nd term is calculated as follows.

 Tr[2¯¯¯¯¯¯λADMΔANΓPΓMN]= 2Tr[¯¯¯¯¯ϵBΓNDMλB¯¯¯¯¯¯λAΓPΓMN] = −12Tr[¯¯¯¯¯¯λAΓM1DMλB]¯¯¯¯¯ϵBΓNΓM1ΓPΓMN −124Tr[¯¯¯¯¯¯λAΓM1M2M3DMλB]¯¯¯¯¯ϵBΓNΓM3M2M1ΓPΓMN (3.8)

where we used a Fierz identity

 χ¯ψ=−122(¯ψΓMχΓM+13!⋅2¯ψΓMNPχΓPNM)1−(−)ψ^Γ2 (3.9)

for Weyl fermions with the same chirality. By using

 ΓM1M2M3ψ¯χΓM1M2M3=0 (3.10)

where and are Weyl fermions with the same chirality, and

 ¯¯¯¯¯¯λAΓPDMλB=12DM(¯¯¯¯¯¯λAΓPλB)+12δBA¯λCΓPDMλC (3.11)

we find

where we also used the equation of motion of gaugino. Collecting these we obtain

 2g26Δ¯¯¯¯¯¯¯SAP= −12Tr[FQRFMN]¯¯¯¯¯ϵAΓRQPMN−4Tr[FPMFMN]¯¯¯¯¯ϵAΓN−Tr[FMNFMN]¯¯¯¯¯ϵAΓP −αTr[(DAB)†FMN]¯¯¯¯¯ϵB(ΓPMN+2δMPΓN) +2∂MTr[¯¯¯¯¯¯λAΓPλB]¯¯¯¯¯ϵBΓM+2Tr[¯¯¯¯¯¯λCΓPDMλC]¯¯¯¯¯ϵAΓM. (3.13)

Integrating out the auxiliary field gives . Then

 2g26Δ¯¯¯¯¯¯¯SAP= −4g26TPM¯¯¯¯¯ϵAΓM−12Tr[FQRFMN]¯¯¯¯¯ϵAεPQRMNLΓL +2∂MTr[¯¯¯¯¯¯λAΓPλB]¯¯¯¯¯ϵBΓM+2Tr[¯¯¯¯¯¯λCΓ[PDM]λC]¯¯¯¯¯ϵAΓM (3.14)

where , is the stress tensor on shell given by

 TMP= 14g26Tr[gMPFQNFQN+4FPNFNM−2¯¯¯¯¯¯λAΓ(MDP)λA]. (3.15)

A supercharge with Sp(1) index is defined by

 QA=∫d5xS0A. (3.16)

In the same argument given in §2, one can show that for a gauge invariant operator . Then local form of supersymmetry algebra of SYM in six dimensions is determined as

 {QB,¯¯¯¯¯¯¯SAP}= (−2iδBATPM+δBAJPM+δBAJ′PM+JBAPM)ΓM (3.17)

where

 JPM= −i4g26Tr[FQRFLN]εPMQRLN, (3.18) J′PM= −i2g26∂NTr[¯¯¯¯¯¯λCΓPMNλC], (3.19) JBAPM= ig26∂MTr[¯¯¯¯¯¯λAΓPλB]. (3.20)

There are several comments. Firstly as in ten dimensional case the contributions of fermions are given by total derivative terms. Secondly the terms in the right-hand side are all conserved, which is consistent with the fact that the SUSY current in the left-hand side is conserved. Especially for , these are off-shell divergenceless. These anti-symmetric tensors are not distinguishable in the algebra (3.17). There also exists non-R symmetric tensor . Those tensors are so-called brane currents [32], which describes extended BPS objects in the theory.

One might ask whether total derivative terms of fermions appearing in the superalgebra are truly physical or not, since they may be absorbed by an improvement transformation preserving SUSY.4 A general study of this was done in four dimensions by using superfield formalism [32]. As a result an improvement transformation keeping SUSY including operators with spin not more than one was determined.5 And a general supercurrent multiplet called -multiplet was classified into several irreducible supercurrent multiplets by whether there exists an improvement transformation to kill a submultiplet inside the -multiplet. To perform this kind of general analysis of supercurrent in the current case, it is important to develop superfield formalism in six dimensions which can determine an improvement transformation including higher spin operators. We leave these problems to future work.

Volume integration of both sides of (3.17) leads to supersymmetry algebra of six dimensional SYM theory as

 {QB,¯¯¯¯¯¯¯QA}= (δBA(−2iPM+ZM+Z′M)+ZBAM)ΓM (3.21)

where we set

 PM=∫d5xT0M,ZM=∫d5xJ0M,Z′M=∫d5xJ′0M,ZBAM=∫d5xJBA0M. (3.22)

are brane charges corresponding to the brane currents mentioned above.

### 3.2 Inclusion of a hyper multiplet

In this subsection we determine supersymmetry algebra of six dimensional SYM including a hyper multiplet. Extension to a multiple case is straightforward. A hyper multiplet consists of two complex scalar fields , , and a chiral fermion which has the opposite chirality to that of gaugino to interact therewith: We consider a case where the hyper multiplet is in the fundamental representation of the gauge group for notational simplicity. Generalization to other representation can be easily done. The supersymmetric Lagrangian of the hyper multiplet is given by

 LH=−DM(qA)†DMqA+12¯¯¯¯ψ/Dψ+εAB(qA)†¯¯¯¯¯¯λBψ−εAB¯ψλAqB+√2(qA)†DABqB (3.23)

and the supersymmetry transformation is determined as

 ΔqA= εAB¯¯¯¯¯ϵBψ,Δ(qA)†=εAB¯ψϵB, (3.24) Δψ= 2εBAΓMϵBDMqA,Δ¯ψ=−2εBA¯¯¯¯¯ϵBΓMDM(qA)†. (3.25)

The variation of the action of a hyper multiplet under the SUSY transformation is computed as

 Δ(∫d6xLH)=∫d6x¯¯¯¯¯¯¯¯SAMhyp∂MϵA (3.26)

where

 ¯¯¯¯¯¯¯SAPhyp=εAB¯¯¯¯ψΓPΓNDNqB−(DNqA)†ψTC6ΓPΓN−2(qA)†¯¯¯¯¯¯λBΓPqB+(qB)†¯¯¯¯¯¯λAΓPqB. (3.27)

Thus the supercurrent is given by

 ¯¯¯¯¯¯¯SAP= 12g26Tr[¯¯¯¯¯¯λAFMNΓPΓMN]+εAB¯¯¯¯ψΓPΓNDNqB−(DNqA)†ψTC6ΓPΓN −2(qA)†¯¯¯¯¯¯λBΓPqB+(qB)†¯¯¯¯¯¯λAΓPqB. (3.28)

Note that can be determined by using .

We can show that the supersymmetry current (3.28) is conserved: on shell. To show this, we need equations of motion of the gauge multiplet

 1g26DNFNM=−1g26¯¯¯¯¯¯λAΓMλA+DMqA(qA)†−qADM(qA)†−12ψT(ΓM)T¯ψT, (3.29) 1g26DM¯¯¯¯¯¯λAΓM=−(εABqB¯ψ+ψTC6(qA)†), (3.30) DAB=−√2g26(qA(qB)†−12δABqC(qC)†), (3.31)

and those of the hyper multiplet

 D2qA+εAB¯¯¯¯¯¯λBψ+√2DABqB=0, (3.32) 12⧸Dψ+εABλBqA=0,−12DM¯ψΓM+εAB(qA)†¯λB=0. (3.33)

We also need to employ another Fierz rearrangement

 χ¯ψ=−14(¯ψχ+12¯ψΓMNχΓNM)1−(−)ψ^Γ2 (3.34)

where are Weyl fermions with different chirality, and a formula

 Tr[¯¯¯¯¯¯λAΓM(¯¯¯¯¯¯λBΓMλB)]=0. (3.35)

Let us determine supersymmetry algebra in six dimensional SYM theory including a hyper multiplet. As seen from the equations of motion above, it is complicated to determine SUSY algebra including fermionic sector, thus we neglect the fermionic part in this paper, which we leave to future work. The variation of supercurrent under the supersymmetry transformation is computed as follows.

 Δ¯¯¯¯¯¯¯SAP= −2TPM¯¯¯¯¯ϵAΓM+14g26Tr[FQRFMN]¯¯¯¯¯ϵAΓPQRMN −4∂M[(qA)†DNqB−12δBA(qC)†DNqC]¯¯¯¯¯ϵBΓPMN (3.36)

where the stress tensor of the bosonic fields is given by

 TMP= 14g26Tr[gMP(FQNFQN+12DABDBA)+4FPNFNM] +2D(M(qA)†DP)qA−gMP∂N((qA)†DNqA). (3.37)

Note that the quartic terms of the complex scalar fields vanish, which is required from consistency with conservation of the supercurrent in the left hand side of the superalgebra. As in the previous sections we can show that . Thus we obtain local form of supersymmetry algebra of six dimensional SYM including a hyper multiplet.

 {QB,¯¯¯¯¯¯¯SAP}= δBA(−2iTPMΓM+JPMΓM)+CBAPQRSΓSRQ (3.38)

where

 CBAPQRS= 2i3∂M[(qA)†DNqB−12δBA(qC)†DNqC]εPMNQRS. (3.39)

Thus supersymmetry algebra in six dimensional SYM including a hyper multiplet is obtained as

 {QB,¯¯¯¯¯¯¯QA}= δBA(−2iPM+ZM)ΓM+YBAMNPΓPNM (3.40)

where we set

 YBAQRS=∫d5xCBA0QRS. (3.41)

### 3.3 6d N=2 superalgebra

In this section we determine supersymmetry algebra of