18 Sep 2015 Supersymmetry Algebra in Super Yang-Mills Theories

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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 = 2 SYM including a hyper multiplet, which leads to the formula of central charge withN 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 = 2 algebra in six dimensions is determined by dimensional§ reduction of ten N 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 conclu- § sion and discussion. Appendix contains a formula of gamma matrix ( A) and convention in six dimensions used in this paper ( B). § §

1 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 1 ﬁelds of SYM in ten dimensions are a gauge ﬁeld AM (M =0, 1, , 9) and a Majorana- Weyl fermion (gaugino) λ, whose chirality we choose as positive.···

T Γˆ10λ = λ, λ = C10λ¯ , (2.1) where ΓM are SO(1,9) gamma matrices, λ¯ = iλ†Γ0,

Γˆ10 =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 T acts only on them. The SYM Lagrangian (density) in ten dimensions is given by

1 1 MN 1 ¯ M 10 = 2 Tr FMNF + λΓ DMλ (2.3) L g10 4 2

where FMN = [DM,DN], DM = ∂M + AM. The action constructed from this Lagrangian is invariant under supersymmetry transformation rule given by 1 ∆A =¯ǫΓ λ, ∆λ = F ΓMNǫ (2.4) M M 2 MN where ǫ is a supersymmetry parameter of Majorana-Weyl fermion satisfying Γˆ10ǫ = ǫ and C ¯ǫT = ǫ. The supersymmetry current is obtained as 10 −

P 1 1 P MN P 1 1 MN P S = 2 Tr λ FMNΓ Γ , S = 2 Tr FMNΓ Γ λ (2.5) g10 2 g10 −2

where the SUSY current satisﬁes SPǫ = ǫSP. To compute the supersymmetry algebra of this theory, we compute variation of the SUSY current under supersymmetry transformation.

2 P ¯ P MN ¯ P MN 2g10∆S =Tr[∆λFMNΓ Γ +2λDM∆ANΓ Γ ]. (2.6)

The 1st term can be calculated as

P MN 1 RQPMN PM N MN P Tr[∆λF¯ MNΓ Γ ]= Tr[FQRFMN]¯ǫΓ 4Tr[F FMN]¯ǫΓ Tr[F FMN]¯ǫΓ − 2 − − (2.7)

1 The gauge ﬁeld in this paper is anti-hermitian.

2 The 2nd term is calculated as follows.

P MN P MN Tr[2λD¯ M∆ANΓ Γ ] =2Tr[¯ǫΓNDMλλ¯Γ Γ ]

1 Q P MN 1 QRS P MN =− Tr[λ¯Γ DMλ]¯ǫΓNΓQΓ Γ + Tr[λ¯Γ DMλ]¯ǫΓNΓSRQΓ Γ 8 3! 1 + Tr[λ¯ΓQRSTUD λ]¯ǫΓ Γ ΓPΓMN , (2.8) 2 5! M N UTSRQ · in which we used ten dimensional Fierz identity

1 1 1 1 ( )ψΓˆ χψ¯ = − ψ¯ΓMχΓ + ψ¯ΓMNPχΓ + ψ¯ΓMNPQRχΓ − − 10 24 M 3! PNM 2 5! RQPNM 2 · (2.9) where ψ, χ are Weyl fermions of the same chirality and we denote the chirality of ψ by ψ M ( ) . By using the equation of motion of gaugino Γ DMλ = 0 and a formula −

M1 M5 Γ ··· ψχ¯ΓM1 M5 = 0 (2.10) ··· where ψ and χ are Weyl fermion with the same chirality, the above can be simpliﬁed as

¯ M2M1 P ¯ P M ¯ P M3M2M1 (2.8) =Tr[λΓM1 DM2 λ]¯ǫΓ + 2Tr[λΓ DMλ]¯ǫΓ + Tr[λΓ M1M2 DM3 λ]¯ǫΓ . (2.11)

Summing up these terms we ﬁnd2

2 2 M 1 RQMN 1 M3 M2M1 2g ∆SP = 4g TPMǫ¯Γ Tr[FQRFMN]¯ǫΓP ∂ Tr[λ¯ΓM M M λ]¯ǫΓP 10 − 10 − 2 − 4 1 2 3

1 M3M2M1 1 N M + ∂M Tr[λ¯ΓPM M λ]¯ǫΓ ∂ Tr[λ¯ΓPMNλ]¯ǫΓ (2.12) 2 3 1 2 − 2 where TPM is the stress tensor given by

1 N MN ¯ TMP = 2 4Tr[FP FNM]+ ηMPTr[FMNF ] 2Tr[λΓ(MDP)λ] (2.13) 4g10 − ¯ 1 ¯ 1 and we used λΓM1M2M3 DMλ = 2 DM(λΓM1M2M3 λ), and X(AYB) := 2 (XAYB + XBYA). The supercharge is deﬁned by

Q = d9xS0. (2.14) Z Under the standard convention of canonical formalism, it can be shown that

∆ = [ iǫQ,¯ ] (2.15) O − O 2 Our result in the fermionic part is diﬀerent from that in [30]. One of the reasons is that the stress tensor given in [30] is not a symmetric one in the fermionic part. However the argument there does not need modiﬁcation since the fermionic part was neglected in other parts of that paper.

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

∂ 1 ¯ 0 Πλ = L = 2 [ λΓ ]. (2.16) ∂(∂0λ) 2g10 −

Under the canonical commutation relation [Πλ,λ]= iδ, where δ is the unit matrix in terms of implicit space, gauge and spinor indices, one can easily show that

∆λ = [ iǫQ,λ¯ ]. (2.17) − On the other hand, the canonical momentum of the gauge ﬁeld is computed as

∂ 1 0M ΠAM = L = 2 F , (2.18) ∂(∂0AM) g10 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 M is given by DM Π = 0, where M 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 M [ΠAM , AN ]= iδN δ. By using this it is not difficult to show that

∆A = [ iǫQ,¯ A ]. (2.19) M − M

We stress that the SUSY variation (2.4) is reproduced for the dynamical gauge fields (AM ) 3 and not for the auxiliary one (A0). 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).

3 This standpoint may be different from one argued in [31], where SUSY algebra of a general four dimensional = 2 SYM of an = 2 vector multiplet was studied. It seems there that the SUSY variation of allN the components ofN the gauge fields was reproduced in Appendix D, which may be incorrect for that of the auxiliary gauge field.

4 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

M M M3M2M1 M2M1 Q, SP = 2iTPMΓ + JPMΓ + JPM M M Γ + iCM M ΓP { } − 1 2 3 1 2 M5M4M3M2M1 + JP ΓM1M2M3M4M5 (2.20) where we deﬁne i N ¯ JPM = −2 ∂ Tr[λΓPMNλ], (2.21) 4g10 1 M3 ¯ CM1M2 = 2 ∂ Tr[λΓM1M2M3 λ], (2.22) − 8g10 i ¯ JPM1M2M3 = 2 ∂M3 Tr[λΓPM1M2 λ], (2.23) 4g10

M5M4M3M2M1 i QRMNM5M4M3M2M1 JP = 2 Tr[FQRFMN]εP , (2.24) − 4g10 with ε01 9 = 1. Note that the contributions of fermions are total derivative terms. Espe- ··· cially we obtain supersymmetry algebra in ten dimensional SYM as

M M M3M2M1 M1M2M3M4M5 Q, Q = 2iPMΓ + ZMΓ + ZM1M2M3 Γ + ZM5M4M3M2M1 Γ { } − (2.25)

M3 ¯ M2M1 where we used ∂ Tr[λΓM1M2M3 λ]¯ǫΓ0 = 0 on shell, and we set

P M = d9xT 0M, ZM = d9xJ 0M, Z Z (2.26) ZM1M2M3 = d9xJ 0M1M2M3 , ZM5M4M3M2M1 = d9xJ 0M5M4M3M2M1 . Z Z 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 SU(2) Sp(1) global symmetry. ≃ The bosonic field contents are a gauge field AM , M = 0, 1, , 5 and the SU(2) triplet A A B A ··· A auxiliary fields D B which satisfy (D B)† = D A, D A = 0. The super partner λ is a Sp(1)-Majorana Weyl fermion satisfying

A A AB B T A Γˆλ =+λ , ε C6(λ ) = λ (3.1)

12 where ε12 = ε = 1, and Γˆ and C6 are a chirality matrix and a charge conjugation in six dimensions, respectively, deﬁned by

Γ=Γˆ 012345, C6 =Γ035. (3.2)

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

1 1 MN 1 A M A 1 A B V = 2 Tr FMN F + λ Γ [DM ,λ ]+ D BD A (3.3) L g6 4 2 2

where FMN = [DM ,DN ], DM = ∂M + AM . The supersymmeric transformation rule is

A A ∆AM =ǫ ΓM λ , 1 ∆λA = F ΓMN ǫA + αDA ǫB, 2 MN B (3.4) 1 ∆DA =α(D λBΓM ǫA δAD λC ΓM ǫC ), B M − 2 B M

A A A AB B T where ǫ is also a symplectic-Majorana Weyl fermion such that Γˆǫ = ǫ and ε C6(ǫ ) = ǫA. Thus the type of SUSY is (1,0). α 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, α is uniquely determined as α = √2. The supersymmetry current of this theory is computed in the same way as in ten dimensions.

A 1 1 A MN A 1 1 MN A SP = 2 Tr λ FMN ΓP Γ , SP = 2 Tr FMN Γ ΓP λ , (3.5) g6 2 g6 −2

A A A A where they are determined so as to satisfy ǫ SP = SP ǫ . Let us compute the SUSY algebra of = 1 SYM consisting of a vector multiplet. N 2 A A MN A MN 2g6∆ǫSP =Tr[∆λ FMN ΓP Γ +2λ DM ∆AN ΓP Γ ]. (3.6)

The 1st term can be calculated as 1 Tr[∆λAF Γ ΓMN ]= Tr[F F ]ǫAΓRQ MN 4Tr[F M F ]ǫAΓN Tr[F MN F ]ǫAΓ MN P − 2 QR MN P − P MN − MN P A B MN M N N M αTr[(D )†F ]ǫ (Γ + δ Γ δ Γ ). (3.7) − B MN P P − P The 2nd term is calculated as follows.

A MN B B A MN Tr[2λ DM ∆AN ΓP Γ ] =2Tr[ǫ ΓN DM λ λ ΓP Γ ] 1 =− Tr[λAΓM1 D λB]ǫBΓ Γ Γ ΓMN 2 M N M1 P 1 Tr[λAΓM1M2M3 D λB]ǫBΓ Γ Γ ΓMN (3.8) − 24 M N M3M2M1 P where we used a Fierz identity

1 1 1 ( )ψΓˆ χψ¯ = − ψ¯ΓM χΓ + ψ¯ΓMNP χΓ − − (3.9) 22 M 3! 2 PNM 2 ·

6 for Weyl fermions ψ, χ with the same chirality. By using

M1M2M3 Γ ψχ¯ΓM1M2M3 = 0 (3.10) where ψ and χ are Weyl fermions with the same chirality, and 1 1 λAΓ D λB = D (λAΓ λ )+ δBλC Γ D λC (3.11) P M 2 M P B 2 A P M we ﬁnd

A MN A B B M C C A M Tr[2λ DM ∆AN ΓP Γ ] =2∂M Tr[λ ΓP λ ]ǫ Γ + 2Tr[λ ΓP DM λ ]ǫ Γ (3.12)

where we also used the equation of motion of gaugino. Collecting these we obtain 1 2g2∆SA = Tr[F F ]ǫAΓRQ MN 4Tr[F M F ]ǫAΓN Tr[F MN F ]ǫAΓ 6 P − 2 QR MN P − P MN − MN P A B MN M N αTr[(D )†F ]ǫ (Γ +2δ Γ ) − B MN P P A B B M C C A M +2∂M Tr[λ ΓP λ ]ǫ Γ + 2Tr[λ ΓP DM λ ]ǫ Γ . (3.13)

A Integrating out the auxiliary ﬁeld gives D B = 0. Then 1 2g2∆SA = 4g2T ǫAΓM Tr[F F ]ǫAε QRMNLΓ 6 P − 6 PM − 2 QR MN P L A B B M C C A M +2∂M Tr[λ ΓP λ ]ǫ Γ + 2Tr[λ Γ[P DM]λ ]ǫ Γ (3.14)

where ε012345 = 1, TPM is the stress tensor on shell given by 1 T = Tr g F F QN +4F N F 2λAΓ D λA . (3.15) MP 4g2 MP QN P NM − (M P ) 6 A supercharge with Sp(1) index is deﬁned by

QA = d5xS0A. (3.16) Z

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

B A B B B B M Q , S =( 2iδ T + δ J + δ J ′ + J )Γ (3.17) { P } − A PM A PM A PM A PM where

i QRLN JPM = 2 Tr[FQRFLN ]εPM , (3.18) − 4g6

i N C C JPM′ = 2 ∂ Tr[λ ΓPMN λ ], (3.19) − 2g6

B i A B JA PM = 2 ∂M Tr[λ ΓP λ ]. (3.20) g6

7 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 con-

served. Especially for JPM ,JPM′ , these are off-shell divergenceless. These anti-symmetric tensors are not distinguishable in the algebra (3.17). There also exists non-R symmetric B tensor JA PM . 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 S 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 isS 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

B A B B M Q , Q =(δ ( 2iP + Z + Z′ )+ Z )Γ (3.21) { } A − M M M A M where we set

M 5 0M 5 0 5 0 B 5 B 0 P = d xT , ZM = d xJ M , Z′ = d xJ ′ M , Z M = d xJ M . (3.22) Z Z M Z A Z A

B ZM ,ZM′ ,ZA M 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 qA, A = 1, 2, and a chiral fermion ψ which has the opposite chirality to that of gaugino to interact therewith: Γˆψ = ψ. 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

A M A 1 AB A B A B A A B = D (q )†D q + ψ/Dψ + ε (q )†λ ψ ε ψλ¯ q + √2(q )†D q (3.23) LH − M 2 − AB B 4The author would like to thank the referee for raising this question. 5 Existence of an improvement transformation to kill a total derivative term is not sufficient to decide the term as unphysical. To decide so, it also requires fields constructing the term to fall off fast enough at spatial infinity.

8 and the supersymmetry transformation is determined as

A AB B A B ∆q =ε ǫ ψ, ∆(q )† = εABψǫ¯ , (3.24) M B A BA B M A ∆ψ =2ε Γ ǫ D q , ∆ψ¯ = 2ε ǫ Γ D (q )†. (3.25) BA M − M The variation of the action of a hyper multiplet under the SUSY transformation is computed as

6 6 A A ∆ d x H = d xS ∂M ǫ (3.26) Z L Z M hyp where

A N B A T N A B B B A B S = ε ψΓ Γ D q (D q )†ψ C Γ Γ 2(q )†λ Γ q +(q )†λ Γ q . (3.27) P hyp AB P N − N 6 P − P P Thus the supercurrent is given by

A 1 A MN N B A T N SP = 2 Tr[λ FMN ΓP Γ ]+ εABψΓP Γ DN q (DN q )†ψ C6ΓP Γ 2g6 − A B B B A B 2(q )†λ Γ q +(q )†λ Γ q . (3.28) − P P A A A A A Note that SP can be determined by using ǫ SP = SP ǫ . M A We can show that the supersymmetry current (3.28) is conserved: ∂ SM = 0 on shell. To show this, we need equations of motion of the gauge multiplet 1 1 1 NM A M A M A A A M A T M T ¯T 2 DN F = 2 λ Γ λ + D q (q )† q D (q )† ψ (Γ ) ψ , (3.29) g6 −g6 − − 2 1 A M B ¯ T A 2 DM λ Γ = (εABq ψ + ψ C6(q )†), (3.30) g6 − A 2 A B 1 A C C D = √2g (q (q )† δ q (q )†), (3.31) B − 6 − 2 B and those of the hyper multiplet

2 A AB B A B D q + ε λ ψ + √2D Bq =0, (3.32)

1 B A 1 M AB A B Dψ + ε λ q =0, D ψ¯Γ + ε (q )†λ =0. (3.33) 2 6 AB −2 M We also need to employ another Fierz rearrangement

1 1 1 ( )ψΓˆ χψ¯ = − (ψχ¯ + ψ¯ΓMN χΓ ) − − (3.34) 4 2 NM 2 where ψ, χ are Weyl fermions with diﬀerent chirality, and a formula

A M B B Tr[λ Γ (λ ΓM λ )]=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

9 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.

A A M 1 A QRMN ∆SP = 2TPM ǫ Γ + 2 Tr[FQRFMN ]ǫ ΓP − 4g6

A B 1 B C C B MN 4∂ [(q )†D q δ (q )†D q ]ǫ Γ (3.36) − M N − 2 A N P where the stress tensor of the bosonic ﬁelds is given by

1 QN 1 A B N TMP = 2 Tr gMP (FQN F + D BD A)+4FP FNM 4g6 2 A A A N A +2D (q )†D q g ∂ ((q )†D q ). (3.37) (M P ) − MP N 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 ∆ = [ iǫAQA, ]. Thus we obtain local form of supersymmetry algebra of six dimensionalO SYM− includingO a hyper multiplet.

QB, SA =δB( 2iT ΓM + J ΓM )+ CB ΓSRQ (3.38) { P } A − PM PM A PQRS where

BPQRS 2i A B 1 B C C PMNQRS C = ∂ [(q )†D q δ (q )†D q ]ε . (3.39) A 3 M N − 2 A N Thus supersymmetry algebra in six dimensional SYM including a hyper multiplet is obtained as

QB, QA =δB( 2iP + Z )ΓM + Y B ΓPNM (3.40) { } A − M M A MNP where we set

Y BQRS = d5xCB0QRS. (3.41) A Z A

3.3 6d =2 superalgebra N In this section we determine supersymmetry algebra of = 2 SYM by performing dimensional reduction from that of ten dimensional SYM, whichN was computed in 2. = 2 § N SYM in six dimensions is constructed by a pair of vector multiplet and hyper multiplet in the adjoint representation. Thus six dimensional = 2 SYM Lagrangian is given N by addition of the Lagrangians of those multiplets, which were derived in the previous subsections. 1 1 1 1 =2 MN A A A B √ A A B N = 2 Tr FMN F + λ Dλ + D BD A + 2(q )†[D B, q ] L g6 4 2 6 2

A M A 1 AB A B B A D (q )†D q + ψ Dψ + ε (q )†[λ , ψ]+ ε ψ¯[λ , q ] . (3.42) − M 2 6 AB

10 Note that the coupling constants of the vector multiplet and the hyper multiplet are the same. Integrating out the auxiliary ﬁeld results in

=2 1 1 MN 1 A A 1 A B N = 2 Tr FMN F + λ Dλ D′ BD′ A L g6 4 2 6 − 2

A M A 1 AB A B B A D (q )†D q + ψ Dψ + ε (q )†[λ , ψ]+ ε ψ¯[λ , q ] (3.43) − M 2 6 AB where A A B 1 A C C D′ = √2 [q , (q )†] δ [q , (q )†] . (3.44) B − − 2 B In order to determine = 2 supersymmetry transformation and show that the La- grangian (3.43) has sixteenN maximal supersymmetry, we perform dimensional reduction for the SYM Lagrangian in ten dimensions. We compactify four directions xm+5, where m = 1, 2, 3, 4. Then Xm = iAm+5 become four real scalar ﬁelds in six dimensions. We − (10) decompose the SO(1,9) gamma matrices denoted by ΓM as

Γ(10) =Γ 1, Γ(10) = Γˆ γ (3.45) M M ⊗ m+5 ⊗ m where ΓM (M =0, , 5) are SO(1,5) gamma matrices and γm are SO(4) gamma matrices, which we realize by··· a chiral expression

0σ ¯m γm = (3.46) σm 0 whereσ ¯i = σi (i =1, 2, 3) are Pauli matrices andσ ¯4 = σ4 = i. Then the chirality matrix and charge conjugation matrix in ten dimensions are computed− as

1 0 iσ2 0 Γˆ10 =Γˆ , C10 = C6 . (3.47) ⊗ 0 1 ⊗ 0 iσ2 − − Therefore the Majorana-Weyl condition (2.1) in ten dimensions reduces to

A T λ+ ˆ A A A AB B λ = A , Γλ = λ , λ = ε C6λ , (3.48) λ ± ± ± ± ± ± − where A =1, 2. This means that a ten-dimensional Majorana-Weyl fermion reduces to two sympletic-Majorana Weyl fermions λA in six dimensions. Then ten dimensional = 1 SYM Lagrangian reduces to ± N

=2 1 1 MN 1 M m 1 m n N = 2 Tr FMN F DM XmD X + [Xm,Xn][X ,X ] L g6 4 − 2 4

1 A M A 1 A M A 1 A A m B A A m B + λ+Γ DM λ+ + λ Γ DM λ + (λ+(¯σm) B[iX , λ ]+ λ (σm) B[iX ,λ+]) . 2 2 − − 2 − − − (3.49)

11 Note that this Lagrangian has manifest SO(4) SU(2) SU(2) symmetry. This SO(4) symmetric Lagrangian (3.49) agrees with (3.43)≃ under the× following identiﬁcation.

A A 1 1 λ+ = λ , λ = ψ, − √2 1 1 1 1 2 2 2 2 (3.50) (q )† q (q )† + q q (q )† q +(q )† X1 = − , X2 = , X3 = − , X4 = . √2i √2 √2i √2 − The = 2 supersymmetry transformation rule boils down to N A A A A ∆AM =ǫ+ΓM λ+ + ǫ ΓM λ , − − A A B A A B ∆Xm =i(ǫ+(¯σm) Bλ ǫ (σm) Bλ+), − − − A 1 MN A M mA B 1 mnA B (3.51) ∆λ+ = FMN Γ ǫ+ iDM XmΓ σ¯ Bǫ [Xm,Xn]σ Bǫ+, 2 − − − 2 A 1 MN A M mA B 1 mnA B ∆λ = FMN Γ ǫ + iDM XmΓ σ Bǫ+ [Xm,Xn]¯σ Bǫ . − 2 − − 2 − where the supersymmetry parameters ǫA satisfy ± T A A A AB B Γˆǫ = ǫ , ǫ = ε C6ǫ . (3.52) ± ± ± ± ∓ ± Therefore the type of SUSY is (1,1). Performing dimensional reduction for ten dimensional SUSY current (2.5), we obtain two supersymmetry currents in six dimensions.

A 1 1 A P MN B n B P M 1 B mn B P S P = 2 Tr λ+FMN Γ Γ iλ (¯σ ) AΓ Γ DM Xn λ+(σ ) A[Xm,Xn]Γ , − g6 2 − − − 2

A 1 1 A P MN B n B P M 1 B mn B P S+P = 2 Tr λ FMN Γ Γ + iλ+(σ ) ADM XnΓ Γ λ (¯σ ) A[Xm,Xn]Γ . g6 2 − − 2 − (3.53)

The = 2 supersymmetry algebra in six dimensions is computed by dimensional reduction N of that of ten dimensional = 1 algebra calculated in 2. To simplify the situation, we ignore the contributions ofN fermions. SUSY charges are §

QA = d5xS0A (3.54) ± Z ±

B B B B and ∆ = i[ǫ+Q + ǫ Q+, ], which is justiﬁed by the ten dimensional result. From (2.20)O we calculate− − the local− formO of SUSY algebra of = 2 SYM in six dimensions. N B A B M B M nm QRS nmB Q , S P = 2iδ ATPM Γ + JPM δ AΓ + C P σ AΓSRQ, (3.55) { − − } − B A mB m B LK l QRST B Q+, S P = 2iTP mσ A + CPKLσm AΓ + C P σl AΓTSRQ, (3.56) { − } − QB, S A = 2iδB T ΓM J δB ΓM Cnm QRSσ¯nmB Γ , (3.57) { + +P } − A PM − PM A − P A SRQ B A mB m B LK l QRST B Q , S P =2iTP mσ¯ A + CPKLσ¯m AΓ C P σ¯l AΓTSRQ, (3.58) { − − } −

12 where we used the equation of motion of the gauge ﬁeld D F M [D Xn,X ]=0, (3.59) M P − P n and we set 1 T = − Tr g (F F NQ +2D X DP Xm [X ,X ][Xm,Xn]) PM 4g2 MP NQ M m − m n 6 + 4(F N F D XmD X ) , (3.60) M NP − M P m i N TP m = 2 ∂N Tr[FP Xm], (3.61) g6 nm QRS i MNQRS n m C P = − 2 εP ∂M Tr[X DN X ], (3.62) 6g6 1 Cm = − ε ∂RTr XmF MN , (3.63) PKL 4g2 PRMNKL 6 QRST 1 NQRST q r m ClP = − 2 εP σqrml∂N Tr[[X ,X ],X ]. (3.64) 72g6 Note that σ = 1. This leads to supersymmetry algebra of six dimensional = 2 SYM. 1234 N B A M B mn B SRQ Q , Q =( 2iPM + ZM )Γ δA + ZQRSσmn AΓ , (3.65) { − −} − B A mB m B NM m B TSRQ Q+, Q = 2iPmσ A + ZMN σm AΓ + ZQRST σm AΓ , (3.66) { −} − QB, QA =( 2iP Z )ΓM δB Zmn σ¯ B ΓSRQ, (3.67) { + +} − M − M A − QRS mn A B A mB m B NM m B TSRQ Q , Q+ =2iPmσ¯ A + ZMN σ¯m AΓ ZQRST σ¯m AΓ , (3.68) { − } − where

P M = d5xT 0M , P m = d5xT 0m, (3.69) Z Z mn 5 mn0 m 5 m0 l 5 l0 Z = d xC MNP , Z = d xC KL, Z = d xC QRST . (3.70) MNP Z KL Z QRST Z 4 Superalgebra in 5d SYM

In this section we study supersymmetry algebra of SYM in ﬁve dimensions. We derive this by dimensional reduction from higher dimensional SUSY algebra studied in the previous sections.

4.1 Vector multiplet We first consider SYM consisting of one vector multiplet. This theory can be obtained by dimensional reduction of six dimensional theory studied in 3.1. We degenerate the fifth direction and the gauge field of this direction becomes a scalar§ field in the adjoint representation, A5 = iϕ, where ϕ takes a real value. We decompose the gamma matrices in six dimensions into five dimensions ones in the following way. Γ = γ σ , Γ = 1 σ . (4.1) µ µ ⊗ 1 5 ⊗ 2 13 Then the six dimensional chirality matrix and the charge conjugation matrix is computed as Γ=ˆ 1 σ , C = C iσ , (4.2) ⊗ 3 6 5 ⊗ 2 A where C5 = iγ03. The Sp(1)-Majorana Weyl fermion λ6d in six dimension constrained by (3.1) reduces− to A A λ A AB B A λ6d = A , λ′ =0, ε C5λ = λ , (4.3) λ′ which is simply the Sp(1)-Majorana condition in five dimensions. From the six dimensional Lagrangian of = 1 SYM of a vector multiplet given by (3.3), we obtain that of five dimensional one asN

1 1 µν 1 µ 1 A B 1 A A 1 A A V = 2 Tr Fµν F DµϕD ϕ + D BD A + λ /Dλ λ [ϕ,λ ] (4.4) L g5 4 − 2 2 2 − 2 µ where g5 is a coupling constant of this theory and D = γ Dµ. The SUSY transformation rule can also be obtained from six dimensional one6 given by (3.4).

A A ∆Aµ =ǫ γµλ , ∆ϕ =ǫAλA, 1 (4.5) ∆λA = F γµνǫA DϕǫA + αDA ǫB, 2 µν − 6 B 1 ∆DA =α D λBγµǫA [ϕ, λB]ǫA δA(D λC γµǫC [ϕ, λC ]ǫC ) , B µ − − 2 B µ − where ǫA is a fermionic supersymmetry parameter satisfying εABC ǫB = ǫA. α is arbitrary − 5 when we consider only a vector multiplet while α = √2 when a hyper multiplet is introduced. The supersymmetry current of the Lagrangian (4.4) is computed by dimensional reduction from (3.5).

A 1 A 1 ρ µν ρ µ Sρ = 2 Tr[λ ( Fµν γ γ Dµϕγ γ )]. (4.6) g5 2 − The supersymmtery algebra of this theory can be computed in the same way. The supersymmetry variation of the SUSY current is computed as follows. ∆SA = 2T ǫAγµ +2iT ǫA ρ − ρµ ρ5 1 A µνσλ A µνσλ A µσνλ + 2 (Tr[Fµν Fσλ]ǫ γρ +4∂σTr[Fµν ϕ]ǫ γρ γλ +2∂µTr[ϕDσϕ]ǫ γρ γλν) 4g5

1 A B B µ 1 ν C C A µ C C A + 2 ∂µTr[λ γρλ ]ǫ γ 2 ∂ (Tr[λ γρµν λ ]ǫ γ + Tr[λ ( γρν)λ ]ǫ ) (4.7) g5 − 2g5 − where γ = i, 01234 − 1 T = Tr g (F F νσ 2D ϕDνϕ)+4(F ν F + D ϕD ϕ) 2λAγ D λA , µρ 4g2 µρ νσ − ν µ νρ µ ρ − (µ ρ) 5 i 1 T = ∂ Tr F νϕ + Tr iλAγ [ϕ,λA] iλAD λA . (4.8) ρ5 g2 ν ρ 4g2 − ρ − ρ 5 5

14 The supercharge is

QA = d4xS0A, (4.9) Z and SUSY transformation is given by ∆ = [ iǫAQA, ], which is already justiﬁed from the higher dimensional result. Thus anti-commutationO − O relation of the supercharge and supercurrent is computed as

B ρA B µ µ λν B µ Q , S =δ ( 2iT γ 2T + j + j′ +(j + j′ )γ + j γ )+ j′ γ (4.10) { } A − ρµ − ρ5 ρ ρ ρµ ρµ ρνλ A ρµ where we set

i µνσλ jρ = 2 Tr[Fµν Fσλ]γρ , (4.11) 4g5 i µνσ jρλ = 2 ∂σTr[Fµν ϕ]γρ λ, (4.12) g5 i µσ jρνλ = 2 ∂µTr[ϕDσϕ]γρ νλ, (4.13) 2g5

B i A B jA′ ρµ = 2 ∂µTr[λ γρλ ], (4.14) g5

i ν C C jρµ′ = 2 ∂ (Tr[λ γρµνλ ], (4.15) − 2g5

i ν C C jρ′ = 2 ∂ Tr[λ γρνλ ]. (4.16) 2g5

Note that jρ is instanton-particle number current. Volume integration of both sides gives SUSY algebra of ﬁve dimensional SYM as

B A B µ λν B µ Q , Q =δ (( 2iP + Z + Z′ )γ 2P + Z + Z′ + Z γ )+ Z′ γ (4.17) { } A − µ µ µ − 5 νλ A µ where

P µ = d4xT 0µ, P 5 = d4xT 05, (4.18) Z Z 4 0 4 0 4 0 Z = d xj , Zλ = d xj λ, Zνλ = d xj νλ, (4.19) Z Z Z B 4 B0 4 0 4 0 Z′ µ = d xj′ µ, Z′µ = d xj′ µ, Z′ = d xj′ . (4.20) A Z A Z Z P 5 is the Kaluza-Klein momentum arising by circle compactiﬁcation of six dimensional SYM and Z is the instanton-particle charge. These are diﬀerent but indistinguishable in the superalgebra.

4.2 Inclusion of a hyper multiplet We can obtain a theory of a hyper multiplet in ﬁve dimensions by dimensional reduction for six dimensional theory studied in 3.2. The chiral fermion in six dimensions denoted § 15 by ψ6d reduces to

ψ′ ψ = , ψ′ =0. (4.21) 6d ψ By using this notation we obtain the Lagrangian of a hyper multiplet from (3.23).

A M A A 2 A 1 1 = D (q )†D q (q )†ϕ q + ψ/Dψ + ψϕψ LH − M − 2 2 AB A B A B A A B + ε (q )†λ ψ ε ψλ q + √2(q )†D q . (4.22) − AB B The SUSY transformation is computed from six dimensional one as

∆qA =εABǫBψ, ∆ψ =2ε (γµǫBD ǫBϕ)qA. (4.23) BA µ − Five dimensional SUSY current is computed as

A 1 A 1 ρ µν ρ µ ν B B Sρ = 2 Tr[λ ( Fµνγ γ Dµϕγ γ )] + εABψ γρ(γ Dν q ϕq ) g5 2 − − − A T ν A T A B B B A B +(D q )†ψ C γ γ (q )†ϕψ C γ 2(q )†λ γ q +(q )†λ γ q . (4.24) ν 5 ρ − 5 ρ − ρ ρ Supersymmetry variation of supercurrent is computed from (3.36) as

∆SA = 2T ǫAγµ +2iT ǫA ρ − ρµ ρ5 1 A µνσλ A µνσλ A µσνλ + 2 (Tr[Fµν Fσλ]ǫ γρ +4∂σTr[Fµν ϕ]ǫ γρ γλ +2∂µTr[ϕDσϕ]ǫ γρ γλν) 4g5

A B 1 B C C B µνσλ 2∂ [(q )†D q δ (q )†D q ]ǫ γ γ − µ ν − 2 A ν ρ λσ A B 1 B C C B µ +4∂ [(q )†ϕq δ (q )†ϕq ]ǫ γ (4.25) µ − 2 A ρ where the stress tensor of the bosonic ﬁelds is given by

1 νσ ν ν Tµρ = 2 Tr gµρ(FνσF 2DνϕD ϕ)+4(Fµ Fνρ + DµϕDρϕ) 4g5 − A A A ν A +2D (q )†D q g ∂ ((q )†D q ), (4.26) (µ ρ) − µρ ν i ν A A A A T = ∂ Tr F ϕ + iD (q )†ϕq i(q )†ϕD q . (4.27) ρ5 g2 ν ρ µ − µ 5 By using ∆ = [ iǫAQA, ], we obtain local supersymmetry algebra of ﬁve dimensional SYM includingO a− hyper multiplet.O

QB, SA =δB( 2iT γµ 2T + j + j γλ + j γλν )+ cB γλσ + cB γλσν (4.28) { ρ } A − ρµ − ρ5 ρ ρλ ρνλ Aρσλ Aρνσλ where

B A B 1 B C C µν c = 2i∂ [(q )†D q δ (q )†D q ]γ , (4.29) Aρσλ − µ ν − 2 A ν ρ σλ B νσλ 2i A B 1 B C C µνσλ c = ∂ [(q )†ϕq δ (q )†ϕq ]γ . (4.30) Aρ 3 µ − 2 A ρ

16 Thus supersymmetry algebra in SYM in ﬁve dimensions is obtained as

QB, QA =δB(( 2iP + Z )γµ 2P + Z + Z γλν)+ Y B γσνµ + Y B γµν (4.31) { } A − µ µ − 5 νλ A µνσ A νµ

where Z,Zµ,Zνλ are given by (4.19), and

P µ = d4xT 0µ, P 5 = d4xT 05,Y Bνµ = d4xcB0νµ,Y Bνσλ = d4xcB0νσλ. Z Z A Z A A Z A (4.32)

Note that one can add a real mass parameter m for the hyper multiplet by giving a vev to the adjoint scalar field ϕ. Modification of SUSY algebra is done by replacing ϕ m+ϕ. → 4.3 5d =2 superalgebra N We study SUSY algebra of maximally supersymmetric Yang-Mills theory in five dimensions in this subsection. This was studied in a different notation in [18]. This theory can be obtained by reducing six dimensional = 2 SYM to five dimensional one. Two Sp(1)- Majorana Weyl fermions in six dimensionsN denoted by λA in 3.3 reduces to ± § A A A λ1 A λ2′ λ+ = A , λ = A (4.33) λ1′ − λ2 with

T A A AB B A AB B A λ1′ = λ2′ =0, ε C5λ1 = λ1 , ε C5λ2 = λ2 . (4.34)

The six dimensional = 2 Lagrangian given by (3.49) reduces to N

=2 1 1 µν 1 µ 1 µ m 1 m 1 m n N = 2 Tr Fµν F DµϕD ϕ DµXmD X + [ϕ,Xm][ϕ,X ]+ [Xm,Xn][X ,X ] L g5 4 − 2 − 2 2 4 1 1 1 1 + λAγµD λA λA[ϕ,λA]+ λAγµD λA + λA[ϕ,λA] 2 1 µ 1 − 2 1 1 2 2 µ 2 2 2 2 1 + ( iλA(¯σ )A [Xm,λB]+ iλA(σ )A [Xm,λB]) . (4.35) 2 − 1 m B 2 2 m B 1

Five dimensional = 2 transformation rule is obtained from (3.51). N A A A A ∆Aµ =ǫ1 γµλ1 + ǫ2 γµλ2 , ∆ϕ =ǫAλA ǫAλA, 1 1 − 2 2 ∆X =iǫA(¯σ )A λB iǫA(σ )A λB, m 1 m B 2 − 2 m B 1 1 1 ∆λA = F γµν ǫA DϕǫA i DX σ¯mA ǫB i[ϕ,X ]¯σmA ǫB [X ,X ]σmnA ǫB, 1 2 µν 1 − 6 1 − 6 m B 2 − m B 2 − 2 m n B 1 1 1 ∆λA = F γµν ǫA+ DϕǫA + i DX σmA ǫB i[ϕ,X ]σmA ǫB [X ,X ]¯σmnA ǫB, 2 2 µν 2 6 2 6 m B 1 − m B 1 − 2 m n B 2 (4.36)

17 T AB B A AB B A where ǫ1, ǫ2 are fermionic SUSY parameters satisfying ε C5ǫ1 = ǫ1 ,ε C5ǫ2 = ǫ2 One can rewrite the Lagrangian (4.35) in a SO(5) symmetric form.− This can be easily− done by dimensional reduction from ten dimensional SYM. Degenerating the five directions xI , where I = 5, 6, 7, 8, 9, we obtain five real scalar fields from the gauge fields of those directions, denoting by X = iA . We decompose the ten dimensional gamma matrices I − I in a way that

Γ = γ 1 σ , Γ = 1 γ σ (4.37) µ µ ⊗ ⊗ 1 I ⊗ I ⊗ 2

where γµ (µ =0, , 4) are SO(1,4) gamma matrices, γI are SO(5) gamma matrices. Then ten dimensional chirality··· matrix and charge conjugation matrix are computed as

Γˆ =1 1 σ , C = C ω iσ (4.38) 10 ⊗ ⊗ 3 10 5 ⊗ ⊗ − 2 where ω = γ68 is an Sp(2) invariant matrix. The Majorana-Weyl condition in ten dimensions (2.1) reduces to

A T λ A A AB B λ = A , λ′ =0, λ = ω C5λ , (4.39) λ′

where A =1, 2, 3, 4. In other words, a ten dimensional Majorana-Weyl fermion reduces to an Sp(2)-Majorana fermion in ﬁve dimensions. Under this notation the ten dimensional SYM Lagrangian reduces to

=2 1 1 µν 1 µ I 1 I J N = 2 Tr FµνF DµXI D X + [XI ,XJ ][X ,X ] L g5 4 − 2 4

1 A µ A 1 A I A B + λ γ D λ λ (γ ) B[X ,λ ] . (4.40) 2 µ − 2 I

This Lagrangian has manifest SO(5) Sp(2) symmetry. To connect the Sp(2)-invariant ≃ Lagrangian (4.40) with (4.35), one has to decompose the Sp(2)-Majorana fermion into two Sp(1)-Majorana ones, which breaks manifest Sp(2) symmetry. Realizing the SO(5) gamma matrices γI by a chiral expression, we can rewrite the Sp(2)-Majorana condition given by (4.39) as A T A λ+ A AB B λ = A , λ = ε C5λ (4.41) λ ± ± ± − where A, B = 1, 2. The Lagrangian (4.40) agrees with that given by (4.35) under an identiﬁcation such that

A A A A 5 λ+ = λ1 , λ = iλ2 , X = ϕ. (4.42) − Five dimensional supersymmetry current can be obtained by dimensional reduction from ten dimensional supersymmetry current (2.5).

A 1 1 A ρ µν I B I B ρ µ 1 I J B ρ IJ B Sρ = 2 Tr Fµνλ γ γ DµX λ (γ ) Aγ γ [X ,X ]λ γ (γ ) A . (4.43) g5 2 − − 2

18 The = 2 supersymmetry algebra in ﬁve dimensions is also computed by dimensional reductionN of that of ten-dimensional supersymmetry algebra, which keeps manifest SO(5) symmetry. We neglect the contributions of fermions for simplicity. From (2.12) we calculate

A A µ B I B 1 λσµν A 1 λσµν I B B ∆Sρ = 2Tρµǫ γ 2iTρI ǫ (γ ) A + 2 Tr[FλσFµν ]ερ ǫ 2 γρ ∂σTr[X Fµν ]ǫ γI Aγλ − − 4g5 − g5 1 λσµν I J B B 1 I J K B ν IJKLM B + 2 γρ ∂µTr[X DνX ]ǫ γIJ Aγλσ + 2 ∂ν Tr[[X ,X ],X ]ǫ γρ ε γML A 2g5 6g5 (4.44) where ε12345 = 1, 1 T = Tr g (F F νσ 2D X DνXI + [X ,X ][XI ,XJ ]) + 4(F νF + D XI D X ) , µρ 4g2 µρ νσ − ν I I J µ νρ µ ρ I 5 i T = ∂ Tr F νX . (4.45) ρI g2 ν ρ I 5 The supercharge with Sp(2) index is

QA = d4xS0A, (4.46) Z and ∆ = i[ǫBQB, ]. Then the local supersymmetry algebra of = 2 SYM is O − O N B A B µ I B B I B λ Q , S = 2iT δ γ +2T (γ ) A + j δ + j γ Aγ { ρ } − ρµ A ρI ρ A ρλ I IJ B λσ LM λσµ B + jρλσγIJ Aγ + jρµσλγ γML A (4.47) where

I i σµν I jρλ = 2 γρλ ∂σTr[X Fµν ], (4.48) − g5 IJ i µν I J jρλσ = 2 γρλσ ∂µTr[X DνX ], (4.49) 2g5 LM i ν IJKLM jρµσλ = 2 ∂ Tr[[XI ,XJ ],XK]γρνµσλε . (4.50) 36g5 Thus supersymmetry algebra is computed as B A µ B I B B I B λ Q , Q = 2iP γ δ +2P (γ ) A + Zδ + Z γ Aγ { } − µ A I A λ I IJ B λσ LM λσµ B + Zλσ γIJ Aγ + Zµσλ γ γML A (4.51) where we set P µ = d4xT 0µ, P I = d4xT 0I , (4.52) Z Z I 4 I 0 IJ 4 IJ 0 LM 4 LM 0 Z = d xj λ, Z = d xj λσ, Z = d xj µσλ. (4.53) λ Z λσ Z µσλ Z This result agrees with that in [18] up to the conventions, except the quadratic and quartic i terms of the scalar fields. In [18] the quartic term remains as Z0, though it always vanishes in our results, which is required for conservation of supercurrent or supercharge. Another I J one is the relative coefficients of the term X DνX mismatch between those results. Note that these mismatches do not affect the analysis done in [18].

19 5 Superalgebra in 4d SYM

In this section we investigate superalgebras in four dimensional SYM by performing dimensional reduction for higher dimensional SYM studied in the previous sections. The torus compactification gives four dimensional = 2 SYM with the kinetic term canonical. Thus strictly speaking our study is not so generalN as to apply to general = 2 SYM, which admit exact analysis to turn out to have rich structure of SUSY QFTN[33, 29] and are to be obtained by Riemann surface compactification of six dimensional (2,0) SCFT describing an M-five brane [34, 35]. However relying on the Lagrangian description it becomes possible to give an explicit expression for a generic form of = 2 superalgebra including fermionic contributions. Especially on an instanton or monopoleN background there exist fermionic zero modes [36, 37, 38, 39, 40], which can a priori affect the central charge formula in Coulomb branch [41]. Our analysis implies that contribution of such a fermionic zero mode vanishes in the superalgebra. In addition our study provides another method to derive extended supersymmetry algebra in four dimensions, which may be simpler than direct computation in canonical formalism especially for the fermionic part [31].

5.1 =2 vector multiplet N First we consider = 2 SYM consisting of an = 2 vector multiplet, which can be obtained by six dimensionalN SYM studied in 3.1.N We compactify six dimensional theory in x4, x5 directions. Degeneration of the two§ tori leads to two real scalar fields from the gauge fields of these direction. We combine them to be a complex scalar field so that A iA φ = 4 − 5 . (5.1) √2 We decompose six dimensional gamma matrices into four dimensional ones by Γ = γ 1, Γ =γ ˆ σ , Γ =γ ˆ σ (5.2) µ µ ⊗ 4 ⊗ 1 5 ⊗ 2 whereγ ˆ = iγ0123 is a chirality matrix in four dimensions. Then six dimensional chirality matrix and charge conjugation matrix can be computed as Γˆ =ˆγ σ , C = C γˆ iσ (5.3) ⊗ 3 6 4 ⊗ 2 where C = iγ . Thus six dimensional symplectic-Majorana Weyl fermion constrained 4 − 03 by (3.1), which we denote by λ6d, is decomposed as follows. A A λ+ A A A AB B T λ6d = A , γλˆ = λ , λ = ε C4(λ ) . (5.4) λ ± ± ± ± ∓ − By using the last condition above, we can always write the fermionic part of the theory only in terms of λA. From (3.3) we obtain the Lagrangian of = 2 SYM as + N

1 A A 1 µν µ 1 A B 1 2 V = 2 Tr λ+[/D, λ+]+ Fµν F Dµφ†D φ + D BD A [φ,φ†] L g4 4 − 2 − 2 T 1 AB A B B T A + ( ε λ C4[φ, λ ]+ εAB(λ ) C4[φ†,λ ]) (5.5) √2 − + + + +

20 where g4 is a coupling constant of this theory. The = 2 supersymmetry transformation N A rule is computed from (3.4). The supersymmetry parameter denoted by ǫ6d is subject to similar decomposition to gaugino in (5.4) so that

A A ǫ+ A A A AB B T ǫ6d = A , γǫˆ = ǫ , ǫ = ε C4(ǫ ) . (5.6) ǫ ± ± ± ± − ∓ − Eliminating ǫA by using the last equation of (5.6) we obtain = 2 SUSY transformation. − N A A A A ∆Aµ =ǫ+γµλ+ + λ+γµǫ+, B T A ∆φ =√2εAB(ǫ+) C4λ+, T A 1 µν A AB B A A B ∆λ = F γ ǫ + √2[D,φ]ε C ǫ [φ,φ†]ǫ + αD ǫ , + 2 µν + 6 4 + − + B + T A B µ A B µ A C T A AC B C ∆D =α(D λ γ ǫ ǫ γ D λ √2ε [φ†, (λ ) ]C ǫ + √2ε [φ, λ ]C ǫ B µ + + − + µ + − BC + 4 + + 4 + T 1 A C µ C C µ C C T D DC D C δ (D λ γ ǫ ǫ γ D λ √2ε [φ†, (λ ) ]C ǫ + √2ε [φ, λ ]C ǫ )), − 2 B µ + + − + µ + − DC + 4 + + 4 + (5.7)

where α is arbitrary when one considers only a vector multiplet though α = √2 when we also consider coupling of a hyper multiplet, which is introduced in the next subsection. The supersymmetry current in this theory is computed from (3.5). The result is

A 1 A ρ µν B T ρ µ A ρ Sρ =Tr[ Fµνλ+γ γ + √2εAB(λ+) C4Dµφ†γ γ λ+γ [φ,φ†]], 2 − (5.8) A 1 µν ρ A µ ρ AB B T ρ B S =Tr[ F γ γ λ √2D φγ γ ε C (λ ) + [φ,φ†]γ λ ]. ρ 2 µν + − µ 4 + + From the supercurrent we obtain supercharge in four dimensions

QA = d3xS0A, (5.9) Z

B B B B then ∆ = i[ǫ+Q + Q ǫ+, ]. Performing dimensional reduction for (3.36) we obtain local formO of− supersymmetry algebraO of four dimensional SYM.

B A B B µ Q , S =(( 2iT + j + j′ )δ + j )γ , (5.10) { ρ } − ρµ ρµ ρµ A A ρµ T B A Q , S =ε (2iT j j′ )C , (5.11) { ρ } AB ρ − ρ − ρ 4 T B A T A A µ T Q , (S ) =(( 2iT j + j′ )δ j )(γ ) , (5.12) { ρ } − ρµ − ρµ ρµ B − B ρµ B A T BA Q , (S ) =ε (2iT j j′ )†C , (5.13) { ρ } ρ − ρ − ρ 4

21 where

1 νσ ν 2 ν T = Tr g (F F 4D φD φ† 2[φ,φ†] )+4(F F +2D φD φ) µρ 4g2 µρ νσ − ν − µ νρ (µ ρ) 4 2(λAγ D λA D λAγ λA) , (5.14) − + ρ µ + − µ + ρ + √ 2 ν 1 AB A B T Tρ = 2 ∂ν Tr[Fρ φ†]+ 2 ε Tr[λ+C4Dρ(λ+) ] − g4 2g4 √ 2 A A A A + 2 (Tr[λ+γρ[φ†,λ+] [φ†, λ+]γρλ+]), (5.15) 4g4 − 1 λν jρµ = 2 ∂λTr[φDνφ φ†Dνφ†]iσρµ , (5.16) − g4 −

√2 λµν jρ = 2 ∂λTr[φ†Fµν ]iσρ , (5.17) − g4 T i CD µ C D jρ′ = 2 ε ∂ Tr[λ+γρµC4λ+ ], (5.18) − g4 B 2i A B 1 B C C jA ρµ = 2 ∂µTr[λ+γρλ+ δA λ+γρλ+], (5.19) g4 − 2 i µ C σ C jρν′ = 2 σρνµσ∂ Tr[λ+γ λ+], (5.20) − g4

with σ0123 = i. Note that in the bosonic terms there exists a brane current which describes − 6 one dimensional object (string) as jρµ. Performing volume integration for both sides, we obtain supersymmetry algebra in four dimensional = 2 SYM. N B A B B µ Q , Q =(( 2iP + Z + Z′ )δ + Z )γ , (5.21) { } − µ µ µ A A µ T B A Q , Q =εAB(2i Z Z′)C4, (5.22) {T } P − − B A T A A µ T Q , (Q ) =(( 2iP Z + Z′ )δ Z )(γ ) , (5.23) { } − µ − µ µ B − B µ B A T BA Q , (Q ) =ε ( 2i † Z† Z′†)C , (5.24) { } − P − − 4 where we set

µ 3 0µ 3 0 3 0 3 0 P = d xT , = d xT , Z = d xj , Zλ = d xj λ, (5.25) Z P Z Z Z B 3 B0 3 0 3 0 Z′ µ = d xj′ µ, Z′ = d xj′ µ, Z′ = d xj′ . (5.26) A Z A µ Z Z This result shows that contributions of fermion zero modes to the superalgebra vanishes. This is because a fermionic zero mode (on an instanton background) is essentially given

6Although improvement transformations keeping = 2 SUSY are not known, those in = 1 were studied in [32], which suggests that Schwinger termsN in a superalgebra can be reabsorbed ifN it behaves suitably at the boundary. Thus since jρµ, jρ, jρµ′ in our superalgebra are Schwinger terms, these may be reabsorbed into an improvement transformation. However, these are not always removable because, for example, jρ measures a background magnetic charge in the Coulomb phase and aﬀects the physical central charge. We leave a problem to clarify whether other Schwinger terms are physical to future works.

22 µν by a shift generated by SUSY transformation [40], λ Fµνγ ǫ0 with ǫ0 a constant spinor, 2 ∼ which implies that the zero mode scales as r− near the boundary r . ∼∞ In particular we obtain the famous formula of central charge as

T QB , QA =2√2ε C (5.27) { } ABZ 4 where 1 1 1 = (i Z Z′). (5.28) Z √2 P − 2 − 2 For example, let us consider SYM with SU(2) gauge group in the Coulomb branch.

A φ = a∗σ3, λ =0, (5.29) where a is a complex number. In electrically and magnetically charged background such that 1 3 i0 1 3 1 ijk ne = 2 d x∂i(if ), nm = d x∂i(i ε fjk), (5.30) g4 Z 4π Z 2

where fµν = Tr[Fµν σ3], one can show that the central charge is computed as = n a + n a (5.31) Z e m D 4πi 7 where aD = τ0a with the holomorphic coupling τ0 = 2 . This gives the same formula as g4 in [33].

5.2 Inclusion of a hyper multiplet In this subsection we study four dimensional = 2 SYM including a hyper multiplet by N dimensional reduction. We use the same notation for two complex scalar ﬁelds qA. A six dimensional chiral fermion ψ6d reduces to ψ ψ6d = − , γψˆ = ψ , (5.32) ψ+ ± ± ± Then the Lagrangian of a hyper multiplet reduces to

A µ A 1 A A B H = Dµ(q )†D q + (ψ+/Dψ+ + ψ /Dψ )+ √2(q )†D Bq L − 2 − − AB A B A A T B A A T A + ε (q )†λ+ψ +(q )†(λ+) C4ψ+ + εABψ λ+q ψ+C4(λ+) q − − − 1 A A + (ψ φψ+ + ψ+φ†ψ ) (q )† φ,φ† q . (5.33) √2 − − − { } The SUSY transformation boils down to

A AB B A T ∆q =ε ǫ+ψ (ǫ+) C4ψ+, (5.34) − − µ B A A A ∆ψ =2(γ εBAǫ+Dµq + √2C4ǫ+φq ), (5.35) − T µ A A B A ∆ψ =2( γ C ǫ D q √2ε ǫ φ†q ). (5.36) + − 4 + µ − BA + 7 The real part of the holomorphic coupling appears once the topological term F F is introduced. ∧ 23 The supercurrent is computed from six dimensional one (3.28). Since the part of a vector multiplet was already computed as (5.8), we have only to compute the part of a hyper multiplet. The result is

A µ B B A T µ A T Sρ hyp =εABψ γργ Dµq + εABψ+γρ( √2φ†)q +(Dµq )†ψ+C4γργ +(q )†√2φ†ψ C4γρ − − − A B B B A B 2(q )†λ γ q +(q )†λ γ q , (5.37) − + ρ + ρ A µ T A T A A µ A Sρ hyp = γ γρC4ψ+ Dµq + γρC4ψ √2φq +(Dµq )†γ γρψ +(q )†√2φγρψ+ − − − B A B B A B 2(q )†γ λ q +(q )†γ λ q . (5.38) − ρ + ρ + Supercharge in four dimensions is given by (5.9). From (3.36) we obtain local supersymmetry algebra of four dimensional SYM including the contribution of a hyper multiplet.

QB, SA =(( 2iT + j )δB +(jB + cB ))γµ, (5.39) { ρ } − ρµ ρµ A A ρµ Aρµ T QB , SA =ε (2iT j )C ε cC C γλσ, (5.40) { ρ } AB ρ − ρ 4 − CB Aρσλ 4 T QB , (SA)T =(( 2iT j )δA (jA + cB ))(γµ)T , (5.41) { ρ } − ρµ − ρµ B − B ρµ Aρµ B A T BA CA C λσ Q , (S ) =ε (2iT j )†C ε (c )†γ C , (5.42) { ρ } ρ − ρ 4 − Bρσλ 4 where

1 νσ ν 2 ν Tµρ = 2 Tr gµρ(FνσF 4DνφD φ† 2[φ,φ†] )+4(Fµ Fνρ +2D(µφDρ)φ) 4g4 − − A A A ν A +2D (q )†D q g ∂ ((q )†D q ), (5.43) (µ ρ) − µρ ν √2 ν √ A A √ A A Tρ =− 2 ∂νTr[Fρ φ†] 2Dρ(q )†φ†q + 2(q )†φ†Dρq , (5.44) g4 − Bρλ A B 1 B C C ρµνλ c =4i∂ [(q )†D q δ (q )†D q ]σ , (5.45) A µ ν − 2 A ν Bρσλ A B 1 B C C ρµσλ c = 2i√2∂ [(q )†φ†q δ (q )†φ†q ]σ . (5.46) A − µ − 2 A This leads to supersymmetry algebra in four dimensional = 2 SYM. N QB, QA =(( 2iP + Z )δB +(ZB + yB ))γµ, (5.47) { } − µ µ A A µ A µ T QB , QA =ε (2i Z)C ε yCµν C γ , (5.48) { } AB P − 4 − CB A 4 νµ T QB , (QA)T =(( 2iP Z )δA (ZA + yB ))(γµ)T , (5.49) { } − µ − µ B − B µ A µ B A T BA CA Cµν Q , (Q ) =ε ( 2i † Z†)C ε (y )†γ C , (5.50) { } − P − 4 − B νµ 4 where we set

B 3 B0 B 3 B0 y µ = d xc µ, y µν = d xc µν. (5.51) A Z A A Z A

Bρλ Bρσλ Due to inclusion of a hyper multiplet there appear brane currents cA ,cA in the local Bλ B form of superalgebra and corresponding brane charges yA ,yA µν in the superalgebra.

24 One can include a complex mass m of a hyper multiplet in this superalgebra by shifting the adjoint scalar ﬁeld in the vector multiplet in a way that φ m + φ. Under this shift √2 A A A A → T T imJ , where J = i(q )†D q iD (q )†q is the U(1) ﬂavor current. Thus the ρ → ρ − ρ ρ ρ − ρ formula of central charge (5.28) is changed as 1 1 = (i + mF Z). (5.52) Z √2 P − 2

where F = d3xJ 0 is the U(1) ﬂavor charge.8 Let us computeR this central charge with SU(2) gauge group in the Coulomb branch. The Kaluza-Klein momentum is computed as

= i√2aN (5.53) P − e where we used the equation of motion of the gauge ﬁeld and set

2 3 i0 Ne = 2 d x∂i(if ). (5.54) g4 Z Remark that the hyper multiplet contributes to the Kaluza-Klein momentum so that the electric charge is twice as great as that in pure = 2 SYM case with the form of central charge ﬁxed.9 The Dirac quantization conditionN requires us to redeﬁne the magnetic charge to be half compared to the pure SYM case.

1 3 1 ijk Nm = d x∂i(i ε fjk). (5.55) 8π Z 2 Then Z is computed as

Z = 2√2a N (5.56) − D m 8πi where aD = τa with τ = 2 . Finally the central charge is obtained as g4 1 = Nea + NmaD + F m (5.57) Z √2 which matches the formula given in [29] with the same normalization of the holomorphic coupling τ. This normalization of the holomorphic coupling is important for the = 2 SU(2) SYM with four flavors to enjoy SL(2, Z) symmetry [29] as well as to obtainN the correct moduli space of = 2 SU(3) SYM with six flavors as the enlarged fundamental region in the upper halfN complex plain, in which the genuine strong coupling limit exists as Imτ 0 [42]. → 8 1 i 1 In multiple flavor case, this changes as = (i + miF Z), where mi is the mass of ith hyper Z √2 P − 2 multiplet and Fi is the flavor U(1) charge of the ith hyper multiplet. 9 The factor two in (5.54) does not depend on the number of hyper multiplets.

25 5.3 =4 superalgebra N In this final subsection we determine superalgebra in = 4 SYM by dimensional reduction for ten dimensional SYM. By compactifying six directionsN x3+m, where m =1, 2, , 6, the gauge fields of these directions become scalar fields, which we denote by X =··· iA . m − 3+m Accordingly we decompose the ten-dimensional gamma matrices as Γ = γ 1, Γ =γ ˆ γ (5.58) µ µ ⊗ m ⊗ m where γm are SO(6) gamma matrices, respectively. Since = 4 SYM has SU(4) R- symmetry, it is convenient to rewrite the SO(6) vector representationN by SU(4) anti- symmetric representation. 1 γ = (γ iγ ), γ = ε (γ )†, (5.59) a4 2 a − a+3 ab abc c4 where a, b, c =1, 2, 3. These satisfy

AB 1 ABCD γAB = γBA, γ = ε γCD =(γAB)† (5.60) − 2 m where A, B, C, D =1, 2, 3, 4. We do the same thing for X . γAB is explicitly realized as

0 ρ˜AB γAB = − , (5.61) ρAB 0 where CD C D C D (ρAB) = δ δ δ δ , (˜ρAB)CD = εABCD. (5.62) A B − B A Then the ten-dimensional chirality matrix and charge conjugation matrix are computed as

γˆ 0 0 C4γˆ Γˆ10 = , C10 = − . (5.63) 0 γˆ C4γˆ 0 − − A ten-dimensional Majorana-Weyl fermion (2.1) is decomposed as

λ+A A T A T λ = A , γλˆ = λ , λ = C4(λ+A) , λ = λ+AC4. (5.64) λ ± ± ± − − − − The SYM Lagrangian in ten dimensions (2.3) reduces to

=4 1 1 µν 1 µ AB 1 AB CD N = 2 Tr Fµν F DµXABD X + [XAB,XCD][X ,X ] L g4 4 − 2 4

T T AB + λ A Dλ A + λ C[iXCD,C (λ D) ] λ C [iX ,λ B] . (5.65) + 6 + + 4 + − +A 4 + The supersymmetry transformation rule is

T T T ∆A =ǫ Aγ λ A ǫ γ λ A , µ + µ + − +A µ + AB ABCD T T T ∆X =ε ǫ C λ D + ǫ AC (λ B) ǫ BC (λ A) , +C 4 + + 4 + − + 4 + 1 µν T BC ∆λ A = F γ ǫ A 2i DXABC (ǫ B) 2[XAB,X ]ǫ C, (5.66) + 2 µν + − 6 4 + − +

26 where we used

ǫ+A A T A T ǫ = A , γǫˆ = ǫ , ǫ = C4(ǫ+A) , ǫ = ǫ+AC4. (5.67) ǫ ± ± ± − − − − Let us perform dimensional reduction for SUSY current. The result is

ρ 1 1 ρ µν BA B ρ µ CA ρ S A = 2 Tr Fµν λ+Aγ γ 2iDµX λ γ γ + 2[XBC,X ]λ+Bγ , (5.68) − g4 2 − −

ρA 1 1 A ρ µν ρ µ BC B ρ S+ = 2 Tr Fµν λ γ γ 2iDµXBAλ+Bγ γ + 2[X ,XCA]λ γ . (5.69) g4 2 − − − ρ ρA B ρB ρ ρ ρB B S A,S+ are determined so as to satisfy ǫ S+ = S Bǫ+B, ǫ+BS B = S+ ǫ . Then the supercharge− with SU(4) index is − − − −

3 0 A 3 0A Q A = d xS A, Q+ = d xS+ , (5.70) − Z − Z B B and ∆ = i[ǫ+BQ B + ǫ Q+, ]. The = 4 superalgebra in four dimensions can be − computedO by− dimensional reduction− O fromN ten dimension as done in ﬁve dimensions. We neglect the contribution of fermions, which is given by total derivative terms and thus vanishes as discussed in the pure = 2 SYM. The local version of supersymmetry algebra of = 4 SYM is N N A µ A µ Q B, Sρ A = 2iTρµδB γ + jB ρµγ , (5.71) { − − } − B BA BA BA λσ Q+, Sρ A = 2iTρ + jρ + jρσλ γ , (5.72) { − } − QB, S A = 2iT δBγµ jB γµ, (5.73) { + ρ+ } − ρµ A − Aρµ A λσ Q B, Sρ+ = 2iTρBA jρBA jBAρσλγ , (5.74) { − } − − − where 1 νσ ν AB AB CD Tµρ = 2 Tr gµρ(FνσF 2DνXABD X + [XAB,XCD][X ,X ]) 4g4 − ν AB + 4(Fµ Fνρ + DµX DρXAB) , (5.75) 2i T = ∂ Tr F νX , (5.76) ρAB g2 ν ρ AB 4 A 4i σ ν CA jB ρµ = 2 σρσνµ∂ Tr[XBCD X ], (5.77) − g4 BA 2 σµν BA jρ = 2 σρ ∂σTr[X Fµν ], (5.78) − g4 BA 4 ν BC DA jρσλ = 2 σρνσλ∂ Tr[[X ,XCD],X ]. (5.79) − 3g4 Volume integration of both sides leads to supersymmetry algebra in = 4 SYM. N A µ A µ Q B, Q A = 2iPµδB γ + ZB µγ , (5.80) { − − } − B BA BA BA λσ Q+, Q A = 2iP + Z + Zσλ γ , (5.81) { − } − QB, QA = 2iP δBγµ ZB γµ, (5.82) { + + } − µ A − Aµ A λσ Q B, Q+ = 2iPBA ZBA ZBAσλγ , (5.83) { − } − − −

27 where we set

P µ = d3xT 0µ, P AB = d3xT 0AB, (5.84) Z Z A 3 A0 BA 3 BA0 AB 3 AB0 Z µ = d xj µ, Z = d xj , Z = d xj λσ. (5.85) B Z B Z λσ Z As an example, let us consider the case of SU(2) gauge group in the Coulomb branch. a X12 = σ3, X13 = X14 =0. (5.86) √2 Then (5.83) is computed as

2 4 Q 1, Q+ = Q 3, Q+ =2√2 (5.87) { − } { − } Z where = n a + n a (5.88) Z e m D 4πi with ne, nm given by (5.30), aD = τ0a, τ0 = 2 . The formula of the central charge with g4 normalization of the holomorphic coupling in = 4 SYM is the same as pure = 2 N N SYM, which is again consistent with the result in [29].

6 Discussion

We have determined supersymmetry algebra of SYM of a vector multiplet in six dimensions including the contribution of fermions, which is given by boundary terms. We have extended this calculation to the case including a hyper multiplet. For SUSY algebra of six dimensional maximally SYM we have carried out dimensional reduction for that in ten dimensions. From six dimensional results we have performed dimensional reduction to determine SUSY algebras of five and four dimensional SYM. From six to five the Kaluza- Klein momentum arising from torus compactification is different from the instanton-particle charge though they are indistinguishable in the superalgebra. And the Kaluza-Klein momentum corresponds to the electric charge part in the famous formula of central charge. We have derived the whole extended supersymmetry algebra as well as the holomorphic coupling constant introduced in [29] against the four dimensional = 2 SYM including fundamental hyper multiplets and = 4 SYM. N N Since we started from SYM in six dimensions with the canonical kinetic term in this paper, the theory obtained by dimensional reduction inherited this property. Computing SUSY algebra of general SYM with the non-canonical kinetic term is left to future work, though the general structure of the algebra will remain unchanged. Especially in five and four dimensions a general Lagrangian contains topological terms such as Chern-Simons term and F F , respectively, which has an extra effect on physics of the theory [43]. It should∧ be possible to determine BPS states in maximally SYM in six dimensions in Higgs branch. In terms of brane picture, maximally SYM in six dimensions is realized on D- five branes, and Higgs branch corresponds to separation thereof. Then BPS states on this branch will correspond to supersymmetric brane configuration realized on the separated

28 D-five branes set up. It would be interesting to clarify relations between those BPS states in six dimensional SYM and those in five dimensional maximally SYM in broken phase, which has close relationship with the (2,0) theory describing M-five branes [18]. We hope to come back to these problems in the future.

Acknowledgments

The author was supported by the Israeli Science Foundation under grant 352/13 and 504/13.

A A formula of gamma matrix

In this appendix, we derive a formula of gamma matrix given in Appendix of [44]. We use notation such that

µ0 µn [µ0 µn] 1 σ µσ µσ n Γ ··· =Γ Γ = ( ) Γ (0) Γ ( ) (A.1) ··· (n + 1)! − ··· σ XSn+1 ∈ where SN is the set of permutation of N elements. Denoting Cn,m,Dn,m by

M1 Mm N1 Nn N1 Nn Γ ··· Γ ··· ΓMm M1 =Cn,mΓ ··· , (A.2) ··· m 1 M1 Mm−1[N1 N2 Nn] N1 Nn ( ) − Γ ··· Γ ··· ΓMm−1 M1 =Dn,mΓ ··· , (A.3) − ··· we can relate Cn,m,Dn,m by

m m+n Cn,m =( ) Cn 1,m +2m( ) Dn 1,m 1, (A.4) − − − − − 1 D = (C +( )mC ). (A.5) n,m 2 n,m − n+1,m From these we ﬁnd

m m+n n+1 Cn,m =( ) Cn 1,m + m( ) Cn 1,m 1 + m( ) Cn,m 1. (A.6) − − − − − − − One can easily check that

C =1, C = D(D 1) (D (m 1)), (A.7) n,0 0,m − ··· − −

where D is an arbitrary dimension. By using (A.6) and (A.7) one can determine Cn,m (and thus Dn,m) inductively. As examples, we determine Cn,m when D = 6, 10. The result of D = 6 matches that given in [44].

29 m =3 120 0 24 0 − m =2 30 10 2 6 − − m =1 6 4 2 0 − m =0 1 1 1 1 Cn,m n =0 n =1 n =2 n =3 Table 1: D = 6.

m =5 30240 0 3360 0 1440 0 − m =4 5040 1008 336 336 48 240 − − m =3 720 288 48 48 48 0 − − m =2 90 54 26 6 6 10 − − m =1 10 8 6 4 2 0 − − m =0 1 1 1 1 1 1 Cn,m n =0 n =1 n =2 n =3 n =4 n =5 Table 2: D = 10.

B Convention in six dimensions

In this appendix we collect convention in six dimensions used in this paper. We realize SO(1,5) matrices satisfying Γ , Γ = 2g , where (g ) = diag( 1, 1, , 1), in two { M N } MN MN − ··· ways. One is Γ = γ σ , Γ = 1 σ (B.1) µ µ ⊗ 1 5 ⊗ 2 where γµ (µ =0, 1, 2, 3) are SO(1,3) matrices realized as

0σ ¯µ′ γµ = , (B.2) σµ′ 0 whereσ ¯′ = σ′ = iσ ,σ ¯′ = σ′ = σ ,σ ¯′ = σ′ = σ ,σ ¯′ = σ′ = i with σ Pauli matrices 0 0 2 1 1 1 2 2 3 3 − 3 i satisfying σiσj = δij + iεijkσk. This realization is useful when we consider dimensional reduction from six dimensions to ﬁve ones. The other is

Γ = γ 1, Γ =γ ˆ σ , Γ =γ ˆ σ (B.3) µ µ ⊗ 4 ⊗ 1 5 ⊗ 2 whereγ ˆ = iγ0123 is a chirality matrix in four dimensions. This is convenient when we do dimensional reduction from six to four. In both cases, we deﬁne the charge conjugation matrix as C6 =Γ035, which satisﬁes

2 T M M T C =1, C∗ = C , C = C , C Γ = (Γ ) C . (B.4) 6 6 6 6 6 6 − 6 In Lorentzian six dimensions there exists a symplectic majorana Weyl spinor. By denoting Sp(1)-Majorana fermion by λA it satisﬁes

T A AB B λ = ε C6λ . (B.5)

30 Two symplectic Majorana fermions ψA, χA satisfy

ψAγ γ χB =( )k+1(χAγ γ ψB δBχDγ γ ψD). (B.6) µ1 ··· γk − µk ··· γ1 − A µk ··· γ1 Especially taking trace in terms of Sp(1) index gives

ψAγ γ χA =( )kχAγ γ ψA. (B.7) µ1 ··· γk − µk ··· γ1 References

[1] L. Brink, J. H. Schwarz, and J. Scherk, “Supersymmetric Yang-Mills Theories,” Nucl.Phys. B121 (1977) 77.

[2] E. Witten, “Phase transitions in M theory and F theory,” Nucl.Phys. B471 (1996) 195–216, hep-th/9603150.

[3] N. Seiberg, “Five-dimensional SUSY ﬁeld theories, nontrivial ﬁxed points and string dynamics,” Phys.Lett. B388 (1996) 753–760, hep-th/9608111.

[4] O. Aharony and A. Hanany, “Branes, superpotentials and superconformal ﬁxed points,” Nucl.Phys. B504 (1997) 239–271, hep-th/9704170.

[5] O. Aharony, A. Hanany, and B. Kol, “Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams,” JHEP 9801 (1998) 002, hep-th/9710116.

[6] O. DeWolfe, A. Hanany, A. Iqbal, and E. Katz, “Five-branes, seven-branes and ﬁve-dimensional E(n) ﬁeld theories,” JHEP 9903 (1999) 006, hep-th/9902179.

[7] F. Benini, S. Benvenuti, and Y. Tachikawa, “Webs of ﬁve-branes and N=2 superconformal ﬁeld theories,” JHEP 0909 (2009) 052, 0906.0359.

[8] O. Bergman, D. Rodrguez-Gmez, and G. Zafrir, “5-Brane Webs, Symmetry Enhancement, and Duality in 5d Supersymmetric Gauge Theory,” JHEP 1403 (2014) 112, 1311.4199.

[9] H.-C. Kim, S.-S. Kim, and K. Lee, “5-dim Superconformal Index with Enhanced En Global Symmetry,” JHEP 1210 (2012) 142, 1206.6781.

[10] D. Bashkirov, “A comment on the enhancement of global symmetries in superconformal SU(2) gauge theories in 5D,” 1211.4886.

[11] M. Taki, “Notes on Enhancement of Flavor Symmetry and 5d Superconformal Index,” 1310.7509.

[12] O. Bergman, D. Rodrguez-Gmez, and G. Zafrir, “Discrete θ and the 5d superconformal index,” JHEP 1401 (2014) 079, 1310.2150.

[13] O. Bergman, D. Rodrguez-Gmez, and G. Zafrir, “5d superconformal indices at large N and holography,” JHEP 1308 (2013) 081, 1305.6870.

31 [14] Y. Tachikawa, “Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories,” PTEP 2015 (2015), no. 4, 043B06, 1501.01031.

[15] G. Zafrir, “Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories,” 1503.08136.

[16] K. Yonekura, “Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories,” 1505.04743.

[17] M. R. Douglas, “On D=5 super Yang-Mills theory and (2,0) theory,” JHEP 1102 (2011) 011, 1012.2880.

[18] N. Lambert, C. Papageorgakis, and M. Schmidt-Sommerfeld, “M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills,” JHEP 1101 (2011) 083, 1012.2882.

[19] Z. Bern, J. J. Carrasco, L. J. Dixon, M. R. Douglas, M. von Hippel, et al., “D=5 maximally supersymmetric Yang-Mills theory diverges at six loops,” Phys.Rev. D87 (2013), no. 2, 025018, 1210.7709.

[20] H.-C. Kim, S. Kim, E. Koh, K. Lee, and S. Lee, “On instantons as Kaluza-Klein modes of M5-branes,” JHEP 1112 (2011) 031, 1110.2175.

[21] H.-C. Kim, J. Kim, and S. Kim, “Instantons on the 5-sphere and M5-branes,” 1211.0144.

[22] N. Seiberg, “Nontrivial ﬁxed points of the renormalization group in six-dimensions,” Phys.Lett. B390 (1997) 169–171, hep-th/9609161.

[23] U. H. Danielsson, G. Ferretti, J. Kalkkinen, and P. Stjernberg, “Notes on supersymmetric gauge theories in ﬁve-dimensions and six-dimensions,” Phys.Lett. B405 (1997) 265–270, hep-th/9703098.

[24] O. J. Ganor and A. Hanany, “Small E(8) instantons and tensionless noncritical strings,” Nucl.Phys. B474 (1996) 122–140, hep-th/9602120.

[25] N. Seiberg and E. Witten, “Comments on string dynamics in six-dimensions,” Nucl.Phys. B471 (1996) 121–134, hep-th/9603003.

[26] M. Duﬀ, H. Lu, and C. Pope, “Heterotic phase transitions and singularities of the gauge dyonic string,” Phys.Lett. B378 (1996) 101–106, hep-th/9603037.

[27] O. J. Ganor, “Six-dimensional tensionless strings in the large N limit,” Nucl.Phys. B489 (1997) 95–121, hep-th/9605201.

[28] E. Witten, “New ’gauge’ theories in six-dimensions,” JHEP 9801 (1998) 001, hep-th/9710065.

[29] N. Seiberg and E. Witten, “Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD,” Nucl.Phys. B431 (1994) 484–550, hep-th/9408099.

32 [30] H. Osborn, “Topological Charges for N=4 Supersymmetric Gauge Theories and Monopoles of Spin 1,” Phys.Lett. B83 (1979) 321.

[31] I. A. Popescu and A. D. Shapere, “BPS equations, BPS states, and central charge of N=2 supersymmetric gauge theories,” JHEP 0210 (2002) 033, hep-th/0102169.

[32] T. T. Dumitrescu and N. Seiberg, “Supercurrents and Brane Currents in Diverse Dimensions,” JHEP 1107 (2011) 095, 1106.0031.

[33] N. Seiberg and E. Witten, “Electric - magnetic duality, monopole condensation, and conﬁnement in N=2 supersymmetric Yang-Mills theory,” Nucl.Phys. B426 (1994) 19–52, hep-th/9407087.

[34] E. Witten, “Solutions of four-dimensional ﬁeld theories via M theory,” Nucl.Phys. B500 (1997) 3–42, hep-th/9703166.

[35] D. Gaiotto, “N=2 dualities,” JHEP 1208 (2012) 034, 0904.2715.

[36] R. Jackiw and C. Rebbi, “Solitons with Fermion Number 1/2,” Phys.Rev. D13 (1976) 3398–3409.

[37] C. R. Nohl, “Bound State Solutions of the Dirac Equation in Extended Hadron Models,” Phys.Rev. D12 (1975) 1840.

[38] C. Callias, “Index Theorems on Open Spaces,” Commun.Math.Phys. 62 (1978) 213–234.

[39] M. Atiyah, N. J. Hitchin, V. Drinfeld, and Y. Manin, “Construction of Instantons,” Phys.Lett. A65 (1978) 185–187.

[40] N. Dorey, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, “The Calculus of many instantons,” Phys.Rept. 371 (2002) 231–459, hep-th/0206063.

[41] E. Witten and D. I. Olive, “Supersymmetry Algebras That Include Topological Charges,” Phys.Lett. B78 (1978) 97.

[42] P. C. Argyres and N. Seiberg, “S-duality in N=2 supersymmetric gauge theories,” JHEP 0712 (2007) 088, 0711.0054.

[43] E. Witten, “Dyons of Charge e theta/2 pi,” Phys.Lett. B86 (1979) 283–287.

[44] T. Kugo and K. Ohashi, “Supergravity tensor calculus in 5-D from 6-D,” Prog.Theor.Phys. 104 (2000) 835–865, hep-ph/0006231.