CC BY
45
UDC 530.1; 539.1
OFF-SHELL INVARIANT SUPER YANG-MILLS WITH GAUGED CENTRAL CHARGES FOR N=D=2
AND N=D=4: "DO WE NEED A CONSTRAINT?"
N. Kawamoto*
Graduate School of Science, Hokkaido University, Sapporo, 060-0810 Japan. E-mail: kawamoto@particle.sci.hokudai.ac.jp
We investigate to derive off-shell invariant twisted super Yang-Mills for N=2 in 2-dimensions and N=4 in 4-dimensions with a central charge by super connection ansatz formalism. We find off-shell invariant N=2 algebra with and without an extra constraint in 2-dimensions. On the other hand in 4-dimensions we find off-shell invariant N=4 twisted SUSY algebra including one central charge always with a constraint.
Keywords: superspace, supersymmetry, Yang-Mills, quantization.
It has already been formulated as a super connection ansatz where N=D=4 twisted super Yang-Mills with a central charge can be formulated at the off-shell level with a constraint [13]. It is an interesting question to ask if we can formulate N=4 super Yang-Mills in 4-dimensions without constraint by imposing the similar ansatz as 2-dimensional N=2 case leading to a super Yang-Mills formulation without constraint.
Throughout of this paper we use Euclidean formulation of SUSY algebra since we have in mind the application of the formulation into lattice SUSY [18-20].
2 Dirac-Kaehler twisted supersymmetry
We first show how N=D=2 twisted SUSY algebra naturally appears from a quantization of gauge theory. Let's first consider a very simple 2-dimensional abelian BF theory:
1 Introduction
It has been a long-standing question: "Can we construct off-shell invariant N=D=4 super Yang-Mills formulation ?" The answer was claimed to be negative especially for the case of R-symmetry SU(4) case where only on-shell invariance was realized [2]. It was, however, claimed later that off-shell invariance was realized for R-symmetry USp(4) case with a central charge [3,4]. However there appeared a constraint in this formulation [5].
There were intensive investigations on N=2 and N=4 SUSY algebra with central charge with a hope that N=D=4 super Yang-Mills can be formulated by superspace formalism [6]. There were also trials by harmonic superspace approach on this question [7,8].
In the analyses of extended SUSY algebra it has been recognized that the quantization of gauge theory leads to a twisted version of SUSY algebra. It was especially shown that N=2 super Yang-Mills in 4-dimensions can be derived by quantizing topological Yang-Mills with instanton gauge fixing [9]. It has been intensively investigated to find a procedure of extending this formulation into N=4 super Yang-Mills formulation [10].
In dealing with N=D=4 supersymmetry algebra we proposed twisted superspace formulation by Dirac-Kaehler twisting procedure [11-13]. This Dirac-Kaehler twist is equivalent to Marcus twist of N=4 in 4-dimensions [14] among other twisting procedures [15,16]. The Dirac-Kaehler twisting procedure, however, has nice generalization to other dimensions. Especially for 2-dimensional N=2 super Yang-Mills with central charge we found super connection ansatz where off-shell invariant super Yang-Mills can be formulated with and without a constraint [17].
*Talk given at "Quantum Field Theory and Gravity (QFTG'14)" (Tomsk, July 28 - August 3, 2014), based on the work in collaboration with K. Asaka, K. Nagata and J. Saito [1].
S = J d2x^u,
(1)
which has the following gauge symmetry:
4 = 0, = dMv. (2)
After the Lorentz gauge fixing: = 0, a quantized
action leads:
S = / ^^ + M-», - ^c],
(3)
which has BRST invariance with nilpotent BRST charge s2 =0. It is interesting to recognize that we can find family of BRST charges sM and s which has the following fermionic symmetry at the on-shell level:
tA stA sm4>A s4>A
t 0 -eMv d v c 0
Uv dv c 0 -evpOP
c 0 iu m 0
c -ib 0 -it
b 0 0
sM} = -idM' ^ sM}
d v
= {s, s} = s2 = {sM, sv} = 0.
(4)
(5)
= d2
xss ^ s m s 2
Mv sMsv (-icc),
tA stA sMtA ntA
t iP -eMvdvc 0
Uv dv c -i^Mv X -evpdpc
c 0 iU m 0
c -ib 0 -it
b 0 9m c -iP
X eMv OmUv 0 -dpUM
P 0 -dpt - eMvdvb 0
where R is the R-symmetry generator and Uo and V5 are considered as central charges. Then twisted super-symmetry algebra with central charges is given by
{s, sM} = Pm , {s, sM} = -eMvPv , {s, s} = 0, s2 = S2 = 2(Uo - V5), {sM, sv} = Smv(Uo + V5).
(9)
On-shell N=D=2 twisted supersymmetry
In fact these family of femionic charges satisfy the following twisted N=D=2 supersymmetry algebra:
3 Twisted superspace and super connection
We introduce a twisted super field which is expanded by super coordinates 9 a corresponding to super charge sa:
What is surprising here is that we can find off-shell invariant N=D=2 supersymmetric action by introducing auxiliary fields A and p:
S = J d2x[epu+ - icd^d^c - iAp]
$(xM, 0a, z) = ^(xM, z) + 9a^a(xm, z) + 1 0a0b tAB (xm, z) +----
(10)
which has the s-exact form of the action with respect to the super charges. It has the following off-shell invariant twisted N=D=2 supersymmetry:
where xM is the space time coordinate and z is a coordinate (parameter) corresponding to a central charge. Twisted supersymmetry transformation is generated by super chages Qa and we introduce super derivative DJ as:
$ = £aQA$, {QI, Dj} =0,
(11)
where £a is super parameters. In order to consider gauge theory in this twisted superspace we introduce super covariant derivative
Vi = DI - iri (I = A),
(12)
Off-shell invariant N=D=2 supersymmetry
We call this fermionic symmetry algebra as twisted supersymmetry since the fermionic charges sa are related with super charges of N=D=2 super symmetry algebra in the following way:
{Qai,Qfj } = 26 ij Y Maf PM, Q ai = (1s + Y% - iY5S)c
(6)
where 7^ and 7s are properly chosen 7-matrices in 2-dimensions [17]. We call this supersymmetry algebra as Dirac-Kaehler twisted supersymmetry algebra [11-13]. This Dirac-Kaehler twisting procedure can be extended into 4-dimensions for N=D=4 supersymmetry algebra [12,13].
We can extend the N=D=2 twisted supersymmetry algebra to include super charges [17]:
{Qai, Qj } = 25 ij 7^ Pp + 2^ Sij Uo + 2y5^ 7iSj Vs, (7) [Qai, R] = iSijQaj, [Uo, any] = [V5, any] = 0, (8)
where r_j can be identified as super connection. We introduce a notation to express the lowest order term with respect to the super coordinates; i.e. $|gA=0 =
= If we generalize the notation of (12) for the gauge covariant derivative with I = ^,z then VJ = DM = d„ - iAp. ~
Let us introduce a table notation of (anti-) commuting relations. For example the twisted supersymmetry algebra with central charge in (9) can be read as
s sM s
s 2 (Uo - V5) Pm 0
sv 6vm (Uo + V5) evpp p
ss 2(Uo - V5)
where a low and a column crossing location of term represents the value of the corresponding anticommutation relation of super charges.
c
2
s
1
4 Super connection ansatz and SUSY transformation
Once SUSY algebra is given, it is straightforward to examine the full closure of the SUSY algebra by super connection formalism [8,10,12, 21]. For an ansatz of given SUSY algebra all the possible combinations of Jacobi identities give a criteria for a consistency of the full algebra.
Let us consider the super connection Ansatz (A) (Table 1), where we included VM and Vz which are defined in {Va, Vb } in the table. For this ansatz N=D=2 twisted supersymmetry algebra with central charge can be read with an identification, Va ^ sa, W = 0, Vz ^ Z, -¿V^ = PM:
{ S, Sm} = PM , {s, Sm} = -eMvPv , {S, s} = 0 ,
S2 = S2 = 2z, {sm, Sv} = ±£Mvz .
We next derive all possible non-trivial relations by using graded Jacobi identities until we don't get any new relation. In other words if we get inconsistent relations from the Jacobi identities we consider that the starting super connection ansatz is not taken correctly.
For example the following graded Jacobi identity is satisfied for boson W, fermion — and fermion x:
[W, {V,x}] + {V, [x, w]} - {x, [W, V]} = 0.
[V, {V, V}] + [V, {V, V}] + [V, {V, V}] = 0,
= ±i^vVW t VW
F
±e„v V VW + 1 VpVff W,
Mv — ± Mv vv rv 1 2 Mvcpa v p V a
G = V W, G = V W
= -VfW,
GM = 2iVMVW - VMW.
SUSY transformations of these component fields can be obtained by taking (anti-)commutator with the field and identifying the original Jacobi identities. For example the following SUSY transformation can be identified as
s(W |) = QW | VW | = p,
DW| = DW| - ¿[r,W]|
(17)
where Wess-Zumino gauge is chosen here: r = 0. In a similar way we can derive all the N=D=2 SUSY transformations for these component fields (Table 2).
We can find off-shell invariant action under this SUSY transformation as:
1 1^.1
S = J d2xTr(±i(D^)2 - 1 F2v - 1D2 ± 24
(18)
(13)
1
^ mv 2' T 2iAM(DMp - eMvDvp) - ¿^{p, p}
- ¿^{p,p}± ¿^{AM,AM}) .
In order to confirm off-shell closure of the algebra we need the following non-trivial constraint:
T D] - {Am, Am} T {p, p} T {p, p} = 0, (19)
which cannot be obtained as one of Jacobi identities. For Abelian case the constraint becomes simple as: dMgM = 0, and can be solved as gM = £MvdvB. We can then obtain off-shell SUSY invariant action without constraint:
As a concrete example of deriving a non-trivial relation is
S:
Jd2xTr(± 1(^)2 - 4F2v - 2D2 ± 2(d,B)
(14)
1
T 2«Am(dfP - eMvdv/5) ,2
(20)
+ e(-^M v Fm v + 2pp + BD + v AMAv 1 ).
where {V, V} = 0. We then obtain the following relation:
[V, -iW + Vz] = -¿VW + ¿G = 0 ^VW = G. (15) Similarly we obtain the following relations:
VmVW = € m v Vv V W,
Fm = -iVf W, Fm = v Vv W,
The constraint (19) cannot be solved for the non-Abelian case in a local way. This example is similar to the N=D=4 super Yang-Mills with R-symmetry USp(4) case where one constraint appears as a extra condition.
5 Off-shell N=D=2 SUSY invariant action without constraint
In order to find other ansatz which doesn't generate a constrained equation like (19), we impose another ansatz as in the following:
We identify component fields of super multiplets as:
W | = vw | = p, VV w | = 5, (16)
vm w | = am, Vz W | = D, Gf | = g».
V V Vv Vv Vz
V 0 0 -i(Vv + Fv ) -iFv ¿G
V5 0 i€vp ( Vp - Fp) ¿G
V ¿M v Vz iFMv
Vm iFMv ¿Gm
Vz 0
N=D=2 Ansatz (B)
2
From this ansatz we obtain the following relations by using graded Jacobi identities:
6 N=D=4 super Yang-Mills formulation with central charges
VF„ = eUv VFv , VuFv + VvFu = Sul,V 0F0
Gp =0, Fp = -iVFp ,
Fp = iV Fp :
Fn
-n5pv(VpFp - G) + -eßv(epaVpFa - G)
2
1
Fpv = VßVFu - VvVFp + i[Fß, Fv] + 2eßvVG, VG = V G = VG + V G = 0, VzFp = i(VMG - e^vVvG), Gp =2 (VpG + eßuVvG).
(21)
In this ansatz we make the following identification of component fields of N=D=2 super multiplet:
Fp| = Фp , VFp| = Xp , VpFv | = -(5pvP + e^vp):
VMVFM| = D.
(22)
Most general N=4 supersymmetry algebra in 4-dimensions can be given by
{Qai,Qßj } = 2C- VC)aßPp
+ 2Caß (C- iU0 + (C-175)ij U5) + 2(75 C )aß (C-1Vo + (C-175)ij V5). (27)
The N=D=4 super charges Qai can be decomposed into twisted super charges:
i 1
Qai = ^ (1s + 7% + 2 Y^ sßv + T SM + 75S)ai. (28)
In this Dirac-Kaehler twisting mechanism the spinor suffix a and the extended SUSY suffix i are rotated by angular momentum generators Jpv and R-symmetry generators Rßv, respectively. In this way the fermionic super charges having spinor suffix change into the twisted super charges having scalar, vector, tensor. suffix which are now rotated by a new rotation generators J'pv. They have the following relation [11-13]:
SUSY transformation of these component fields can be obtained as in the previous example - see Table 3. In this ansatz the super charges have the following nilpotent nature:
J11.v - Jpv + Rpv .
(29)
The N=D=4 Dirac-Kaehler twisted SUSY algebra corresponding to (27) is given by
{s sp} = {5, sp} = Pp ,
„2 _ ~2
s± = 0, (s± — si ± is2).
(23)
G = ae^vV-VFv , G = -aV+VFp,
(24)
S :
/1 1 1
d2xTr{2(Dмфv)2 + -F|2v + 2D2 - ipD+Xi
sS = sS = s±S = 0.
{sp, s pa } 5pvpa Pv 1 {sp, s pa } epvpa Pv ■
sp} = {5, sp} = {s, spv} = ^ spv} = 0 ,
In order to derive off-shell SUSY invariant action it is convenient to find a s-exact form of an action. To find this type of action we recognized that among the relations in (21) the relations on G and G are crucial to be solved. We actually found a solution of VG = VsG = 0
2s2
2s2 = Uo + V5 ,
where a is a constant. We then found a off-shell SUSY invariant action:
{sp, Sv} = {Sp, Sv} = 5pv(Uo - V5),
{s,S} = U5 + Vo , {sm,Sv} = 5pv(Us - Vo),
{ Spv, Spa } = 5^vpa (Uo + V5 ) ^pvpa (U5 + Vo),
where 5pvpa = 5pp5va - 5pa5vp and epvpa is Euclidean e-tensor.
We first investigate the case of ansatz where no central charge is inserted (Table 4). Graded Jacobi identities for this ansatz lead the following relations:
VW = V W = Vpv W = 0, VpF = V pF = 0,
- 5pvVF = VpFv + ^vFp , 5pvVVF = VpFv + VvFp ,
liv V ± V v I v vx l
v\ ^iivpa V pFa
- ipepv DpXv - -[ф|, Фv ]2 - ia-1G|C?|}, (25)
where G| and G| are fermionic fields. The action satisfies N=2 SUSY invariance at the off-shell level without constraint:
V[pFv] epv pa V pFa
V F =- 5 X7F + e V F
v pv1- p upvpa v 1 a i cpvpa v 1 at
Fp = -iVFp , Fp = iV Fp ,
~ VpW = 2VFp ,
VpW = -2^Fp ,
F,
pv
- 2 (5pv^F - öpvpa VpFa ) ,
Fpv = - 2(5pv VF + 5pvpa VV pFa ) ,
(26) Fp
i5pvpa VFa iepvpa VFa .
We define component fields of N=4 super multiplets as FM| = , W| = A, F| = B , VFM| = AM , V FM| = AM ,
1
N=4 SUSY transformation of these component fields are given by Tables 8, 9. For the off-shell closure of the above SUSY algebra we need the following constraint:
VFv 1 = vp + , VFv 1 = Vp + 2 WpPa. + [<^, HM] t 2{p, p } + 2{AM, À M}
2
We can then obtain on-shell closed SUSY transformation of these fields as shown in Tables 5, 6. Then we obtain on-shell invariant N=4 super Yang-Mills action without central charge having SU(4) R-symmetry:
/1 1 1
dVTr{4(D^v )2 + 1 F2,, + 8 D+AD-B
+ 1D-AD+B - ^]2 + 116[A, B]2
- iAu(D+P - d-pmv - [B, au]) - iÀ(D-p 1 i
- 2eUvpa D+Ppa ) + ip [A,P] - 8 eUvpa a{pmv, Ppa H .
We now investigate the super connection ansatz of N=D=4 with a central charge given by Table 7.
The corresponding twisted SUSY algebra with a central charge Z is given by
SU} {à , À , j^U spa} ¿^vpa Pv ,
{à U, spa } euvpa Pv ,
{S sU} = {S , SU} = {S suv} = {À suv} = 0 ,
2s2 = 2s2 =0, {SM,Sv} = {SM,Sv} = 0, {s, s } = Z,
{sU, À v} Z , {suv, spa } euvpaZ,
where Z = U5 for + and Z = V0. Graded Jacobi identities lead:
V(MFv) = V Vw , V(U Fv) = —V VW, VFM = ± 1VMW, Vfm = T1VMW,
V W = - e VF VrFi = e VF
v uvpa v px a ? V [u vj c^vpa V p*- a i
V[uFv] euvpa VpFa ,
V F = — ^ VF + e VF
v P uvpa v ± a \ c^vpa v ± at
Fu = -iVFM , Fi = ¿VFM , FMv = -iVvFM ,
FUv iV vFU , Fuvp i(^uvpa VFa + e^vpaV Fa) ,
Fuv = V[MVFvj + i[Fu, Fv], G = G = GMv = 0, GM = 2VFM , GM = —2VFM ,
gu = 2(VGM + V gu) ,
ZFM = 2 (VGM — V gu) , ZW = 2iVuFu + 2VVw
We define component fields of N=4 super multiplets:
Fui = , W| = A, VFM| = AM , V FM| = AM ,
VMFv | = p + PMv , VMFv | = p + 11 eMvpaPpa ,
VVWi = H, Gui = , VzFui =
1 i 1 T 1 w{pmv,pp.} ± 2D+D-A ± ^[A, H] = 0. (30)
Off-shell twisted N=4 SUSY invariant action in this case is given by
S = jd4xTr( 1 dm^v + 1 F2v ± ±(4 + H) + H(¿dm^m + 1H) - 1(dm^)2 - 2ipD+ am - 2ipd-Am - 2ipMv (D-Av
+ 1 D+Aff) - 2iA{AM, am} - 4[^, ^]2),
where the constraint (30) is crucial to prove the offshell invariance of SUSY for this action.
We now try another N=D=4 twisted SUSY ansatz which has similarity with the N=2 Ansatz (B) leading no constraint in 2-dimensions (Table 10). Graded Jacobi identities lead:
V(MFv) = V VW, VFM = ± 1VMW,
V(MFv) = — V VW, .1, 2
V FM = TxVMW,
VUv euvpa VpFa , V[uFvj euvpa VpFa ,
eF
c^vpa v p± a
Mv
V [MFv]
V F =— ^ VF + € VF
v MvF p ^Mvpa v F a + €Mvpa v F a ,
Fm = -iVFf , Fm = iVFf , Fmv = -iVvFm ,
FMv iVvFm , FMvp i(^Mvpa VFa + €MvpaVFa) ,
Fmv = V[mVFv]+ i[Ff,Fv], G = G = Gmv = 0, Gm = 2VFm , Gm = -2VFm , Gm =2(VGm + VGm) ,
ZFm = 2(VGm -VGm) , ZW = 2«VmFm + 2VVW
We define component fields of N=D=4 super multiplets as
FmI = ^m , W| = A, VFm| = Am , VFm| = 5m ,
VfFv | = ¿mv p + Pmv , V mFv | = ¿mv p + 1 €uvpa Ppa ,
VVW| = H, GmI = gM , ZFmI = Hm , ZW| = K.
Twisted SUSY transformation of these component fields are given by Tables 11, 12, 13. For a closure
of SUSY algebra for this ansatz we again need the following constraint:
iDp9M - [t m, Hp] - 2DmDmA - ^[t m, [t m, A]]
T 4[A, K] T {P, P} T & P} T t{PMV, PMV}
-{Xm,Xm}-{Àm,Àm} = 0.
(31)
For the case of one central charge we have tried all possible super connection ansatz. We have found out two possible consistent Ansatz (B) and (C) but for both cases we need a constraint equation for the off-shell closure of N=D=4 twisted algebra with a central charge. Off-shell N=D=4 twisted SUSY invariant action for this Ansatz (C) can be given by
S :
J d4xTr(TDptvDptv - 1 F2v ± 1(gM - D pA)
T T(Hm - i[A, tM])2 - 8K2 - 2iP(D mXm - [t m, Xp]) - 2iP(D pXp + [tp, Xp]) - 2iPpv(DpXv
+ [tp,Xv ]) i^M vaf P M v (D
Xs f - [t
Xf ])
± iA({XM, Xp} + {Xp, sm}) + - [t m, tv]2 ). (32)
7 Equivalence of the Ansatz (B) and Ansatz
(C)
Consider the general Ansatz of N=D=4 (Tables 14, 15). Define the following new connections and curvatures:
vnew = -U-iV + V), Vnew = —(v - iV), 22
1
1
VMew = — (iV M + Vm) , vMew = — (Vm + iV m) ,
1
V2
ynew = 1 ( i^ _ 1, Y7 ) V Mv = AT( iv Mv 0^M vpa v pa):
X n
ne
iXo , X m
iXM , X5
(33)
-iX5 (34)
Xnew = -iXo, x;new = iXp, X5new = -iX5
It turns out that this Ansatz (D) and the ansatz given by the above new system has exactly the same
form. Surprisingly the Ansatz (B) and Ansatz (C) have exactly the same relations as this new and old system. In other words these ansatz are essentially the same and thus both of cases naturally need the constraint relation for the N=4 SUSY closure.
8 Conclusion and Discussions
In 2-dimension we found two types of super connection ansatz which realize off-shell closure of N=2 twisted SUSY including central charge with and without constraint. Off-shell twisted SUSY invariant actions are found for each ansatz. On the other hand in 4-dimension we examined two possible ansatz of N=4 twisted SUSY algebra with a central charge and we found that both of ansatz having similarity with the 2-dimensional ansatz need a constraint equation for the off-shell closure of N=4 twisted SUSY algebra. In fact we found that these two ansatz are essentially equivalent to each other.
We have thus investigated a possibility of off-shell twisted N=D=4 invariant super Yang-Mills formulation without a constraint by super connection formalism. As far as N=4 twisted SUSY algebra with one super charge is concerned a constraint is inevitable for the off-shell closure of the algebra. We consider that this may be related to the fact that 10-dimensional N=1 super Yang-Mills theory can be formulated only at the on-shell level. N=D=4 super Yang-Mills can be dimensionally reduced from the 10-dimensional N=1 formulation and the on-shell nature could be kept invariant in the dimensionally reduced formulation and thus may lead to off-shell closure but with a constraint in 4-dimensions.
Acknowledgements
I would like to thank my collaborators, K. Asaka, A. D'Adda, I. Kanamori, J. Kato, A. Miyake, K. Nagata, J. Saito, T. Tsukioka and Y. Uchida for useful discussions and fruitful collaborations. Thanks are also due to I. L. Buchbinder and V. Epp for useful comments and kind hospitality in Tomsk. This work was supported in part by Japanese Ministry of Education, Science, Sports and Culture under the grant number 22540261.
2
V V Vv Vv Vz
V -iW + Vz 0 -iVv -iFv iG
V Vz - iW ievp Vp -iFv iG
VM (iW + Vz ) -iF ^ Mv iGM
V -iF Mv iGM
Vz 0
Table 1. N=D=2 Ansatz (A)
s sM s Z
^ Av Av P P D gv P -iAv i (gv - Dv 2D i1 e F ± 4mv x mv TiDMAM evpDpp - Av] Am ±i^MvP T ieMvP ± 2 ^Mv D + 2 FMv -1 (gM + DM^) i eMv (gv + Dv ieMvDvP - iDMP eM^ evpDp [^P] ± eMv [^,P] P ievp Ap - i dvp(gp - Dp^) T 4 eMv Fmv 2D TieMv DMAv -evp(DpP + Ap]) D gv -iDvP + ievpDp/5 - i[^, Av] TiDMAM + #,P] TieMv DMAv + #,P] ±DMgM T DmDm^ ±2i{AM,AM} + #, D] ±DpFvp - 2evp{Ap, P} -2{Av,P} + #, Dv
Table 2. N=D=2 SUSY transformation for Ansatz (A)
s SM s Z
^v Av Av P P D Av -iAv 0 i[D+, D-] - D - i epff [D+,D+] -iD+Ap 1 (¿MV P + eMv /5) 2 ¿Mv P + 2 eMvP + i ¿MvG| 2eMvG| AMv 1 (VM G| - eMv Vv G|) 1 (VM G| + eMv Vv G|) i (D+P - eMvD-P) - i (DmG| - eMv Dv G|) evp Ap ievp Ap 0 i epff [D-,D-] - i [D+, D-] - D 1 (VvG| - evpVpG|) i (Vv G| + evpVpG|) -1(D-G| - evpD+G|) 1 (VzG| - epffVpVffG|) 1 (Vz G| + epff VpVff G|) - i (D-VpG| + epff D+VCT G|) i{P,G|} + i{P,G|} -i({G|,G|} + {G|, gG|})
Table 3. N=D=2 SUSY transformation for Ansatz (B)
V V Vp Vp Vpff Vp
V 0 -iW -i(Vp + Fp) 0 0 - iFp
V 0 0 -i(Vp - Fp) 0 -iFp
vm 0 ^¿MpF ^Mvpo" (Vv Fv ) -iF îfmp
VM 0 ieMvpo" ( Vv + Fv) -iF îfmp
Vmv ieMvpCTW -iF Mvp
Vm -iF "
Table 4. N=D=4 Ansatz without central charge (A)
s s
Ap Xp Suvpa Aa + tuvpaAa
AP -iXp iXp iSuvpa Aa ituvpaAa
A 0 0 0
B —2p 2p tuvpa p pa
xp 0 2 D-A 1 t D+ A 2 cuvpa J-/a Jrx
~xp 2 D+A 0 1 S D- A 2 ußvpa^ a
P — 2 (Da^a + 1 [A,B]) 0 2 [D-,D-]
P 0 — 2 (Da<pa — 1 [A, B]) 4 tuv pa [D+, D+]
Ppa — 2 [D+,D+] 4 tuvpa [D— , D— ] 2 Suva^ Spaßj [D— , D+] — 2 Suvpa (Da^a + 2 [A B])
Table 5.
Su Su
^p Sup + pup Supp + 2 tupaß Paß
Ap —iS^p + ip^p iSupp 22 tupaß paß
A —2ÀM 2AM
B 0 0
Ap 2 [D+,D-] + 2 Sßp(Da <pa + 1 [A, B]) 4 W [D+,D+]
X — 4tupaß [D-,D-] — 2 [D—,D+] + 2 Sßp(Da^a — 1 [A, B])
p 0 — 2 D— B
p 2 D+ B 0
P pa 1 t D— B 2 tuvpa Dv B 1 S D+ R 2 SuvpaDv B
Table 6.
V V Vp Vp Vpa Vp Vz
V 0 Vz —i(Vp + Fp ) 0 0 —iFp iG
V 0 0 —i(Vp — Fp) 0 —iFp_ iG
Vu 0 Sßp(TVz — iW ) iSuvpa (Vv Fv ) —iFJl p iGu
VM 0 ituvpa (Vv + Fv ) —iF up iGu
VMv —t V cuvpa v z -iF (,± uvp iG uv
VM —iFupp iGu
Vz 0
Table 7. N = D=4 Ansatz with a central charge (B)
s s SM àm
^p Àp Ap ¿mp P + Pmp ¿mpP + 2 eMpaßPaß
Ap —¿Ap ¿Àp -¿¿mpP + ¿Pmp ¿¿MpP 2 ^Mpaß Paß
A' P tÀm ±A[p]
Ap 0 g+ i[D+,D-] - ¿mpH' 4 eMpaß [D+,D+]
Ap -g- 0 - 4 eMpaß[D- D-] -2 [D-, D+] + ¿mp(2[D-,D+] + H')
P H ' 0 0 Tg+ - ¿D-A'
P 0 -1 [D- ,D+] - H' ±g- + ¿D+A' 0
Pp^ - i D+, D+] Ê^VpCT (Tg+ - ¿D- A') ¿Mvpa (±g- + ¿D+A')
H ' 0 -»D-ÀM -¿D+P _ 22 ^Mvp^ D+ppct
T( I eMvaßD+Paß -iD-P + ¿[A, Ap]) 0 -¿¿MpD-Àv + ¿D-àm -¿D-Ap - ¿eMpaßD+Aß
g- 0 ±(iD-Ppff -¿D+p + ¿[A,À p]) ¿eMpaß D-Aß + ¿D+Ap ¿¿mpD+Av - ¿D+Am
Table 8. N = D=4 SUSY transformation with a central charge for Ansatz (B)
sMV Z
^p ¿Mvpa + ^Mvp^ Hp
Ap ¿¿Mvpa ¿^Mvp^ gp
A' 2 ^Mvp^ Pp^ 2 [D-,D+] + H
Ap g<7 T(-«D-À + 2 D+PpCT + ¿[A, Am ])
Ap ¿Mvpa g+ T(-«D+P + ¿D-Pmv + ¿[A, Am])
p 2 [D-, D- ] -»d-Àm
p 44[D+, D+] -»d+am _
Ppa 22 ¿mv«7¿p^ßY [Da , D+ ] + ¿Mvp^ H -¿êmvpctD- ACT - ¿D+^Av]
H' »D-Av] »D-g- +2»{am,am}
(-¿D+P + ¿D-PCTa + ¿[A, ACT]) T»( 2 D-D+D- + ÊMvpa {Av ,Ppff } +
2{p,A} + D-H ' + 2[A',g+])
g- T¿мvpст (-¿D-s + 2e^YaßD + Paß + ¿[A, ACT]) T»( 2 D-D+D+ + 2{Av ,pmv }+ 2{P, àm} + D+H' + 2[A',g-])
Table 9. N = D=4 SUSY transformation with a central charge for Ansatz (B)
V V Vp Vp Vpff Vp Vz
V Vz - »W 0 -»Vp »Fp 0 -»Fp »G
V Vz - »W -»Fp -»Vv 0 -»Fp »G
vm ¿¿MV Vz 0 ¿¿Mvpa Vv + ¿^Mvp^ Fv - »FMP »Gm
vm ¿¿MpVz ¿^Mvpa Vv + ¿¿Mvpa Fv - »FMP »Gm
vmv ¿Mvpa (Vz - »W ) »FMvp »GMV
vm -»FMp »Gm
Vz 0
Table 10. N = D=4 Ansatz with a central charge (C)
s s
0p xf ap
AP iXp -iAp
A' TP ±P
ap H' g'
ap -g' H'p
P - 5 dm0m ±K'
P TK' - i D^
Pp a — 2ep aaß F-ß - 5 D[p Fpa - 5 £ paaß Da0ß
K' ± i (DMAM T5(DMAM - VI)
g' ± "1 (D pP - 1 £p aaß Da Paß T5 (DpP - DaPpa + p, p]
-№p,Pl - [^a, Pp a ] ± 2[A',A p ]) + paaß [0a, Paß ] ± 2[A', Ap ])
T § (DpP - Da Pp a + [0p,p] T 5 (DpP - 1 £ paaß D a Paß
+ "£paaß [0a , Paß] ± 2[A', Ap]) -№p,P] - [0a ,Pp a ] ± 2[A',Pp])
Table 11.
sM sM
0p Sm p P + Pm p SM p p + 2 £M paß Paß
A -iSMp P+ 5 eMp aß Paß iSßpP - iPßp
A' 1 • -AM • A[P]
A 1£Mpaß F-ß + 5 Sßp Dv 0v - 5 D(^p) F+p ± SMpK' + 5£M paßDa0ß
A -F+p ± SM pK' + 1 £M paßDa0ß - "£m paßFaß + SM p 5Dv0v - f D(M0p)
P - 5 DM0M T § gM
P ±gM 2HM
Pp a ± " ^Mvpa Hv T 2 £Mvpa gv ±22£MvpaHv ± f SMvpagv
K' ± 5 (DMp - 1 £Mvpa Dv Pp a - ^ P] - [0v, Pßv]) T"1 (DMP - DvPmv + [0M,p] + 1 £Mv pa [0v,Ppa])
g' - 5 (Smp(Dvpv + [0v, Av]) - D(mAAp) -5(-Smp(DvAv - [0v, A]) + D(ßAp)
+ [0[M , Ap]] - £ßpaß (DaAß + [0a, Aß ])) + Ap]] + £Mpaß (DaA ß - [0a, Aß]))
H' -5(Smp(DvAv - [0v, Av]) + D[ßAp] -5(Smp(DvAv + [0v, Av]) + D[MAp]
+ [0(m , Ap)] + £Mpaß (DaAß - Aß])) -[0(м, Ap)] + £M paß (DaAß + [0a, Aß]))
Table 12.
Z
0p 5Mvp a Xa + paXa Hp
AP i5Mvpa Xa + i^^vpa Xa gp
A' Tpappa 2 'K
xp 5 Mv p a Ha + ^^v pa ga Ti(DpP - DaPpa + [0p, P]
+1 ^-paaß [0a, Paß])
xp 5Mvpa ga ^p,vpa Ha Ti(DpP - 2 tpaaß Da Paß - [0p, P] [0a, Ppa])
P 2 e^vpa Fpa - 2d[m0v] -%(dpxp - [0p, Xp] - i[A P])
p F-v + 2 ^ßvpa Dp0a -i(DpXp +[0p,Xp] - [A, p])
Ppa ^Mvaj 5paßj Faß + 22 5Mvaj 5paßj D(a0ß) -i(D[MXv] + [0M Av]]
2 5 MV pa Da0a T ^mvpa K +^Mvpa (DpXa - [0p, Xa]) - [A, Pmv])
K' ±2 (D[MAv] - [0[m,Xv]] + t-Mvpa (DpXa ±i({X,X} + {A,A }- Dpg'p +[0p,H'p]
± 25Mvpa (Da p - 1 ea^aß DyPaß - [0a , P] ±[A,K'])
g' Ti(- 1 Da Fpa ± DpK' - 2 [0a ,Dp0a ]
+ [0rho ,K ])) - [0a,Paa] ± [A, Xa ]) ± 2 ^Mvpa (Da P + {P, Xp} + {P , Xp} + {Xa , Ppa }
-DaPaa + [0a, P] + 1 ea7aß 0, Paß ] ± [A, Xa ]) + 2 ipaaß { A a, Paß })
H ± " 5Mvpa (DaP - DaPaa + [0a, P] Ti(-2 Da Da 0p + § [0a , [0p ,0a ]]
+1 eajaß [0y , Paß ] ± [A, Xa ]) T "2 eMvpa (Da P T[0p,K']+ {p,Xp}-{p,Xp}
- 1 ^-a^aß Dy Paß - [0a, P] - [0a,Paa] ± [A, A a ]) { A a, Ppa } + 2 epaaß {Xa, Paß })
Table 13.
V V vp vp
V Xo + X5 X5 + xo -i(Vp + iXp) -X
V Xo + X5 X' , -i(Vp - iXp)
vm vm Vmv Vm 5mp(x0 - x5) 5Mp (X5 - X0 ) 5mp(x0 - X5)
Table 14. Ansatz (D)
Vpa vp Vz
V 0 -iFp iG
V 0 -iFp iG
vm i5Mvpa (Vv iXv ) ^MvpaXv —iF irMP 'Gm
vm i^Mvpa (Vv + iXv ) 5Mvpa Xv -iF mp i&M
Vmv 5Mvpa (X0 + X5 ) - eMvpa (X5 + X0 ) -iF 1,1 Mvp iG Mv
VM -iF ~ Mp iGm
Vz 0
Table 15. Ansatz (D)
References
[1] K. Asaka, N. Kawamoto, K. Nagata, and J. Saito, to appear.
[2] M. F. Sohnius, Nucl. Phys. B136 (1978) 461.
[3] M. F. Sohnius, Nucl. Phys. B138 (1978) 109; P. Fayet, Nucl. Phys. B149 (1979) 137.
[4] M. F. Sohnius, K. S. Stelle, and P. C. West, Phys. Lett. 92B (1980) 123; Nucl. Phys. B173 (1980) 127.
[5] W. Siegel and M. Rocek, Phys. Lett. B105 (1981) 275;
J. Hassoun, A. Restuccia, J. G. Taylor, and Peter. C. West, Nucl. Phys. B243 (1984) 423;
C. T. Card, P. R. Davis, A. Restuccia, and J. G. Taylor, Phys. Lett. B146 (1984) 199.
[6] B. de Wit, V. Kaplunovsky, J. Louis, and D. Lust, Nucl. Phys. B451 (1995) 53, [arXiv:hep-th/9504006];
P. Claus, B. de Wit, M. Faux, B. Kleijin, R. Siebelink, and P. Termonia, Phys. Lett. B373 (1996) 81, [arXiv:hep-th/9512143];
I. Gaida, Phys. Lett. B373 (1996) 89, [arXiv:hep-th/9512165];
A. Hindawi, B. A. Ovrut, and D. Waldram, Nucl. Phys. B392 (1997) 85, [arXiv:hep-th/9609016];
I. Buchbinder, A. Hindawi, amd B. A. Ovrut, Phys. Lett. B413 (1997) 79, [arXiv:hep-th/9706216];
R. Grimm, M. Hasler, and C. Herrmann, Int. J. Mod. Phys. A13 (1998) 1805, [arXiv:hep-th/9706108];
N. Dragon, E. Ivanov, S. M. Kuzenko, E. Sokatchev, and U. Theis, Nucl. Phys. B538 (1999) 441, [arXiv:hep-th/9805152].
[7] A. Galperin, E. A. Ivanov, S. Kalitsyn, V. Ogievetsky, and E. S. Sokatchev, Class. Quant. Grav. 1 (1984) 469.
[8] I. L. Buchbinder, O. Lechtenfelt, and I. B. Samsonov, Nucl. Phys. B802 (2008) 208, [arXiv:hep-th/0804.3063];
D. V. Belyaev and I. B. Samsonov, JHEP 1104 (2011) 112, [arXiv:hep-th/1103.5070];
I. L. Buchbinder and N. G. Pletnev, Nucl.Phys. B877 (2013) 936-955, [arXiv:hep-th/1307.6300].
[9] E. Witten, Comm. Math. Phys. 117 (1988) 353; Comm. Math. Phys. 118 (1988) 411; L. Baulieu and I. M. Singer, Nucl. Phys. Proc. Supple. 15B (1988) 12;
R. Brooks, D. Montano, and J.Sonnenschein, Phys. Lett. B214 (1988) 12;
J. M. F. Labastida and M. Pernici, Phys. Lett. B212 (1988) 56; Phys. Lett. B213 (1988) 319;
J. M. F. Labastida and C. Lozano, Nucl. Phys. B502 (1997) 741, [arXiv:hep-th/9709192].
[10] M. Alvarez and J. M. F. Labastida, Nucl. Phys. B437 (1995) 356, [arXiv:hep-th/9404115]; Phys. Lett. B315 (1993) 251, [arXiv:hep-th/9305028].
[11] N. Kawamoto and T. Tsukioka, Phys. Rev. D61 (2000) 105009, [arXiv:hep-th/9905222];
J. Kato, N. Kawamoto, and Y. Uchida, Int. J. Mod. Phys. A19 (2004) 2149, [arXiv:hep-th/0310242].
[12] J. Kato, N. Kawamoto, and A. Miyake, Nucl. Phys. B721 (2005) 229, [arXiv:hep-th/0502119];
J. Kato and A. Miyake, Mod. Phys. Lett. A21 (2006) 2569, [arXiv:hep-th/0512269]; JHEP 0903:087 (2009), [arXiv:hep-th/1104.1252].
[13] J. Saito, Soryushironkenkyu (Kyoto), 111(2005) 117, [arXiv:hep-th/0512226].
[14] N. Marcus, Nucl. Phys. B452 (1995) 331, [arXiv:hep-th/9506002];
M. Blau and G. Thompson, Nucl. Phys. B492 (1997) 545, [arXiv:hep-th/9612143].
[15] J. P. Yamron, Phys. Lett. B213 (1988) 325.
[16] C. Vafa and E. Witten, Nucl. Phys. B431 (1994) 3, [arXiv:hep-th/9507050].
[17] K. Asaka, J. Kato, N. Kawamoto, A. Miyake, Prog. Theor. Exp. Phys. 113B03 (2013), [arXiv:hep-th/1309.4622].
[18] A. D'Adda, I. Kanamori, N. Kawamoto and K. Nagata, Nucl. Phys. B707 (2005) 100, [arXiv:hep-lat/0406029]; Phys. Lett. B633 (2006) 645, [arXiv:hep-lat/0507029]; Nucl. Phys. B798 (2008) 168, [arXiv:hep-lat/0707.3533].
[19] A. D'Adda, N. Kawamoto and J. Saito, Phys.Rev. D81 (2010) 065001, [arXiv:hep-th/0907.4137]
[20] A. D'Adda, I. Kanamori, N. Kawamoto and J. Saito, JHEP 1203 (2012) 043, [arXiv:hep-lat/1107.1629];
A. D'Adda, A. Feo, I. Kanamori, N. Kawamoto and J. Saito, JHEP 1009 (2010) 059, [arXiv:hep-lat/1006.2046].
[21] B. Milewsky, Nucl. Phys. B217 (1983) 172.
Received Ц.11.20Ц
Н. Кавамото
СУПЕРСИММЕТРИЧНАЯ ТЕОРИЯ ПОЛЯ ЯНГА-МИЛЛСА С КАЛИБРОВАННЫМ ЦЕНТРАЛЬНЫМ ЗАРЯДОМ ВНЕ МАССОВОЙ ОБОЛОЧКИ ДЛЯ N=0=2 И N=0=4:
«НУЖНЫ ЛИ НАМ СВЯЗИ?»
Мы исследуем вывод твистованной суперсимметричной теории поля Янга-Миллса с центральным зарядом вне массовой оболочки для N=2 в двух измерениях и N=4 в четырех измерениях, используя формализм суперсвязностей. В двух измерениях мы находим N=2 инвариантную вне массовой оболочки супералгебру без дополнительной связи. Однако в четырех измерениях мы находим, что твистованная N=4 супералгебра с одним центральным зарядом всегда содержит связь.
Ключевые слова: суперпространство, суперсимметрия, Янг-Миллс, квантование.
Кавамото Н., доктор, профессор. Университет Хоккайдо.
Саппоро, 060-0810 Япония.
E-mail: kawamoto@particle.sci.hokudai.ac.jp
