Суперполевые модели на s^2 и s^3 Текст научной статьи по специальности «Математика»

udc 530.1; 539.1

SUPERFIELD MODELS ON S2 AND S3

I. B. Samsonov

INFN, Sezione di Padova, 35131 Padova, Italy. Tomsk Polytechnic University, pr. Lenina 30, 634050 Tomsk, Russia.

E-mail: [email protected]

We develop superfield models for constructing classical actions of various models with rigid supersymmetry on S2 and S3. We introduce superspaces based on supercoset manifolds SU(2|1)/U(1) and SU(2|1)/[U(1) X U(1)]. We show that models on S3 with extended supersymmetry can have different versions which are invariant under different supersymmetry groups. Among the models with extended supersymmetry on S3 we consider the N = 4 and N = 8 SYM theories, Gaiotto-Witten and ABJM models as well as their analogs on S2.

Keywords: superspace, supercoset, super Yang-Mills model.

1 Introduction and summary

Recently, there have been a surge of interest to su-persymmetric field theories with rigid supersymmetry on curved manifolds with topology of sphere. In [1] it was demonstrated that such models are in the core of the so-called supersymmetric localization method which allows one to compute various quantum observables in these models exactly, beyond the perturbation theory. Although the supersymmetry plays the crucial role in this method, all applications were given using the component field formulations of field models in which the supersymmetry is not manifest. It is natural to expect that the localization method can be properly extended and applied to superfield models which have explicit supersymmetry by construction. As a first step towards this goal we develop appropriate su-perfield methods for constructing classical Lagrangians of various models with rigid supersymmetry on S2 and S3.

Our consideration is based on the supercoset spaces SU(2|1)/U(1) and SU(2|1)/[U(1) x U(1)] which contain the spheres S3 and S2 as their bosonic bodies. We introduce gauge and matter superfields in these superspaces and construct classical actions using standard methods of quantum field theory in a curved superspace with four supercharges [2]. In 3d case we denote this supersymmetry as N =2 while for two-dimensional models it is N = (2, 2). Gauge and matter field theories with this supersymmetry on

S3

were

constructed for the first time in [3,4] and on S2 in [5,6] using standard component field methods. The authors of these papers found many new exact results for these models on the quantum level.

One of the main results of our consideration is the application of superfield methods for constructing classical actions of various models with extended super-symmetry both on S3 and S2, such as the N = 4 and

N = 8 SYM theories, Gaiotto-Witten and ABJM. For three-dimensional models with extended supersymmetry we show that there exist different supergroups with the same number of fermionic generators and containing S3 in the bosonic sector. As a consequence, there are different versions of models with extended super-symmetry on S3 which reduce to the same model in flat space limit. In the two-dimensional case there is no such ambiguity. Superfield classical actions of these theories are given explicitly together with the hidden supersymmetry transformations.

The present contribution is based essentially on our papers [7,8].

2 Superspaces as coset realizations

The analogs of two- and three-dimensional Euclidian Poinare groups are SU(2|1) and SU(2|1) x SU(2) for models on S2 and S3, respectively. Therefore, it is natural to introduce superspaces as the following coset spaces

2d

SU (211) U(1) x U (1)

3d

SU (2|1)

"UÔT

(1)

The superalgebra su(2|1) is spanned by four bosonic generators Ma, (a = 1, 2, 3) and R and four Grassmann-odd generators Qa, Qa, a = 1, 2. There are the following non-trivial commutation relations among these generators:

2i

[Ma, Mb] = -£abcMc , r

[Ma,Qa] = - 1(Ya)^Qp , [Ma,Qa] = -^Ya^QP , { Qa,Qp } = Yaap Ma + 1eap R,

[R, Qa] — — Qa , [R, Qa] — Qa .

(2)

Let us introduce superspace coordinates ZM = (xm, (9M), where m takes values from one to d with d = 2 for S2 and d = 3 for S3 case. The geometry of the supercosets (1) is entirely encoded in the supervielbein one-forms

EA

dzM Ema(z)

E A

(Ea,Ea,Ea).

(3)

Using the commutation relations (2), the expressions of the supervielbens can be found explicitly in a given coordinate system. Then, it is straightforvard to find the inverse supervielbein Eam and construct the co-variant derivatives Da = (Da, Da, Da ). The algebra (2) defines (anti)commutation relations among these derivatives. In 3d case they are

[Da, Db] — — r-£abcMc , 2r

[Da, Da] — — 2r (Ya^Dß , [Da, Da] — — 2r (Ya)a^ß ,

{Da, Dß} — ^Da — 1 YaßMa + ^aßR, {Da, Dß} — {Da, Dß} — 0 ,

while for the case of S2 we find [Da, Db] — r2ÊabM, [Da, Da] — — 2r (Ya)aDß , [Da, Da] — — 2r (Ya)a^ß , {Da, Dß } — «Yaß Da + 1Ya ß M + -^aß R ,

{Da, Dß} — {Da, Dß} — 0 .

S — ddxd26d26 EC

3 Superfield actions with minimal supersym-metry

3.1 Gauge superfield

Gauge theory in superspace is described by gauge superfield connections VA which covariantize the superspace derivatives, Va = Da + VA. The gauge connections obey superspace constraints which correspond to the following deformations of (anti)commutation relations (4) and (5), 3d:

{Va, Vp} = {Va, Vp} =0 ,

{Va, Vp} = iYaapVa - 2YSpMa + 1 £apR + ¿£apG ,

[Va, Vb] = - ^Mab + iFab , 2r

[Va, Va] = - (Ya)aVp - (7a ^p ,

[Va, Va] = -(7a)a^p + (Ya ^p , (9)

2d:

(4)

{Va, Vp} = {Va, VP} =0 ,

{Va, Vp} = i7apVa + 1 YapM + ^apR

+i£ap G + Yap H, [Va, V5] = ^£abM + iFab , [Va, Va] = - 2r (Ya)aVp - (Ya )QWp , [Va, Va] = -2r (Ya)a^p + (Ya )QWp . (10)

Here Wa (Wa) are (anti)chiral superfield strengths while G and H are linear,

V aWß — 0, V2G — V2H — 0, V 2G — V2 H — 0.

(5)

(11)

Here M = M3.

Any action in full superspace has the following form

(6)

where L is a superfield Lagrangian and E is the Berezinian of the supervielbein,

E = BerEMA . (7)

The latter has the following important property

J ddxd2dd2d E = 0 . (8)

As a consequence, the supercosets (1) have vanishing volume.

These superfield strengths can be expressed in terms of unconstrained gauge potential V as

G = 2Da(e-VDaeV), Wa = -4D2(e-VDaeV),

H = - 2Y3aßDa (e-VDpeV). (12)

Gauge transformation for V has standard form

V . ¿Ä V —¿A /io\

e —>■ e e e , (13)

where A and A are covariantly (anti)chiral local gauge parameters.

Superfield Yang-Mills action has the same form both on S2 and S3,

4

(14)

Ssym = —¡2tr j daxd20d2<9EG2 ,

where g2 is the gauge coupling constant of mass dimension [g] = 1/2 in 3d and [g] = 1 in 2d.

In 3d, one can consider also the Chern-Simons ac-

tion

ik

SCS = -—try dt J daxd29d20 E

xVa(e-tVVaetV)e-tV3tetV . (15)

For integer k this action is invariant under large gauge transformations and is topological. In 2d this action is not topological any more, but represents a BF-type interaction of component fields.

An interesting feature of 2d gauge theories is the possibility to construct the following gauge invariant action for the gauge superfield

SBF = k tr J dt J d2xd2dd2d E

x (Y3DDa(e-tVVetV)e-tVdt,etV , (16)

where k is a coupling constant of mass dimension -1. One can check that in components this action contains only BF-type interaction of fields and has no dynamical degrees of freedom.

3.2 Chiral superfield

By definition, the chiral superfield $ and its anti-chiral counterpart $ obey

Vn$ = 0 , Vn$ = 0 .

R$ = -q$ , R$ = q$ .

S = 4 / ddxd26d26 E $eV $

V

Vaea

2r. (YT^ .

(20)

Models with extended supersymmetry should involve extra Killing spinors, say r/a and f)a. However, there is a sign ambiguity in the Killing spinor equation for these spinors,

i

Vana = ± 2r (YV .

(21)

(17)

In general, such superfields can be charged under R generator,

It is important to note that 3d Killing spinors with opposite signs in (21) are independent while in 2d they are related to each other by the Y3 matrix. Thus, su-persymmetric field theories on S3 with extended su-persymmetry can have different versions which differ in the number of positive and negative Killing spinors. In this section we consider various such models, starting with the case of the N =4 SYM model.

4.1 N = 4 SYM with SU(2) x SU(2) R-symmetry

The N = 4 gauge multiplet in 3d (or N = (4, 4) in 2d) consists of the N =2 gauge superfield V and a chi-ral superfield $. The chiral superfield, in principle, can have arbitrary R-charge. Extra supersymmetry should transform these N = 2 superfields into each other.

In this subsection we consider extra supersymme-tries generated by Killing spinors which obey the Killing spinor equation with the same sign as (20). Such a Killing spinor appears as a component of chiral superfield Y,

y = a + ea na + e2b,

(22)

(18)

Classical action for the chiral superfield minimally interacting with gauge superfield V in some representation reads

(19)

where a and b are some constants corresponding to the parameters of internal symmetry group. The form of transformations of the superfields V and $ is fixed uniquely from the requirement of closure of such transformations to a supergroup:

ATV

i(Y£ - Y $),

+2 J ddxd2d E W ($) + 2y ddxd2d E W($),

where W($) is a superpotential. Note that for models on

S2

or S3 the R-charge of the superpotential is fixed to be —2 to balance the R-charge of the chiral measure.

In conclusion of this section we point out that su-perfield classical actions on S2 and S3 for gauge and matter superfields have a simple form which is very similar to usual flat space actions. They respect the full symmetry under the SU(2| 1) group by construction.

4 Models with extended supersymmetry

Superfield models constructed in the previous section are invariant under minimal supersymmetry generated by four Killing spinors ea and ea which obey

Sr$ = VaGVaY + -GY .

(23)

Together with the manifest supersymmetry group, these transformations form the group SU(2|2) x SU(2). The classical SYM action with this supersymmetry reads

Ssym4 = — 4tr J ddxd2dd2d E[G2 — e-V$eV$ r1

+ q dtVa(e-tVVaetV)e-tVdtetV 1 . (24)

2r J)

A novel feature of this action is the appearance of the last perm which has the form of the Chern-Simons action. This term is present for q = 0 and drops out in the flat limit r ^ to.

We point out that in the 2d case the classical action (24) describes the SYM model with N = (4, 4) supersymmetry.

i

r

4.2 N =4 SYM with U (1) x U (1) R-symmetry

Now let us consider Killing spinors na which obey (21) with minus sign. Using such a Killing spinor we construct the following transformations of N =2 su-perfields V and $

AnV = (0a - 1026»а)паФ - ГПаФ , r

¿П Ф = -¿^VaG,

¿П Ф = wfVaG.

(25)

One can check that these transformations (together with the manifest supersymmetry) form a supergroup SU(2|1) x SU(2|1) only if the R-charge of the chiral superfield is fixed as

RФ = - Ф,

RФ = Ф.

S = - tr J d3xd20d2<9 E(G2 - e_vФevФ).

(27)

S S

N=8

SYM

Sym Scs

Sym + Scs + spot, 4

— —2"tr j ddxd20d2<9 E(G2 - e_v Ф^ Ф,) : /•1 /•

4

tr

3rg2 JQ

xDa(e

S

pot

dt J ddxd20d2<9 E tv^ tv

)e

-tv DaetV )e

dtet

- 3g2 trj ddxd20E £ijfcФЛФд, Фй] + c.c.

RФi = - -Ф,.

г 3 '

R<ï>г

2Ф « 3

We point out that the action (29) describes the N = (8, 8) SYM model on S2 with the same form of the hidden supersymmetry transformations (29).

4.4 Gaiotto-Witten theory

In flat space the Gaiotto-Witten theory was introduced in [9]. In the N =2 superspace it is described by two N =2 gauge superfields V and V corresponding to two different gauge groups and two chiral superfields (a hypermultiplet) X+ and X_ in the bifundamental representation. The classical superfield action for this model on S3 reads

Sgw

Scs [V] - Scs[V] + Sx ,

(31)

(26)

The classical N = 4 SYM action with this supersym-metry has simple form

This action has U (1) x U (1) R-symmetry and does not possess an S2 analog.

4.3 N =8 SYM

Here we consider only the N =8 SYM model with Killing spinors obeying (20) with the same sign. In this case the N = 8 multiplet consists of the N = 2 gauge superfield V and an SU(3) triplet of chiral superfields Ф., i = 1, 2, 3. These superfields transform among each other by means of the extra N =6 supersymmetry as

STV = iT^ - iT, (28)

= VaGDaTi + 3rGYi + 2jV2(TjФк),

_. _ _ 2 - 1

Sтф = -VaGDaTг - — GTг - ^£ijkV2(Tj Фк),

where Ti is a triplet of superfield parameters of the form (22). The invariant N = 8 SYM action appears to be

SX = 4 try ddxd20d2 0 E(X+eV X+e_V

+X_e_VX_eV), (32)

where Scs is the Chern-Simons action (15). We find that this action is invariant under the following hidden supersymmetry transformation

AV = S X+X_ + £X_X+ , av = S x_X+ + SX+ X_,

<X± = ±V2(YXT), (33)

where X± and X± are covariantly (anti)chiral super-

fields, X+ = e_v X

ÎY+ — X+, ÎY— —

o-v

X_ = X_. In (33) the superfield parameter Y has the form (22) while S is related to the latter as follows

DaS= - ^ Da Y , k

DaS= - ^ Da Y . (34)

(29)

Note that these transformations correspond to the case of Killing spinors obeying (20) with the same sign which form together N = 4 supersymmetry. Note also that the action (31) has a two-dimensional analog which we refer to as the Gaiotto-Witten model on S2. The latter possesses N = (4,4) supersymmetry.

4.5 ABJM model

Finally, let us construct the classical action of the ABJ(M) theory [10-12] on S3. This model is similar to the Gaoitto-Witten theory, but it involves two hyper-multiplets, (X+i, Xj ), i = 1, 2, where X+i and Xj are chiral superfields in the bi-fundamental representation of the gauge group. We find the following generalization of this action on S3:

Sabjm = Sx =

Scs [V] - Scs[V] + Sx + Spot,

(35)

-v

Similarly to the N = 4 SYM model, this action has the Chern-Simons term. However, R-charges of all chiral superfields are fixed now

S

pot

(30)

4tr y ddxd20d2<9 E(X+ evX+ie

+X_e_vX_,eV) , 4ni С

—— tr / ddxd20 E(X+iXiX+jXj -Xi X+iXj X+j ) + c.c.

e

This action is invariant under the following hidden su- In two-dimensional case the action (35) corresponds persymmetry to a reduction of the ABJM model to S2 which pos-

8in . sesses N = (6,6) supersymmetry.

av = -—(rj x+i x- + tj x- x+ ),

к

fyj vi v T j у у ^ Acknowledgements

к • • • -

Ay = — — (YjiX- X+j + Yj X+ X-)

. = y2(Y jX ) This work was partially supported by the Padova

+ 2( i jj , University Project CPDA119349 and by the INFNSpe-

¿X+ = V (Yij). (36) cial Initiative ST&FI. Work of I. B. S. was also sup-

Here X±i and X±i are covariantly (anti)chiral super- ported by the RFBR grants No. 12-02-00121, 13-02-

fields while Yj is a quartet of chiral superfield param- 90430 and 13-02-91330 and by the LRSS grant No.

eters each of which is similar to (22). 88.2014.2.

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Received 14.11.2014

if. В. Самсонов СУПЕРПОЛЕВЫЕ МОДЕЛИ HA S2 И S3

Развиваются суперполевые методы для построения классических действий различных моделей с глобальной супер-симметриеи на S2 и S3. Вводятся суперпространства, основанные на факторпространствах вида SU(2|1)/U(1) и SU(211)/[U(1) х U(1)]. Показывается, что модели с расширенной суперсимметрией на S3 имеют различные неэквивалентные версии, обладающие одинаковым числом суперсимметрий, но основанные на различных супералгебрах. В частности, построены классические действия для N = 4 и N = 8 суперсимметричных моделей Янга-Миллса, а также теории Гайотто-Виттена и ABJM на S3. Для всех этих случаев рассмотрены аналогичные модели на S2.

Ключевые слова: суперпространство, суперподе Янга-Миллса.

Самсонов И. В., доктор физико-математических наук. Томский политехнический университет. Пр. Ленина, 30, 634050 Томск, Россия. Национальный институт ядерной физики. Via Marzolo 8, 35131 Padova, Италия. E-mail: [email protected]