# (1,0) superconformal models in six dimensions

**ABSTRACT** We construct six-dimensional (1,0) superconformal models with non-abelian

gauge couplings for multiple tensor multiplets. A crucial ingredient in the

construction is the introduction of three-form gauge potentials which

communicate degrees of freedom between the tensor multiplets and the Yang-Mills

multiplet, but do not introduce additional degrees of freedom. Generically

these models provide only equations of motions. For a subclass also a

Lagrangian formulation exists, however it appears to exhibit indefinite metrics

in the kinetic sector. We discuss several examples and analyze the excitation

spectra in their supersymmetric vacua. In general, the models are

perturbatively defined only in the spontaneously broken phase with the vev of

the tensor multiplet scalars serving as the inverse coupling constants of the

Yang-Mills multiplet. We briefly discuss the inclusion of hypermultiplets which

complete the field content to that of superconformal (2,0) theories.

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**ABSTRACT:**We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0) supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings.05/2012; - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We first review self-dual (chiral) gauge field theories by studying their Lorentz non-covariant and Lorentz covariant formulations. Then, we construct a non-Abelian self-dual two-form gauge theory in six dimensions with a spatial direction compactified on a circle. This model reduces to the Yang-Mills theory in five dimensions for a small compactified radius R. This model also reduces to the Lorentz-invariant Abelian self-dual two-form theory when the gauge group is Abelian. The model is expected to describe multiple 5-branes in M-theory. We will discuss its decompactified limit, covariant formulation, BRST-antifield quantization and other generalizations.06/2012; - SourceAvailable from: Hitoshi Nishino[Show abstract] [Hide abstract]

**ABSTRACT:**We present a self-dual non-Abelian N=1 supersymmetric tensor multiplet in D=2+2 space-time dimensions. Our system has three on-shell multiplets: (i) The usual non-Abelian Yang-Mills multiplet (A_\mu{}^I, \lambda{}^I) (ii) A non-Abelian tensor multiplet (B_{\mu\nu}{}^I, \chi^I, \varphi^I), and (iii) An extra compensator vector multiplet (C_\mu{}^I, \rho^I). Here the index I is for the adjoint representation of a non-Abelian gauge group. The duality symmetry relations are G_{\mu\nu\rho}{}^I = - \epsilon_{\mu\nu\rho}{}^\sigma \nabla_\sigma \varphi^I, F_{\mu\nu}{}^I = + (1/2) \epsilon_{\mu\nu}{}^{\rho\sigma} F_{\rho\sigma}{}^I, and H_{\mu\nu}{}^I = +(1/2) \epsilon_{\mu\nu}{\rho\sigma} H_{\rho\sigma}{}^I, where G and H are respectively the field strengths of B and C. The usual problem with the coupling of the non-Abelian tensor is avoided by non-trivial Chern-Simons terms in the field strengths G_{\mu\nu\rho}{}^I and H_{\mu\nu}{}^I. For an independent confirmation, we re-formulate the component results in superspace. As applications of embedding integrable systems, we show how the {\cal N} = 2, r = 3 and {\cal N} = 3, r = 4 flows of generalized Korteweg-de Vries equations are embedded into our system.Nuclear Physics B 10/2012; 863(3):510–524. · 4.33 Impact Factor

Page 1

arXiv:1108.4060v1 [hep-th] 19 Aug 2011

MFA-11-36

(1,0) superconformal models

in six dimensions

Henning Samtlebena, Ergin Sezginb, Robert Wimmera

aUniversit´ e de Lyon, Laboratoire de Physique, UMR 5672, CNRS et ENS de Lyon,

46 all´ ee d’Italie, F-69364 Lyon CEDEX 07, France

bGeorge P. and Cynthia W. Mitchell Institute

for Fundamental Physics and Astronomy

Texas A&M University, College Station, TX 77843-4242, USA

Abstract

We construct six-dimensional (1,0) superconformal models with non-abelian

gauge couplings for multiple tensor multiplets. A crucial ingredient in the

construction is the introduction of three-form gauge potentials which com-

municate degrees of freedom between the tensor multiplets and the Yang-

Mills multiplet, but do not introduce additional degrees of freedom. Gener-

ically these models provide only equations of motions. For a subclass also

a Lagrangian formulation exists, however it appears to exhibit indefinite

metrics in the kinetic sector. We discuss several examples and analyze the

excitation spectra in their supersymmetric vacua. In general, the models

are perturbatively defined only in the spontaneously broken phase with the

vev of the tensor multiplet scalars serving as the inverse coupling constants

of the Yang-Mills multiplet. We briefly discuss the inclusion of hypermul-

tiplets which complete the field content to that of superconformal (2,0)

theories.

Page 2

Contents

1 Introduction

2

2Non-abelian tensor fields in six dimensions

2.1 Minimal tensor hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Extended tensor hierarchy . . . . . . . . . . . . . . . . . . . . . . . . .

4

4

7

3 Superconformal field equations

3.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Minimal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Extended model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 Adding hypermultiplets . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Supersymmetric vacua and excitation spectrum . . . . . . . . . . . . .

3.6 A model with adjoint tensor multiplets . . . . . . . . . . . . . . . . . .

8

9

10

12

13

13

15

4 Action

4.1

4.2

4.3

4.4

4.5

16

16

17

18

20

22

Conditions for existence of an action . . . . . . . . . . . . . . . . . . .

The action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Multiplet structure of excitations . . . . . . . . . . . . . . . . . . . . .

Example: SO(5) gauge group . . . . . . . . . . . . . . . . . . . . . . .

Example: Nilpotent gauge group . . . . . . . . . . . . . . . . . . . . .

5 Conclusions 23

A Conventions25

1

Page 3

1 Introduction

One of the discoveries of the seminal analysis in [1] is the existence of interacting quan-

tum field theories in five and six dimensions. Of particular interest are six-dimensional

(2,0) superconformal theories which are supposed to describe the low energy limit of

multiple coincident M5 branes.

However, no Lagrangian description for these theories is known and it is in general

believed that no such formulation exists: The M/string theory origin implies that these

theories have no free (dimensionless) parameter, which would allow a parametrization

to weak coupling and thus make the existence of a Lagrangian description plausible.

This conclusion was also drawn from symmetry properties which imply that tree level

amplitudes have to vanish [2]. In addition, these (2,0) theories consist of chiral tensor

multiplets and so far it has often been considered as impossible to define non-abelian

gauge couplings for such multiplets.

Regarding the first aspect the situation is similar to that of multiple M2 branes, as

it was before the recent developments that were triggered by the discovery of the three

dimensional N = 8 superconformal BLG model [3, 4]. The meaning of this N = 8

model in the M/string theory context is rather unclear, but subsequently a N = 6

superconformal theory (ABJM model) was formulated for an arbitrary number of M2

branes [5]. The decisive observation in [5] is that an orbifold compactification of the M

theory/supergravity background provides a dimensionless, though discrete, parameter

k which allows a parametrization to weak coupling and thus also a Lagrangian formu-

lation. The orbifold compactification breaks N = 8 supersymmetry down to N = 6

except for k = 1,2, where the theory is strongly coupled. The N = 6 ABJM model has

the same field content as the N = 8 multiplet and it has been argued that monopole

operators enhance the supersymmetry to N = 8 for k = 1,2 [6, 7] (for U(2) gauge

group see [8, 9]).

We take here an analogous route. Instead of focusing on (2,0) supersymmetry we

construct (1,0) superconformal models for interacting multiple tensor multiplets. One

major obstacle, the nonabelian gauging of the (self dual) tensor fields, is resolved by the

introduction of a tensor hierarchy [10, 11, 12] which besides the Yang-Mills gauge field

and the two-form gauge potentials of the tensor multiplets contains also three-form

gauge potentials. We therefore have an extended tensor gauge freedom with p = 0,1,2

p-form gauge parameters.

We then formulate essentially unique supersymmetry transformations for the vari-

ous fields, where we find a suitable extension of the structures introduced in [13]. While

in [13] the 2-form potential is a singlet, here it carries a representation of the local gauge

group, which is facilitated by the introduction of a 3-form potential that mediates cou-

plings between the tensor and vector multiplets. While the brane interpretation of our

models requires further investigation, it is worth mentioning that the field content of

the model in [13] is known to arise in the worldvolume description of D6 branes stretch

between NS fivebranes [14, 15, 16, 17, 18]. The closure of the supersymmetry algebra

2

Page 4

into translations and extended tensor gauge transformations puts the system on-shell

with a particular set of e.o.m. For example the tensor multiplet field strength has to

satisfy its self-duality condition and the Yang-Mills field strength is related to the field

strength of the three-form potentials by a first-order duality equation. Consequently,

the three-form gauge potentials do not introduce additional degrees of freedom. They

communicate degrees of freedom between the tensor multiplets and the Yang-Mills mul-

tiplet. We also describe the extension of the tensor hierarchy to higher p-form gauge

potentials and briefly discuss the inclusion of hypermultiplets which complete the field

content to that of superconformal (2,0) theories.

Consistency of the tensor hierarchy imposes a number of conditions on the possible

gauge groups and representations. We discuss several solutions of these conditions.

Generically these models provide only equations of motions, but for a subclass also a

Lagrangian formulation exists. In particular we find a Lagrangian model with SO(5)

gauge symmetry. However, the existence of a Lagrangian description necessarily im-

plies indefinite metrics for the kinetic terms. It is at the moment not clear if the

resulting ghost states can be decoupled with the help of the large extended tensor

gauge symmetry. This and other questions regarding the quantization of the theory

we have to leave for a further investigation. A general feature of all considered cases

is that the models are perturbatively defined only in the spontaneously broken phase

with the vev of the tensor multiplet scalars serving as inverse coupling constants of the

Yang-Mills multiplets.

To write down a Lagrangian for a self dual field strength is in general a formidable

task. For a single M5 brane, in which case the e.o.m. are known [19], this has been

done in [20, 21].

We consider these difficulties to be of a different category than

finding a superconformal non-abelian theory. When we formulate a Lagrangian we

understand that the first order duality equations are consistently imposed in addition

to the second order Lagrangian e.o.m., just as in the democratic formulation of ten-

dimensional supergravity [22].

Finally we want to comment on some recent attempts and proposals for the descrip-

tion of the (2,0) theory. The low energy description of the theory when compactified

on a small circle is expected to be given by the maximal supersymmetric Yang-Mills

theory in five dimensions. Recent attempts tried to basically rewrite five-dimensional

Yang-Mills theory in six dimensions [23, 24] or introduced non-abelian gaugings at the

cost of locality [25]. Furthermore, it was recently proposed that the (2,0) theory is

identical to five-dimensional super Yang-Mills theory for arbitrary coupling or compact-

ification radius [26, 27]. It is not clear yet how one could obtain Yang-Mills theories in

five dimensions from the models presented here (even when including hypermultiplets).

Clearly a mechanism more complicated than a trivial dimensional reduction has to be

considered.

The paper is organized as follows: in section 2 we present the general non-abelian

hierarchy of p-forms in six dimensions. We show that all couplings are parametrized

in terms of a set of dimensionless tensors that need to satisfy a number of algebraic

3

Page 5

consistency constraints. In particular, we find that non-abelian charged tensor fields

require the introduction of St¨ uckelberg-type couplings among the p-forms of different

degree. In section 3, we extend the non-abelian vector/tensor system to a supersym-

metric system. Closure of the supersymmetry algebra puts the system on-shell and we

derive the modified field equations for the vector and tensor multiplets. In particular,

we obtain the first-order duality equation relating vector fields and three-form gauge

potentials. In section 3.4 we sketch the extension of the model upon inclusion of hyper-

multiplets and gauging of their triholomorphic isometries. In section 3.5 we derive the

general conditions for maximally supersymmetric vacua and compute the fluctuation

equations by linearizing the equations of motion around such a vacuum. Finally, we

give in section 3.6 an explicit example with an arbitrary compact gauge group and

tensor fields transforming in the adjoint representation.

Section 4 presents the additional conditions on the couplings in order to allow for

a Lagrangian formulation. We give the full action in section 4.2. In section 4.3 we

calculate the fluctuation equations induced by the action and show that the degrees

of freedom arrange in the free vector and self-dual tensor multiplet as well as in cer-

tain ‘non-decomposable’ combinations of the two. We illustrate the general analysis

in sections 4.4 and 4.5 with two explicit models that provide solutions to the con-

sistency constraints with compact gauge group SO(5) and nilpotent gauge group N8,

respectively. Finally, we summarize our findings in section 5.

2 Non-abelian tensor fields in six dimensions

In this section, we present the general (non-abelian) couplings of vectors and anti-

symmetric p-form fields in six dimensions. While the standard field content of the

ungauged theories falls into vector and tensor multiplets, it is a general feature of

these theories that the introduction of gauge charges generically requires the introduc-

tion of and couplings to three-form potentials. The specific couplings can be derived

successively and in a systematic way by building up the non-abelian p-form tensor hi-

erarchy, as worked out in [10, 11, 12], see also [28, 29, 30] for some applications to the

specific 6D context. Rather than going again step by step through the derivation of

the general couplings, we directly present the final result as parametrized by a set of

constant tensors (generalized structure constants) that need to satisfy a system of al-

gebraic consistency equations (generalized Jacobi identities). In section 2.1 we present

the couplings for the minimal field content required to introduce non-abelian couplings

between vector and tensor fields. In section 2.1, we extend the system to include also

four-form gauge potentials.

2.1 Minimal tensor hierarchy

The basic p-form field content of the theories to be discussed is a set of vector fields Ar

and two-form gauge potentials BI

µ,

µν, that we label by indices r and I, respectively. In

4

Page 6

addition, we will have to introduce three-form gauge potentials that we denote by Cµνρr.

The fact that three-form potentials are labeled by an index r dual to the vector fields

is in anticipation of their dynamics: in six dimensions, these fields will be the on-shell

duals to the vector fields. For the purpose of this section however, the dynamics of

these fields is not yet constrained, the construction of the tensor hierarchy remains

entirely off-shell, and the indices ‘r’ and ‘r’ might be taken as unrelated. Similarly,

throughout this section, the self-duality of the field strength of the two-form gauge

potentials, which is a key feature of the later six-dimensional dynamics, is not yet an

issue.

The full non-abelian field strengths of vector and two-form gauge potentials are

given as

Fr

µν

≡ 2∂[µAr

ν]− fstrAs

µAt

ν+ hr

IBI

µν,

HI

µνρ

≡ 3D[µBI

νρ]+ 6dI

rsAr

[µ∂νAs

ρ]− 2fpqsdI

rsAr

[µAp

νAq

ρ]+ gIrCµνρr, (2.1)

in terms of the antisymmetric structure constants fstr= f[st]r, a symmetric d-symbol

dI

of different degree.1Covariant derivatives are defined as Dµ ≡ ∂µ− Ar

action of the gauge generators Xron the different fields given by Xr·Λs≡ −(Xr)tsΛt,

Xr· ΛI≡ −(Xr)JIΛJ, etc. The field strengths are defined such that they transform

covariantly under the set of non-abelian gauge transformations

rs= dI

(rs), and the tensors gIr, hr

Iinducing St¨ uckelberg-type couplings among forms

µXr with an

δAr

µ

= DµΛr− hr

IΛI

µ,

∆BI

µν

= 2D[µΛI

ν]− 2dI

rsΛrFs

µν− gIrΛµν r,

∆Cµνρr

= 3D[µΛνρ]r+ 3bIrsFs

[µνΛI

ρ]+ bIrsHI

µνρΛs+ ... ,(2.2)

where we have introduced the compact notation

∆BI

µν

≡ δBI

≡ δCµνρr− 3bIrsBI

µν− 2dI

rsAr

[µδAs

ν],

∆Cµνρr

[µνδAs

ρ]− 2bIrsdI

pqAs

[µAp

νδAq

ρ]. (2.3)

The ellipsis in the last line of (2.2) represent possible terms that vanish under projection

with gIr. This system is completely defined by the choice of the invariant tensors gIr,

hr

p-form gauge fields can be used to gauge away some of the p-forms, giving mass to

others by the St¨ uckelberg mechanism. However, for the general analysis of couplings,

we find it the most convenient to work with the uniform system (2.2) and to postpone

possible gauge fixing to the analysis of particular models.

Consistency of the tensor hierarchy requires that the gauge group generators in the

various representations are parametrized as

I, bIrs, dI

rs, and frst. It is obvious from (2.2) that the shift symmetry action on the

(Xr)st

(Xr)IJ

= −frst+ dI

= 2hs

rsht

I,

IdJ

rs− gJsbIsr, (2.4)

1We use canonical dimensions such that a p-form has mass dimension p and as a result all constant

tensors fstr, dI

I, are dimensionless.

rs, gIr, hr

5

Page 7

in terms of the constant tensors appearing in the system. The second relation exposes

an important feature of the tensor hierarchy: tensor fields can be charged under the

gauge group only if either hr

vanishing St¨ uckelberg-type couplings in the field strengths (2.1). This corresponds to

the known results [31, 32] that in absence of such couplings and the inclusion of addi-

tional three-form gauge potentials, the free system of self-dual tensor multiplets does

not admit any non-abelian deformations. On the other hand, the first relation of (2.4)

shows that in presence of hr

tion’ (Xr)stare not just given by the structure constants but acquire a modification,

symmetric in its indices (rs).

Furthermore, consistency of the system, i.e. covariant transformation behavior of

the field strengths (2.1) under the gauge transformations (2.2) requires the constant

tensors to satisfy a number of algebraic consistency constraints. A first set of con-

straints, linear in f, g, h, is given by

Ior gIrare non-vanishing, i.e. they require some non-

I, the gauge group generators in the ‘adjoint representa-

2?dJ

rtbIsu+ 2dK

r(udI

v)s− dI

rsdJ

uv

?hs

?hu

J

= 2fr(usdI

v)s− bJsrdJ

uvgIs,

?dJ

rsbIut+ dJ

rubKstδJ

IJ

= frsubIut+ frtubIsu+ gJubIurbJst, (2.5)

and ensures the invariance of the d- and the b-symbol under gauge transformations.

The remaining constraints are bilinear in f, g, h and take the form

f[pqufr]us−1

3hs

IdI

u[pfqr]u

hr

= 0 ,

IgIs

= 0 ,

frsthr

I− dJ

rsht

Jhr

I

= 0 ,

gJshr

KbIsr− 2hs

Ihr

KdJ

rs

= 0 ,

− frtsgIt+ dJ

rths

JgIt− gItgJsbJtr

= 0 . (2.6)

They may be understood as generalized Jacobi identities of the system: together with

(2.5) they ensure the closure of the gauge algebra according to

[Xr,Xs] = −(Xr)stXt, (2.7)

for the generators (2.4), as well as gauge invariance of the tensors f, g and h. The

first equation of (2.6) shows that the standard Jacobi identity is modified in presence

of a non-vanishing hI

restrictive, it admits rather non-trivial solutions and we will discuss explicit examples

of solutions in sections 3.6, 4.4, and 4.5, below. The system admits different abelian

limits with frst= 0 = gIrand either hr

(2.5), (2.6) are trivially satisfied. A slightly more general solution is given by vanishing

hr

particular choice dI

the Yang-Mills multiplet to an uncharged self-dual tensor multiplet as described in [13].

r. Even though the set of constraints (2.5), (2.6) looks highly

Ior dI

rsvanishing, in which the constraints

I= 0 = gIrwith frstrepresenting the structure constants of a Lie algebra. With the

rs= dIδrs, the vector-tensor system then reduces to the coupling of

6

Page 8

The covariant field strengths (2.1) satisfy the modified Bianchi identities

D[µFr

D[µHI

νρ]

=

1

3hr

IHI

µνρ,

νρσ]

=

3

2dI

rsFr

[µνFs

ρσ]+1

4gIrH(4)

µνρσ r, (2.8)

where the non-abelian field strength H(4)

the second equation. In turn, its Bianchi identity is obtained from (2.8) as

µνρσ r of the three-form potential is defined by

D[µH(4)

νρστ]r

= −2bIrsFs

[µνHI

ρστ]+ ... ,(2.9)

where the ellipsis represents possible terms that vanish under projection with gIr. We

finally note that the general variation of the field-strengths is given by

δFr

µν

= 2D[µδAr

ν]+ hr

I∆BI

µν,

δHI

µνρ

= 3D[µ∆BI

νρ]+ 6dI

rsFr

[µνδAs

ρ]+ gIr∆Cµνρr,

δH(4)

µνρσ r

= 4D[µ∆Cνρσ]r− 6bIrsFs

[µν∆BI

ρσ]+ 4bIrsHI

[µνρδAs

σ]+ ... , (2.10)

again with the ellipsis representing possible terms that vanish under projection with gIr.

2.2 Extended tensor hierarchy

The field content introduced in the last section were the p-forms Ar

which in particular we have defined their non-abelian field strengths. Strictly speak-

ing, in the entire system, only a subset of the three-form potentials have appeared,

defined by projection with the tensor gIras gIrCµνρr. As it turns out, this trunca-

tion is precisely the ‘minimal field content’ required in order to write down an action

and/or define a consistent set of equations of motion. Off-shell on the other hand, the

tensor hierarchy may be extended to the full set of three-form potentials, which then

necessitates the introduction of four-form gauge potentials, etc.

For later use, we present in this section the results of the general tensor hierarchy for

the four-form gauge potentials which we denote by C(4)

H(5)

µ, BI

µν, Cµνρr, for

µνρλαwith covariant field strength

α . The full version of the Bianchi identity (2.9) then reads

D[µH(4)

νρστ]r

= −2bIrsFs

[µνHI

ρστ]+1

5krαH(5)

µνρστ α, (2.11)

where now the field strength H(5)

α itself satisfies the Bianchi identity

D[µH(5)

νρλστ] α

=

10

3cαIJHI

[µνρHJ

λστ]−5

2ct

αsFs

[µνH(4)

ρλστ]t+ ··· ,(2.12)

up to terms vanishing under projection with the tensor krα. The new constant tensors

krα, cαIJ, and ct

αsare constrained by the relations

krαcαIJ

= hs

[IbJ]rs,krαct

αs= frst− bIrsgIt+ dI

rsht

I,gKrkrα= 0 ,(2.13)

7

Page 9

which extend the constraints (2.5), (2.6). As a consistency check, we note that equa-

tions (2.5), (2.6) imply the orthogonality relations

gKrhs

[IbJ]rs

= 0 ,

gKr?frst− gItbIrs+ ht

IdI

rs

?

= 0 , (2.14)

showing that (2.13) does not imply new constraints among the previous tensors. Fur-

thermore, consistency of the extended system requires an additional relation among b-

and d-symbol to be satisfied

bJr(sdJ

uv)

= 0 ,(2.15)

as also noted in [28]. The new tensor gauge transformations take the form

∆Cµνρr

= 3D[µΛνρ]r+ 3bIrsFs

[µνΛI

ρ]+ bIrsHI

µνρΛs− krαΛµνρα,

∆C(4)

µνρσ α

= 4D[µΛνρσ]α− 8cαIJH[I

[µνρΛJ]

σ]+ 6ct

αsFs

[µνΛρσ]t

+ ct

αsH(4)

µνρσ tΛs+ ... ,(2.16)

where the first equation completes the corresponding transformation law of (2.2) and

the second transformation is given up to terms that vanish under projection with the

tensor krα. Accordingly, the general variation of the non-abelian field strengths from

(2.11), (2.12) is given by

δH(4)

µνρσ r

= 4D[µ∆Cνρσ]r− 6bIrsFs

[µν∆BI

ρσ]+ 4bIrsHI

[µνρδAs

σ]+ krα∆C(4)

µνρσ α,

δH(5)

µνρστ α

= 5D[µ∆C(4)

νρστ] α− 10ct

αsFs

µν∆Cρστ] t− 20cαIJH[I

[µνρ∆BJ]

στ]

− 5ct

αsδAs

[µH(4)

νρστ] t+ ... .(2.17)

Continuing along the same line, the tensor hierarchy can be continued by introducing

five-form and six-form potentials together with their field strengths and non-abelian

gauge transformations.For the purpose of this paper we will only need the vec-

tor/tensor system up to the four-form gauge potentials given above.

3 Superconformal field equations

In the previous section we have introduced the tensor hierarchy for p-form gauge po-

tentials (p = 1,2,3) with the associated generalized field strengths (2.1) and Bianchi

identities (2.8). Gauge covariance w.r.t. the extended tensor gauge symmetry (2.2)

implies a number of conditions on the (dimensionless) invariant tensors and generators

of the gauge group (2.4)–(2.6), but otherwise does not contain any information about

the dynamics of theses fields.

The aim of this section is to complete the bosonic fields of the tensor hierarchy

into supersymmetry multiplets in order to obtain a non-abelian superconformal model

8

Page 10

for the (1,0) vector and tensor multiplets. With the given (bosonic) field content of

the tensor hierarchy (2.1), a supersymmetric tensor hierarchy will contain Yang-Mills

multiplets (Ar

I, respectively. The index i = 1,2 indicates the Sp(1) R-symmetry, the field Yij

denotes the auxiliary field of the off-shell vector multiplets. In addition one has to

accommodate within this structure the three-form potential Cµνρrwhose presence was

crucial in the last section in order to describe non-abelian charged tensor fields.

µ,λir,Yij r), and tensor multiplets (φI,χiI,BI

µν), labeled by indices r and

3.1Supersymmetry

The coupling of a single (1,0) self-dual tensor multiplet to a Yang-Mills multiplet was

introduced in [13] and as a first step we give the necessary generalization for a non-

abelian coupling of an arbitrary number of these tensor multiplets. To this end, we

introduce supersymmetry transformations such that they close into translations and

the extended tensor gauge symmetry (2.2) according to

[δǫ1,δǫ2] = ξµ∂µ+ δΛ+ δΛµ+ δΛµν, (3.1)

with field dependent transformation parameters for the respective transformations.

These parameters are given by

ξµ

Λr

ΛI

≡

= −ξµAr

= −ξνBI

= −ξρCρµν r− bIrsΛsBI

1

2¯ ǫ2γµǫ1,

µ,

νµ+ dI

µrsΛrAs

µ+ ξµφI,

µν−2

Λµν r

3bIrpdI

qsΛsAp

[µAq

ν],(3.2)

as will be shown shortly. With dI

ing algebra of [13].2The supersymmetry transformations for the Yang-Mills multiplet

are given by

rs= α′dIδrs, bIrs= 0, this reproduces the correspond-

δAr

µ

= −¯ ǫγµλr,

δλir

=

1

8γµνFr

µνǫi−1

2Yij rǫj+1

4hr

IφIǫi,

δYij r

= −¯ ǫ(iγµDµλj)r+ 2hr

I¯ ǫ(iχj)I. (3.3)

Here the generalization w.r.t. the transformations for the off-shell pure Yang-Mills

multiplet is parametrized by the constant tensor hr

of the tensor multiplets on the r.h.s. of the transformations. These additional terms

are necessary for the supersymmetry algebra to close to the generalized tensor gauge

symmetry (3.1), (3.2). E.g. the last term in δλiris required to produce the proper δΛµ

action in the commutator of supersymmetries on the vector field Ar

last term in δYij rensures the proper closure of the supersymmetry algebra on λir. It

Iand brings in the fields (φI,χiI,BI

µν)

µ. Likewise, the

2Note that in canonical dimensions, the tensor dI

rsis dimensionless.

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Page 11

then comes as a non-trivial consistency check, that the variation of this last term is

precisely what is needed for closure of the algebra on Yij r. Even though, fields from

the tensor multiplets appear in these transformation rules, the Yang-Mills multiplet by

itself, using the necessary tensor multiplet transformations, still closes off-shell.

Next we give the supersymmetry transformations of the tensor multiplet

δφI

δχiI

∆BI

= ¯ ǫχI,

=

= −¯ ǫγµνχI,

1

48γµνρHI

µνρǫi+1

4γµDµφIǫi−1

2dI

rsγµλir¯ ǫγµλs,

µν

∆Cµνρr

= −bIrs¯ ǫγµνρλsφI, (3.4)

where we have used the same notation (2.3) for general variation introduced in the

tensor hierarchy. We also note that γµνρǫiacts as a self-duality projector such that

only HI +

r.h.s. of these transformations has been generalized by the introduction of the general

d-symbol, and the inclusion of covariant field strengths and derivatives on the now

charged fields of the tensor multiplets. In particular, the important new ingredient in

these transformation rules is the three-form potential Cµνρrwhich is contained in the

definition of HI

(2.10). Its presence has been vital in establishing the non-abelian bosonic vector-tensor

system in the last section, and similarly, its presence turns out to be indispensable for

closure of the supersymmetry algebra here. To group it with the tensor multiplet in

(3.4) is a mere matter of convenience; with the same right it might be considered as a

member of the gauge multiplet (indeed, as mentioned before by its dynamics the three-

form potential will be the dual of the vector fields Ar

transformation (3.4), mixing Yang-Mills and tensor multiplet fields, displays its dual

role as a messenger between these two multiplets. Note that we have given in (3.4)

the supersymmetry transformation for the uncontracted three-form Cµνρr, although all

the explicit couplings only contain the contracted expression gK rCµνρr. We will come

back to this difference in the following.

Closure of the supersymmetry algebra on the tensor multiplet according to (3.1) is

now rather non-trivial and heavily relies on the extra terms arising from variation of

the three-form potential. In particular, the algebra closes only on-shell on the tensor

multiplets. In the search for new model or theory such a property may be considered

as feature that provides a certain uniqueness. We will discuss these equations and their

individual origin now in detail.

µνρ, see (A.1), is actually alive in δχiI. W.r.t. the couplings discussed in [13], the

µνρand contributing to its supersymmetry transformation according to

µ). The form of its supersymmetry

3.2Minimal model

We first investigate the equations of motion resulting from supersymmetrization of the

bosonic field content of the minimal tensor hierarchy of section 2.1. In particular, this

model includes only the projected subset gK rCµνρrof three-form gauge potentials. The

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Page 12

resulting tensor multiplet field equations are given by

HI −

µνρ

= −dI

rs¯λrγµνρλs,

γσDσχiI

=

1

2dI

rsFr

στγστλis+ 2dI

rsYij rλs

j+?dI

rshs

J− 2bJsrgIs?φJλir,

DµDµφI

= −1

2dI

rs

?Fr

µνFµν s− 4Yr

ijYij s+ 8¯λrγµDµλs?

?¯λrχJ− 3dI

− 2?bJsrgIs− 8dI

rshs

J

rshr

Jhs

KφJφK. (3.5)

The first equation, which imposes a self duality condition on the three-from field

strength, originates in the closure of supersymmetry on the associated two-form poten-

tial BI

field equation is obtained by the supersymmetry transformation of the χiI- equation.

The fact that the tensor fields are charged under the gauge group has rather non-

trivial consequences, namely supersymmetry variation of the field equations (3.5) in

turn implies the following first-order equations of motion for the Yang-Mills multiplets

µν. The closure on δχiIgives the fermionic equations of motion while the scalar

gKrbIrs

?Ys

µνφI− 2¯λsγµνχI?

?φIγµDµλs

ijφI− 2¯λs

(iχI

j)

?

= 0 ,

gKrbIrs

?Fs

=

1

4!εµνλρστgKrH(4)λρστ

r

,

gKrbIrs

i+1

2γµλs

iDµφI?

= gKrbIrs

?1

4Fs

+3

µνγµνχI

i+

1

24HI

µνργµνρλs

i− Ys

ijχj I

2hs

JφIχJ

i+1

3dI

uvγµλu

i¯λsγµλv?

.

(3.6)

The first equation is the algebraic equation for the auxiliary field Yij r, while the second

equation provides the anticipated duality of vector fields and three-form potentials by

relating their respective field strengths. In particular, derivation of this equation and

use of the Bianchi identity (2.9) gives rise to a standard second-order equation of Yang-

Mills type for the vector fields Ar

inferred from closure of the supersymmetry algebra on the three-form gauge potentials

gKrCµνρr. The appearance of the Yang-Mills dynamics (3.6) from supersymmetry of

the tensor field equations (3.5) is in strong contrast to the model of [13] (in which

effectively gKr= 0, and the tensor field are not charged) where the vector fields remain

entirely off-shell or can alternatively be set on-shell with field equations that do not

contain the tensor multiplet fields. Moreover, in the model of [13], an algebraic equation

analogous to the first equation of (3.6) is excluded by the appearance of an anomaly

in its supersymmetry variation (see also [33]). We should stress that in the present

model, such anomalies are actually absent due to the particular Fierz identities (A.6),

(A.7) in combination with the identity (2.15). I.e. the quartic fermion terms in the

supersymmetry variation of (3.6) cancel precisely, which yields a strong consistency

check of the construction.

To summarize, the system of equations of motion (3.5), (3.6) consistently trans-

forms into itself under supersymmetry. It describes a novel system of supersymmetric

µ. Equivalently, the first two equations of (3.6) can be

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Page 13

non-abelian couplings for multiple (1,0) tensor multiplets in six dimensions. The equa-

tions of motion contain no dimensionful parameter and hence the system is at least

classically (super)-conformal. A crucial ingredient to the model are the three-form

gauge potentials Cµνρrwhich are related by first-order duality equations to the vector

fields of the theory and thus do not constitute new dynamical degrees of freedom. This

is similar to the situation of Chern-Simons matter theories in the context of multiple

M2 branes [5], [3]. The actual model depends on the explicit choice of the gauge group

and representations and the associated invariant tensors of the gauge group which have

to satisfy the conditions (2.4)–(2.6). The task that remains is to find explicit solutions

for these constraints. We will discuss different examples in sections 3.6, 4.4 and 4.5

below.

3.3Extended model

The above described model represents the minimal field content and equations of mo-

tion, required for closure of the supersymmetry algebra and the supersymmetry of the

equations of motions. In particular, it relies on the projected subset gK rCµνρrof three-

form gauge potentials. Just as for the bosonic tensor hierarchy in section 2.2, one may

seek to extend the above supersymmetric system to the full set of three-form gauge

potentials. With the supersymmetry transformation of general Cµνρrgiven by (3.4),

closure of the supersymmetry algebra leads to the following uncontracted equations

bIrs

?Ys

µνφI− 2¯λsγµνχI?

?φIγµDµλs

ijφI− 2¯λs

(iχI

j)

?

= 0 ,

bIrs

?Fs

=

1

4!εµνλρστH(4)λρστ

r

,

bIrs

i+1

2γµλs

iDµφI?

= bIrs

?1

+ hs

4Fs

µνγµνχI

i+

1

24HI

µνργµνρλs

i− Ys

ijχj I+

J

?2φIχJ

i−1

2φJχI

i

?+1

3dI

uvγµλu

i¯λsγµλv?

,

(3.7)

In order to have this system close under supersymmetry it is necessary to introduce also

a four-form gauge potential. Consequently the tensor hierarchy has to be continued one

step further as described in section 2.2. The resulting supersymmetry transformation

of the four-form potential is

∆C(4)

µνρσ α

= 2cαIJφ[I¯ ǫγµνρσχJ], (3.8)

Furthermore, supersymmetry of the field equations (3.7) induces the first-order field

equations

1

5!εµνρλστkrαH(5)µνρλσ

α

= 2krα?cαIJ

?φIDµφJ− 2¯ χIγµχJ?− ct

µνρσ α is given by a first-order duality equations,

αubJtv¯λuγµλv?

. (3.9)

This shows that the dynamics of C(4)

which relates these four-form potentials to the Noether current of some underlying

global symmetry. In particular, this first-order equation ensures that the four-form

gauge potentials do not constitute new dynamical degrees of freedom.

12

Page 14

3.4 Adding hypermultiplets

Another possible extension of the supersymmetric model presented above is the inclu-

sion of hypermultiplets. As is well known, global supersymmetry requires the hyper-

scalars to parametrize a hyper-K¨ ahler manifold Mh, more precisely superconformal

symmetry requires Mhto be a hyper-K¨ ahler cone. The above presented non-abelian

theories can be extended to include gaugings of isometries on the hyper-K¨ ahler cone

along the lines of [34, 35, 36], from which the additional couplings and in particular the

resulting scalar potential can be inferred. While we defer the details of this extension

to another publication, here we only sketch a few relevant elements of the construc-

tion. Within in the above construction, gauging of triholomorphic isometries on the

hyper-K¨ ahler cone is achieved by introducing an embedding tensor ϑrαthat encodes

the coupling of vector fields Ar

algebraic conditions

µto hyper-K¨ ahler isometries Kαand is subject to the

fprsϑsα

= fβγαϑpβϑrγ,hr

Iϑrα= 0 , (3.10)

with the structure constants fαβγof the algebra of hyper-K¨ ahler isometries. On the

other hand, in the presence of hypermultiplets, the vector multiplet equations of motion

(3.7) allow for a consistent modification, in particular in the Y -field equation as

bIrs

?Ys

ijφI− 2¯λs

(iχI

j)

?

= krαPij

α, (3.11)

with the constant tensor krαfrom (2.13), and the moment maps Pij

triholomorphic hyper-K¨ ahler isometries. It is only by means of this algebraic equation

for Ys

the existence of an action eventually leads to the identification

αassociated with the

ijthat the hyperscalars enter the tensor multiplet field equations. Further requiring

krα

= ϑrα, (3.12)

i.e. relates the gauging of hyper-K¨ ahler isometries to a modification of the vector and

tensor multiplet field equations.

3.5 Supersymmetric vacua and excitation spectrum

We study now supersymmetric vacua for the minimal model of section 3.2 and the

excitation spectrum in such vacua, i.e. the linearized field equations. The algebraic

equation for the vector field strength, the second equation in (3.6), indicates that

the expectation value of the tensor multiplet scalar φIserves as an (inverse) coupling

constant. This notion will become more evident in the subsequent sections where we

discuss models which provide a Lagrangian. Consequently, the perturbative analysis is

limited to the spontaneously broken phase where φIhas a (large) expectation value.

The Killing spinor equations of the theory (4.3) are obtained from (3.3), (3.4)

0

!≡ δλir=

1

8γµνFr

µνǫi−1

2Yij rǫj+1

4hr

IφIǫi,

0

!≡ δχiI=

1

48γµνρHI +

µνρǫi+1

4γµDµφIǫi, (3.13)

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Page 15

and characterize solutions that preserve some fraction of supersymmetry. These equa-

tions show that a Lorentz-invariant solution preserving all supersymmetries corre-

sponds to setting the scalar fields to constant values φI

0satisfying

φI

0hr

I

= 0 , (3.14)

and setting all other fields to zero. Expanding the scalar fluctuations as φI≡ φI

ϕIand imposing the condition (3.14) one obtains at the linearized level for the field

equations (3.5), (3.6) the system:

0+

(dBI+ gIrCr)−= 0 ,NI

rYr

ij= 0 ,

/ ∂ χiI+ 2NI

rλir= 0 ,NI

rdAr− gIr ∗dCr= 0 ,

✷ϕI− NI

r∂ · Ar= 0 ,NI

r/ ∂λir= 0 ,(3.15)

where we have defined the matrices

Krs ≡ φI

0bIrs,NI

r ≡ gIsKsr. (3.16)

and used that NI

tion (3.14).

rhr

J= 0, by the first identity in (2.14) and the susy vacuum condi-

Unbroken gauge symmetry.

(3.14) the vector gauge transformations ΛrXr are broken down to the subgroup of

transformations ΛrXrwhich satisfy

For a generic supersymmetric vacuum which satisfies

XrJIφJ

0= − NI

r

!= 0, (3.17)

where the index r labels the subset of unbroken generators (2.4). The rest of the

extended tensor gauge symmetry (2.2) remains intact. Consequently, in the case that

the gauge group is not completely broken, the matrix NI

the matrix Krs, always has some null-directions. The fluctuation equations (3.15)

show that for these null-directions the fields of the corresponding vector multiplets

drop out of this perturbative analysis. This is nothing else than the above mentioned

observation that the perturbative analysis is valid only in the spontaneously broken

phase and that the unbroken sector of the Yang-Mills multiplet is (infinitely) strongly

coupled and perturbatively not visible. This part of the spectrum decouples and should

be integrated out for a proper treatment.

In general it is rather difficult to break the gauge symmetry completely with a

single scalar field. The addition of hypermultiplets as sketched in section 3.4 may

offer additional possibilities in this directions. This is for example comparable with the

situation of N = 2 SQCD, for which mixed Coulomb-Higgs phases with vev’s for vector

multiplet and hypermultiplet scalars exist where the theory is completely higgsed. In

such a case there would be regions in the moduli space of vacua where the complete

spectrum of the models discussed here is perturbatively accessible. For the extended

models of section 3.3 on the other hand, the coupling of the Yang-Mills multiplet is

given by the matrix Krswhich may have less null directions than the matrix NI

r, and for invertible gIralso

r.

14

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