(1,0) superconformal models in six dimensions
ABSTRACT We construct sixdimensional (1,0) superconformal models with nonabelian
gauge couplings for multiple tensor multiplets. A crucial ingredient in the
construction is the introduction of threeform gauge potentials which
communicate degrees of freedom between the tensor multiplets and the YangMills
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
YangMills multiplet. We briefly discuss the inclusion of hypermultiplets which
complete the field content to that of superconformal (2,0) theories.

Article: CFD based aerodynamic modeling to study flight dynamics of a flapping wing micro air vehicle
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ABSTRACT: The demand for small unmanned air vehicles, commonly termed micro air vehicles or MAV's, is rapidly increasing. Driven by applications ranging from civil searchandrescue missions to military surveillance missions, there is a rising level of interest and investment in better vehicle designs, and miniaturized components are enabling many rapid advances. The need to better understand fundamental aspects of flight for small vehicles has spawned a surge in high quality research in the area of micro air vehicles. These aircraft have a set of constraints which are, in many ways, considerably different from that of traditional aircraft and are often best addressed by a multidisciplinary approach. Fastresponse nonlinear controls, nanostructures, integrated propulsion and lift mechanisms, highly flexible structures, and low Reynolds aerodynamics are just a few of the important considerations which may be combined in the execution of MAV research. The main objective of this thesis is to derive a consistent nonlinear dynamic model to study the flight dynamics of micro air vehicles with a reasonably accurate representation of aerodynamic forces and moments. The research is divided into two sections. In the first section, derivation of the nonlinear dynamics of flapping wing micro air vehicles is presented. The flapping wing micro air vehicle (MAV) used in this research is modeled as a system of three rigid bodies: a body and two wings. The design is based on an insect called Drosophila Melanogaster, commonly known as fruitfly. The mass and inertial effects of the wing on the body are neglected for the present work. The nonlinear dynamics is simulated with the aerodynamic data published in the open literature. The flapping frequency is used as the control input. Simulations are run for different cases of wing positions and the chosen parameters are studied for boundedness. Results show a qualitative inconsistency in boundedness for some cases, and demand a better aerodynamic data. The second part of research involves preliminary work required to generate new aerodynamic data for the nonlinear model. First, a computational mesh is created over a 2D wing section of the MAV model. A finite volume based computational flow solver is used to test different flapping trajectories of the wing section. Finally, a parametric study of the results obtained from the tests is performed.01/2012;  SourceAvailable from: Christian Saemann[Show abstract] [Hide abstract]
ABSTRACT: We present and discuss explicit solutions to the nonabelian selfdual string equation as well as to the nonabelian selfduality equation in six dimensions. These solutions are generalizations of the 't HooftPolyakov monopole and the BPST instanton to higher gauge theory. We expect that these solutions are relevant to the effective description of M2 and M5branes.12/2013; 89(6).  SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: Very few AdS_6 x M_4 supersymmetric solutions are known: one in massive IIA, and two IIB solutions dual to it. The IIA solution is known to be unique; in this paper, we use the pure spinor approach to give a classification for IIB supergravity. We reduce the problem to two PDEs on a twodimensional space Sigma. M_4 is then a fibration of S^2 over Sigma; the metric and fluxes are completely determined in terms of the solution to the PDEs. The results seem likely to accommodate nearhorizon limits of (p,q)fivebrane webs studied in the literature as a source of CFT_5's. We also show that there are no AdS_6 solutions in elevendimensional supergravity.06/2014;
Page 1
arXiv:1108.4060v1 [hepth] 19 Aug 2011
MFA1136
(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, F69364 Lyon CEDEX 07, France
bGeorge P. and Cynthia W. Mitchell Institute
for Fundamental Physics and Astronomy
Texas A&M University, College Station, TX 778434242, USA
Abstract
We construct sixdimensional (1,0) superconformal models with nonabelian
gauge couplings for multiple tensor multiplets. A crucial ingredient in the
construction is the introduction of threeform 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 YangMills 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
2Nonabelian 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 sixdimensional
(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 nonabelian
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 YangMills gauge field
and the twoform gauge potentials of the tensor multiplets contains also threeform
gauge potentials. We therefore have an extended tensor gauge freedom with p = 0,1,2
pform 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 2form potential is a singlet, here it carries a representation of the local gauge
group, which is facilitated by the introduction of a 3form 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 onshell
with a particular set of e.o.m. For example the tensor multiplet field strength has to
satisfy its selfduality condition and the YangMills field strength is related to the field
strength of the threeform potentials by a firstorder duality equation. Consequently,
the threeform gauge potentials do not introduce additional degrees of freedom. They
communicate degrees of freedom between the tensor multiplets and the YangMills mul
tiplet. We also describe the extension of the tensor hierarchy to higher pform 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
YangMills 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 nonabelian 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 YangMills
theory in five dimensions. Recent attempts tried to basically rewrite fivedimensional
YangMills theory in six dimensions [23, 24] or introduced nonabelian gaugings at the
cost of locality [25]. Furthermore, it was recently proposed that the (2,0) theory is
identical to fivedimensional super YangMills theory for arbitrary coupling or compact
ification radius [26, 27]. It is not clear yet how one could obtain YangMills 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 nonabelian
hierarchy of pforms 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 nonabelian charged tensor fields
require the introduction of St¨ uckelbergtype couplings among the pforms of different
degree. In section 3, we extend the nonabelian vector/tensor system to a supersym
metric system. Closure of the supersymmetry algebra puts the system onshell and we
derive the modified field equations for the vector and tensor multiplets. In particular,
we obtain the firstorder duality equation relating vector fields and threeform 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 selfdual tensor multiplet as well as in cer
tain ‘nondecomposable’ 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 Nonabelian tensor fields in six dimensions
In this section, we present the general (nonabelian) couplings of vectors and anti
symmetric pform 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 threeform potentials. The specific couplings can be derived
successively and in a systematic way by building up the nonabelian pform 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 nonabelian couplings
between vector and tensor fields. In section 2.1, we extend the system to include also
fourform gauge potentials.
2.1 Minimal tensor hierarchy
The basic pform field content of the theories to be discussed is a set of vector fields Ar
and twoform gauge potentials BI
µ,
µν, that we label by indices r and I, respectively. In
4
Page 6
addition, we will have to introduce threeform gauge potentials that we denote by Cµνρr.
The fact that threeform 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 onshell
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 offshell, and the indices ‘r’ and ‘r’ might be taken as unrelated. Similarly,
throughout this section, the selfduality of the field strength of the twoform gauge
potentials, which is a key feature of the later sixdimensional dynamics, is not yet an
issue.
The full nonabelian field strengths of vector and twoform 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 dsymbol
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 nonabelian gauge transformations
rs= dI
(rs), and the tensors gIr, hr
Iinducing St¨ uckelbergtype 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
pform gauge fields can be used to gauge away some of the pforms, 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 pform 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¨ uckelbergtype 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 threeform gauge potentials, the free system of selfdual tensor multiplets does
not admit any nonabelian 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 nonvanishing, 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 bsymbol 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 nonvanishing hI
restrictive, it admits rather nontrivial 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 YangMills multiplet to an uncharged selfdual 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 vectortensor 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 nonabelian field strength H(4)
the second equation. In turn, its Bianchi identity is obtained from (2.8) as
µνρσ r of the threeform 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 fieldstrengths 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 pforms Ar
which in particular we have defined their nonabelian field strengths. Strictly speak
ing, in the entire system, only a subset of the threeform 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. Offshell on the other hand, the
tensor hierarchy may be extended to the full set of threeform potentials, which then
necessitates the introduction of fourform gauge potentials, etc.
For later use, we present in this section the results of the general tensor hierarchy for
the fourform 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 dsymbol 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 nonabelian 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
fiveform and sixform potentials together with their field strengths and nonabelian
gauge transformations.For the purpose of this paper we will only need the vec
tor/tensor system up to the fourform gauge potentials given above.
3 Superconformal field equations
In the previous section we have introduced the tensor hierarchy for pform 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 nonabelian 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 YangMills
multiplets (Ar
I, respectively. The index i = 1,2 indicates the Sp(1) Rsymmetry, the field Yij
denotes the auxiliary field of the offshell vector multiplets. In addition one has to
accommodate within this structure the threeform potential Cµνρrwhose presence was
crucial in the last section in order to describe nonabelian 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) selfdual tensor multiplet to a YangMills 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 YangMills 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 offshell pure YangMills
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.
9
Page 11
then comes as a nontrivial 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 YangMills multiplet by
itself, using the necessary tensor multiplet transformations, still closes offshell.
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 selfduality projector such that
only HI +
r.h.s. of these transformations has been generalized by the introduction of the general
dsymbol, 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 threeform potential Cµνρrwhich is contained in the
definition of HI
(2.10). Its presence has been vital in establishing the nonabelian bosonic vectortensor
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 YangMills 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 threeform 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 nontrivial and heavily relies on the extra terms arising from variation of
the threeform potential. In particular, the algebra closes only onshell 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 threeform gauge potentials. The
10
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 threefrom field
strength, originates in the closure of supersymmetry on the associated twoform 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 firstorder equations of motion for the YangMills 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 threeform 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 secondorder equation of Yang
Mills type for the vector fields Ar
inferred from closure of the supersymmetry algebra on the threeform gauge potentials
gKrCµνρr. The appearance of the YangMills 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 offshell or can alternatively be set onshell 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
11
Page 13
nonabelian 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 threeform
gauge potentials Cµνρrwhich are related by firstorder 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 ChernSimons 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 threeform 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 fourform 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 fourform potential is
∆C(4)
µνρσ α
= 2cαIJφ[I¯ ǫγµνρσχJ], (3.8)
Furthermore, supersymmetry of the field equations (3.7) induces the firstorder field
equations
1
5!εµνρλστkrαH(5)µνρλσ
α
= 2krα?cαIJ
?φIDµφJ− 2¯ χIγµχJ?− ct
µνρσ α is given by a firstorder duality equations,
αubJtv¯λuγµλv?
. (3.9)
This shows that the dynamics of C(4)
which relates these fourform potentials to the Noether current of some underlying
global symmetry. In particular, this firstorder equation ensures that the fourform
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 hyperK¨ ahler manifold Mh, more precisely superconformal
symmetry requires Mhto be a hyperK¨ ahler cone. The above presented nonabelian
theories can be extended to include gaugings of isometries on the hyperK¨ 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
hyperK¨ ahler cone is achieved by introducing an embedding tensor ϑrαthat encodes
the coupling of vector fields Ar
algebraic conditions
µto hyperK¨ ahler isometries Kαand is subject to the
fprsϑsα
= fβγαϑpβϑrγ,hr
Iϑrα= 0 , (3.10)
with the structure constants fαβγof the algebra of hyperK¨ 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 hyperK¨ 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 hyperK¨ 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)
13
Page 15
and characterize solutions that preserve some fraction of supersymmetry. These equa
tions show that a Lorentzinvariant 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 nulldirections. The fluctuation equations (3.15)
show that for these nulldirections 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 YangMills 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 CoulombHiggs 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 YangMills 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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