264 Md. 307 (1972)
286 A.2d 115
KATZ ET AL.
v.
HOLSINGER, AN INFANT ETC. ET AL.
[No. 174, September Term, 1971.]
Court of Appeals of Maryland.
Decided January 20, 1972.
The cause was argued before HAMMOND, C.J., and BARNES, McWILLIAMS, FINAN, SMITH and DIGGES, JJ.
Thomas G. Andrew, with whom were Rollins, Smalkin, Weston & Andrew on the brief, for appellants.
Richard R. Beauchemin, with whom were Arnold, Beauchemin & Huber on the brief, for appellees.
BARNES, J., delivered the opinion of the Court.
The appellants, Francis Katz and Clara I. Griffin, the owners and landlords of a second floor apartment with a second story porch located at 1239 West Baltimore Street, Baltimore City (the subject property)  the defendants *309 below, appeal from a judgment obtained in the Baltimore City Court (Liss, J.) on May 18, 1971, by the appellees and plaintiffs below, Zella Mae Holsinger, an infant, by Nellie Esther Holsinger (tenant of the second floor apartment), her mother and next friend, and Mrs. Holsinger, for $64,000 and $7,500, respectively, with interest from February 10, 1971 (the date of the verdict), and costs. The verdict for Mrs. Holsinger under Count Two of the declaration for medical expenses and loss of services of her infant daughter was for $15,000, which, however, was reduced to $7,500 by a remittitur granted by the trial judge and accepted by Mrs. Holsinger. The principal questions presented to us are whether the lower court erred (1) in not directing a verdict for the defendants in that (a) there allegedly was no legally sufficient evidence that the defendant landlords had contracted, for a consideration, to repair the tenants' porch railing, (b) the failure of the landlords to repair the porch railing was not the proximate cause of the accident, and (c) the tenant had waived any contractual right to require the landlords to repair the porch railing and (2) in various instructions to the jury (a) in regard to permanent injury and (b) in regard to the landlords' duty to repair. In declining the motion of the landlords for a judgment n.o.v. or, in the alternative, for a new trial, Judge Liss filed a helpful written memorandum. Finding no error, we will affirm the judgments.
The jury could have found the following from the testimony. In 1960 Mrs. Holsinger and her husband became tenants of the subject property. There was no written lease. The rent was paid at first by the week. After the husband died on June 6, 1966, the rent was paid by the month. The only means of access to the subject property was by a wooden stairway in the rear of the premises leading to a porch on the second floor from which access was given to the second floor apartment itself. The porch was made of wood (as were the steps), with a wooden railing and a number of wooden balusters attached to the wooden railing and to the porch floor. The *310 landlords occupied the first floor as a men's store operated by Mr. Katz. The evidence offered on behalf of the plaintiffs indicated that at the time the subject property was rented in 1960, the landlords  through Mr. Katz  agreed to make the repairs to the subject property as a condition of the tenancy. During the time the husband was living, whenever he made repairs to the subject property, the landlords paid him for making those repairs. Indeed, Mr. Katz testified on cross-examination that the landlords "did all of the repairs to Mrs. Holsinger's apartment, inside and outside as part of her tenancy" and that "always throughout her tenancy anything to be done, we always did." He further stated that he did not expect Mrs. Holsinger to make any repairs to the porch.
Mrs. Holsinger testified that she noticed, approximately a week and one-half prior to July 21, 1967, the day of the accident, that two balusters were missing. She tried to put them back in place but was not able to do this because they were rotten. Mrs. Rebecca Nellie Evans, a neighboring tenant, noticed that the balusters were missing about three weeks before the day of the accident. About a week before the accident, she told Mr. Katz about the situation and told him that if he did not nail boards across the remaining balusters, "we are going to find little Zella Mae laying [sic] on the concrete down below." Mrs. Evans stated that Mr. Katz replied that he was "going to take care of it just as soon as he got time." The day before the accident, Mrs. Evans told Mrs. Griffin that she had some nails and a hammer and that she was going to "put a board across there to keep her (the baby Zella, then two years old) from falling out." According to Mrs. Evans, Mrs. Griffin said that she was not to do this "because Francis [Katz, the other landlord] is going to have it fixed."
On the morning of the accident, Mr. Katz, accompanied by a carpenter, Joseph Dotterweich, went to the porch to repair it. Because other workmen were on the roof *311 of the building making repairs to it, no repairs were then made, even though Mr. Katz admitted on cross-examination that "it was too serious a thing to put off."
Later on July 21, 1967, the infant plaintiff, Zella, after having been given a piece of pie by her mother, and while her mother was cutting a piece of pie for herself, suddenly went out of the door of the apartment and fell through the hole in the porch railing to the concrete pavement below. She suffered a fracture to her skull and to both wrists. Approximately six months after the accident, Zella suffered convulsions. She underwent electro-encephalogram tests at the University Hospital which revealed certain brain damage resulting in Zella's becoming an epileptic.
Additional facts will be given when the questions are later discussed. In considering the facts upon the motion of the defendants for a directed verdict or for a motion for judgment n.o.v., we resolve any conflicts of fact in favor of the plaintiffs and give them the benefit of any reasonable inferences deducible from the facts favorable to them. Bosley v. Grand Lodge of Ancient Free & Accepted Masons of Md., 263 Md. 303, 316, 283 A.2d 587, 596 (1971) and cases therein cited.
We will discuss the questions raised in the order already indicated.

1 (a)
The landlords correctly pointed out to us that at common law there was no implied covenant to repair and no warranty of the fitness of the leased premises for occupancy. As Judge (later Chief Judge) Hammond stated for the Court in Farley v. Yerman, 231 Md. 444, 448, 190 A.2d 773, 775 (1963):
"Although at common law there was no implied covenant to repair and no warranty of the fitness for occupancy of leased premises, a tenant now may maintain an action for injuries sustained as a result of an uncorrected defect in *312 rented quarters if there was a contractual obligation to repair the particular defect and a reasonable opportunity to correct it. Under these circumstances the landlord has the obligation to use reasonable care to make the needed repairs with reasonable diligence; and if he does not, wilfully or negligently, and harm ensues as a result, there arises a tort liability, subject to the usual rules as to proof of causation and the absence of contributory negligence on the part of the tenant. McKenzie v. Egge, 207 Md. 1, 6-7."
We cited Farley with approval and followed it in Sacks v. Pleasant, 253 Md. 40, 251 A.2d 858 (1969). We cited Sacks with approval in Kleiman v. Mono of Maryland, Inc., 254 Md. 548, 255 A.2d 393 (1969). See 1 Tiffany, Landlord and Tenant § 87, p. 134 (3rd ed. 1939); Restatement (Second) of Torts § 357, p. 241 (1965); 51C C.J.S. Landlord & Tenant § 366 (1), page 923.
The landlords contend that there was insufficient evidence that they had contracted, for a consideration, to make repairs to the subject property, including repairs to the porch. We do not agree with this contention. The testimony, not only of the tenant, but also by one of the landlords, was sufficient to present a jury issue in regard to the making of an agreement by the landlords with the tenants to repair the subject property as part of the original oral lease and supported by the consideration of that original leasing of the premises, as well as in regard to the landlord having sufficient notice of the particular defect and a reasonable opportunity to repair it.

1 (b)
The landlords next contend that, assuming for the argument that an enforceable contract to repair exists, nevertheless the failure of the landlords to repair the porch railing was not a proximate cause of the accident on the theory that the mother's allegedly negligent failure *313 to prevent the infant Zella from going out on the porch  although not imputed to the infant from the fact of parenthood or custodianship by virtue of the provisions of Code (1969 Replacement Vol.) Art. 75, § 2  was so extraordinary in the situation as to constitute a superseding cause of Zella's injuries, relying, in part, on the Restatement (Second) of Torts, supra, § 447, comment g at p. 480, as well as on Farley v. Yerman, supra, 231 Md. 444, 190 A.2d 773 (1963) and Barnes v. Housing Authority, 231 Md. 147, 189 A.2d 100 (1963).
In our opinion, Farley v. Yerman is dispositive of this issue contrary to the contention of the landlords.
As in the present case, Farley involved a young child (four years of age) of a tenant. The landlord urged that his negligence, if any, was merely passive and potential while the failure of the parents to make physical arrangements which would have prevented access to the flames of a gas log or to keep their children far enough away from them to prevent the accident was the moving and effective cause of the harm which made their negligence the only proximate cause of the injury. Judge (now Chief Judge) Hammond, stated for the Court:
"We think the rule urged upon us is not applicable here. The landlord not only knew of the danger in the abstract but in the context of the almost continual presence of very young children in close proximity to the danger. He cannot disclaim realization that the parents might not be able to do more than they did, or that they would not do more, or that their acting as they did was not to be deemed so unusual or extraordinary as to be unreasonable or, finally, that the foreseeable acts or omissions of the parents would be but normal responses to a situation created by his own conduct.
"All this being so, any negligence of the parents was not a superseding cause of the harm *314 which the landlord's negligent conduct was a substantial factor in bringing about. Restatement, Torts, Sec. 447. See also Sec. 452."
* * *
"In McKenzie v. Egge, supra, (at page 11 of 207 Md.) the Court, in rejecting the landlord's claim that the adult tenant was guilty of contributory negligence or assumption of risk, as a matter of law, in using the defective and dangerous porch of which she had complained and which collapsed and injured her, adopted Sec. 473 of Restatement, Torts: `If the defendant's negligence has made the plaintiff's exercise of a right or privilege impossible unless he knowingly exposes himself to a risk of bodily harm, the plaintiff is not guilty of contributory negligence in so doing unless the risk is unreasonable.' Miller v. Graff, 196 Md. 609, held that a child of four cannot be guilty of contributory negligence. Cf. State, use of Taylor v. Barlly, 216 Md. 94, 101-102. Under Code (1957), Art. 75, Sec. 2, any negligence of her parents would not be imputable to the infant, Virginia Farley, from the fact of their parenthood." (231 Md. at 449-50, 190 A.2d at 775-76.)
In the present case, the landlords not only knew that the balusters were missing and of the theoretical danger involved in this defective condition, but Mr. Katz  one of the landlords  was specifically warned of the actual danger to the two year old infant Zella prior to the accident. He must necessarily have realized that a parent cannot watch an active infant every moment and that in an instant she might leave her mother, go on the porch contrary to the mother's instructions and fall through the open hole in the porch railing. The acts or omissions of the mother in the instant case were not "so unusual or extraordinary as to be unreasonable" and the foreseeable acts or omissions by the mother "would be but the normal *315 responses" to a situation created by the failure of the landlords to repair the porch railing.

1 (c)
Little need be said in regard to the contention of the landlords that, in any event, the tenant had waived any contractual right to have the landlords repair the porch railing because of the testimony of Mrs. Holsinger that sometimes Mr. Katz when asked to make repairs sometimes did them and sometimes he did not do them. This does not, of itself, indicate any waiver of the obligation, but rather a possible breach of the obligation by the landlords. There was evidence, as we have previously observed, that at times Mrs. Holsinger's husband would make the necessary repairs and the landlords would pay him later for doing this. It was at least a jury issue in regard to whether there had been any "waiver" of the obligation of the landlords to repair, and the jury verdict for the plaintiffs concludes this issue against the landlords.

2.
The landlords challenge the propriety of certain instructions of the lower court.

(a)
Judge Liss in his charge to the jury stated in regard to damages:
"In assessing damages you may consider these factors, the physical condition of the plaintiff before the accident as compared with her physical condition following the accident as a consequence of the injuries which you find she may have sustained, any mental and physical pain and suffering which the plaintiff has sustained as a result of the injuries, the extent, if any, to which the injuries are permanent in character, the medical expenses which have been incurred, and will and may in the future be incurred, for *316 treatment of the injuries as a result of the accident.
"Now, after considering all of these elements, if you find for the plaintiff  I point out to you that you do not arrive at this portion of the case until you have first been satisfied by a fair preponderance of the evidence that the defendant was negligent, and that his negligence was the proximate cause of the accident, but if you so find, then in assessing damages you should allow such damages as will fairly and justly compensate the plaintiff for the injuries, losses, and disabilities, which she has sustained as the result of the accident."
The landlords excepted to this instruction because they contended that there was no legally sufficient evidence in the case "to show a continuing requirement of treatment or medication for the child, and there is no evidence in the case legally sufficient to show that there is any permanent disability, or any permanent injury as a result of the accident."
Dr. Frank Schuster, a specialist in neurology and who treated Zella, testified that after Zella had suffered convulsions some six months after the accident, she underwent electro-encephalogram tests and pneumo-encephalogram tests at the University Hospital. In reviewing these tests, as well as the medical records at Franklin Square Hospital where Zella was treated immediately after the accident and after examining Zella, herself, he arrived at a diagnosis. He stated:
"Well, it certainly led us to feel as strongly as we possibly could we were dealing with some form of post-traumatic seizure disorder."
Dr. Schuster had previously testified:
"Q. As a result of the EEG, Doctor, what is the significance of the findings as you have related them? A. Well, the significance showed there was definite slow wave activity on the left *317 side, which would be considered abnormal for that child's age at that time.
"Q. Do you mean brain damage, Doctor? A. It would be interpreted as indicative of brain damage, yes."
He stated later that in considering Zella's history:
"Therefore, although I cannot be certain I must state that the probability exists that the trauma and subsequent seizure disorder could be related. That is about as much as I could really say about the case I stated."
Still later, Dr. Schuster testified:
"Q. Doctor, do you have an opinion that you can state with reasonable medical probability as to whether or not these injuries are calculated to be permanent?
"(Mr. Andrew) Objection.
"(The Court) Well, let's get the answer first. Do you have an opinion, Doctor? A. As to the probability of its resulting from the injury?
"Q. (Mr. Beachemin) Its being permanent? A. Being permanent. I think because of the clinical changes on the air study that there is a good chance there may be some permanent injury, although I cannot be sure, and the only way one would ever really know is to do a follow-up air study later on, which is not a procedure that one would do. There's a definite hazard in doing a repeat air study. And just to find out what is going on, unless you run into trouble of course  unless there are clinical changes."
* * *
"Q. Doctor, from your history you have taken, and your examination of this child, and your review of the hospital records, can you state *318 whether or not with reasonable medical probability this child can be classified as an epileptic?
A. If you use those terms?
"Q. Yes, that term. A. Yes, sir.
"Q. And could you state, Doctor, whether that is a permanent condition? A. That is difficult to really state, as I say we have a long period yet to go during puberty, and so forth, to see whether or not this reoccurs. Now, I would say that 50 per cent of the patients who have these seizures, whatever the cause may be, will have them for the rest of their lives, but another 50 per cent may not have them, and would disappear."
In deciding whether the jury should be instructed that there was no evidence from which it could find that Zella had suffered permanent injuries the trial court should resolve in her favor all conflicts in the evidence, should assume the truth of all evidence and all inferences that may naturally and legitimately be deduced from the evidence which tend to support her right to recover for permanent injuries. Baer Brothers v. Keller, 208 Md. 556, 558, 119 A.2d 410, 411 (1956).
Applying this approach to the facts, we are of the opinion that the trial court properly declined to charge the jury that there was no legally sufficient evidence that Zella had suffered permanent injuries. Dr. Schuster's testimony was, in our opinion, more certain and definite in regard to the causal connection between the trauma resulting from the accident and Zella's injuries than was the testimony of the attending physician who testified for the plaintiff in Hughes v. Carter, 236 Md. 484, 204 A.2d 566 (1964) which this Court held to be sufficient to warrant the submission of causal connection to the jury.
There was also, in our opinion, sufficient evidence of permanency of Zella's injuries. From Dr. Schuster's testimony, it appears that Zella, with reasonable medical *319 probability, can be classed as an epileptic. He testified that it would be necessary to wait until puberty to see whether epileptic seizures occur and that 50% of the patients who have them, have them for the rest of their lives. Under all of the circumstances, it was for the jury to determine whether Zella's injuries were permanent. The trial court properly instructed the jury in this regard.

(b)
Finally, the landlords contend that the trial court incorrectly charged the jury in regard to the duty of the landlords to repair. The trial court instructed the jury on this matter, as follows:
"Under the Common Law there is no duty on the part of the landlord to make repairs to a premises which he rents to a tenant unless there is some specific agreement by him to make repairs, but there can be a duty and duty will arise on his part under certain circumstances, even if there is not even a written or specific agreement to make repairs. If there is in fact a defective condition in the premises such as the condition which has been testified to in this case, and if the evidence shows that the landlord was given notice that this condition existed, and if the landlord agrees to repair that condition, and on the strength of that promise the tenant remains on the premises and continues to stay there and pay the rent, but if the landlord fails, after having been given reasonable notice and having had a reasonable opportunity to make the repairs, if he fails to do so, and as a consequence the tenant is injured because of that failure, then this would constitute an obligation on the part of the landlord, and would render him negligent for such failure."
The landlords excepted to this portion of the charge on the grounds that (1) there was no evidence that the *320 tenant remained on the premises relying on the promise to repair and there was no showing of the payment of rent between the time of the promise to repair and the date of the accident and (2) there was no evidence of any consideration to support the promise to repair.
We have already considered these questions under the discussions in regard to the denial by the trial court of the landlords' motions for a directed verdict and for a judgment n.o.v., or, in the alternative, for a new trial. As we have indicated, there was, in our opinion, sufficient evidence of consideration for the promise of the landlords to repair. There was no prejudicial error in the trial court's instruction in this regard.
Judgment affirmed, the appellants to pay the costs.


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package com.eucalyptus.autoscaling.common.internal.activities;

import java.util.Date;
import javax.persistence.Column;
import javax.persistence.Embeddable;
import javax.persistence.Temporal;
import javax.persistence.TemporalType;
import com.eucalyptus.crypto.util.Timestamps;

/**
 *
 */
@Embeddable
public class ActivityCause {

  @Temporal( TemporalType.TIMESTAMP)
  @Column(name = "metadata_timestamp", nullable = false, updatable = false)
  private Date timestamp;

  @Column(name = "metadata_cause", nullable = false, updatable = false)
  private String cause;

  public ActivityCause() {
  }

  public ActivityCause( final String cause ) {
    this( new Date(), cause );
  }

  public ActivityCause( final Date timestamp,
                        final String cause ) {
    this.timestamp = timestamp;
    this.cause = cause;
  }

  public Date getTimestamp() {
    return timestamp;
  }

  public void setTimestamp( final Date timestamp ) {
    this.timestamp = timestamp;
  }

  public String getCause() {
    return cause;
  }

  public void setCause( final String cause ) {
    this.cause = cause;
  }

  public String toString() {
    return "At " + Timestamps.formatIso8601Timestamp( getTimestamp() ) + " " + getCause() + ".";
  }
}


---
bibliography:
- 'Master.bib'
---

[**Ergin Sezgin and Linus Wulff**]{}

[[*George and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,\
Texas A&M University, College Station, TX 77843-4242, USA*]{}]{}

We determine the off-shell $N=1$ supersymmetry transformation rules for a tensor-Yang-Mills system in which the tensor field transforms in a nontrivial representation of the Yang-Mills group, and there is an additional vector multiplet in the same representation. We then show that this system describes a massive tensor multiplet coupled to Yang-Mills in which the additional vector multiplet is eaten by the tensor field. Next, we construct an action which consists of four separately off-shell supersymmetric pieces and find that integrating out the auxiliary fields gives rise to an infinite number of higher order couplings of the scalar field belonging to the tensor multiplet. We describe the dualization of the massive tensor to a massive vector, thereby ending up with an on-shell supersymmetric nonabelian massive vector multiplet coupled to an off-shell Yang-Mills multiplet which we refer to as the supersymmetric Proca-Yang-Mills system. This description of massive vectors is contrasted with the standard one obtained through the Higgs mechanism.

Introduction
============

In this paper, we construct $N=1, D=4$ off-shell supersymmetric couplings of $n_V$ vector multiplets and $n_T$ tensor multiplets in such a way that the combined system is invariant under a semi-simple gauge group $G$ and the tensor fields transform in a non-trivial representation of $G$. The bosonic sector of this construction is based on the tensor hierarchy formalism [@deWit:2005hv] and it implies that, up to a number of tensor multiplets which are singlets of $G$, the system describes an off-shell Yang-Mills multiplet coupled to an off-shell massive tensor multiplet. The case of a single massive tensor, singlet under $G$, has already been constructed by other methods in [@D'Auria:2004sy; @Louis:2004xi], and partial results have been obtained for the nonabelian case in [@Nishino:2012sm]. In addition to an off-shell invariant containing the kinetic terms, we construct two additional off-shell invariants, one of which contains a mass term and another containing a topological “theta” term for the 2-form potential. We find that the elimination of the Yang-Mills multiplet auxiliary fields leads to an infinite number of terms in even powers of the tensor multiplet scalar fields. This nonpolynomial structure is due to the nonabelian nature of the massive tensors, and these terms disappear in the abelian case. We also show how the full off-shell $N=1, D=4$ supersymmetry transformations for the tensor-Yang-Mills system follow from a similar system in $D=6$ with $N=(1,0)$ supersymmetry [@Samtleben:2011fj; @Samtleben:2012mi] upon dimensional reduction on a 2-torus followed by a truncation.

It is known that a massive tensor field in $4D$ is on-shell dual to a massive vector field. This dualization has been carried out for the system where all 2-form potentials are singlets of $G$ [@D'Auria:2004sy; @Louis:2004xi]. Here, we generalize the dualization procedure to the case in which the 2-form potentials are in a nontrivial representation of $G$, thereby ending up with the couplings of on-shell nonabelian massive vector multiplets to off-shell Yang-Mills multiplets, which we will refer to as the supersymmetric Proca-Yang-Mills system.

It is interesting to compare this model with the supersymmetric Yang-Mills-Higgs system in which the massive vector multiplet arises by means of Higgs mechanism. The latter models contain also massive scalar multiplets. We show that these can’t be truncated while preserving supersymmetry. Thus, the Proca-Yang-Mills system presented here provides an alternative Higgsless model for a description of massive vectors coupled to Yang-Mills. However, the tree-level scattering amplitude grows with energy for the longitudinal components of the massive vectors which means break-down in unitarity at high energies, unlike the situation in the Yang-Mills-Higgs model where the additional couplings of the Higgs scalars restore unitarity [@Bell:1973ex; @LlewellynSmith:1973ey; @Cornwall:1974km]. Motivated by an attempt to fix the tree-level unitarity problem in a bosonic Higgsless model where new couplings involving a 2-form potential are considered [@Lahiri:2011ic], in Appendix B we extend the four parameter supersymmetric tensor-Yang-Mills system described in Section 3 by introducing two new off-shell invariants, which is possible if we take the 2-form potential in the adjoint representation. In this extended model new couplings relevant to the unitarity issue arise though a further study is required to determine whether they are sufficient to restore tree-level unitarity.

The tensor hierarchy and supersymmetry
======================================

Consider the coupling of $n_V$ vector and $n_T$ tensor off-shell $N=1, D=4$ supermultiplets consisting of fields && { A\_\^r, \^r,D\^r } , r=1,...,n\_V ,\
&& { B\_\^I, \^I,\^I } ,I=1,...,n\_T , where the spinors are Majorana. The full covariant non-abelian field strengths are [@deWit:2005hv] $$\begin{aligned}
{\cal F}_{\mu\nu}^r &\equiv&
2 \partial_{[\mu} A_{\nu]}^r - f_{st}{}^r A_\mu^s A_\nu^t + h^r{}_I\,B_{\mu\nu}^I
\;,\nonumber\\[.5ex]
{\cal H}_{\mu\nu\rho}^I &\equiv& 3 D_{[\mu} B_{\nu\rho]}^I +
6 \, d^I{}_{rs}\,  A_{[\mu}^r \partial^{\vphantom{r}}_\nu A_{\rho]}^s
- 2 f_{pq}{}^s d^I{}_{rs}\, A_{[\mu}^r A_\nu^p A_{\rho]}^q\ ,
\label{defF}\end{aligned}$$ in terms of the antisymmetric structure constants $f_{st}{}^r=f_{[st]}{}^r$, a symmetric $d$-symbol $d^I{}_{rs}=d^I{}_{(rs)}$, and the tensors $h^r{}_I$ inducing general Stückelberg-type couplings among forms of different degree. We use canonical dimensions such that a $p$-form has mass dimension $p$ and as a result all constant tensors $f_{st}{}^r$, $d^I{}_{rs}$ and $h^r{}_I$, are dimensionless. The covariant derivatives are defined as $D_\mu \equiv \partial_\mu - A_\mu^r X_r$ with an action of the gauge generators $X_r$ on the different fields given by $X_r \cdot \Lambda^s \equiv - (X_{r})_{t}{}^s \Lambda^t$, $X_r \cdot \Lambda^I \equiv - (X_{r})_{J}{}^I \Lambda^J$, etc. The field strengths (\[defF\]) are defined such that they transform covariantly under the set of non-abelian gauge transformations [@deWit:2005hv] $$\begin{aligned}
\delta A_\mu^r &=& D_\mu \Lambda^r - h^r{}_I \Lambda_\mu^I
\;,\nonumber\\[.5ex]
\delta B_{\mu\nu}^I &=& 2 d^I{}_{rs}\,A_{[\mu}^r \,\delta A_{\nu]}^s2 + D_{[\mu} \Lambda_{\nu]}^I -2\, d^I{}_{rs} \,\Lambda^r {\cal F}_{\mu\nu}^s\ .
\label{gaugesym}\end{aligned}$$ This vector/tensor gauge system is completely defined by the choice of the invariant tensors $h^r{}_I$, $d^I{}_{rs}$, and $f_{rs}{}^t$. Consistency of the tensor hierarchy, i.e. covariance of the field strengths (\[defF\]) requires that the gauge group generators in the various representations are given by [@deWit:2005hv] $$\begin{aligned}
(X_{r})_{s}{}^t &=&(X^{\rm V}_{r})_{s}{}^t~\equiv~ -f_{rs}{}^t + h^t{}_I\,d^I{}_{rs}\ ,
\nonumber\\
(X_{r})_{I}{}^J &=& (X^{\rm T}_{r})_{I}{}^J~\equiv~ 2\, d^J{}_{rs} h^s{}_I\ .
\label{genpar}\end{aligned}$$ Further constraints follow from closure of the algebra (or generalized Jacobi identities) $[X_r, X_s] = -(X_{r})_{s}{}^t \,X_t$ and gauge invariance of the tensors $h^r{}_I$ and $d^I{}_{rs}$. Upon the use of these invariance conditions together with (\[genpar\]), the generalized Jacobi identities take the equivalent form [@deWit:2005hv; @Samtleben:2011fj; @Samtleben:2012mi] $$\begin{aligned}
{}
f_{rs}{}^t h^r{}_I - d^J{}_{rs}\,h^t{}_J h^r{}_I &=& 0\ ,
\label{c2}
{\\[0.2cm]}
f_{[pq}{}^u f_{r]u}{}^s - \frac13 h^s{}_I\, d^{I}{}_{u[p} f_{qr]}{}^u &=& 0\ ,
\label{c3}
{\\[0.2cm]}
h^s{}_I h^r{}_K \, d^J{}_{rs} &=& 0 .
\label{c4}\end{aligned}$$

For an arbitrary matrix $h^r{}_I$, we can choose a basis in the space of vector and two-form tensor fields according to a split $A_\mu^r \longrightarrow \{A_\mu^\alpha , A_\mu^a\}$ and $B_{\mu\nu}{}^I \longrightarrow \{B_{\mu\nu}{}_{a'} , B_{\mu\nu}{}^a\}$, such that the matrix $h^r{}_I$ takes diagonal form [@Samtleben:2012mi] $$\begin{aligned}
h^r{}_I &=&
 \left(
\begin{array}{cc}
h^\alpha{}^{b'} & h^\alpha{}_b \\
h^{a}{}^{b'} & h^a{}_b
\end{array}
\right)
~=~
\left(
\begin{array}{cc}
0 & 0 \\
0 & \delta^a_b
\end{array}
\right)\ ,
\label{r1}\end{aligned}$$ with indices $a=1, \dots, {\rm rank}(h)$ and indices $\alpha$, and $a'$ labeling the complement of the spaces of vector and tensor fields, respectively. In this basis the constraints (\[c2\]) – (\[c3\]) translate as follows: the components $f_{\alpha\beta}{}^\gamma$ are the structure constants of a Lie algebra $\mathfrak{g}$, satisfying standard Jacobi identities. Moreover, we find that [@Samtleben:2012mi] $$\begin{aligned}
 f_{ab}{}^c &=& 0\ ,\quad f_{ab}{}^\alpha=0\ ,\quad
f_{a\alpha}{}^\beta = 0\ ,\qquad f_{\alpha a}{}^b =-\frac12\,(T_\alpha)_a{}^b\ ,
\nn\\
d^c{}_{ab} &=& 0\ ,\qquad d^b{}_{\alpha a} = \frac12\,(T_\alpha)_a{}^b\ ,\quad
\label{r4}\end{aligned}$$ where $T_\alpha$ are the generators of the Lie algebra $\mathfrak{g}$ in some representation ${\cal R}$, i.e.$[T_\alpha, T_\beta] = f_{\alpha\beta}{}^\gamma \,T_\gamma$, with dimension ${\rm dim} \,{\cal R}={\rm rank}(h)$ . Together, we deduce that the generators (\[genpar\]) take the form [@Samtleben:2012mi] $$\begin{aligned}
X_\alpha^{\rm V} &=&
 \left(
\begin{array}{cc}
-f_{\alpha\beta}{}^\gamma &\;\;\; -f_{\alpha\beta}{}^b +d^b{}_{\alpha\beta} \\
0 & (T_\alpha)_a{}^b
\end{array}
\right)\ ,
\qquad X_a^{\rm V} ~=~ 0\ ,
\label{genV}\end{aligned}$$ in the vector sector, and $$\begin{aligned}
X_\alpha^{\rm T} &=&
 \left(
\begin{array}{cc}
 0 & 0  \\
2d_{b'}{}_{\alpha a}&(T_\alpha)_a{}^b
\end{array}
\right)\ , \qquad X_a^{\rm T} ~=~ 0\ ,
\label{genT}\end{aligned}$$ in the tensor sector. Thus, the generalized Jacobi identity reduces to $[X_\alpha, X_\beta] = f_{\alpha\beta}{}^\gamma \,X_\gamma$. To summarize, we have solved the original system of constraints by an explicit choice of basis and without any loss of generality with [(\[r1\])]{} and [(\[r4\])]{}, where all non-vanishing tensors are invariant under the action of the generators (\[genV\]) and (\[genT\]), forming the Lie algebra $\mathfrak{g}$ with structure constants $f_{\alpha\beta}{}^\gamma$.

So far the solution of the constraints reviewed above is general. At this point, we will restrict to the case of a [*semi-simple*]{} Lie algebra ${\mathfrak{g}}$ and [*non-trivial*]{} representations $T_\alpha$. In this case, by proper choice of basis, the matrices $X_\alpha$ from (\[genV\]), (\[genT\]) can be taken to be block-diagonal, i.e.$$\begin{aligned}
f_{\alpha\beta}{}^b &=&0 ~=~d^b{}_{\alpha\beta}\ ,\qquad
d_{b'}{}_{\alpha a}~=~0\ ,
\label{css}\end{aligned}$$ and the generators take the form $$\begin{aligned}
X_\alpha^{\rm V} &=&
 \left(
\begin{array}{cc}
-f_{\alpha\beta}{}^\gamma & 0 \\
0 & (T_\alpha)_a{}^b
\end{array}
\right)\ ,
\qquad
X_\alpha^{\rm T} ~=~
 \left(
\begin{array}{cc}
 0& 0 \\
0 & (T_\alpha)_a{}^b
\end{array}
\right)\end{aligned}$$ which shows that the representation ${\cal R}'$ carried by the the 2-form potentials $B_{\mu\nu a'}$ is trivial.

In summary, the only non-vanishing components of the $h,f,d$ tensors are $$\begin{aligned}
 f_{\alpha\beta}{}^\gamma\ ,\quad h^a{}_b=\delta^a_b\ , \quad
 f_{\alpha a}{}^b =-\frac12\,(T_\alpha)_a{}^b\ ,\quad
 d^b{}_{\alpha a} = \frac12\,(T_\alpha)_a{}^b\ ,\quad
d_{c'\alpha\beta}=d_{c'}\eta_{\alpha\beta}\ ,
\label{a1}\end{aligned}$$ where $f_{\alpha\beta}{}^\gamma$ and $\eta_{\alpha\beta}$ are the structure constants and Cartan-Killing form of a semi-simple Lie algebra $\mathfrak{g}$, respectively, and $d_{c'}$ are arbitrary constants. The resulting model has the fields $$\begin{aligned}
(A_\mu^\alpha, \lambda^\alpha,D^\alpha)\ ,\qquad
(A_\mu^a, \lambda^{a}, D^a)\ ,\qquad  (B_{\mu\nu}^a, \chi^{a}, \phi^a)\ ,
\label{fc1}\end{aligned}$$ and a set of gauge singlet tensor multiplet fields $(B_{\mu\nu a'}, \chi_{a'}, \phi_{a'})$ where the indices $(\alpha, a,a')$ label the adjoint representation, an irreducible representation ${\cal R}$ and singlets of $\mathfrak{g}$, respectively. The field equations for the singlet tensor multiplet of fields can be easily deduced from the results for non-abelian tensor multiplets, and therefore we shall leave them out.

The explicit bosonic field strengths are now given by[^1] $$\begin{aligned}
{\cal F}_{\mu\nu}^\alpha &=&
2 \partial_{[\mu} A_{\nu]}^\alpha - f_{\beta\gamma}{}^\alpha A_\mu^\beta A_\nu^\gamma ~\equiv~ { F}_{\mu\nu}^\alpha\ ,
\label{def1}\\[.5ex]
{\cal F}_{\mu\nu}^a &=&  2 D_{[\mu} A_{\nu]}^a -  (T_{\alpha})_{b}{}^a A_{[\mu}^\alpha A_{\nu]}^b
+ B_{\mu\nu}^a ~\equiv~ {\mathcal B}_{\mu\nu}^a\ ,
\label{def2}\\[.5ex]
{\cal H}_{\mu\nu\rho}^a &=& 3 D_{[\mu} {\mathcal B}_{\nu\rho]}^a\ .
\label{FBH}\end{aligned}$$ The vector and tensor gauge-transformations are given by $$\begin{aligned}
\delta A^\alpha_\mu&=&D_\mu\Lambda^\alpha\ ,
\nonumber\\
\delta A^a_\mu&=&D_\mu\Lambda^a-\Lambda^a_\mu\ ,
\nonumber\\
\delta B^a_{\mu\nu}&=&2D_{[\mu}\Lambda^a_{\nu]}-T_{\alpha b}{}^a\left(\Lambda^\alpha\mathcal B^b_{\mu\nu}+\Lambda^bF^\alpha_{\mu\nu}-A^\alpha_{[\mu}\delta A^b_{\nu]}
-A^b_{[\mu}\delta A^\alpha_{\nu]}\right)\ .\end{aligned}$$ This implies that the massive tensor field $\mathcal B$ transforms as a Yang-Mills covariant field strength $$\delta\mathcal B^a_{\mu\nu}=-T_{\alpha b}{}^a\Lambda^\alpha\mathcal B^b_{\mu\nu}\ .$$ The scalar fields $\phi^a$ transform the same way as $\mathcal B^a_{\mu\nu}$ under the gauge transformations, and their covariant derivative takes the form D\_\^a=\_\^a+A\_\^T\_[b]{}\^a\^b . \[eq:Dphi\]

Turning to supersymmetry, the results summarized above for the bosonic tensor-Yang-Mills system shows that the coupling problem we started with amounts to constructing the coupling of the following multiplets &&   (A\_\^,\^,D\^) , 2 && ([B]{}\_\^a, \^[a]{}, \^a, \^[a]{}, D\^a) . Note that the tensor field $B_{\mu\nu}^a$ has eaten the vector $A_\mu^a$ to become the massive tensor $\mathcal B_{\mu\nu}^a$ through the Stückelberg mechanism.

The supersymmetry transformation rules of the off-shell Yang-Mills multiplet are well known and are given by $$\begin{aligned}
\delta A_\mu^\alpha&=&-{\bar\epsilon}\gamma_\mu\lambda^\alpha\ ,
\nonumber\\
\delta\lambda^\alpha&=&\frac{1}{8}F_{\mu\nu}^\alpha\,\gamma^{\mu\nu}\epsilon+\frac{i}{2}D^\alpha\,\gamma_5\epsilon
\nonumber\\
\delta D^\alpha&=&\frac{i}{2}{\bar\epsilon}\gamma_5 {\slashed D} \lambda^\alpha\ .
\label{YM}\end{aligned}$$ As for the supersymmetry transformation rules for the massive tensor multiplet, so far they have been determined fully only for an abelian version of the multiplet. Here we present the full result for the nonabelian case, by which we mean that all the members of the multiplet, including the tensor field, carry the same and nontrivial representation of the semi-simple gauge group $G$. In addition to employing the Yang-Mills covariant field strengths described above, and covariantizing derivatives of the fermions as well by employing the Yang-Mills gauge field $A_\mu^\alpha$, we find that we need higher order fermionic terms in the transformation rule for the tensorino in which the Yang-Mills gaugino fields appear explicitly. To determine the full result, we establish the off-shell closure of the supersymmetry algebra on all the members of the massive tensor-Yang-Mills multiplet, and we thus find[^2] $$\begin{aligned}
\delta\mathcal B^a_{\mu\nu}&=&2{\bar\epsilon}\gamma_{[\mu}D_{\nu]}\lambda^a-\bar\epsilon\gamma_{\mu\nu}\chi^a
\nonumber\\
\delta\phi^a&=&{\bar\epsilon}\chi^a
\nonumber\\
\delta\chi^a&=&\frac{1}{24}\mathcal H^a_{\mu\nu\rho}\,\gamma^{\mu\nu\rho}\epsilon
+\frac{1}{4}{\slashed D} \phi^a\,\epsilon
-\frac{1}{2}T_{\alpha b}{}^a\,(\gamma^\mu\lambda^{(\alpha})\,{\bar\epsilon}\gamma_\mu\lambda^{b)}
+\frac{1}{4}T_{\alpha b}{}^a\,(\gamma^\mu\gamma_5\epsilon)\,{\bar\lambda}^\alpha\gamma_\mu\gamma_5\lambda^b
\nonumber\\
\delta\lambda^a&=&\frac{1}{8}\mathcal B^a_{\mu\nu}\,\gamma^{\mu\nu}\epsilon
+\frac{i}{2}D^a\,\gamma_5\epsilon+\frac{1}{4}\phi^a\,\epsilon
\nonumber\\
\delta D^a&=&\frac{i}{2}{\bar\epsilon}\gamma_5 {\slashed D}\lambda^a
-\frac{i}{2}{\bar\epsilon}\gamma_5\chi^a\ .
\label{eq:tensorYM}\end{aligned}$$ Though we will only work in terms of $\mathcal B_{\mu\nu}^a$ we give for completeness the supersymmetry transformation of $B^a_{\mu\nu}$ and $A^a_\mu$: $$\begin{aligned}
\delta B^a_{\mu\nu}&=&
-{\bar\epsilon}\gamma_{\mu\nu}\chi^a
-T_{\alpha b}{}^aA^\alpha_{[\mu}\,{\bar\epsilon}\gamma_{\nu]}\lambda^b
-T_{\alpha b}{}^aA^b_{[\mu}\,{\bar\epsilon}\gamma_{\nu]}\lambda^\alpha\nonumber\\
\delta A^a_\mu&=&-{\bar\epsilon}\gamma_\mu\lambda^a\ .\end{aligned}$$ In Appendix A, we describe an alternative derivation of (\[eq:tensorYM\]) by a dimensional reduction from six dimensions, followed by a consistent truncation, of $N=(1,0)$ superconformal tensor-Yang-Mills system described in [@Samtleben:2011fj; @Samtleben:2012mi]. It is also useful to note that in the abelian limit of the supersymmetry transformation [(\[eq:tensorYM\])]{}, the $\epsilon\lambda^2$ terms in $\delta\chi^a$ are absent and all the covariant derivatives become ordinary partial derivatives.

Action for nonabelian massive tensor-Yang-Mills system
======================================================

To write an action we must introduce also a massive tensor multiplet transforming in the conjugate representation, unless the representation happens to be self-conjugate. These fields will be denoted by lower indices, and thus we have ([B]{}\_[a]{}, \_a, \_a, \_a, D\_a) . \[cmtv\] The supersymmetry transformations of these are obtained from those given above by hermitian conjugation according to the rules: V\^a = (V\_a)\^ ,(T\_[b]{}\^a)\^= -T\_[a]{}\^b , where $V_a$ is a generic field. We now wish to construct a Lagrangian for the massive tensor coupled to Yang-Mills. The off-shell $N=1$ super Yang-Mills Lagrangian is the standard one. We construct three new off-shell invariants involving the massive tensor multiplet and its interaction with the off-shell Yang-Mills. The final result is $$\mathcal L =\mathcal L_{\mathrm{bos}}+\mathcal L_{\mathrm{ferm}}\ ,
\label{master}$$ where $$\begin{aligned}
\mathcal L_{\mathrm{bos}} &=&-\frac{1}{4}F^\alpha_{\mu\nu}F_\alpha^{\mu\nu}+2D^\alpha D_\alpha
-\frac{1}{6}\mathcal H^a_{\mu\nu\rho}\mathcal H_a^{\mu\nu\rho}
-D_\mu\phi^aD^\mu\phi_a
\nn{\\[0.2cm]}
&&{}
-\frac{1}{2}T_{\alpha b}{}^a \left(\phi^b\mathcal B^{\mu\nu}_a \mathcal F^\alpha_{\mu\nu}
+8\phi_aD^bD^\alpha -(a\leftrightarrow b) \right)
\nn{\\[0.2cm]}
&&{}
-\frac{1}{2} m^2 \left(\mathcal B^a_{\mu\nu}\mathcal B_a^{\mu\nu}+2\phi^a\phi_a-8D^aD_a\right)
\nn{\\[0.2cm]}
&&{}
-\frac{1}{8} \theta m^2\left(\varepsilon^{\mu\nu\rho\sigma}\mathcal B^a_{\mu\nu}\mathcal B_{\rho\sigma\,a}
-16\phi^aD_a + h.c.\right)\ ,
\label{eq:bos}\end{aligned}$$ and $$\begin{aligned}
\mathcal L_{\mathrm{ferm}} &=&-2{\bar\lambda}^\alpha{\slashed D}\lambda_\alpha
-4{\bar\chi}^a {\slashed D} \chi_a-4m^2 {\bar\lambda}^a{\slashed D}\lambda_a
\nn{\\[0.2cm]}
&&
+4m^2\left( {\bar\lambda}^a\chi_a +{\bar\lambda}_a\chi^a \right)
+4i\theta m^2\left( {\bar\lambda}^a\gamma_5\chi_a +{\bar\lambda}_a\gamma_5\chi^a\right)
\nn{\\[0.2cm]}
&&
-T_{\alpha b}{}^a
\Big[
F^\alpha_{\mu\nu}\,{\bar\lambda}^b\gamma^{\mu\nu}\chi_a
+\mathcal B^b_{\mu\nu}\,{\bar\lambda}^\alpha\gamma^{\mu\nu}\chi_a
+\frac{1}{3}\mathcal H_{a\,\mu\nu\rho}\,{\bar\lambda}^\alpha\gamma^{\mu\nu\rho}\lambda^b
\nonumber\\
&&{}
-6\phi^b\,{\bar\chi}_a\lambda^\alpha
+2D_\mu\phi_a\,{\bar\lambda}^\alpha\gamma^\mu\lambda^b
-4\phi_a\,{\bar\lambda}^b {\slashed D} \lambda^\alpha
+4iD^b\,{\bar\lambda}^\alpha\gamma_5\chi_a
\nonumber\\
&&{}
+4iD^\alpha\,{\bar\lambda}^b\gamma_5\chi_a
-(a\leftrightarrow b)
\Big]
\nonumber\\
&&{}
-2T_{(\alpha b}{}^aT_{\beta) a}{}^c
\left(
2{\bar\lambda}^b\gamma_5\gamma^\mu\lambda^\beta\,{\bar\lambda}_c\gamma_\mu\gamma_5\lambda^\alpha
+{\bar\lambda}^b\gamma_5\gamma^\mu\lambda_c\,{\bar\lambda}^\beta\gamma_\mu\gamma_5\lambda^\alpha
\right)\ ,
\label{eq:ferm}\end{aligned}$$ where $m\neq0$ is a mass parameter and $\theta$ is a dimensionless constant. The parts of the total Lagrangian consisting of the terms that depend only on $(A_\mu^\alpha,\lambda^\alpha,D^\alpha)$, those proportional to $m^2$ and $\theta m^2$ and the remainder, are separately off-shell supersymmetric and Yang-Mills gauge invariant. It should not be too difficult to write the action in superspace but we will not attempt to do this here. The abelian version of this action (coupled to additional vectors and scalars) was considered in [@Louis:2004xi].

It is also worth observing that if we set $m^2=0$, then the total Lagrangian does not contain a kinetic term for $\lambda^a$ and the ${\mathcal B}$ field equation $$D^\rho\mathcal H^a_{\mu\nu\rho}-T_{\alpha b}{}^a\phi^bF^\alpha_{\mu\nu}=0$$ gives an unwanted constraint, as can be seen by taking its divergence, which yields [^3] $$T_{\alpha b}{}^a
\left(
F^{\nu\rho\,\alpha}\mathcal H^b_{\mu\nu\rho}
-2D^\nu(\phi^bF^\alpha_{\mu\nu})
\right)
=0\ .$$ This problem is avoided by letting $m^2\ne 0$, in which case the ${\mathcal B}$ field equation now implies a constraint on $D^\mu\mathcal B_{\mu\nu}$ which is a typical subsidiary condition for massive fields (recall the standard Proca equation which implies that the vector is divergence free). Taking $m^2\ne 0$ also furnishes a kinetic term for $\lambda^a$.

Integrating out the auxiliary fields
====================================

In this section we will show that when integrating out the auxiliary fields in [(\[master\])]{} we obtain an on-shell supersymmetric action which involves an infinite expansion in even powers of $\phi$. The equations of motion of the $D$’s are $$\begin{aligned}
m^2D^a&=&-\frac12\theta m^2\phi^a-T_{\alpha b}{}^a\left(\phi^b\,D^\alpha+i{\bar\lambda}^\alpha\gamma_5\chi^b\right)\ ,
\label{aux1}\\
m^2D_a&=&-\frac12\theta m^2\phi_a+T_{\alpha a}{}^b\left(\phi_b\,D^\alpha+i{\bar\lambda}^\alpha\gamma_5\chi_b\right)\ ,
\label{aux2}\\
D_\alpha&=&T_{\alpha b}{}^a\left(\phi_a\,D^b-\phi^b\,D_a+i{\bar\lambda}^b\gamma_5\chi_a
-i{\bar\lambda}_a\gamma_5\chi^b\right)\ .
\label{aux3}\end{aligned}$$ Using the first two equations in the last one gives && D\^= -[G]{}\^T\_[b]{}\^a ( \_aT\_[c]{}\^b[|]{}\^\_5\^c +\^bT\_[a]{}\^c[|]{}\^\_5\_c -m\^2[|]{}\^b\_5\_a +m\^2[|]{}\_a\_5\^b ) ,\[aux4\] where [ $$\fbox{
\rule[0.7cm]{0pt}{0pt}
${\mathcal G}^{\alpha\beta}= \left(\delta_{\alpha\beta}
+\frac{2}{m^2}\phi^b\,(T_{(\alpha}T_{\beta)})_b{}^a\phi_a\right)^{-1} \label{eq:aux2}$
\rule[-0.45cm]{0pt}{0pt}
}$$ ]{}

The representation matrices are defined such that $T_{(\alpha}T_{\beta)}=\frac{1}{2 d_R}\delta_{\alpha\beta}+\frac{i}{2}d_{\alpha\beta\gamma}T^\gamma$, where $d_R$ is the dimension of the representation, so that for an $SO(N)$ gauge group for example, for which $d_{\alpha\beta\gamma}=0$, we have ${\mathcal G}_{\alpha\beta}=(1+ (m^2 d_R)^{-1} \phi^2)^{-1}\,\delta_{\alpha\beta}$.

Using the expressions for the auxiliary fields in [(\[aux1\])]{}-[(\[eq:aux2\])]{} in the Lagrangian (\[master\]) we obtain, after some algebra, the on-shell Lagrangian $$\mathcal L=\mathcal L_{\mathrm b}+\mathcal L_{2\mathrm f}+\mathcal L_{4\mathrm f}\ ,
\label{MTV}$$ where the bosonic terms are $$\begin{aligned}
\mathcal L_{\mathrm b}&=&
-\frac{1}{6}\mathcal H^a_{\mu\nu\rho}\mathcal H_a^{\mu\nu\rho}
-\frac{1}{4}F^\alpha_{\mu\nu}F_\alpha^{\mu\nu}
-D_\mu\phi^aD^\mu\phi_a
-\frac12 m^2\mathcal B^a_{\mu\nu}\mathcal B_a^{\mu\nu}
-M^2\phi^a\phi_a
\nonumber\\
&&{}
-\frac14\theta m^2\varepsilon^{\mu\nu\rho\sigma}\mathcal B^a_{\mu\nu}\mathcal B_{\rho\sigma\,a}
-\frac{1}{2}T_{\alpha b}{}^a\left(\phi^bF^\alpha_{\mu\nu}\mathcal B^{\mu\nu}_a-\phi_aF^\alpha_{\mu\nu}\mathcal B^{\mu\nu\,b}\right)\ ,\end{aligned}$$ with the $M^2$ defined by [ $$\fbox{
\rule[0.7cm]{0pt}{0pt}
$M^2=m^2(1+\theta^2)\,.$
\rule[-0.45cm]{0pt}{0pt}
}$$ ]{} The quadratic fermion terms are $$\begin{aligned}
\mathcal L_{2\mathrm f}&=&
-4{\bar\chi}^a {\slashed D} \chi_a
-4m^2{\bar\lambda}_a {\slashed D} \lambda^a
-2{\bar\lambda}^\alpha {\slashed D}\lambda_\alpha
+4m^2\left[{\bar\lambda}^a(1+i\theta\gamma_5)\chi_a+{\bar\lambda}_a(1+i\theta\gamma_5)\chi^a\right]
\nonumber\\
&&{}
-T_{\alpha b}{}^a
\Big[
-6\phi^b\,{\bar\chi}_a\lambda^\alpha
+\frac{1}{3}\mathcal H_{a\,\mu\nu\rho}\,{\bar\lambda}^\alpha\gamma^{\mu\nu\rho}\lambda^b
+\mathcal B^b_{\mu\nu}\,{\bar\lambda}^\alpha\gamma^{\mu\nu}\chi_a
+F^\alpha_{\mu\nu}\,{\bar\lambda}^b\gamma^{\mu\nu}\chi_a
\nonumber\\
&&{}
+2D_\mu\phi_a\,{\bar\lambda}^\alpha\gamma^\mu\lambda^b
-4\phi_a\,{\bar\lambda}^b {\slashed D}\lambda^\alpha
+2i\theta\phi_a\,{\bar\lambda}^\alpha\gamma_5\chi^b
-(a\leftrightarrow b)
\Big]\end{aligned}$$ and the quartic fermion terms are $$\begin{aligned}
\mathcal L_{4\mathrm f}&=&
-2(T_{(\alpha}T_{\beta)})_b{}^a
\left(
2{\bar\lambda}^b\gamma_5\gamma^\mu\lambda^\beta\,{\bar\lambda}_a\gamma_\mu\gamma_5\lambda^\alpha
+{\bar\lambda}^b\gamma_5\gamma^\mu\lambda_a\,{\bar\lambda}^\beta\gamma_\mu\gamma_5\lambda^\alpha
\right)
-\frac{4}{m^2}(T_\alpha T_\beta)_b{}^a\,{\bar\lambda}^\alpha\gamma_5\chi^b\,{\bar\lambda}^\beta\gamma_5\chi_a
\nonumber\\
&&{}
+\frac{2}{m^4}{\mathcal G}^{\beta\delta}\,T_{\beta b}{}^a T_{\delta c}{}^d
\Big(
T_{\alpha a}{}^e\phi^b\,{\bar\lambda}^\alpha\gamma_5\chi_e
+T_{\alpha e}{}^b\phi_a\,{\bar\lambda}^\alpha\gamma_5\chi^e
+m^2{\bar\lambda}_a\gamma_5\chi^b
-m^2{\bar\lambda}^b\gamma_5\chi_a
\Big)
\nonumber\\
&&
\cdot\Big(
T_{\gamma f}{}^c\,\phi_d\,{\bar\lambda}^\gamma\gamma_5\chi^f
+T_{\gamma d}{}^f\,\phi^c\,{\bar\lambda}^\gamma\gamma_5\chi_f
+m^2{\bar\lambda}_d\gamma_5\chi^c
-m^2{\bar\lambda}^c\gamma_5\chi_d
\Big)\ ,\end{aligned}$$ where ${\mathcal G}^{\alpha\beta}$ defined in [(\[eq:aux2\])]{} is the source of nonpolynomial interactions of the scalar fields with the fermions. The elimination of the auxiliary fields have given rise to the $\theta^2$ dependent term in ${\mathcal L}_{\mathrm b}$ and all the terms proportional to inverse powers of $m$ in $\mathcal L_{2\mathrm f}$ and $\mathcal L_{4\mathrm f}$. All the terms of the form ${\mathcal G} \times \psi^2$ have canceled, where $\psi$ denotes the fermions.

The action of the Lagrangian [(\[MTV\])]{} is invariant under supersymmetry transformations [(\[YM\])]{} and [(\[eq:tensorYM\])]{}, with the substitutions [(\[aux1\])]{}-[(\[eq:aux2\])]{}. If we set $\theta=0$ and truncate to cubic order in fields, the Lagrangian up to field redefinitions[^4] agrees with the Lagrangian given in [@Nishino:2012sm], where the Lagrangian as well as the supersymmetry transformations were constructed only to this order.

Dualization to a massive vector
===============================

It is known that a single massive tensor is dual to a massive vector field. The dualization was described in superspace in [@Louis:2004xi]. Here we will perform the corresponding dualization in the non-abelian case. The first step is to implement the fact that $\mathcal H=D\mathcal B$ by adding to the Lagrangian $$\mathcal L_{\mathrm{constr.}}=
\frac16 M\varepsilon^{\mu\nu\rho\sigma}C^a_\mu(3D_\nu\mathcal B_{\rho\sigma\,a}-\mathcal H_{\nu\rho\sigma\,a})
+ h.c.\ ,$$ where $C^a_\mu$ and $C_{\mu\,a}$ are Lagrange multiplier fields enforcing the constraint and the coefficient is chosen for later convenience. In the new Lagrangian $${\mathcal L'}={\mathcal L}+\mathcal L_{\mathrm{constr.}}\ ,
\label{L2}$$ where ${\mathcal L}$ is given in [(\[master\])]{}, $\mathcal H$ is treated as an independent field. The equation of motion of $\mathcal B^a$ now reads [ $$\fbox{
\rule[0.7cm]{0pt}{0pt}
$
\mathcal B^a_{\mu\nu}
=
M^{-1}\left(\theta G^a_{\mu\nu}+ *G^a_{\mu\nu}\right)
- M^{-2}\,T_{\alpha b}{}^a\left(
\phi^bF^\alpha_{\mu\nu}
-\theta\phi^b*F^\alpha_{\mu\nu}
-2{\bar\lambda}^\alpha\gamma_{\mu\nu}(1-i\theta\gamma_5)\chi^b
\right)\ ,\label{dr}$
\rule[-0.45cm]{0pt}{0pt}
}$$ ]{} where $G=DC$ is the Yang-Mills covariant field strength of $C$ and the Hodge dual is given by $*G_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}$. The equation of motion of $\mathcal H$ reads [ $$\fbox{
\rule[0.7cm]{0pt}{0pt}
$
\mathcal H^a_{\mu\nu\rho}=M\varepsilon_{\mu\nu\rho\sigma}C^{\sigma\,a}-2T_{\alpha b}{}^a\,{\bar \lambda}^\alpha\gamma_{\mu\nu\rho}\lambda^b\,.    \label{eq:HC} $
\rule[-0.45cm]{0pt}{0pt}
}$$ ]{} The off-shell Lagrangian [(\[master\])]{} as well as the on-shell one [(\[MTV\])]{} give the same equations for ${\mathcal B}_{\mu\nu}$ and $\mathcal H_{\mu\nu\rho}$. Therefore, we can substitute back these results in either Lagrangian. For the convenience of having the Yang-Mills system off-shell, we use the expressions for $\mathcal B$ and $\mathcal H$ in [(\[master\])]{}, together with the expression for $D^a$ [(\[aux1\])]{}, thereby obtaining the full Lagrangian for the on-shell super Proca system coupled to off-shell Yang-Mills given by: =[L’]{}\_[b]{}+[L’]{}\_[2f]{}+[L’]{}\_[4f]{} , \[L22\] where the bosonic terms are $$\begin{aligned}
{\mathcal L'}_{\mathrm b}&=&-\frac{1}{4}F^\alpha_{\mu\nu}F_\alpha^{\mu\nu}
-\frac{1}{2}G^a_{\mu\nu}\left(G^{\mu\nu}_a-\theta*G^{\mu\nu}_a\right)-D_\mu\phi^aD^\mu\phi_a
-M^2C^a_\mu C_a^\mu-M^2\phi^a\phi_a+2D^\alpha D_\alpha
\nonumber\\
&&{}
-\frac{1}{2M}T_{\alpha b}{}^a\,F_{\mu\nu}^\alpha\left(
\theta \phi^b G_a^{\mu\nu} -\phi_a*G^{\mu\nu b}
-\theta \phi_a G^{\mu\nu b} +\phi^b*G^{\mu\nu}_a \right)
\nonumber\\
&&{}
-\frac{1}{2M^2}(T_\alpha T_\beta)_b{}^a \phi_a\phi^b F^\alpha_{\mu\nu}
\Big(F^{\mu\nu\,\beta} -\theta *F^{\mu\nu\,\beta}\Big)
+\frac{4}{m^2}(T_\alpha T_\beta)_b{}^a\,\phi_a\,\phi^b\,D^\alpha\,D^\beta\ ,
\label{Lbos}\end{aligned}$$ the quadratic fermion terms are $$\begin{aligned}
{\mathcal L'}_{2\mathrm f}&=&
-2{\bar\lambda}^\alpha{\slashed D}\lambda_\alpha-4{\bar\chi}^a{\slashed D}\chi_a
-4m^2{\bar\lambda}_a{\slashed D}\lambda^a
+4m^2\left({\bar\lambda}^a(1+i\theta\gamma_5)\chi_a+{\bar\lambda}_a(1+i\theta\gamma_5)\chi^a\right)
\nonumber\\
&&{}
+\frac{1}{M}T_{\alpha b}{}^a
\Big(
iG_a^{\mu\nu}\,{\bar\lambda}^\alpha\gamma_{\mu\nu}\gamma_5(1-i\theta\gamma_5)\chi^b
-6M\phi_a\,{\bar\lambda}^\alpha\chi^b
-2iM\theta\phi_a\,{\bar\lambda}^\alpha\gamma_5\chi^b
\nonumber\\
&&{}
-2iM^2C_a^\mu\,{\bar\lambda}^\alpha\gamma_\mu\gamma_5\lambda^b
-2MD_\mu\phi_a\,{\bar\lambda}^\alpha\gamma^\mu\lambda^b
+4M\phi_a\,{\bar\lambda}^b{\slashed D}\lambda^\alpha
\nonumber\\
&&{}
-MF^\alpha_{\mu\nu}\,{\bar\lambda}^b\gamma^{\mu\nu}\chi_a
-4iMD^\alpha\,{\bar\lambda}^b\gamma_5\chi_a
-(a\leftrightarrow b)
\Big)
\nonumber\\
&&{}
+\frac{1}{M^2}(T_\alpha T_\beta)_b{}^a
\Big(
\phi_aF^\beta_{\mu\nu}\,{\bar\lambda}^\alpha\gamma^{\mu\nu}(1-i\theta\gamma_5)\chi^b
+\phi^bF^\alpha_{\mu\nu}\,{\bar\lambda}^\beta\gamma^{\mu\nu}(1-i\theta\gamma_5)\chi_a
\Big)
\nonumber\\
&&{}
+\frac{4i}{m^2}(T_\alpha T_\beta)_b{}^a
\Big(
\phi_a\,D^\beta\,{\bar\lambda}^\alpha\gamma_5\chi^b
+\phi^b\,D^\alpha\,{\bar\lambda}^\beta\gamma_5\chi_a
\Big)\end{aligned}$$ and the quartic fermion terms are $$\begin{aligned}
{\mathcal L'}_{4\mathrm f}&=&
-\frac{2}{M^2}(T_\alpha T_\beta)_b{}^a
\Big[
2(1+\theta^2){\bar\lambda}^\beta\gamma_5\chi_a\,{\bar\lambda}^\alpha\gamma_5\chi^b
+{\bar\lambda}^\alpha\gamma_{\mu\nu}(1-i\theta\gamma_5)\chi^b\,{\bar\lambda}^\beta\gamma^{\mu\nu}\chi_a
\\
&&{}
+M^2{\bar\lambda}^b\gamma_\mu\gamma_5\lambda^\alpha\,{\bar\lambda}_a\gamma^\mu\gamma_5\lambda^\beta
-M^2{\bar\lambda}^b\gamma^\mu\gamma_5\lambda^\beta\,{\bar\lambda}_a\gamma_\mu\gamma_5\lambda^\alpha
+M^2{\bar\lambda}^b\gamma_5\gamma^\mu\lambda_a\,{\bar\lambda}^\beta\gamma_\mu\gamma_5\lambda^\alpha
\Big]\ .\nonumber\end{aligned}$$ Equation (\[eq:HC\]) which expresses $\mathcal H$ in terms of $C$ can be used to determine the supersymmetry transformation of the massive vector field $C$. One finds $$\begin{aligned}
\delta C^a_\mu
&=&
\frac{im^2}{M}{\bar\epsilon}\gamma_\mu\gamma_5(1+i\theta\gamma_5)\lambda^a
+\frac{i}{M}{\bar\epsilon}\gamma_5D_\mu\chi^a
+\frac{i}{M}T_{\alpha b}{}^a\phi^b\,{\bar\epsilon}\gamma_\mu\gamma_5\lambda^\alpha
\nonumber\\
&&{}
+\frac{i}{M}{\bar\epsilon}\gamma_\mu\gamma_5(\gamma^\nu D_\nu\chi^a-m^2\lambda^a+\ldots)\ ,
\label{dC}\end{aligned}$$ where the last term is proportional to the equation of motion for $\chi$ and, since in the dualized model the supersymmetry algebra will be on–shell we will simply drop this term. The other supersymmetry transformations are obtained by using the expressions for $\mathcal B$ and $\mathcal H$ in terms of the vector field $C$ in the supersymmetry algebra (\[eq:tensorYM\]). The supersymmetry transformations which leave this action invariant are therefore the standard ones for the off-shell Yang-Mills fields (\[YM\]) together with the on-shell supersymmetry transformation of the massive vector multiplet $$\begin{aligned}
\delta C^a_\mu
&=&
\frac{im^2}{M}{\bar\epsilon}\gamma_\mu\gamma_5(1+i\theta\gamma_5)\lambda^a
+\frac{i}{M}{\bar\epsilon}\gamma_5D_\mu\chi^a
+\frac{i}{M}T_{\alpha b}{}^a\phi^b\,\bar\epsilon\gamma_\mu\gamma_5\lambda^\alpha
\nonumber\\
\delta\phi^a&=&{\bar\epsilon}\chi^a
\nonumber\\
\delta\chi^a&=&
\frac{i}{4}M C^a_\mu\,\gamma^\mu\gamma_5\epsilon
+\frac{1}{4}D_\mu\phi^a\,\gamma^\mu\epsilon
-\frac{1}{2}T_{\alpha b}{}^a\,(\gamma^\mu\lambda^{(\alpha})\,{\bar\epsilon}\gamma_\mu\lambda^{b)}
-\frac{1}{4}T_{\alpha b}{}^a\,(\gamma^\mu\gamma_5\epsilon)\,{\bar\lambda}^\alpha\gamma_\mu\gamma_5\lambda^b
\nonumber\\
\delta\lambda^a&=&
\frac{i}{8M}G^a_{\mu\nu}\,\gamma^{\mu\nu}\gamma_5(1-i\theta\gamma_5)\epsilon
-\frac{1}{8M^2}T_{\alpha b}{}^a\left(\phi^bF^\alpha_{\mu\nu}-2{\bar\lambda}^\alpha\gamma_{\mu\nu}\chi^b\right)\,\gamma^{\mu\nu}(1-i\theta\gamma_5)\epsilon
\nonumber\\
&&{}
-\frac{i}{2m^2}T_{\alpha b}{}^a\left(\phi^b\,D^\alpha+i{\bar\lambda}^\alpha\gamma_5\chi^b\right)\,\gamma_5\epsilon
+\frac{1}{4}\phi^a\,(1-i\theta\gamma_5)\epsilon
\ .\end{aligned}$$ As a byproduct of the dualization of a massive tensor we have shown how to couple a massive vector, transforming in a non-trivial representation of the gauge-group $G$, to Yang-Mills. Note that this construction does not involve any scalar multiplets and is therefore quite different from the usual approach via the Higgs mechanism. A more detailed analysis of the differences will be presented in the next section. Note also that if we integrate out the auxiliary field of the Yang-Mills multiplet $D^\alpha$ using equation (\[aux4\]) and (\[eq:aux2\]) we get an infinite series of terms involving powers of the scalar $\phi$, both in the supersymmetry transformations and in the action. If we had started with a completely on-shell formulation we would therefore only have been able to construct the model order by order in these terms.

It should be possible to find a completely off-shell description of this system. To do this one should introduce the appropriate auxiliary fields to make the massive vector multiplet off-shell. These consist of one complex and one real scalar. It should then be possible to formulate this model in superspace. We leave this problem for the future.

Comparison to supersymmetric Higgs model
========================================

The supersymmetric Proca-Yang-Mills in the previous section contains only the massive vector multiplet and the Yang-Mills multiplet, and in particular no scalar multiplets. Here we shall compare this model with the supersymmetric Yang-Mills-Higgs model which does contain scalar multiplets. We will see that, except in the abelian case, the truncation of the scalar multiplets without breaking supersymmetry is not possible. Therefore the two models represent completely distinct ways of coupling massive vectors to super Yang-Mills.

To be concrete we will analyze the specific example of breaking $SU(n+1)\rightarrow SU(n)\times U(1)$ ($n>1$) via a supersymmetric version of the Higgs mechanism, which arises for example in supersymmetric GUTs [@Dimopoulos:1981zb]. The starting point is the (renormalizable) Lagrangian $$\begin{aligned}
\mathcal L_{\mathrm{Higgs}}&=&
\mathrm{Tr}\Big[
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-2\bar\lambda{\slashed D}\lambda
+\left(\frac14 m\bar\psi\psi_L+\frac{1}{2}z\,\bar\psi\psi_L+\mathrm{h.c.}\right)
-|f|^2 -\frac12 D^2
\nn{\\[0.2cm]}
&&
-\frac{1}{2}D_\mu z D^\mu z^*
-\frac{1}{2}\bar\psi\gamma^\mu D_\mu\psi_L
+2i[z,\bar\psi]\lambda_R
-2i\bar\lambda[\psi_L,z^*]
\Big]\ ,
\label{ymh}\end{aligned}$$ where $$\begin{aligned}
\label{eq:feqn}
f^*=mz+zz-\frac{1}{n+1}\mathrm{Tr}(zz)\ ,\qquad  D=\frac12[z,z^*]\ ,\end{aligned}$$ and all fields are in the adjoint representation of $SU(n+1)$. The supersymmetry transformations are given by $$\begin{aligned}
\delta A_\mu&=&-{\bar\epsilon}\gamma_\mu\lambda\ ,
\nonumber\\
\delta\lambda&=&\frac{1}{8}F_{\mu\nu}\,\gamma^{\mu\nu}\epsilon+\frac{i}{2}D\,\gamma_5\epsilon
\nonumber\\
\delta z&=&\frac{1}{2}\bar\epsilon\psi_L
\nonumber\\
\delta\psi_L&=&D_\mu z\,\gamma^\mu\epsilon_R+f\,\epsilon_L
\label{eq:scalarsusy}\end{aligned}$$ where $\psi_L=\frac12(1+\gamma_5)\psi$. The conditions for a supersymmetric vacuum are the vanishing of the $f$ and $D$-terms. We can write a field in the adjoint representation as $z=z_i{}^j$ where $i,j=0,\ldots,n$ are indices of the (anti-)fundamental representation of $SU(n+1)$. With this notation we will consider the supersymmetric vacuum solution given by $$\langle z_i{}^j\rangle=\frac{1}{n-1}m\delta_i^j-\frac{n+1}{n-1}m\delta_i^0\delta^j_0\,,$$ which corresponds to the breaking $SU(n+1)\rightarrow SU(n)\times U(1)$. Next we expand the scalar field around this vacuum as $$z_i{}^j=\langle z_i{}^j\rangle+\hat z_i{}^j\,.$$ Under the symmetry breaking the adjoint scalar $z_i{}^j$ splits up as $z_i{}^j=(z_0{}^0,z_0{}^b,z_a{}^0,z_a{}^b)$ where $a,b=1,\ldots,n$ is an $SU(n)$ (anti-)fundamental index. An infinitesimal gauge-transformation of the broken components of $(z+z^*)_i{}^j$ is given by $$\delta(z+z^*)_0{}^a=i\Lambda_0{}^i(z+z^*)_i{}^a-i(z+z^*)_0{}^i\Lambda_i{}^a=\frac{2i(n+1) m}{n-1}\Lambda_0{}^a+\ldots$$ and similarly for $(z+z^*)_a{}^0$. This means that the real part of $z_0{}^a$ and $z_a{}^0$ can be gauged away by a suitable gauge-transformation $\Lambda_a{}^0$ and $\Lambda_0{}^a$. Therefore by performing a finite gauge-transformation $$U(\xi)=e^{i\xi^aT_a{}^0+i\xi_aT_0{}^a}\,,$$ where $T_0{}^a$ and $T_a{}^0$ are the broken generators which don’t preserve the vacuum, we can bring the scalar to the form $$z_i{}^j=
\left(
\begin{array}{cc}
\frac{-n}{n-1}m +\hat z&i\phi^b\\
i\phi_a&\frac{1}{n-1}m\delta_a^b-\frac{1}{n}\delta_a^b\hat z+\hat z_a{}^b
\end{array}
\right)\,.
\label{eq:zgauge}$$ Note that the scalars $\phi_a$ are now real, in the sense that $(\phi_a)^*=\phi^a$, the imaginary part being eaten by the off-diagonal vectors which become massive. The off-diagonal part of the $SU(n+1)$ gauge field is massive and denoted by C\_\^a = (A\_)\_0\^a ,C\_[a]{}= (A\_)\_a\^0 . Using this notation and [(\[eq:zgauge\])]{} in [(\[ymh\])]{}, the resulting bosonic Lagrangian is \_[bos]{} &=&-14 ( (F\^)\_a\^b + 2iC\^[\[]{}\_a C\^[\]\^b]{}) ( (F\_)\_b\^a + 2iC\_[b]{}C\_\^a)-12 G\^\_a G\_\^a 2 && -14 ( F\^ + 2iC\^[\[a]{} C\^[\]]{}\_a ) ( F\_ + 2iC\_\^b C\_[b]{}) -|f|\^2 -12 D\^2 2 && -|\_z-C\_\^a\_a+C\_[a]{}\^a|\^2 -|D\_z\_a\^b-\_a\^b\_z-C\_[a]{}\^b+C\_\^b\_a|\^2 2 && -|D\_\^a+mC\_\^a-zC\_\^a+C\_\^bz\_b\^a|\^2 2 && -|D\_\^a-mC\_\^a+z\^\*C\_\^a+C\_\^bz\^\*\_b\^a|\^2 , \[ymhbos\] where $F_{\mu\nu a}{}^b$ and $F_{\mu\nu}$ are the standard $SU(n)$ and $U(1)$ field strengths, while G\_\^a = 2D\_[\[]{}C\_[\]]{}\^a=2\_[\[]{}C\_[\]]{}\^a+2iA\_[\[]{}C\_[\]]{}\^a+2iC\_[\[]{}\^c(A\_[\]]{})\_c\^a . is the field strength of the massive vectors.

In order to make the comparison to the model of massive vectors considered in the previous section we need to truncate the extra scalar multiplets. These consist of the singlet $\hat z$ and the $SU(n)$ adjoint $\hat z_a{}^b$. To see if this truncation is consistent with supersymmetry we should look at the corresponding supersymmetry transformations coming from (\[eq:scalarsusy\]). These supersymmetry transformations have to be accompanied by a compensating gauge-transformation with (we put $\psi_0{}^a \equiv \chi^a$ and $\psi_a{}^0 \equiv \chi_a$) $$\begin{aligned}
\Lambda_a{}^0=-\frac{i}{2}R_a{}^b\,\bar\epsilon\chi_b
\qquad
\Lambda_0{}^a=\frac{i}{2}\bar\epsilon\chi^b\,R_b{}^a
\label{eq:compensating}\end{aligned}$$ where $$R_a{}^b=
\left(
\frac{2(n+1)}{n-1}\,m\delta_b^a
-\frac{n+1}{n}(\hat z+\hat z^*)\delta_b^a
+\hat z_b{}^a
+\hat z^*_b{}^a
\right)^{-1}$$ in order to preserve the gauge (\[eq:zgauge\]). The full supersymmetry transformations for the Yang-Mills multiplet $(A_{\mu a}{}^b,\lambda_a{}^b)$, the massive Wess-Zumino multiplets $(\hat z_a{}^b, \psi_a{}^b)$ and $(\hat z,\psi)$, and the massive vector multiplet $(C_\mu^a, \phi^a, \chi^a, \lambda^a)$ can be read off from [(\[eq:scalarsusy\])]{} by using [(\[eq:zgauge\])]{} and the compensating gauge transformations of the fields with gauge parameter given by [(\[eq:compensating\])]{}. In particular, the supersymmetry transformations of the scalar fields take the form $$\begin{aligned}
\delta\hat z&=&\frac{1}{2}\bar\epsilon\psi_L
-\frac{i}{2}\phi^a\,R_a{}^b\,\bar\epsilon\chi_b
-\frac{i}{2}\bar\epsilon\chi^b\,R_b{}^a\,\phi_a
\nonumber\\
\delta\hat z_a{}^b&=&\frac{1}{2}\bar\epsilon\psi_{La}{}^b
+\frac{i}{2}R_a{}^c\,\bar\epsilon\chi_c\,\phi^b
+\frac{i}{2}\phi_a\,\bar\epsilon\chi^c\,R_c{}^b
-\mathrm{trace}\ .\end{aligned}$$ In a truncation where the scalar fields $(\hat z,{\hat z}_a{}^b)$ are set to zero, we have $R_a{}^b\sim\delta_a^b$, and consequently the vanishing of the variations $\delta\hat z$ and $\delta{\hat z}_a{}^b$ requires that $\chi_a\,\phi^b+\phi_a\,\chi^b=0$. This represents an unacceptable constraint on the fields of the massive vector multiplet. Therefore we conclude that it is not possible (except in the abelian case) to truncate out the additional scalars of the Higgs model and therefore to compare it to the model considered in this paper. In the abelian case the comparison is trivial since all the complicated non-linear terms disappear. The model obtained in this paper by dualizing a massive tensor therefore gives a very different description of massive vectors coupled to super Yang-Mills than that obtained through the Higgs mechanism.

Comparing the bosonic Lagrangians [(\[Lbos\])]{} with [(\[ymhbos\])]{} shows another key difference. As a test of the tree-level unitarity, considering the process $CC\to CC$ scattering, we see that the model based on [(\[ymhbos\])]{} furnishes the vertices $C^2A$, $CCz$ and $C^4$, all of which contribute at the tree level, giving a unitary result in the sense that upon the substitution $C_\mu \to k_\mu/m$ for the longitudinal components of the incoming and outgoing massive vector bosons, all the terms that grow with energy cancel each other out [@Bell:1973ex; @LlewellynSmith:1973ey; @Cornwall:1974km; @Horejsi:1993hz]. In the model based on [(\[Lbos\])]{}, however, there is only the $C^2 A$ vertex contributing to the same tree-level amplitude which then grows with energy for the longitudinal components. Whether this problem with unitarity can be cured by further extension of our model, or its interpretation as an effective field theory, remains to be investigated.

Motivated by a bosonic Higgsless model where this issue has been studied [@Lahiri:2011ic], in Appendix B we extend the four parameter supersymmetric tensor-Yang-Mills model by introducing two new off-shell invariants, which are possible if we take the 2-form potential in the adjoint representation.

Conclusions
===========

In coupling $n_V$ vector multiplets to $n_T$ tensor multiplets such that gauge symmetry based on a semi-simple group is realized and that the tensor fields transform in a nontrivial representation of the gauge group, we have seen that the vector fields split into two sets; one belonging to the adjoint representation of the gauge group, and the other that is eaten by as many tensor fields which become massive. Thus we end up with off-shell supersymmetric coupling of a Yang-Mills multiplet to off-shell massive tensor multiplet carrying a nontrivial representation of the gauge group. The massive tensor multiplet also contains scalar fields which are in the same representation. A noteworthy feature of the resulting model is that the elimination of the auxiliary fields needed for the off-shell closure of the algebra gives rise to nonpolynomial couplings of scalar fields to the fermionic bilinears even though the kinetic terms for the scalars is an ordinary one not involving any curved scalar manifold.

Dualization of the massive tensor to a massive vector produces a model in which an on-shell massive nonabelian vector multiplet is coupled to an off-shell Yang-Mills multiplet. The two arbitrary parameters of the model are the mass $M$ and the angle $\theta$. The $\theta \to 0$ limit exists but the $M\to 0$ limit is singular. In the abelian limit the $M\to 0$ limit exists but the resulting Lagrangian is not supersymmetric.

The bosonic sector of the nonabelian model differs from a bosonic Yang-Mills-Higgs system in which the Yang-Mills group $G$ is broken down to $H$ with massive vector fields in representations of $H$, and presence of scalar multiplets. In the supersymmetric Proca-Yang-Mills system, however, we start with a Yang-Mills group H from the beginning and couple it to massive vector multiplets only without introducing any scalar multiplets. We showed that a consistent supersymmetric truncation of the scalar multiplets in the Yang-Mills-Higgs system is not possible, showing that the two models are genuinely distinct. In Section 6, we discussed the issue of breakdown in tree-level unitarity in the Proca-Yang-Mills system, which is of course absent in the Yang-Mills-Higgs system. In Appendix B, we presented a 2-parameter extension of the model which has implications for solving this problem though the full study of this extended model remains to be done.

Possible generalizations of the massive tensor-Yang-Mills system presented here are as follows. The tensor hierarchy can be extended to include the 3-form and 4-form potentials. Non-semisimple groups may be considered for completeness. Coupling to scalar multiplets would generalize the results of [@Louis:2004xi] where such couplings in the case of the abelian model have been given. Coupling of the nonabelian model to supergravity can be carried out and that would generalize the result of [@Mukhi:1979wc] where a single massive vector multiplet, and [@Nishino:2012sm] where a truncated version of our full non-abelian model have been coupled to supergravity. Finally, the computation of the full tree-level amplitude for the scattering of massive vectors in the extended model given in Appendix B, and its implications for unitarity is an interesting open problem.

Acknowledgements {#acknowledgements .unnumbered}
================

We thank B. Dutta, Tianjun Li, W. Schulgin, H. Samtleben and R. Wimmer for useful discussions. This research is supported in part by NSF grant PHY-0906222.

Dimensional reduction from $D=6$
================================

The supersymmetry transformations of the $D=4$ $N=1$ tensor-YM system considered in this paper can be obtained from the corresponding $N=(1,0)$ supersymmetric system in $D=6$ [@Samtleben:2012mi]: $$\begin{aligned}
\delta \phi^a &=& \bar\epsilon\chi^a\,,\nonumber\\
\delta {\mathcal B}_{\mu\nu}^a &=& 2 \, \bar\epsilon\,\gamma_{[\mu}D_{\nu]} \lambda^a -\bar\epsilon\gamma_{\mu\nu}\chi^a\,,
\nonumber\\
\delta \lambda^{i\,a} &=& \frac{1}{8}\,\gamma^{\mu\nu}\mathcal B^a_{\mu\nu} \epsilon^i
-\frac{1}{2}\,Y^{ij\,a}\epsilon_j + \frac14  \phi^a \epsilon^i \,,
\nonumber\\
\delta\chi^{i\,a} &=&\frac{1}{48}\gamma^{\mu\nu\rho}\,\mathcal H_{\mu\nu\rho}^a\,\epsilon^i
+\frac{1}{4}\,\gamma^\mu D_\mu\phi^a \epsilon^i
-\frac{1}{2}(T_{\alpha})_b{}^a\, \gamma^\mu\lambda^{i\,(\alpha}\, \bar\epsilon\gamma_\mu \lambda^{b)}\,,
\nonumber\\
\delta Y^{ij\,a}&=&-{\bar\epsilon}^{(i}\gamma^\mu D_\mu\lambda^{j)a}+2\,\bar\epsilon^{(i} \chi^{j) a}\,.\end{aligned}$$ where $i=1,2$ and the spinors are symplectic-Majorana-Weyl. The $D=4$ system is obtained by making the following ansatz $$\begin{aligned}
\sigma_2\epsilon &=& \gamma_5 \epsilon\ ,\quad \tau_2\epsilon=\gamma_5\epsilon\,,
\quad \sigma_2\lambda = \gamma_5 \lambda\ ,\quad \tau_2\lambda=\gamma_5\lambda\,,
\quad \sigma_2\chi = \gamma_5 \chi\ ,\quad \tau_2\chi=-\gamma_5\chi\,,
\nonumber\\
A_4&=&0\ , \quad A_5=0\ ,\quad {\mathcal B}_{\mu 4}=0\ ,\quad {\mathcal B}_{\mu 5}=0\,,
\quad {\mathcal B}_{45} \equiv \varphi\ , \quad Y_i{}^j= (i\tau_2)_i{}^j\,D\,,\end{aligned}$$ where we have suppressed the representation indices, $D$ is a real scalar, the Pauli $\tau$-matrices act on the R-symmetry indices and the Pauli $\sigma$-matrices act on the $SO(2)$ doublet indices where $SO(2)$ is contained in $SO(5,1)\sim SO(3,1)\times SO(2)$. We also use the convention by which the 6D Dirac $\Gamma$-matrices are $\Gamma^\mu=\gamma^\mu\times 1\ , \Gamma^4=\gamma^5\times \sigma^3$ and $\Gamma^5=\gamma_5\times\sigma^1$. The resulting $N=1, D=4$ supersymmetry transformation rules are $$\begin{aligned}
\delta\phi^a&=&\bar\epsilon\chi^a\,,
\nonumber\\
\delta\varphi^a&=&i\bar\epsilon\gamma_5\chi^a\,,
\nonumber\\
\delta\mathcal B_{\mu\nu}^a &=&2\bar\epsilon\,\gamma_{[\mu}D_{\nu]}\lambda^a
-\bar\epsilon\gamma_{\mu\nu}\chi^a\,,
\nonumber\\
\delta\chi^a&=&
\frac{1}{48}\mathcal H_{\mu\nu\rho}^a\,\gamma^{\mu\nu\rho}\epsilon
+\frac{i}{8}D_\mu\varphi^a\,\gamma^\mu\gamma_5\epsilon
+\frac{1}{4}D_\mu\phi^a\,\gamma^\mu\epsilon
-\frac{1}{2}(T_\alpha)_b{}^a\,\gamma^\mu\lambda^{(\alpha}\,\bar\epsilon\gamma_\mu\lambda^{b)}\,,
\nonumber\\
\delta \lambda^a &=& \frac{1}{8}\mathcal B^a_{\mu\nu}\,\gamma^{\mu\nu}\epsilon
+\frac{i}{4}\varphi^a\,\gamma^5\epsilon
+\frac{i}{2}D^a\,\gamma_5\epsilon
+\frac{1}{4}\phi^a\,\epsilon\,,
\nonumber\\
\delta D^a&=&\frac{i}{2}\bar\epsilon \gamma_5\gamma^\mu D_\mu\lambda^a
-i\bar\epsilon\gamma_5\chi^a\,.\end{aligned}$$ This is not yet the off-shell system that we want since there is an extra scalar $\varphi^a$. It turns out that the closure of this algebra requires, among other things, that $$D^\mu\varphi^a
=
\frac{1}{6}\varepsilon^{\mu\nu\rho\sigma}\mathcal H^a_{\nu\rho\sigma}
-2i(T_\alpha)_b{}^a\,{\bar\lambda}^b\gamma^\mu\gamma_5\lambda^\alpha\ .$$ Using this fact in the expression for $\delta\chi^a$ and performing the shift $D^a\rightarrow D^a-\frac{1}{2}\varphi^a$ we obtain the algebra $$\begin{aligned}
\delta\phi^a&=&\bar\epsilon\chi^a\,,
\nonumber\\
\delta\mathcal B_{\mu\nu}^a &=&2\bar\epsilon\,\gamma_{[\mu}D_{\nu]}\lambda^a
-\bar\epsilon\gamma_{\mu\nu}\chi^a\,,
\nonumber\\
\delta\chi^a&=&
\frac{1}{24}\mathcal H_{\mu\nu\rho}^a\,\gamma^{\mu\nu\rho}\epsilon
+\frac{1}{4}D_\mu\phi^a\,\gamma^\mu\epsilon
-\frac{1}{2}(T_\alpha)_b{}^a\,(\gamma^\mu\lambda^{(\alpha})\,\bar\epsilon\gamma_\mu\lambda^{b)}\,,
+\frac{1}{4}(T_\alpha)_b{}^a\,(\gamma^\mu\gamma_5\epsilon)\,{\bar\lambda}^b\gamma_\mu\gamma_5\lambda^\alpha
\nonumber\\
\delta\lambda^a&=&\frac{1}{8}\mathcal B^a_{\mu\nu}\,\gamma^{\mu\nu}\epsilon
+\frac{i}{2}D^a\,\gamma_5\epsilon
+\frac{1}{4}\phi^a\,\epsilon\,,
\nonumber\\
\delta D^a&=&\frac{i}{2}\bar\epsilon \gamma_5\gamma^\mu D_\mu\lambda^a
-\frac{i}{2}\bar\epsilon\gamma_5\chi^a\,.\end{aligned}$$ This coincides with the algebra in (\[eq:tensorYM\]).

An extension of the model
=========================

In Section 6, we compared our Higgsless model based on [(\[master\])]{} for the description of supersymmetric Proca-Yang-Mills system with the supersymmetric Yang-Mills-Higgs system and we noted the presence of divergences in the tree level amplitude for the scattering of the longitudinally polarized massive vectors in the Higgsless model. One attempt in constructing a Higgsless model that avoids this problem is based on the Lagrangian [@Lahiri:2011ic] = [Tr]{} ( -14 F\^2 +1[12]{}H\^2+ m BF ) , \[badL\] where $F=dA+A^2$ and the 2-form potential has the field strength $H=dB+ [A,B]$. Diagonalization of the kinetic terms gives a massive vector boson and the potential divergences in the scattering of two such longitudinally polarized vectors turn out to cancel out [@Lahiri:2011ic]. An important shortcoming of this model not addressed in [@Lahiri:2011ic] is that the divergence of the the field equation for the 2-form potential gives an unacceptable constraint; see the discussion around (3.6) above, where this type constraint is avoided by introducing an explicit mass term for the 2-form potential. Nonetheless, given that new couplings relevant for the unitarity issue arise due the $B\wedge F$ term in (\[badL\]), here we shall extend our supersymmetric Proca-Yang-Mills model by considering the case when the massive supermultiplet is taken to be in the adjoint representation of the gauge group. In this case, the following two invariants can be added to the action (\[master\]) (where needed we use subscripts $1,2$ to distinguish fields of the Yang-Mills and tensor multiplet respectively) L\_[extra]{} &=& m [Tr]{} ( B\_F\^ -8|\_1+8|\_1[D]{}\_2 -8D\_1D\_2) 2 && +m[Tr]{} (\^B\_F\_ -16i|\_1\_5+8D\_1) . Adding this Lagrangian to that given in [(\[master\])]{} gives the total bosonic Lagrangian $$\begin{aligned}
{\mathcal L''}_{\mathrm{bos}}&=& {\rm Tr} \Bigl(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
-\frac{1}{6}\mathcal H_{\mu\nu\rho}\mathcal H^{\mu\nu\rho}
-\frac12 \mathcal B_{\mu\nu}
\left( m^2 \mathcal B^{\mu\nu}+\theta m^2*\mathcal B^{\mu\nu}-2m\alpha F^{\mu\nu} -4m\beta*F^{\mu\nu}\right)
\nn{\\[0.2cm]}
&&{}
-D_\mu \phi D^\mu\phi-m^2 \phi^2
+4m\phi\left(2\beta D_1+\theta m D_2\right)
+2\left(D_1^2-4m\alpha D_1D_2 +2 m^2D_2^2\right)
\nn{\\[0.2cm]}
&& -\phi[F_{\mu\nu},{\mathcal B}^{\mu\nu}]+8\phi[D_1,D_2] \Bigr) \ .\end{aligned}$$ Integrating out the auxiliary fields gives, modulo the fermions, $$\begin{aligned}
D_1 &=& 2m(\alpha D_2 -\beta \phi)+2[\phi,D_2]
\label{d1}\\
m^2D_2 &=&-\frac12 \theta m^2 \phi-[\phi,D_1]+m\alpha D_1\ .
\label{d2}\end{aligned}$$ Solving for $D_1^\alpha = 2 {\rm Tr}\,(T^\alpha D_1) $, up to fermions, gives $$\begin{aligned}
D_1^\alpha
&=&
-m \left(\theta\alpha+2\beta\right) \left(\delta_{\alpha\beta}-2\alpha^2\delta_{\alpha\beta}
+\frac{2}{m^2}f_{\alpha\gamma}{}^\delta f_{\beta\delta}{}^\kappa\,\phi^\gamma\phi_\kappa\,\right)^{-1} \phi^\beta\ ,
\label{d2sol}\end{aligned}$$ and substitution into [(\[d2\])]{} gives $D_2$. Note that, unlike in the case of the model given in section 3, here the elimination of the auxiliary fields generate a potential.

Upon dualizing the massive 2-form potential to a massive vector field, the resulting bosonic terms in the Lagrangian take the form $$\begin{aligned}
\mathcal L'_{\mathrm{bos}}&=& {\rm Tr}\,\Big[
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
+\frac{m^2}{2M^2}(\alpha^2+4\theta\alpha\beta-4\beta^2)F_{\mu\nu} F^{\mu\nu}
-M^2 C_\mu C^\mu
\nonumber\\
&&{}
-\frac{m^2}{2M^2}(\theta \alpha^2-4\alpha\beta-4\theta\beta^2)F_{\mu\nu}*F^{\mu\nu}
-\frac{1}{2}G^{\mu\nu}\left(G_{\mu\nu}-\theta *G^{\mu\nu}\right)
\nonumber\\
&&{}
+\frac{m}{M}F^{\mu\nu}\Big[(\theta\alpha-2\beta)G_{\mu\nu}
+(\alpha+2\theta\beta)*G_{\mu\nu}\Big]
-D_\mu\phi D^\mu\phi-m^2\phi^2
\nonumber\\
&&{}
+2D_1^2+8m\beta\,\phi\,D_1-8m\alpha\,D_1\,D_2+4\theta m^2\phi D_2+4m^2D_2^2
\nonumber\\
&&{}
+\frac1M\phi\,[*\, G_{\mu\nu},F^{\mu\nu}-\theta * F^{\mu\nu}]+8\phi [D_1,D_2]\,\Big]
\nonumber\\
&&{}
-\frac{1}{4M^2} f_{\alpha\gamma}{}^\kappa f_{\beta\kappa}{}^\delta\,\phi_\delta\phi^\gamma F^{\mu\nu\,\alpha}(F^\beta_{\mu\nu}-\theta*F^\beta_{\mu\nu})\ .\end{aligned}$$ Substituting for $D_1$ and $D_2$ the expressions given in [(\[d2sol\])]{} and [(\[d2\])]{} yields a potential that vanishes in the absence of the parameters $\alpha$ and $\beta$. Considering the tree level amplitude for the $CC\to CC$ scattering, we observe that, in addition to the $C^2 A$ vertex which contributes to the tree graph, the introduction of the $(\alpha,\beta)$ parameters gives rise to a mixing $AC$ which together with the usual $A^3$ and $A^4$ couplings, gives rise to new contributions. However, a detailed computation and analysis of the tree level $CC\to CC$ scattering amplitude remains to be performed.

[^1]: In the particular case, when ${\cal R}$ is the adjoint representation, this gauge structure was considered in [@Chu:2011fd].

[^2]: In our conventions $\eta_{\mu\nu}={\rm diag}(-1,+1,+1,+1)$ and $\gamma_{\mu\nu\rho\sigma}=i\varepsilon_{\mu\nu\rho\sigma}\gamma_5$. A crucial Fierz identity is $(C\gamma_\mu)_{\alpha(\beta}\,(C\gamma^\mu)_{\gamma\delta)}=0$, where $C$ is the anti-symmetric charge conjugation matrix with $C^2=-1$. More generally we have $
    \psi{\bar\chi}
    =
    -\frac{1}{4} {\bar\psi}\chi
    -\frac{1}{4}{\bar\psi}\gamma_5\chi\,\gamma_5
    +\frac{1}{4}{\bar\psi}\gamma^\mu\chi\,\gamma_\mu
    -\frac{1}{8}{\bar\psi}\gamma^{\mu\nu}\chi\,\gamma_{\mu\nu}
    +\frac{1}{4}{\bar\psi}\gamma^\mu\gamma_5\chi\,\gamma_\mu\gamma_5\,.$

[^3]: Note that using the equation of motion for the Yang-Mills gauge field, $$\begin{aligned}
    D_\nu F_\alpha^{\mu\nu}
    +T_{\alpha b}{}^a
    \Big(
    D_\nu(\phi^b\mathcal B^{\mu\nu}_a-\phi_a\mathcal B^{\mu\nu\,b})
    +\phi^bD^\mu\phi_a
    -\phi_aD^\mu\phi^b
    +\frac{1}{2}\mathcal B^b_{\nu\rho}\mathcal H_a^{\mu\nu\rho}
    -\frac{1}{2}\mathcal B_{\nu\rho\,a}\mathcal H^{\mu\nu\rho\,b}
    \Big)
    =0\ ,\end{aligned}$$ does not help.

[^4]: The required field redefinitions are $
    \rho^i\rightarrow2\sqrt2\,i(m\,\lambda^a+m^{-1}\,T_{\alpha b}{}^a\phi^b\lambda^\alpha),\
    \varphi\rightarrow\sqrt2\,\phi\,,\quad H_{\mu\nu}\rightarrow\sqrt2\,m\mathcal B_{\mu\nu}\,,\quad\chi\rightarrow2\sqrt2\,i\chi\,,\quad\lambda^I\rightarrow-2i\lambda^\alpha
    $ together with $g\rightarrow m$, $D_\mu\rightarrow\frac{1}{m}D_\mu$.


Q:

How to remove Hive table bucketing

I mistakenly executed the following ALTER statement with table stocks:
ALTER TABLE stocks
CLUSTERED BY (exchange, symbol)
INTO 48 BUCKETS;

How to undo this command?

A:

If table is managed, then make it external:
ALTER TABLE stocks SET TBLPROPERTIES('EXTERNAL'='TRUE');

Describe table and note it's location, serde properties and check it is EXTERNAL:
describe formatted stocks;

Then drop table and create again specifying location, data will remain for EXTERNAL table:
   DROP TABLE stocks;

    CREATE EXTERNAL TABLE stocks(
            columns definition)
        STORED AS TEXTFILE --use the same DDL
        location '/table_location_path'; --use the same path from describe table

Alternatively you can create table with another name pointing to the same location, check it works, make first table EXTERNAL, drop it, rename second table using
ALTER TABLE tablename RENAME TO stocks.


Wednesday, January 6, 2016

Dow Theory Update for January 6: US Stocks remain under bear market spell

Joshua Brown, of the Reformed Broker, sees global bear market

Trends have not changed since my last post of December
24th. Market action, though, seems to confirm the picture I have
been depicting during December: Be cautious with US stocks, and, after all,
gold and silver miners might not be so weak after all. I am writing before the close.

Following the primary bear
market signal of Friday 11th December, stocks made lower confirmed lows on
12/18/2015 and 1/6/2016 (today). Hence, any secondary (bullish) reaction is to
be counted starting from these last recorded lows. Hitherto, no secondary
reaction has materialized yet. Here you have an updated chart.

Primary bear market in full gear

Furthermore, there is global
headwind for US stocks. Joshua Brown, quoting Meb Faber, and UBS, believes that
a worldwide primary bear market is underway. Here you have the link to his succinct
and interesting post entitled “The Global Bear Market Has Already Begun”

SIL has violated its 9/10/2015
closing low (last primary bear market low) unconfirmed by GDX. Both ETF miners
are under a strong secondary reaction (displayed by the red rectangles on the
chart below).

We have to wait for GDX to
confirm. Until then we cannot declare a new primary bear market. The longer it
takes for GDX to confirm, the better the odds for the primary bull market to
survive. However, price action is king. Since mid-November GDX has refused to
confirm.

I see that gold and silver
miners ETFs seem to be weathering the storm afflicting US (and world) stocks.
Given pervasive weakness all around, their relative strength might be
indicative that strong hands are quietly accumulating such stocks.

About Me

Disclaimer/Disclosure

This blog (dowtheoryinvestment.com) is strictly a personal journal applying my interpretation of Dow Theory principles to the action of the stock market and my musings about investment and trading in general. This blog is intended solely for entertaining, illustrative or informational purposes. I am not a registered investment advisor and neither the information nor the opinions expressed should be construed as a solicitation to buy or sell any stock, option, ETF, mutual fund, currency, commodity, or any other security. I am unaware of any readers personal circumstance, financial condition, risk tolerance or goals and objectives, so nothing read here should be considered advice suitable for them. Anyone reading this blog does so with the understanding that this is strictly meant as an analytical exercise and does not proffer actionable advice in any way, shape or form. Trading and investing always entail risk and possible loss of funds and should only be undertaken after appropriate due diligence by the trader/investor and after consulting a registered investment adviser.


Optimization of anti-pseudomonal antibiotics for cystic fibrosis pulmonary exacerbations: I. aztreonam and carbapenems.
Acute pulmonary exacerbations (APE) in cystic fibrosis (CF) are associated with loss of lung function that may require aggressive management with intravenous antibiotics. The aim of this review is to provide an evidence-based summary of pharmacokinetic/pharmacodynamic (PK/PD), tolerability, and efficacy studies utilizing aztreonam and anti-pseudomonal carbapenems (i.e., doripenem, imipenem-cilastatin, and meropenem) in the treatment of an APE, and to identify areas where further study is warranted. The current dosing recommendations in the United States and Europe for aztreonam are lower than the literature supported dosing range of 200-300 mg/kg/day divided every 6 hr, maximum 8-12 g/day. In vitro, PK/PD, and tolerability studies show the potential of doripenem 90 mg/kg/day divided every 8 hr, infused over 4 hr, maximum 6 g/day in the treatment of APE. Imipenem-cilastatin 100 mg/kg/day divided every 6 hr, maximum 4 g/day and meropenem 120 mg/kg/day divided every 8 hr, maximum 6 g/day have been shown to be tolerable and effective in the treatment of APE. With availability issues of new anti-pseudomonal agents and a large percentage of CF patients will not regain their lung function following an APE, we suggest the need to determine optimization of aztreonam and meropenem dosing in CF, as well as to determine the clinical efficacy of doripenem in the treatment of APE. The usefulness of imipenem-cilastatin may be limited due to the rapid development of resistance.


Q:

Manipulating lists with uneven amounts of entries

So I have a list like so
list={{a},{b,c},{d,e,f},{g,h}}

and the basic logic I want is to be able to manipulate each of the nested lists individually.
So If I wanted to take the average of each list I would do something like
avglist=Table[(1/3) Sum[list[[n,j]],{j,3}],{n,4}]

But the problem with this is when the program would call list[[1,2]]
So the gist is I am trying to write a program that could average/in general manipulate each of the nested lists of uneven lengths without having to know how many entries are in the list with the most entries.
I can't just add zeroes into the nested lists with fewer entries because if I want to manipulate them first then average it would affect the average.
Thanks

A:

If in general you want to "do the same thing to each element of a list" then think Map.
list = {{a}, {b, c}, {d, e, f}, {g, h}};
Map[Mean, list]

(* Out[2]= {a, (b + c)/2, 1/3 (d + e + f), (g + h)/2} *)

Replace Mean with a function of your own definition if necessary.
myF[v_] := v.v + Length[v] - First[v];
Map[myF, list]

(* Out[4]= {1 - a + a^2, 2 - b + b^2 + c^2, 3 - d + d^2 + e^2 + f^2, 2 - g + g^2 + h^2} *)


Q:

Span tag with display:inline rendering as display:block

I have a span tag like this

<s:textfield name="trnfr" id="trnfr"/>

  <img src="./images/cal.gif" border="0"
  alt='<s:text name="Co.Cal" />' />

It should render as

but it is rendering like this

when i viewed the source in the browser it is being rendered like this

Dont know whats going wrong.
Note: I got the correct image by changing display:block to display:inline dynamically in the browser.

A:

You aren't able to easily override in your CSS because the element is being set to display:block inline (within your HTML), likely by the framework you are using (hard to say without further detail).
As such you will either need to prevent this from being set, remove it (i.e. using JS) or override it in your CSS.
To override in your CSS, you will need to use !important, e.g:
#trnfrm{
  display:inline!important;
}

That said, note that the use of !important is not recommended. Ideally you should look to fix the root cause.

When an !important rule is used on a style declaration, this
  declaration overrides any other declaration made in the CSS, wherever
  it is in the declaration list. Although, !important has nothing to do
  with specificity.  Using !important is bad practice because it makes
  debugging hard since you break the natural cascading in your
  stylesheets.


/*
 * Copyright (C) 2013 Apple Inc. All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY APPLE INC. ``AS IS'' AND ANY
 * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL APPLE INC. OR
 * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
 * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
 * PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
 * OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */

#pragma once

#include <stddef.h>

namespace JSC {

typedef size_t SourceID;
static const SourceID noSourceID = 0;

typedef size_t BreakpointID;
static const BreakpointID noBreakpointID = 0;

} // namespace JSC


<?php

namespace MangoPay;

/**
 * Class represents pagination information
 */
class Pagination extends Libraries\Dto
{
    /**
     * Page number
     * @var int
     */
    public $Page;

    /**
     * Number of items per page
     * @var int
     */
    public $ItemsPerPage;

    /**
     * Number of total pages
     * @var int
     */
    public $TotalPages;

    /**
     * Number of total items
     * @var int
     */
    public $TotalItems;

    /**
     * Array with links to navigation.
     * All values optional. Format:
     * array(
     *      first => http url
     *      prev => http url
     *      next => http url
     *      last => http url
     * )
     * @var array
     */
    public $Links = array();

    /**
     * Construct
     * @param int $page Number of page
     * @param int $itemsPerPage Number of items on one page
     */
    public function __construct($page = 1, $itemsPerPage = 10)
    {
        $this->Page = $page;
        $this->ItemsPerPage = $itemsPerPage;
    }
}


Minister of Fire2.

NULL

Now they aren't sure which should be done first. We thought the surgery but now all the doctors need to get their heads together to determine which is the most pressing. Some good news though and that is concerning the pancreas. They seem to think it might be okay now. She took the ride to the hospital and back with no problem and for sure she gets a little stronger each day.

Minister of Fire2.

NULL

Now they aren't sure which should be done first. We thought the surgery but now all the doctors need to get their heads together to determine which is the most pressing. Some good news though and that is concerning the pancreas. They seem to think it might be okay now. She took the ride to the hospital and back with no problem and for sure she gets a little stronger each day.

Click to expand...

That's good news Dennis! Good thing Judy is ornery my friend Tell Judy I said hello and best wishes!

Guest2.

NULL

Now they aren't sure which should be done first. We thought the surgery but now all the doctors need to get their heads together to determine which is the most pressing. Some good news though and that is concerning the pancreas. They seem to think it might be okay now. She took the ride to the hospital and back with no problem and for sure she gets a little stronger each day.

Mooderator2.

NULL

Staff Member

Good to hear she is feeling better. Hope the good news continues for Judy and you.

PE Alderlea T6 - the gentle giant and a Jotul 602, the little stove that could“Sometimes I wonder whether the world is being run by smart people who are putting us on, or by imbeciles who really mean it.” - Mark Twain -
"A poor worker always blames his tools." - Dad

Minister of Fire2.

NULL

A bit of frustration setting in. Poor Judy has some terrible arthritis. Had to get off the medicine and that means she hurts all over. Doctors not much help on this one yet. I just called our family doctor to see if he has any ideas. It really hurts to see her like this but we're still holding on here.

Minister of Fire2.

NULL

I should have added that she did get outdoors today. Walked almost to the road and back. Yes, it did tire her a lot. She doesn't like that a bit because before this all happened she had been walking 2-3 miles per day besides her daily exercises. Needless to say, this family will have some tough days ahead just trying to get back in shape. lol

Guest2.

NULL

I should have added that she did get outdoors today. Walked almost to the road and back. Yes, it did tire her a lot. She doesn't like that a bit because before this all happened she had been walking 2-3 miles per day besides her daily exercises. Needless to say, this family will have some tough days ahead just trying to get back in shape. lol

Click to expand...

Tell her to keep her head high. The nice thing is it's early yet, heck it ain't even spring yet. Lots of time for her to ease back into the walking/excersize stuff......
Thank God she's doing as well as she is......I'm looking forward to meeting both of you this summer, if we get out that way!

Minister of Fire2.

NULL

A bit of frustration setting in. Poor Judy has some terrible arthritis. Had to get off the medicine and that means she hurts all over. Doctors not much help on this one yet. I just called our family doctor to see if he has any ideas. It really hurts to see her like this but we're still holding on here.

Guest2.

NULL

A bit of frustration setting in. Poor Judy has some terrible arthritis. Had to get off the medicine and that means she hurts all over. Doctors not much help on this one yet. I just called our family doctor to see if he has any ideas. It really hurts to see her like this but we're still holding on here.

Click to expand...

Krill Oil capsules, is suppose to be good for your joints. Pineapple,,, and not from the can.. Sorry to hear that Judy is in such pain Dennis..

Minister of Fire2.

NULL

Backwoods Savage, sorry to hear about the setbacks they can be quite frustrating! We will continue to pray for both of you but its good that she is stubborn and will keep fighting through this situation.

Minister of Fire2.

NULL

This thread has probably went long enough so I'll just say that Judy is a bit better today. We appreciate all the prayers and kind words and will continue to appreciate them. I'll give some updates if and when things change much and say the next hurdle is on March 20.