arXiv subjects
Publications and source records attributed to P. S. Howe.
The $α'^2$ deformation of D=10 SYM is the natural generalisation of the $F^4$ term in the abelian Born-Infeld theory. It is shown that this deformation can be extended to $α'^4$ in a way which is consistent with supersymmetry. The latter requires the presence of higher-derivative and commutator terms as well as the symmetrised trace of the Born-Infeld $α'^4$ term.
We perform a careful investigation of which p-form fields can be introduced consistently with the supersymmetry algebra of IIA and/or IIB ten-dimensional supergravity. In particular the ten-forms, also known as "top-forms", require a careful analysis since in this case, as we will show, closure of the supersymmetry algebra at the linear level does not imply closure at the non-linear level. Consequently, some of the (IIA and IIB) ten-form potentials introduced in earlier work of some of us are discarded. At the same time we show that new ten-form potentials, consistent with the full non-linear supersymmetry algebra can be introduced. We give a superspace explanation of our work. All of our results are precisely in line with the predictions of the E(11) algebra.
The question of whether BPS invariants are protected in maximally supersymmetric Yang-Mills theories is investigated from the point of view of algebraic renormalisation theory. The protected invariants are those whose cohomology type differs from that of the action. It is confirmed that one-half BPS invariants ($F^4$) are indeed protected while the double-trace one-quarter BPS invariant ($d^2F^4$) is not protected at two loops in D=7, but is protected at three loops in D=6 in agreement with recent calculations. Non-BPS invariants, i.e. full superspace integrals, are also shown to be unprotected.
We discuss various approaches to the problem of determining which supersymmetric invariants are permitted as counterterms in maximally supersymmetric super Yang--Mills and supergravity theories in various dimensions. We review the superspace non-renormalisation theorems based on conventional, light-cone, harmonic and certain non-Lorentz covariant superspaces, and we write down explicitly the relevant invariants. While the first two types of superspace admit the possibility of one-half BPS counterterms, of the form $F^4$ and $R^4$ respectively, the last two do not. This suggests that UV divergences begin with one-quarter BPS counterterms, i.e. $d^2 F^4$ and $d^4 R^4$, and this is supported by an entirely different approach based on algebraic renormalisation. The algebraic formalism is discussed for non-renormalisable theories and it is shown how the allowable supersymmetric counterterms can be determined via cohomological methods. These results are in agreement with all the explicit computations that have been carried out to date. In particular, they suggest that maximal supergravity is likely to diverge at four loops in D=5 and at five loops in D=4, unless other infinity suppression mechanisms not involving supersymmetry or gauge invariance are at work.
The superspace geometry relevant to the heterotic string is reviewed from the point of view of the off-shell supermultiplet structure of $N=1,d=10$ supergravity. The anomaly-modified seven-form Bianchi identity is analysed at order $\a'^3$ and shown to admit a complete solution. The corresponding $\a'^3$ deformation of the dimension-zero torsion tensor is derived and shown to obey the appropriate cohomological constraint.
The superform construction of supersymmetric invariants, which consists of integrating the top component of a closed superform over spacetime, is reviewed. The cohomological methods necessary for the analysis of closed superforms are discussed and some further theoretical developments presented. The method is applied to higher-order corrections in heterotic string theory up to $\a'^3$. Some partial results on $N=2,d=10$ and $N=1,d=11$ are also given.
A superspace formulation of IIB supergravity which includes the field strengths of the duals of the usual physical one, three and five-form field strengths as well as the eleven-form field strength is given. The superembedding formalism is used to construct kappa-symmetric SL(2,R) covariant D-brane actions in an arbitrary supergravity background.
A kappa-symmetric action for coincident D-branes is presented. It is valid in the approximation that the additional fermionic variables, used to incorporate the non-abelian degrees of freedom, are treated classically. The action is written as a Bernstein-Leites integral on the supermanifold obtained from the bosonic worldvolume by adjoining the extra fermions. The integrand is a very simple extension of the usual Green-Schwarz action for a single brane; all symmetries, except for kappa, are manifest, and the proof of kappa-symmetry is very similar to the abelian case.
A covariant version of the non-abelian Dirac-Born-Infeld-Myers action is presented. The non-abelian degrees of freedom are incorporated by adjoining to the (bosonic) worldvolume of the brane a number of anticommuting fermionic directions corresponding to boundary fermions in the string picture. The proposed action treats these variables as classical but can be given a matrix interpretation if a suitable quantisation prescription is adopted. After gauge-fixing and quantisation of the fermions, the action is shown to be in agreement with the Myers action derived from T-duality. It is also shown that the requirement of covariance in the above sense leads to a modified WZ term which also agrees with the one proposed by Myers.
In addition to superconformal symmetry, (1,1) supersymmetric two-dimensional sigma models on special holonomy manifolds have extra symmetries that are in one-to-one correspondence with the covariantly constant forms on these manifolds. The superconformal algebras extended by these symmetries close as W-algebras, i.e. they have field-dependent structure functions. It is shown that it is not possible to write down cohomological equations for potential quantum anomalies when the structure functions are field-dependent. In order to do this it is necessary to linearise the algebras by treating composite currents as generators of additional symmetries. It is shown that all cases can be linearised in a finite number of steps, except for G_2 and SU(3). Additional problems in the quantisation procedure are briefly discussed.
A study of (1,1) supersymmetric two-dimensional non-linear sigma models with boundary on special holonomy target spaces is presented. In particular, the consistency of the boundary conditions under the various symmetries is studied. Models both with and without torsion are discussed.
The Green-Schwarz action for an open superstring with additional boundary fermions, representing Chan-Paton factors, is studied at the classical level. The boundary geometry is described by a bundle, with fermionic fibres, over the super worldvolume of a D-brane together with a map from the total space into the type II target superspace. This geometry is constrained by the requirement of kappa-symmetry on the boundary together with the use of the equations of motion for the fermions. There are two constraints which are formally similar to those that arise in the abelian case but which differ because of the dependence on the additional coordinates. The model, when quantised, would be a candidate for a fully kappa-symmetric theory of a stack of coincident D-branes including a non-abelian Born-Infeld sector. The example of the D9-brane in a flat background is studied. The constraints on the non-abelian field strength are shown to be in agreement with those derived from the pure spinor approach to the superstring. A covariant formalism is developed and the problem of quantisation is discussed.
The M2-brane is studied from the perspective of superembeddings. We review the derivation of the M2-brane dynamics and the supergravity constraints from the standard superembedding constraint and we discuss explicitly the induced d=3, N=8 superconformal geometry on the worldvolume. We show that the gauged supermembrane, for a target space with a U(1) isometry, is the standard D2-brane in a type IIA supergravity background. In particular, the D2-brane action, complete with the Dirac-Born-Infeld term, arises from the gauged Wess-Zumino worldvolume 4-form via the brane action principle. The discussion is extended to the massive D2-brane considered as a gauged supermembrane in a massive D=11 superspace background. Type IIA supergeometry is derived using Kaluza-Klein techniques in superspace.
Higher-order invariants and their role as possible counterterms for supergravity theories are reviewed. It is argued that N=8 supergravity will diverge at 5 loops. The construction of $R^4$ superinvariants in string and M-theory is discussed.
The properties of gauge-invariant composite operators and their correlation functions in N=4 SYM are discussed in the analytic superspace formalism. A complete classification of the different types of operators in the theory is given. Operators can be either protected or unprotected according to whether they do not or do have anomalous dimensions, and the analytic superspace formalism allows one to identify which type a given operator is in a straightforward manner. A simple discussion is given of the behaviour of reducible multiplets at threshold. It is pointed out that there is a class of ``semi-protected'' operators which do not have anomalous dimensions but which do not necessarily have non-renormalised three-point functions when the other two operators in the correlator are protected, although two-point functions of such operators are non-renormalised. A complete discussion of superconformal invariants in analytic superspace is given. The paper includes a modified discussion of the transformation rules of analytic superfields which clarifies the $U(1)_Y$ properties of operators and correlation functions and, in particular, explicit examples are given of three-point correlation functions which violate this symmetry. A tensor, $\cE$, invariant under $SL(n|m)$ but not under $GL(n|m)$, is introduced and used in the discussion of $U(1)_Y$ and in the construction of invariants.
The construction of supersymmetric invariant integrals is discussed in a superspace setting. The formalism is applied to D=4, N=4 SYM and used to construct the F^2, F^4 and (F^5 + \del^2 F^4) terms in the effective action of coincident D-branes. The results are in agreement with those obtained by other methods. A simple derivation of the abelian \del^4 F^4 invariant is given and generalised to the non-abelian case. We also find some double-trace invariants. The invariants are interpreted in terms of superconformal multiplets: the F^2 and F^4 terms are given by one-half BPS multiplets, the (F^5+\del^2F^4) arises as a full superspace integral of the Konishi multiplet K and the abelian \del^4 F^4 term comes from integrating the fourth power of the field strength superfield. Counterparts of the abelian invariants are exhibited for the D=6,(2,0) tensor multiplet and the D=3, N=8 scalar multiplet. The method is also applied to D=4, N=8 supergravity. All invariants in the linearised theory (with SU(8) symmetry) which arise from partial superspace integrals are constructed.
The superembedding formalism is used to study correction terms to the dynamics of the M2 brane in a flat background. This is done by deforming the standard embedding constraint. It is shown rigorously that the first such correction occurs at dimension four. Cohomological techniques are used to determine this correction explicitly. The action is derived to quadratic order in fermions, and the modified $\k$-symmetry transformations are given.
A theoretical analysis of higher-order corrections to D=11 supergravity is given in a superspace framework. It is shown that any deformation of D=11 supergravity for which the lowest-dimensional component of the four-form $G_4$ vanishes is trivial. This implies that the equations of motion of D=11 supergravity are specified by an element of a certain spinorial cohomology group and generalises previous results obtained using spinorial or pure spinor cohomology to the fully non-linear theory. The first deformation of the theory is given by an element of a different spinorial cohomology group with coefficients which are local tensorial functions of the massless supergravity fields. The four-form Bianchi Identities are solved, to first order and at dimension $-{1/2}$, in the case that the lowest-dimensional component of $G_4$ is non-zero. Moreover, it is shown how one can calculate the first-order correction to the dimension-zero torsion and thus to the supergravity equations of motion given an explicit expression for this object in terms of the supergravity fields. The version of the theory with both a four-form and a seven-form is discussed in the presence of the five-brane anomaly-cancelling term. It is shown that the supersymmetric completion of this term exists and it is argued that it is the unique anomaly-cancelling invariant at this dimension which is at least quartic in the fields. This implies that the first deformation of the theory is completely determined by the anomaly term from which one can, in principle, read off the corrections to all of the superspace field strength tensors.