Full Superspace Research Articles

M5-brane in the superspace approach

Motivated by Sen's spacetime prescription for the construction of theories with self-dual field strengths, we present a rigid superspace Lagrangian describing noninteracting tensor multiplets living on a stack of M5-branes and containing all the physical constraints on the fields, yielding the on-shell matching of the degrees of freedom. The geometric superspace approach adopted here offers a natural realization of superdiffeomorphisms and is particularly well suited for the coupling to supergravity. However, within this formulation the (anti-)self-duality property of the 3-form field strengths is lost when the superspace Lagrangian is trivially restricted to spacetime. We propose two main paths to address this issue: a first-order superspace extension of Sen's spacetime results, which, once trivially restricted to spacetime, yields all the dynamical equations including the (anti-)self-duality constraint on the 3-form field strengths, and a possible way to obtain a full superspace description of the theory, based on integral forms.

Circumnavigating collinear superspace

In this paper, we extend the collinear superspace formalism to include the full range of mathcal{N} = 1 supersymmetric interactions. Building on the effective field theory rules developed in a companion paper — Navigating Collinear Superspace [1] — we construct collinear superspace Lagrangians for theories with non-trivial F- and D-term auxiliary fields. For (massless) Wess-Zumino models, the key ingredient is a novel type of Grassmann-valued supermultiplet whose lowest component is a (non-propagating) fermionic degree of freedom. For gauge theories coupled to charged chiral matter, the key ingredient is a novel type of vector superfield whose lowest component is a non-propagating gauge potential. This unique vector superfield is used to construct a gauge-covariant derivative; while such an object does not appear in the standard full superspace formalism, it is crucial for modeling gauge interactions when the theory is expres sed on a collinear slice. This brings us full circle, by showing that all types of mathcal{N} = 1 theories in four dimensions can beconstructed in collinear superspace from purely infrared considerations. We speculate that supersymmetric theories with mathcal{N} > 1 could also be implemented using similar collinear superspace constructions.

Linearised actions for mathcal{N} -extended (higher-spin) superconformal gravity

The off-shell actions for mathcal{N} -extended conformal supergravity theories in three dimensions were formulated in [1, 2] for 1 ≤ mathcal{N} ≤ 6 using a universal approach. Each action is generated by a closed super three-form which is constructed in terms of the constrained geometry of mathcal{N} -extended conformal superspace. In this paper we initiate a program to recast these actions (and to formulate their higher-spin counterparts) in terms of unconstrained gauge prepotentials as integrals over the full superspace. We derive transverse projection operators in mathcal{N} -extended Minkowski superspace and then use them to construct linearised rank-n super-Cotton tensors and off-shell mathcal{N} -extended superconformal actions. We also propose off-shell gauge-invariant actions to describe massive higher-spin super-multiplets in mathcal{N} -extended supersymmetry. Our analysis leads to general expressions for identically conserved higher-spin current multiplets in mathcal{N} -extended supersymmetry.

On integrability of strings on symmetric spaces

In the absence of NSNS three-form flux the bosonic string on a symmetric space is described by a symmetric space coset sigma-model. Such models are known to be classically integrable. We show that the integrability extends also to cases with non-zero NSNS flux (respecting the isometries) provided that the flux satisfies a condition of the form H_{abc}H^{cde}~R_{ab}^{de}. We then turn our attention to the type II Green-Schwarz superstring on a symmetric space. We prove that if the space preserves some supersymmetry there exists a truncation of the full superspace to a supercoset space and derive the general form of the superisometry algebra. In the case of vanishing NSNS flux the corresponding supercoset sigma-model for the string is known to be integrable. We prove that the integrability extends to the full string by augmenting the supercoset Lax connection with terms involving the fermions which are not captured by the supercoset model. The construction is carried out to quadratic order in these fermions. This proves the integrability of strings on symmetric spaces supported by RR flux which preserve any non-zero amount of supersymmetry. Finally we also construct Lax connections for some supercoset models with non-zero NSNS flux describing strings in AdS(2,3) x S(2,3) x S(2,3) x T(2,3,4) backgrounds preserving eight supersymmetries.

Wave function of the universe from a matrix-valued first-order formalism

In this paper, the Wheeler–DeWitt equation in full superspace formalism will be written in a matrix-valued first-order formalism. We will also analyze the Wheeler–DeWitt equation in minisuperspace approximation using this matrix-valued first-order formalism. We will note that this Wheeler–DeWitt equation, in this minisuperspace approximation, can be expressed as an eigenvalue equation. We will use this fact to analyze the spacetime foam in this formalism. This will be done by constructing a statistical mechanical partition function for the Wheeler–DeWitt equation in this matrix-valued first-order formalism. This will lead to a possible solution for the cosmological constant problem.

Null polygonal Wilson loops in full superspace

We compute the one-loop expectation value of light-like polygonal Wilson loops in super-Yang–Mills theory in full superspace. When projecting to chiral superspace, we recover the known results for the tree-level next-to-maximally-helicity-violating scattering amplitude. The one-loop maximally-helicity-violating amplitude is also included in our result but there are additional terms which do not immediately correspond to scattering amplitudes. We finally discuss different regularizations and their Yangian anomalies.

Two-loop effective potentials in general [formula omitted], [formula omitted] chiral superfield model

We study local superspace contributions to the low-energy effective action in general chiral three-dimensional superfield model. The effective Kähler and chiral potentials are computed in an explicit form up to the two-loop order. In accordance with the non-renormalization theorem, the ultraviolet divergences appear only in the full superspace while the effective chiral potential receives only finite quantum contributions in the massless case. As an application, the two-loop effective scalar potential is found for the three-dimensional N=2 supersymmetric Wess–Zumino model.

Orthogonality of Hermite polynomials in superspace and Mehler type formulae

In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry 𝒢<O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m)×Sp(2n) are considered. After an overview of the results for O(m) and 𝒢, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proved. Next, Hermite polynomials in a full superspace with O(m)×Sp(2n)-symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced, which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schrödinger operators in superspace. Subsequently, a Mehler formula for the full superspace is obtained, thus yielding an eigenfunction decomposition of the super Fourier transform. Finally, the new results for the Sp(2n)- and O(m)×Sp(2n)-symmetry are compared with the results in the different types of symmetry.

New supersymmetric higher-derivative couplings: full N = 2 superspace does not count!

An extended class of N = 2 locally supersymmetric invariants with higher-derivative couplings based on full superspace integrals, i s constructed. These invariants may depend on unrestricted chiral supermultiplets, on vect or supermultiplets and on the Weyl supermultiplet. Supersymmetry is realized off-shell. A non-renormalization theorem is proven according to which none of these invariants can con tribute to the entropy and electric charges of BPS black holes. Some of these invariant s may be relevant for topological string deformations.

A note on the UV behaviour of maximally supersymmetric Yang–Mills theories

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 (F4) are indeed protected while the double-trace one-quarter BPS invariant (d2F4) 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.

Different representations for the action principle in 4D 𝒩 = 2 supergravity

Within the superspace formulation for four-dimensional = 2 matter-coupled supergravity developed in arXiv:0805.4683, we elaborate two approaches to reduce the superfield action to components. One of them is based on the principle of projective invariance which is a purely = 2 concept having no analogue in simple supergravity. In this approach, the component reduction of the action is performed without imposing any Wess-Zumino gauge condition, that is by keeping intact all the gauge symmetries of the superfield action, including the super-Weyl invariance. As a simple application, the c-map is derived for the first time from superfield supergravity. Our second approach to component reduction is based on the method of normal coordinates around a submanifold in a curved superspace, which we develop in detail. We derive differential equations which are obeyed by the vielbein and the connection in normal coordinates, and which can be used to reconstruct these objects, in principle in closed form. A separate equation is found for the super-determinant of the vielbein E = Ber(EMA), which allows one to reconstruct E without a detailed knowledge of the vielbein. This approach is applicable to any supergravity theory in any number of space-time dimensions. As a simple application of this construction, we reduce an integral over the curved = 2 superspace to that over the chiral subspace of the full superspace. We also give a new representation for the curved projective-superspace action principle as a chiral integral.

Formula omitted] super-Yang–Mills NMHV loop amplitude in superspace

Here we construct N=4 super-Yang–Mills 6 point NMHV loop amplitude (amplitudes with three minus helicities) as a full superspace form, using the SU(4)R anti-commuting spinor variables. Amplitudes with different external particle and cyclic helicity ordering are then just a particular expansion of this fermionic variable. We have verified this by explicit expansion obtaining amplitudes with two gluinos calculated before. We give results for all gluino A(Λ−Λ−Λ−Λ+Λ+Λ+) and all scalar A(ϕϕϕϕϕϕ) scattering amplitudes. A discussion of using MHV vertex approach to obtain these amplitudes are given, which implies a possible simplification for general loop amplitudes.

Integral invariants inN= 4 SYM and the effective action for coincident D-branes

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.

Nonlinear realization of partially broken N=2 superconformal symmetry in four dimensions

Abstract We investigate the nonlinear realization of spontaneously broken D=4 N=2 superconformal symmetry. We particularly study Nambu-Goldstone degrees of freedom for the partial breaking of N=2 superconformal symmetry down to N=1 super-Poincare symmetry, where we get the chiral NG multiplet of dilaton and the vector NG multiplet of NG fermion of broken Q-supersymmetry. Evaluating the covariant derivatives and supervielbeins for the chiral as well as the full superspace, we obtain the nonlinear effective lagrangians.

Superspace measures, invariant actions, and component projection formulae for (2,2) supergravity

In the framework of the prepotential description of superspace two-dimensional (2,2) supergravity, we discuss the construction of invariant integrals. In addition to the full superspace measure, we derive the measure for chiral superspace, and obtain the explicit expressions for going from superspace actions to component actions. We consider both the minimal U A (1) and the extended U V(1) ⋇ U A(1) theories.

The realizations of local supersymmetry

We show that the realizations of local supersymmetry in the full superspace reduce to realizations in the Wess-Zumino gauge in a manifestly covariant way. The natural frame of this covariant Wess-Zumino gauge is the one of the new theta variables of Wess and Zumino. With this formalism we compute the invariant densities of the chiral superspace in a geometrical way. We show also how to compute the transformation law of the nonlinear realizations of local supersymmetry from the transformation law of the new superspace.

Superfield perturbation theory in harmonic superspace.

Superfield perturbation methods are developed for N = 2 supersymmetric theories in harmonic superspace with central charges. To illustrate our methods, explicit examples of loop calculations are given; the self-energies of the gauge hypermultiplet are computed at the one-loop level, giving the expected one-loop ..beta.. function. It is also shown that the effective action may be written as integrals over a single d/sup 8/theta in full superspace (but not in analytic subspace). The implication of this result is also briefly discussed.

On extensions of supersymmetry algebras in d=4 and 11

SUSY algebras, in d=4 and 11, are analyzed, extended by the addition of both spinorial and bosonic central charge generators. It is found that only limited extensions are possible, on use of Jacobi identities. Casimirs, irreps, and spin reduction are discussed, and the problems of constructing actions for SUSY and supergravity theories as full superspace actions are considered.

A USp(8)-symmetric full superspace version of N=8 supergravity

The authors present another full superspace version of N=8 supergravity which has 56 extra central charge dimensions. Torsion and curvature constraints are given whose solution only depends on the N=8 hypermultiplet. An action obtained by projecting the full volume element down onto a 14-dimensional subspace is shown to lead to the correct equations of motion for N=8 supergravity. The model may be constructed so as to have an explicit local USp(8) symmetry.

A full superspace form of N=8 supergravity

We construct a full superspace form of N=8 supergravity by means of constraints imposed on the superspace through vanishing of certain torsions. These constraints are shown to be consistent and to lead to the total geometry being described by the N=8 hypermultiplet with one degenerate central charge. An action is given as the volume of the maximal set of Grassman variables and fourteen extra bosonic coordinates, and shown to lead to the correction equations of motion for N=8 supergravity.