The SuperConformal theory in three space-time dimensions with SO(16) R-symmetry, 128 bosons and 128 fermions cannot sustain interactions. This result is obtained using both light-cone superspace techniques which rely on algebraic consistency, and co-variant methods which rely on SO(16) Fierz identities which fail to produce the desired algebra.

The low-energy effective action for the N = 4 super Yang-Mills on the Coulomb branch is known to include an SO(6)-invariant Wess-Zumino (WZ) term for the six scalar fields. For each maximal, non-anomalous subgroup of the SU(4) R-symmetry, we find a four-dimensional form of the WZ term with this subgroup being manifest. We then show that a recently proposed expression for the four-derivative part of the effective action in N = 4 USp(4) harmonic superspace yields the WZ term with manifest SO(5) R-symmetry subgroup. The N = 2 SU(2) harmonic superspace form of the effective action produces the WZ term with manifest SO(4) x SO(2). We argue that there is no four-dimensional form of the WZ term with manifest SU(3) R-symmetry, which is relevant for N = 1 and N = 3 superspace formulations of the effective action.

We develop \( \mathcal{N} = 4 \) d = 4 bi-harmonic superspace and use it to derive a novel form for the low-energy effective action in \( \mathcal{N} = 4 \) super Yang-Mills theory. We solve the \( \mathcal{N} = 4 \) supergauge constraints in this superspace in terms of analytic superfields. Using these superfields, we construct a simple functional that respects \( \mathcal{N} = 4 \) supersymmetry and scale invariance. In components, it reproduces all on-shell terms in the four-derivative part of the \( \mathcal{N} = 4 \) SYM effective action; in particular, the F 4/X 4 and Wess-Zumino terms. The latter comes out in a novel SO(3) × SO(3)-invariant form.

We develop N = 4 d = 4 bi-harmonic superspace and use it to derive a novel form for the low-energy effective action in N = 4 super Yang-Mills theory. We solve the N = 4 supergauge constraints in this superspace in terms of analytic superfields. Using these superfields, we construct a simple functional that respects N = 4 supersymmetry and scale invariance. In components, it reproduces all on-shell terms in the four-derivative part of the N = 4 SYM effective action; in particular, the F-4/X-4 and Wess-Zumino terms. The latter comes out in a novel SO(3)xSO(3)-invariant form.

Boundary conditions in supergravity on a manifold with boundary relate the bulk gravitino to the boundary supercurrent, and the normal derivative of the bulk metric to the boundary energy-momentum tensor. In the 3D N = 1 setting, we show that these boundary conditions can be stated in a manifestly supersymmetric form. We identify the Extrinsic Curvature Tensor Multiplet, and show that boundary conditions set it equal to (a conjugate of) the boundary supercurrent multiplet. Extension of our results to higher-dimensional models (including the Randall-Sundrum and Horava-Witten scenarios) is discussed.

The light-cone superspace version of the d = 3, N = 8 superconformal theory of Bagger, Lambert and Gustavsson (BLG) is obtained as a solution to constraints imposed by OSp(2, 2|8) superalgebra. The Hamiltonian of the theory is shown to be a quadratic form of the dynamical supersymmetry transformation.

Maximally supersymmetric mass deformation of the Bagger-Lambert-Gustavsson (BLG) theory corresponds to a {non-central} extension of the d=3 N=8 Poincare superalgebra (allowed in three dimensions). We obtain its light-cone superspace formulation which has a novel feature of the dynamical supersymmetry generators being {cubic} in the kinematical ones. The mass deformation picks a quaternionic direction, which breaks the SO(8) R-symmetry down to SO(4)xSO(4). The Hamiltonian of the theory is shown to be a quadratic form of the dynamical supersymmetry transformations, to all orders in the mass parameter, M, and the structure constants, f^{a b c d}.

Maximally supersymmetric theories can be described by a single scalar superfield in light-cone superspace. When they are also (super)conformally invariant, they are uniquely specified by the form of the dynamical supersymmetry. We present an explicit derivation of the light-cone superspace form of the dynamical supersymmetry in the cases of ten- and four-dimensional super-Yang-Mills, and the three-dimensional Bagger-Lambert-Gustavsson theory, starting from the covariant formulation of these theories.

We present a supersymmetric version of the two-brane Randall-Sundrum scenario, with arbitrary brane tensions T1 and T2, subject to the bound |T1,2| ≤ √ −6Λ5, where Λ5 < 0 is the bulk cosmological constant. Dimensional reduction gives N = 1, D = 4 supergravity, with cosmological constant Λ4 in the range 1 2Λ5 ≤ Λ4 ≤ 0. The case with Λ4 = 0 requires T1 = −T2 = √ −6Λ5. This work unifies and generalizes previous approaches to the supersymmetric Randall-Sundrum scenario. It also shows that the Randall-Sundrum fine-tuning is not removed by supersymmetry. bagger@jhu.edu belyaev@pha.jhu.edu

We construct rigidly supersymmetric bulk-plus-boundary actions, both in x-space and in superspace. For each standard supersymmetric bulk action a minimal supersymmetric bulk-plus-boundary action follows from an extended F- or D-term formula. Additional separately supersymmetric boundary actions can be systematically constructed using co-dimension one multiplets (boundary superfields). We also discuss the orbit of boundary conditions which follow from the Euler-Lagrange variational principle.

We present a supersymmetric version of the two-brane Randall-Sundrum scenario, with arbitrary brane tensions T1 and T2, subject to the bound |T1,2| ≤ √ −6Λ5, where Λ5 < 0 is the bulk cosmological constant. Dimensional reduction gives N = 1, D = 4 supergravity, with cosmological constant Λ4 in the range 1 2Λ5 ≤ Λ4 ≤ 0. The case with Λ4 = 0 requires T1 = −T2 = √ −6Λ5. This work unifies and generalizes previous approaches to the supersymmetric Randall-Sundrum scenario. It also shows that the Randall-Sundrum fine-tuning is not a consequence of supersymmetry. bagger@jhu.edu belyaev@pha.jhu.edu

Using the simple setting of 3D N = 1 supergravity, we show how the tensor calculus of supergravity can be extended to manifolds with boundary. We present an extension of the standard F-density formula which yields supersymmetric bulk-plus-boundary actions. To construct additional separately supersymmetric boundary actions, we decompose bulk supergravity and bulk matter multiplets into co-dimension one submultiplets. As an illustration we obtain the supersymmetric extension of the York-Gibbons-Hawking extrinsic curvature boundary term. We emphasize that our construction does not require any boundary conditions on off-shell fields. This gives a significant improvement over the existing orbifold supergravity tensor calculus.

To construct rigidly or locally supersymmetric bulk-plus-boundary actions, one needs an extension of the usual tensor calculus. Its key ingredients are the extended (F-, D-, etc.) density formulas and the rule for the decomposition of bulk multiplets into (co-dimension one) boundary multiplets. Working out these ingredients for d = 4 N = 1 Poincare supergravity, we discover the special role played by R-symmetry (absent in the d = 3 N = 1 case we studied previously). The U(1) A R-symmetry has to be gauged which leads us to extend the old-minimal set of auxiliary fields S, P, A(mu) by a U(1) A compensator a. Our results include the "F + A" density formula, the "Q + L + A" formula for the induced supersymmetry transformations (closing into the standard d = 3 N = 1 algebra) and demonstration that the compensator a is the first component of the extrinsic curvature multiplet. We rely on the superconformal approach which allows us to perform, in parallel, the same analysis for new-minimal supergravity.

We point out a limitation of the existing supergravity tensor calculus on the $S^1/Z_2$ orbifold that prevents its use for constructing general supersymmetric bulk-plus-brane actions. We report on the progress achieved in removing this limitation via the development of ``supersymmetry without boundary conditions.''

We show how the (globally supersymmetric) model of Mirabelli and Peskin can be formulated in the boundary (``downstairs'' or ``interval'') picture. The necessary Gibbons-Hawking-like terms appear naturally when using (codimension one) superfields. This formulation is free of the δ(0) ambiguities of the orbifold (``upstairs'') picture while describing the same physics since the boundary conditions on the fundamental domain are the same. The (natural) boundary conditions follow from the variational principle and form a closed orbit under supersymmetry variation. They reduce to the ``odd =0'' boundary conditions in the absence of bulk-boundary coupling. We emphasize that the action is supersymmetric without the use of any boundary conditions in the off-shell formulation (but some boundary conditions are necessary for on-shell supersymmetry!).

Supersymmetric bulk-brane coupling in Horava-Witten and Randall-Sundrum scenarios, when considered in the orbifold (``upstairs'') picture, enjoys similar features: a modified Bianchi identity and a modified supersymmetry transformation for the ``orthogonal'' part of the gauge field. Using a toy model with a 5D vector multiplet in the bulk (like in Mirabelli-Peskin model, but with an odd gauge field Am), we explain how these features arise from the superfield formulation. We also show that the corresponding construction in the boundary (``downstairs'') picture requires introduction of a special ``compensator'' (super)field.

We construct the action and transformation laws for bulk five-dimensional AdS supergravity coupled to one or two brane-localized Goldstone fermions. The resulting bulk-plus-brane system gives a model-independent description of brane-localized supersymmetry breaking in the Randall-Sundrum scenario. We explicitly reduce the action and transformation laws to spontaneously broken four-dimensional supergravity (in AdS, Minkowski or dS space).

Bulk supergravity on a manifold with boundary must be supplemented by boundary conditions that preserve local supersymmetry. This "downstairs" picture has certain advantages over the equivalent "upstairs" picture, expressed in terms of orbifolds. In particular, Scherk-Schwarz supersymmetry breaking can be described much more simply in the downstairs picture. Nevertheless, physics on the fundamental domain can always be lifted upstairs, so long as fields are allowed to be discontinuous across the boundary. In this talk we apply these considerations to five-dimensional supergravity in a warped Randall-Sundrum background.