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$F^4$) are indeed protected while the double-trace one-quarter BPS invariant (d²F⁴$d^2{F}^4$) is not protected at two loops in D=7$D=7$, but is protected at three loops in D=6$D=6$ in agreement with recent calculations. Non-BPS invariants, i.e. full superspace integrals, are also shown to be unprotected.     

Structure of supergravity

It is shown how to construct actions and kinetic multiplets out of scalar multiplets of supergravity. The multiplets containing the scalar and the conformal tensor are obtained and yield the gauge actions of Einstein and Weyl supergravity. The complete supersymmetric R² and Bel-Robinson invariants are also obtained as well as general n-loop counterterms.     

Ultraviolet divergences in non-renormalizable supersymmetric theories

We present a pedagogical review of our current understanding of the ultraviolet structure of N = (1,1) 6D supersymmetric Yang–Mills theory and of N = 8 4D supergravity. These theories are not renormalizable, they involve power ultraviolet divergences and, in all probability, an infinite set of higherdimensional counterterms that contribute to on-mass-shell scattering amplitudes. A specific feature of supersymmetric theories (especially, of extended supersymmetric theories) is that these counterterms may not be invariant off shell under the full set of supersymmetry transformations. The lowest-dimensional nontrivial counterterm is supersymmetric on shell. Still higher counterterms may lose even the on-shell invariance. On the other hand, the full effective Lagrangian, generating the amplitudes and representing an infinite sum of counterterms, still enjoys the complete symmetry of original theory. We also discuss simple supersymmetric quantum-mechanical models that exhibit the same behaviour.     

Higher order invariants in extended supergravity

On-shell linearized extended supergravity is presented in superspace for all N. The formalism is then used in the construction of higher order invariants which may serve as counterterm lagrangians. It is shown that three-loop counterterms exist for N ? 3 and (N ? 1)-loop counterterms for N ? 4. In the full non-linear theory, the presence of a global non-compact symmetry group for N ? 4 does not allow a simple extension of the (N ? 1)-loop term, but N-loop counterterms may be constructed.     

Auxiliary fields and ultraviolet divergences in supergravity

The relation between the supergravity counterterms and the coupling of auxiliary fields to matter is investigated. As a result we determine the infinities due to one and two internal-matter loops for the supersymmetric Yang-Mills-Einstein system. Furthermore, the couplings of the auxiliary fields to the massive scalar multiplet are obtained through order ?², and the contribution to the counterterm Lagrangian trilinear in the auxiliary fields is given.     

AdS 5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

We use the formalism of generalized geometry to study the generic supersymmetric AdS ₅ solutions of type IIB supergravity that are dual to ${\mathcal{N}=1}We use the formalism of generalized geometry to study the generic supersymmetric AdS ₅ solutions of type IIB supergravity that are dual to N=1{\mathcal{N}=1} superconformal field theories (SCFTs) in d = 4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of K?hler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.     

An off-shell superspace R2-action in six dimensions

The superspace constraints corresponding to the two inequivalent versions of the 40 + 40 off-shell Weyl multiplet in six dimensions are given. A supersymmetric R²-action (including quartic fermion terms) for one of the two Weyl multiplets is constructed. A crucial role in this construction is played by the observation that there exists a formal relationship between the Weyl and Yang-Mills multiplets. Both the supersymmetric R²-action and the supersymmetric Yang-Mills F²-action coupled to the Weyl multiplet are reformulated as superspace actions.     

Coupling the gauge-invariant supersymmetric non-linear sigma model to supergravity

The gauge-invariant supersymmetric non-linear sigma model is coupled to N = 1 supergravity. The results are expressed in the language of Kahler geometry. They lead to a deeper understanding of the connections between supergravity, R-invariance and the Fayet-Iliopoulos D-term. They also provide a foundation for phenomenological studies of supergravity theories.     

Superspace gauge theories with central charges

The superspace constraints leading to off-shell formulations of extended supersymmetric Yang-Mills theories for N = 2 and 4 are generalised to N = 6 in the abelian case. This leads to an on-shell representation of linearized N = 6 supergravity.     

Counterterms in extended supergravities

Geometrical invariants respecting all necessary symmetries of the theory are shown to exist, starting from the 8th (4th) loop approximation in N = 8 (N = 4) on-shell supergravity. 3-loop counterterms are presented on a linearized level for N = 4 and N = 8 theories. The corresponding 3-loop non-linear invariants are discussed.     

Supergraphity: (II). Manifestly covariant rules and higher-loop finiteness

We continue our discussion of the background field formalism in supersymmetric theories, deriving new covariant Feynman rules for chiral superfields. As a result, we obtain improved power-counting rules for both simple and extended supersymmetry which can be used to make the following statements: If the corresponding extended superfield formalism exist, (a) N=2 supersymmetric Yang-Mills theory is finite beyond one loop, (b) N=4 Yang-Mills is finite at all loops, and (c) N=8 supergravity is finite through six loops. We also find that in simple super-Yang-Mills the radiative corrections to the Fayet-Iliopoulos (“D”) term, which are known to vanish for higher loops, also vanish automatically at one loop for arbitrary couplings.     

0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

Effective action in higher derivative (Super)gravities

'The one-loop effective action (EA) with an accuracy up to linear curvature terms ind=4R ²-gravity, conformal gravity, andN=1,d=4 conformal supergravity on the backgroundR ₄×T_4–k,k=1, 2, 3 is calculated. (Here,R _k is thek-dimensional curved space, T_n is then-dimensional torus). The one-loop EA in multidimensionalR ²-gravity and ind=10 conformal supergravity on the backgroundR ₄ ×T _d–4 is also obtained. The mechanism of inducing the Einstein gravity from the EA of considered theories of higher derivative (super)gravity is presented.We are grateful to I. L. Bukhbinder for the numerous discussions of considered questions.     

ℵ0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ℵ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ℵ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

Higher derivative modifications ofD=10 superspace Yang-Mills theories

We extend the superspace treatment of higher derivative supersymmetric Yang-Mills theories, emphasizing the role played by manifestD=10 Lorentz covariance and supersymmetry invariance. As an example, the superspace formulation of the effective action is considered for the massless fields in the open superstring, as well as in theSO(32) andE ₈×E₈ superstring theories. We show that there exists a unique modification of thef-tensor supercurrent which can provide the embedding of theO(α'³) string corrections in the slope parameter expansion for the Yang-Mills sector inD=10 superspace. The new correction term is relevant for the understanding of nonperturbative effects.     

New minimal higher derivative supergravity coupled to matter

We consider quadratic invariants in the new minimal formulation of N = 1, 4D supergravity, and obtain a simpler form of the Gauss-Bonnet combination. When added to the Einstein lagrangian, these new multiplets propagate different states than those in the old minimal formulation. In particular, the R + αR² theory is shown to be equivalent to standard supergravity coupled to a massive vector multiplet. Our result shows the non-uniqueness of minimal supersymmetrizations of higher derivative theories.     

Supergravity symmetries in superspace

We show the extension of the Forgacs-Manton Killing symmetry for gauge theories, to solutions, extended to superspace, of the d = 11 supergravity theory.     

Gauge theory in deformed $$ \mathcal{N} $$ = (1, 1) superspace

We review the non-anticommutative Q-deformations of = (1, 1) supersymmetric theories in four-dimensional Euclidean harmonic superspace. These deformations preserve chirality and harmonic Grassmann analyticity. The associated field theories arise as a low-energy limit of string theory in specific backgrounds and generalize the Moyal-deformed supersymmetric field theories. A characteristic feature of the Q-deformed theories is the half-breaking of supersymmetry in the chiral sector of the Euclidean superspace. Our main focus is on the chiral singlet Q-deformation, which is distinguished by preserving the SO(4) ∼ Spin(4) “Lorentz” symmetry and the SU(2) R-symmetry. We present the superfield and component structures of the deformed = (1, 0) supersymmetric gauge theory as well as of hypermultiplets coupled to a gauge superfield: invariant actions, deformed transformation rules, and so on. We discuss quantum aspects of these models and prove their renormalizability in the Abelian case. For the charged hypermultiplet in an Abelian gauge superfield background we construct the deformed holomorphic effective action. The text was submitted by the authors in English.     

A Geometric Construction of the Exceptional Lie Algebras <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript> and <Emphasis Type="Italic">E</Emphasis><Subscript>8</Subscript>

We present a geometric construction of the exceptional Lie algebras F ₄ and E ₈ starting from the round spheres S ⁸ and S ¹⁵, respectively, inspired by the construction of the Killing superalgebra of a supersymmetric supergravity background.