Galilean conformal and superconformal symmetries

Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schr?dinger, Galilean conformal, and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c???? from relativistic conformal algebraO(d+1, 2) (d-number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D = 4 superconformal algebra the Galilei conformal superalgebra (GCSA) is obtained, in the formulation using complexWeyl supercharges.     

Dilogarithm identities in conformal field theory and group homology

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin (equivalent to an identity found earlier by Lewin) can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2×2 real matrices viewed as adiscrete group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic K-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2×2 real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with a summary of a number of open conjectures on the mathematical side.To Professor C. N. Yang for his 70^th birthdayThis work was partially supported by grants from the Statens Naturvidenskabelige Forskningsraad and the Paul and Gabriella Rosenbaum Foundation     

A new unified field theory based on the conformal group

We develop here a new unified theory of the electromagnetic and gravitational field, based on a six-dimensional generalization of Maxwell's equations; additional space-time coordinates are interpreted only as mathematical tools in order to obtain a linear realization of the four-dimensional conformal group.     

Contact transformations and conformal group. I. Relativistic theory

We find and identify the contact group of plane hyperbolic conformal geometry as a step to a better understanding of conformal invariance in physics.     

Contact transformations and conformal group. II. Non-relativistic theory

We find the contact group of non-relativistic accelerated systems. A comparison with the relativistic situation is made.     

Modular geometry of superconformal field theory

Two-dimensional superconformal field theory is discussed as geometry on universal supermoduli space, generalizing the conformal case. The compactification divisor of supermoduli space consists of super-Riemann surfaces with spin and super nodes, which are expressed algebraically. The local behavior of the super partition function on universal supermoduli space is described.     

Gauge fixing problem in the conformal QED

The gauge fixing problem in the conformal (spinor and scalar) QED is examined. For the analysis, we generalize Dirac's manifestly conformal-covariant formalism. It is shown that the (vector and matter) fields must obey a certain mixed (conformal and gauge) type of transformation law in order to fix the local gauge symmetry preserving the conformal invariance in the Lagrangian.     

Gauge natural formulation of conformal gravity

We consider conformal gravity as a gauge natural theory. We study its conservation laws and superpotentials. We also consider the Mannheim and Kazanas spherically symmetric vacuum solution and discuss conserved quantities associated to conformal and diffeomorphism symmetries.     

Gauging the conformal group

It is demonstrated that Kibble’s method of gauging the Poincaré group can be applied to the gauging of the conformal group. The action of the gauge transformations is the action of general spacetime diffeomorphisms (or coordinate transformations) combined with a local action of an 11-parameter subgroup of SO(4,2). Because the translational subgroup is not an invariant subgroup of the conformal group the appropriate generalisation of the derivative of a physical field is not a covariant derivative in the usual sense, but this does not lead to any inconsistencies.     

Classical origin of quantum group symmetries in Wess-Zumino-Witten conformal field theory

We elucidate the way by which the quantum group symmetries of the WZW models arise within the canonical formalism of the classical field theory.     

Spin models and superconformal Yang–Mills theory

We apply novel techniques in planar superconformal Yang–Mills theory which stress the role of the Yangian algebra. We compute the first two Casimirs of the Yangian, which are identified with the first two local Abelian Hamiltonians with periodic boundary conditions, and show that they annihilate the chiral primary states. We streamline the derivation of the R-matrix in a conventional spin model, and extend this computation to the gauge theory. We comment on higher-loop corrections and higher-loop integrability.     

Gauge theory and bags

The surface dependent phase factor $\text{P}\text{(}\text{exp}\text{∫}\text{F}{}_{\mathrm{\mu \nu }}{}^1\text{d}\text{σ}{}^{\mathrm{\mu \nu }}$) is introduced and its variational derivatives are investigated. Under certain assumptions it is shown to satisfy a differential equation which coincides with the membrane equation.     

Tree-level recursion relation and dual superconformal symmetry of the ABJM theory

We propose a recursion relation for tree-level scattering amplitudes in three-dimensional Chern-Simons-matter theories. The recursion relation involves a complex deformation of momenta which generalizes the BCFW-deformation used in higher dimensions. Using background field methods, we show that all tree-level superamplitudes of the ABJM theory vanish for large deformations, establishing the validity of the recursion formula. Furthermore, we use the recursion relation to compute six-point and eight-point component amplitudes and match them with independent computations based on Feynman diagrams or the Grassmannian integral formula. As an application of the recursion relation, we prove that all tree-level amplitudes of the ABJM theory have dual superconformal symmetry. Using generalized unitarity methods, we extend this symmetry to the cut-constructible parts of the loop amplitudes.