Alternative approaches to maximally supersymmetric field theories

The central objective of this work is the exploration and application of alternative possibilities to describe maximally supersymmetric field theories in four dimensions: N=4 super Yang-Mills theory and N=8 supergravity. While twistor string theory has been proven very useful in the context of N=4 SYM, no analogous formulation for N=8 supergravity is available. In addition to the part describing N=4 SYM theory, twistor string theory contains vertex operators corresponding to the states of N=4 conformal supergravity. Those vertex operators have to be altered in order to describe (non-conformal) Einstein supergravity. A modified version of the known open twistor string theory, including a term which breaks the conformal symmetry for the gravitational vertex operators, has been proposed recently. In a first part of the thesis structural aspects and consistency of the modified theory are discussed. Unfortunately, the majority of amplitudes can not be constructed, which can be traced back to the fact that the dimension of the moduli space of algebraic curves in twistor space is reduced in an inconsistent manner. The issue of a possible finiteness of N=8 supergravity is closely related to the question of the existence of valid counterterms in the perturbation expansion of the theory. In particular, the coefficient in front of the so-called R counterterm candidate has been shown to vanish by explicit calculation. This behavior points into the direction of a symmetry not taken into account, for which the hidden on-shell E7(7) symmetry is the prime candidate. The validity of the so-called double-soft scalar limit relation is a necessary condition for a theory exhibiting E7(7) symmetry. By calculating the doublesoft scalar limit for amplitudes derived from an N=8 supergravity action modified by an additional R counterterm, one can test for possible constraints originating in the E7(7) symmetry. In a second part of the thesis, the appropriate amplitudes are calculated employing the low-energy limit of string theory, and the double-soft limit relation is indeed shown to hold. However, if the modified action has E7(7) symmetry, the singlesoft scalar limit of any amplitude should vanish. This not being the case suggests that the E7(7) symmetry is broken by the R 4 counterterm. Finally, the Grassmannian formulation of N=4 SYM is investigated in a third part of the thesis. Any amplitude in N=4 SYM theory can be expressed as a linear combination of certain infrared (IR) divergent integrals. Being known as leading singularities, the coefficients of these integrals completely determine the structure of an amplitude. From field-theory calculations it is known that the leading singularities are not independent, but are subject to a set of so-called IR equations. The alternative Grassmannian formulation is conjectured to describe the leading singularities as certain linear combinations of residues of a multidimensional complex integral. These residues are not independent but are related by generalized residue theorems (GRTs), which are multidimensional generalizations of Cauchy’s theorem. Indeed, expressing the leading singularities known from field-theory calculations in terms of these residues supports the conjecture that the IR equations can be derived from GRTs. Here it is shown that GRTs in the Grassmannian formulation do not only give rise to IR equations, but to a larger set of constraints, which can be derived by considering the dual conformal anomaly of one-loop amplitudes. Explicit maps between GRTs and both, dual conformal constraints and IR equations, are deduced and discussed.