The full one-loop supersymmetric effective potential for the Wess-Zumino model is calculated using superfield techniques. This includes the Kähler potential and the auxiliary field potential, of which the former was originally computed in 1993 while the latter is derived for the first time. In the purely bosonic sector our results match those of older component field calculations.

This thesis examines low-energy effective actions of supersymmetric quantum field theories. These actions contain information about the low-energy field content and dynamics of quantum field theories and are essential for understanding their phenomenological and theoretical properties. In chapters 2 to 5, the covariant background field method is used to investigate quantum corrections to sectors of a variety of supersymmetric field theories at 1 and 2 loops. We start by looking at the background field quantisation of a general N=1 super-Yang-Mills theory, rederiving the well-known 1 loop finiteness conditions. This is followed by a reexamination of the effective potential of the Wess-Zumino model, focusing on a derivation of the full auxiliary fields' potential. Next, the 2 loop Euler-Heisenberg effective action is constructed for N=1 supersymmetric quantum electrodynamics; its renormalisation properties and self-dual limit are studied. The final action studied is the 2 loop Kahler potential for beta-deformed N=4 super-Yang-Mills. This sector is purely a product of the deformation and its finiteness is demonstrated in a general background before examining two special cases. Chapter 6 studies spontaneously broken supersymmetry and the pure Goldstino action. A general approach to constructing explicit field redefinitions is used to relate all known models of the Goldstino and to study their nonlinear supersymmetries. This approach is also used to construct the most general pure Goldstino action and to examine its supersymmetry transformations. Finally, a new embedding of the Goldstino into a complex linear superfield is presented. Its interactions to matter and gravity are examined and compared to existing Goldstino superfield constructions.

We propose a Goldstino model formulated in terms of a constrained complex linear superfield. Its comparison to other Goldstino models is given. Couplings to supersymmetric matter and supergravity are briefly described.

This paper constructs an exact field redefinition that maps the Akulov-Volkov action to that recently studied by Komargodski and Seiberg in arXiv:0907.2441. It is also shown that the approach advocated in arXiv:1003.4143v2 and arXiv:1009.2166 for deriving such a relationship is inconsistent.

Starting from the Akulov-Volkov (AV) action, we compute a finite-dimensional Lie group G of all field transformations of the form λ → λ′ = λ + O(λ 3) which preserve the functional structure of low-energy Goldstino-like actions. Associated with G is its twelve-parameter subgroup H of trivial symmetries of the AV action. The coset space G/H is naturally identified with the space of all Goldstino models. We then apply our construction to study the properties of five different Goldstino actions available in the literature. Making use of the most general field redefinition derived, we find explicit maps between all five cases. In each case there is a twelve-parameter freedom in these maps due to trivial symmetries inherent in the Goldstino actions. Finally, by using the pushforward of the AV supersymmetry, we find theoff-shell nonlinear supersymmetry transformations of the other five actions and compare to those normally associated with these actions.

Euler's disk is a toy described at http://www.eulersdisk.com. Aspects of its motion are modelled as an ideal disk rolling on a horizontal plane. In the final stages of Euler disk motions, the disk is nearly flat to the plane. Asymptotic approximations to the frequency of finite amplitude oscillations on steady (non-dissipative) rolling motions of the Euler disk are described. There are two different approximations which are appropriate in different limits. When the parameters are such that both apply, the formulae for the frequency agree: this appears to be new and simple. The material has been used in teaching; the teaching, and related, materials are available via the web [Keady, Math2200 Lecture Handouts, UWA]. References M. Batista. The nearly horizontally rolling of a thick disk on a rough plane. of a thick rigid disk Regular and Chaotic Dynamics 13 (4) (2008), 344--354. doi:10.1134/S1560354708040084 H. Caps, S. Dorbolo, S. Ponte, H. Croisier and N. Vandewalle. Rolling and slipping motion of Euler's disk. Physical Review E 69 (6) (2004), 056610. doi:10.1103/PhysRevE.69.056610 R. H. Cushman and J. J. Duistermaat. Nearly flat falling motions of the rolling disk. Regular and Chaotic Dynamics 11 (1) (2006) 31--60. doi:10.1070/RD2006v011n01ABEH000333 G. Keady. MATH2200 Lecture Handouts. The rolling of a disk on a horizontal plane. Part I, General; and Part II, Euler disk motions. http://school.maths.uwa.edu.au/ keady/papers.html H. K. Moffatt. Euler's disk and its finite-time singularity. Nature 404 (2000) 833. doi:10.1038/35009017 A. H. Nayfeh and D. T. Mook. Nonlinear Oscillations (Wiley: 1979). O. M. O'Reilly. The dynamics of rolling disks and sliding disks. Nonlinear Dynamics 10 (1996) 287--305. M. Srinivasan and A. Ruina. Rocking and rolling: a can that appears to rock might actually roll. Phys. Rev. E 78 (6) (2008) 066609. doi:10.1103/PhysRevE.78.066609 A. A. Stanislavsky and K. Weron. Nonlinear oscillations in the rolling motion of Euler's disk. Physica D 156 (2001) 247--259. doi:10.1016/S0167-2789(01)00281-0 J. L. Synge and B. A. Griffith. Principles of Mechanics. (McGraw-Hill, 3rd ed, 1959). W. T. Thomson. Introduction to Space Dynamics. (Dover: 1986 reprinting of Wiley 1963). R. Villanueva and M. Epstein. Vibrations of Euler's disk. Phys. Rev. E 71 (7) (2005) 066609. doi:10.1103/PhysRevE.71.066609 E. T. Whittaker. A treatise on the analytical dynamics of particles and rigid bodies. (CUP, 2nd ed, 1917).

In N = 2 superconformal field theories the Kahler potential is known to be tree level exact. The beta-deformation of N = 4 SU(N) SYM reduces the amount of supersymmetry to N = 1, allowing for non-trivial, superconformal loop corrections to the Kahler potential. We analyse the two-loop corrections on the Coulomb branch for a complex deformation. For an arbitrary chiral field in the Cartan subalgebra we reduce the problem of computing the two-loop Kahler potential to that of diagonalising the mass matrix, we then present the result in a manifestly superconformal form. The mass matrix diagonalisation is performed for the case of the chiral background that induces the breaking pattern SU(N) --> SU(N-2) x U(1)(2). Then, for the gauge group SU(3), the Kahler potential is explicitly computed to the two-loop order.

The famous AdS/CFT conjecture implies the existence of new nonrenormalisation the- orems for the structure of low-energy effective actions in extended supersymmetric theories. The testing of these implications requires the explicit calcul ation of higher-loop corrections to the effec- tive action for a variety of supersymmetric field theories. T wo sectors of the low-energy supersym- metric effective action that are particularly interesting and amenable to calculations are the Kähler potential and that of the Euler-Heisenberg type. This talk will present recent results from some two-loop calculations made in β -deformed supersymmetric Yang-Mills theory. Some new observations will also be given on the analytic structure of the sunset graph that is a key component of any two-loop effective potential calculation.

The two-loop Euler-Heisenberg-type effective action for = 1 supersymmetric QED is computed within the background field approach. The background vector multiplet is chosen to obey the constraints DαWβ = D(αWβ) = const, but is otherwise completely arbitrary. Technically, this calculation proves to be much more laborious as compared with that carried out in hep-th/0308136 for = 2 supersymmetric QED, due to a lesser amount of supersymmetry. Similarly to Ritus' analysis of spinor and scalar QED, the two-loop renormalisation is carried out using proper-time cut-off regularisation. A closed-form expression is obtained for the holomorphic sector of the two-loop effective action, which is singled out by imposing a relaxed super self-duality condition.