Supersymmetric Yang-Mills theories without anti-commuting variables

Abstract

Supersymmetric field theories can be studied via an alternate approach using purely bosonic variables. In this method, a transformation (Nicolai map) of the bosonic fields exists for supersymmetric gauge theories such that the Jacobian of the map is same as the product of fermion and ghost determinants. This thesis investigates the development of supersymmetric Yang-Mills theories without anti-commuting variables, presenting them as entirely bosonic theories. We derived the second order map (perturbatively in the coupling constant) in the Landau gauge for all pure supersymmetric Yang-Mills theories. This approach yields the well-known old relation that supersymmetric Yang-Mills theories can exist only in D = 3, 4, 6, 10 space-time dimensions. We investigated this formalism to the third order in the coupling constant using the rigorous R prescription. While working on the order g^3 map, we discovered a simpler map through trial and error, also to the third order that works only in space-time dimension six. The existence of two maps at order g^3 in six dimensions highlights the uniqueness of the map and the formalism. In this approach, correlation functions and scattering amplitudes can be calculated using the inverse map. The light-cone gauge is useful for studying scattering amplitudes as the spinor helicity variables appear naturally in this gauge. We studied the Nicolai map approach in the light-cone gauge for supersymmetric Yang-Mills theory and computed the map perturbatively to order g^2. With the physical helicity fields, we obtained two maps at the second order in coupling and discussed the problems related to the uniqueness of these maps.