A partial differential equation (PDE) equates with zero an expression that depends on a collection of ‘arguments’, x, some ‘coefficient’ functions, c(x), and a collection of ‘sought-for’ functions, f(x), and their derivatives. A collection of PDEs is said to be a coupled system if the same f(x)'s (and/or their derivatives) appear in several of the equations, so that solving one equation depends on the solutions of another.
In general, it may be possible to eliminate one of the f(x)'s from the system, by using a few of the equations and/or their derivatives. This then results in a coupled system of (typically fewer, and possibly higher order) PDEs, involving fewer f(x)'s. It may also be possible to iterate such elimination's so that the iteration terminates with a system of PDEs (perhaps a single PDE) involving only one of the f(x)'s - this is called the integrability system (equation).
This iterated elimination process can always be done if the original coupled system was linear, i.e., if each summand in each equation contained at most one of the f(x)'s, and powers thereof. Then, the coupled system may be represented in a matrix form as M|f> = |s>, where |f> is a column vector the components of which are the various f(x)'s, M is a matrix the elements of which are the various differential operators pre-multiplied by the appropriate c(x)'s, and |s> is a column vector the elements of which are the summands in the original PDEs which do not involve the f(x) or their derivatives. All such systems can be turned, through any integral transform, into an inhomogeneous system of linear algebraic equations. The iterated elimination process is then simply (inverse) transform of the Gaussian elimination process.
Supersymmetric vacua are states, |*>, in a quantum field theory which are annihilated by the supersymmetry charges Q and their Hermitian conjugates, Q+. The states |*> may be expanded in multinomials over fermionic fields; since fermionic fields anticommute, they are nilpotent and the expansion terminates. The coefficients in this terminated multinomial series are undetermined functions, f(x), of the bosonic fields, x. As Q and Q+ can always be represented as first order differential operators with respect to the fields (bosonic and fermionic), the system of equations Q|*> = 0 = Q+|*> yields a coupled system of 1st order (linear) PDEs for the functions f(x). As shown above, all such systems can, in principle, solved. In practice, the size of the system grows exponentially with the number of fields involved.
Explicit solutions for the supersymmetric vacua are multiply interesting. Firstly, the entire Hilbert space of a supersymmetric model can be reconstructed from the supersymmetric vacua. In string theory, the correlation functions calculated with the supersymmetric vacua of the underlying world-sheet field theory are the couplings of the effective spacetime field theory. These, in turn, determine the moduli space geometry and parameter space dynamics in string theory.
© Tristan Hübsch, 2008