Vacuum, in general, refers to a minimal energy state of a system. ‘Local’ vacua simply refer to states the energy of which is a local minimum, with respect to small variations of the various parameters in the system; the energy of ‘global’ vacua is at the global minimum. In general supersymmetric sigma models, the Hamiltonian, 2H=Tr{Q,Q+}, is positive definite and so non-negative, and the lowest energy, <H>=0, is achieved by states that are annihilated by the supersymmetry generators, Q, Q+, i.e., by supersymmetric states.
Since the supersymmetry generators may be written as (first order) differential operators in terms of the fields of the supersymmetric field theory, supersymmetric vacua, |*>, are defined by the coupled system of partial differential equations: Q|*> = 0 = Q+|*>. Such states are relatively easy to find in simpler Landau-Ginzburg orbifolds and the corresponding gauged linear sigma models; in the general case, only their number and certain general features may be determined.
The supersymmetric vacua of the underlying world-sheet field theory corresponds to the light particles of the effective spacetime theory. Therefore, the correlation functions calculated with these vacua are the couplings of the corresponding particles. The expectation values of these light particles parametrize both the geometry of the moduli space and also determine the dynamics of the particles determining the parameter space.
Finally, mirror and duality transforms take one superstring model into another. They are increasingly better understood as generated by certain corresponding transformations in the underlying world-sheet field theory, although they were first identified by their action on the light particles, and so on supersymmetric vacua.
© Tristan Hübsch, 2000