Abstract
We use ideas of generalized self-duality conditions to construct real scalar field theories in (1 + 1)-dimensions with exact self dual sectors. The approach is based on a pre-potential U that defines the topological charge and the potential energy of these theories. In our algebraic method to construct the required pre-potentials we use the representation theory of Lie groups. This approach leads naturally to an infinite set of degenerate vacua and so to topologically non-trivial self-dual solutions of these models. We present explicit examples for the groups SU(2), SU(3) and SO(5) and discuss some properties of these solutions.
Highlights
The approach is based on a pre-potential U that defines the topological charge and the potential energy of these theories
The self-dual or BPS solutions arise in theories in which the topological charge has an integral representation and so it has a topological charge density
The construction of self-dual sectors for scalar field theories in (1 + 1)-dimensions that we present in this paper is based on the methods of [6], and can be summarized as follows: suppose one has a topological charge Q with an integral representation such that its density can be split into the sum of the products of two quantities as where Aα and Aα are functionals of the scalar fields φa, a = 1, 2, . . . r, and of their first space derivatives ∂xφa, but not of higher derivatives of these fields
Summary
The construction of self-dual sectors for scalar field theories in (1 + 1)-dimensions that we present in this paper is based on the methods of [6], and can be summarized as follows: suppose one has a topological charge Q with an integral representation such that its density can be split into the sum of the products of two quantities as dx Aα Aα,. Summarising, we see that the first order self-duality equations (2.11) alone imply the Euler-Lagrange equations corresponding to the static energy functional E for the fields φa and any possible extra fields that the matrix ηab can depend on. Note that this fact had already been encoded in the construction presented above, between equations (2.1) and (2.5), since the fields φa which appear in (2.2) and (2.4) can be any fields that the quantities Aα and Aα depend on. Which is, the same as (2.8), and the bound is saturated by the self-dual solutions of (2.11)
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