Self-dual forms: action, Hamiltonian and compactification

Abstract

It has been shown that, by adding an extra free field that decouples from the dynamics, one can construct actions for interacting 2n-form fields with self-dual field strengths in 4n+2 dimensions. In this paper we analyze canonical formulation of these theories, and show that the resulting Hamiltonian reduces to the sum of two Hamiltonians with independent degrees of freedom. One of them is free and has no physical consequence, while the other contains the physical degrees of freedom with the desired interactions. For the special cases of chiral scalars in two dimensions and chiral two form fields in six dimensions, we discuss compactification of these theories respectively on a circle and a two dimensional torus, and show that we recover the expected properties of these systems, including S-duality invariance in four dimensions.1

Geometrical aspects of an Abelian (2,0) action

“…We also clarified some aspects of the hamiltonian analysis. First, we showed that H (s) and H (g) correspond precisely to the Π ± variables introduced in [34]. The fact that Π + describes a non-unitary decoupled sector of the theory is consistent with the fact that H (s) is self-dual with respect to η.…”

“…In this paper we studied the six-dimensional action put forward in [34]-and its (2,0) supersymmetric completion [1]-clarifying many of its unconventional features. This formulation aims to encode the dynamics of a chiral 2-form in 6D into a "2-form" B and an ⋆ η -self-dual "3-form" H. Although all of our analysis is performed for chiral 2-forms in six-dimensions we hope that the techniques we developed can be readily applied to other dimensions.…”

“…holds for any two self-dual three-forms H 1 , H 2 . We note that in [33,34] the following notation is employed…”

“…As a result, B and H are not standard differential forms, a fact that is also reflected in their non-standard transformation properties under diffeomorphisms. It turns out that S H encodes on-shell the degrees of freedom carried by-not one buttwo free 2-forms with self-dual field strength: in the hamiltonian formulation, it can be shown that the theory contains an unphysical sector (with a wrong-sign kinetic term) that explicitly decouples from the physical one [34]. Thus one expects the physical sector to correctly describe the physics of free chiral 2-forms on generic manifolds.…”

“…In addition to the trivial gauge redundancy given by the shift of B by a closed 2-form, this action is also invariant under the following gauge transformation [34]:…”