The Critical Ising Model on a Möbius Strip

We study the two-dimensional critical Ising model on a Möbius strip based on a duality relation between conformally invariant boundary conditions. By using a Majorana fermion field theory, we obtain explicit representations of crosscap states corresponding to the boundary states. We also discuss the duality structure of the partition functions. 1 typeset using PTPTEX.sty The continuum limit of the two-dimensional critical Ising model with boundaries is one of the simplest systems realized by field theories with boundaries. 1), 2) The presence of boundaries gives rise to new effects which depend on boundary conditions. Simple but important boundary conditions for Ising spins are those of the fixed and free ones. These boundary conditions are conformal invariant and are related by duality. A model with such boundary conditions has been analyzed in Ref. 1) using a boundary conformal field theory 3) with the central charge c = 1 2 . However, only a model defined on a cylinder has been considered to this time. The purpose of this paper is to study the critical Ising model on a Möbius strip based on the duality relation between conformally invariant boundary conditions. We use a free massless Majorana fermion field theory as used in Ref. 4) to compute loop amplitudes of superstring theories with open strings. We obtain explicit representations of crosscap states corresponding to the boundary states for conformally invariant boundary conditions and discuss the duality structure of these crosscap states. We also show that the periodic Möbius partition function with the free boundary condition is constructed from those of the periodic cylinder and antiperiodic cylinder with free boundary conditions. We now consider the continuum limit of the critical Ising model on a Möbius strip of width L and boundary length R. We define coordinates x, y as 0 ≤ x ≤ L and 0 ≤ y ≤ R. When we take the coordinate x to be Euclidean time, the system is described by the Hamiltonian HR for the periodic space 0 ≤ y ≤ R. The partition function is then given by 4) Z = 〈B| e−L2HR|C〉 , (1) where |B〉 is a boundary state placed at x = 0 and |C〉 is a crosscap state placed at x = L 2 . In the form (1) the partition function is a function of the modular parameter τ̃ = i R + 1 2 . To represent the system concretely, we introduce free massless Majorana fermion fields ψ(x, y) and ψ(x, y). In the picture (1) the fields have standard oscillator modes ψn and ψn for the periodic space 0 ≤ y ≤ R. 5) In terms of these modes the Hamiltonian is HR = 2π R ∑