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Finitized Conformal Spectra of the Ising Model on the Klein Bottle and Moebius Strip
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Finitized Conformal Spectra of the Ising Model
on the Klein Bottle and Mo?bius Strip
C. H. Otto Chui? and Paul A. Pearce?
Department of Mathematics and Statistics
University of Melbourne
Parkville, Victoria 3010, Australia
May, 2001
Abstract
We study the conformal spectra of the critical square lattice Ising model on
the Klein bottle andMo?bius strip using Yang-Baxter techniques and the solution
of functional equations. In particular, we obtain expressions for the finitized
conformal partition functions in terms of finitized Virasoro characters. This
demonstrates that Yang-Baxter techniques and functional equations can be used
to study the conformal spectra of more general exactly solvable lattice models
in these topologies. The results rely on certain properties of the eigenvalues
which are confirmed numerically.
Key Words: Conformal field theory; Ising model; klein bottle; mo?bius strip.
1 Introduction
There has been much recent progress on understanding general conformal boundary
conditions for rational theories on the cylinder [1] and torus [2] and their relation [3]
to integrable boundary conditions on the lattice. It is therefore of some interest to
extend this understanding to other topologies such as the Klein bottle and Mo?bius
strip. However, while much is known [4, 5, 6] about conformal partition functions
in these topologies from the viewpoint of string theory, very little is known about
integrable lattice boundary conditions on the Klein bottle and Mo?bius strip. Indeed,
?e-mail: C.Chui@ms.unimelb.edu.au
?e-mail: P.Pearce@ms.unimelb.edu.au
1
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u u
λ?u λ?u
u u
λ?u λ?u
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∧ ∧
u u
λ?u λ?u
u u
λ?u λ?u
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Figure 1: Klein bottle (left) Mo?bius strip (right). The boundaries are joined ac-
cording to the orientations indicated by the arrows. The arrows at the bottom left
corners on each face weight indicate its orientation.
the lattice Yang-Baxter techniques which
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